STARK POINTS AND p-ADIC ITERATED INTEGRALS ATTACHED TO MODULAR FORMS OF WEIGHT ONE

HENRI DARMON, ALAN LAUDER AND VICTOR ROTGER

To our families

Abstract. Let E be an elliptic curve over ℚ and let ϱ_{♭} and ϱ_{♯} be odd two-dimensional Artin representations for which ϱ_{♭} ⊗ϱ_{♯} is self-dual. The progress on modularity achieved in the last decades ensures the existence of normalised eigenforms f, g and h of respective weights 2, 1 and 1, giving rise to E, ϱ_{♭}, and ϱ_{♯} via the constructions of Eichler-Shimura and Deligne-Serre. This article examines certain p-adic iterated integrals attached to the triple (f, g, h), which are p-adic avatars of the leading term of the Hasse-Weil-Artin L-series L(E, ϱ_{♭} ⊗ϱ_{♯}, s) when it has a double zero at the center. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on E—referred to as Stark points—which are defined over the number field cut out by ϱ_{♭} ⊗ ϱ_{♯}. This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weight one forms. It is proved when g and h are binary theta series attached to a common imaginary quadratic field in which p splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing p-adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell-Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark-Heegner points attached to Shintani-type cycles on H_{p} ×H), and extensions of ℚ with Galois group a central extension of the dihedral group D_{2n} or of one of the exceptional subgroups A_{4}, S_{4}, and A_{5} of PGL_{2}(ℂ).

Contents

Introduction 1 1. Overconvergent weight one forms and Gross-Stark units 11 2. Iterated integrals and the Garrett-Hida p-adic L-function 17 3. Heegner points and theta series of imaginary quadratic fields 23 4. Stark-Heegner points and theta series of real quadratic fields 40 5. Stark points over A_{4}, S_{4} or A_{5} extensions and exotic weight one forms 46 6. Stark points attached to odd two-dimensional Artin representations 56 7. Stark points over cyclotomic fields and Eisenstein series 57 Appendix A. Digest of numerical examples 61 References 62

Introduction

The purpose of this article is to formulate an analogue of the Stark conjecture on units in number fields in the setting of global points on elliptic curves. We prove this “elliptic Stark conjecture” in some cases and provide numerical evidence in its support in many others. It is offered as a more constructive alternative to the Birch and Swinnerton-Dyer (BSD) conjecture, since it often allows the efficient analytic computation of p-adic logarithms of global points. In contrast, the BSD conjecture (and its p-adic variant formulated by Mazur, Tate and Teitelbaum) gives the (p-adic) heights of such points, from which it appears computationally more difficult to recover the points themselves [Si]. Our conjecture gives a unifying perspective on the two currently known constructions of global points on elliptic curves over ℚ—Heegner points and the still conjectural Stark-Heegner points attached to real quadratic cycles on H_{p} × H—and suggests a possible global underpinning for the latter. It also yields conjectural constructions and explicit formulae, in situations of rank one and two, for global points over cyclotomic fields, abelian extensions of quadratic fields which are not necessarily anticyclotomic, and extensions of ℚ with Galois group a central extension of A_{4}, S_{4} or A_{5}.

Let E be an elliptic curve over ℚ with Galois representation V_{p}(E) := H_{et}^{1}(Eℚ̄, ℚ_{p}(1)), and let

ϱ : G_{ℚ}⟶Aut(V_{ϱ}) ≃ GL_{n}(L), dim_{L}(V_{ϱ}) = n ≥ 1

be an Artin representation—i.e., a continuous Galois representation factoring through a finite quotient Gal (H/ℚ) of G_{ℚ}—with coefficients in some finite extension L ⊂ ℂ of ℚ. The Hasse-Weil-Artin L-series L(E, ϱ, s) of E twisted by ϱ is the L-function of the compatible system V_{p}(E) ⊗ V_{ϱ} of p-adic representations of G_{ℚ}. It is defined on the right half-plane Re(s) > 3/2 by an absolutely convergent Euler product of degree 2n and is expected to admit an analytic continuation to the whole complex plane. Under this assumption, the analytic rank of the pair (E, ϱ) is defined to be

r_{an}(E, ϱ) := ord_{s=1}L(E, ϱ, s).

A natural Galois equivariant refinement of the Birch and Swinnerton-Dyer conjecture predicts that

r_{an}(E, ϱ) =^{?} r(E, ϱ),  (1)

where r(E, ϱ) is the multiplicity of V_{ϱ} as a subrepresentation of E(H)⊗L when ϱ is irreducible, defined for general ϱ by

r(E, ϱ) := dim_{L} Hom_{G_{ℚ}}(V_{ϱ}, E(H) ⊗ L).

Equality (1) has been proved when r_{an}(E, ϱ) = 0 and (in roughly chronological order)

I. ϱ is the odd self-dual two-dimensional Galois representation induced from a ring class character of an imaginary quadratic field, by a series of works building on the breakthroughs of Gross-Zagier and Kolyvagin;

II. ϱ is a Dirichlet character, by the work of Kato [Kato]; III. ϱ is an odd irreducible two-dimensional Galois representation satisfying some mild

restrictions, by [BDR2]; IV. ϱ is an irreducible constituent of the tensor product ϱ_{♭} ⊗ϱ_{♯} of two odd irreducible two-

dimensional Galois representations which is self-dual and satisfies some other mild restrictions, by [DR2].

The setting where r_{an}(E, ϱ) > 0 raises the challenge of constructing non-zero elements in

E(H)_{L}^{ϱ} := ∑_{φ} φ(V_{ϱ}), where φ runs over a basis of Hom_{G_{ℚ}}(V_{ϱ}, E(H) ⊗ L).

Devising strategies for constructing such global points is arguably the most fundamental issue surrounding the Birch and Swinnerton-Dyer conjecture. Yet very little is known in this direction beyond the setting alluded to in item I above, where the methods of Gross, Zagier and Kolyvagin can also be exploited to establish (1) when r_{an}(E, ϱ) = 1, by using Heegner points (as recalled in Ch. 3 below) to construct the predicted non-trivial elements of E(H)_{L}^{ϱ}.

STARK POINTS AND p-ADIC ITERATED INTEGRALS 3

As explained in the survey [BCDDPR], the techniques used to analyze settings II, III and IV can be organised under the common rubric of “Euler systems of Garrett-Rankin-Selberg type”. The present work considers the same settings, but in the less well-understood scenario of “analytic rank ≤ 2.” More precisely, it will be assumed that ϱ is an irreducible constituent of the tensor product ϱ_{♭} ⊗ ϱ_{♯}, where ϱ_{♭} and ϱ_{♯} are odd, two-dimensional, but not necessarily irreducible, Artin representations satisfying the self-duality condition

χ := det(ϱ_{♭})^{−1} = det(ϱ_{♯}),

and that r_{an}(E, ϱ) = 1 or 2. The primary goal of this paper is to propose a conjectural p-adic analytic formula relating global points in E(H)_{L}^{ϱ} to“p-adic iterated integrals” attached to an appropriate triple of modular forms, and to prove it in some cases.

The following general notations and background are required to formulate the “elliptic Stark conjecture” (Conjecture ES) below. For any positive integer N, any Dirichlet character η : (ℤ/Nℤ)^{×} → ℚ̄^{×} and any field L, write S_{k}(N, η)_{L} ⊆ M_{k}(N, η)_{L} for the spaces of classical cusp forms and modular forms of weight k, level N and character η with Fourier coefficients in F. (When η is suppressed from the notation, it is to be understood that η is the trivial character.)

Thanks to Wiles’ proof of the Shimura-Taniyama conjecture, the elliptic curve E is known to be modular: there is a normalised weight two newform f ∈ S_{2}(N_{f})_{ℚ}, where N_{f} is the conductor of E, satisfying

L(f, s) = L(E, s).

The work of Khare and Wintenberger [KW] on Serre’s modularity conjecture likewise implies that any odd, two-dimensional Artin representation arises from a classical (cuspidal or Eisenstein) normalized newform of weight one. More precisely, denoting by N_{g} and N_{h} the Artin conductors of ϱ_{♭} and ϱ_{♯}, there are weight one newforms g ∈ M_{1}(N_{g}, χ^{−1})_{L} and h ∈ M_{1}(N_{h}, χ)_{L} satisfying

L(g, s) = L(ϱ_{♭}, s), L(h, s) = L(ϱ_{♯}, s).

Henceforth, it will be convenient to use ϱ_{g} and ϱ_{h} to designate ϱ_{♭} and ϱ_{♯}, to write

ϱ_{gh} := ϱ_{g} ⊗ ϱ_{h} : Gal (H/ℚ)⟶SL_{4}(L),

and to denote by V_{g}, V_{h} and V_{gh} the L-vector spaces underlying ϱ_{g}, ϱ_{h} and ϱ_{gh}, respectively.

When either g or h is an Eisenstein series, the Rankin-Selberg method yields the analytic continuation and functional equation for the L-function L(E, ϱ_{gh}, s) = L(f ⊗ g ⊗ h, s) relating its values at s and 2 − s. This has been extended to the general case where both g and h are cusp forms, by a result of Garrett. The self-duality condition on ϱ_{gh} implies that the root number ε(E, ϱ_{gh}) that appears in this functional equation is ±1, and determines the parity of r_{an}(E, ϱ_{gh}). Furthermore, it can be written as a product of local root numbers

ε(E, ϱ_{gh}) = ∏_{v|N·∞} ε_{v}(E, ϱ_{gh}), (where N := lcm(N_{f}, N_{g}, N_{h})),

and the local sign ε_{∞}(E, ϱ_{gh}) is always equal to 1. The following hypothesis on the local root numbers ε_{v}(E, ϱ_{gh}) at the non-archimedean places, which is discussed in further detail in Ch. 2, shall be made throughout this article:

Hypothesis A. (Local sign hypothesis) For all finite places v|N, the local signs ε_{v}(E, ϱ_{gh}) are equal to +1. In particular, the same is true of the global root number, and hence L(E, ϱ_{gh}, s) vanishes to even order at s = 1.

This hypothesis on the local sign is often satisfied: for instance, whenever gcd(N_{f}, N_{g}, N_{h}) = 1. Under Hypothesis A, the method of Garrett-Rankin-Selberg relates the central critical value L(f ⊗ g ⊗ h, 1) to the values of the trilinear form

I : S_{2}(N)_{ℂ} × M_{1}(N, χ^{−1})_{ℂ} × M_{1}(N, χ)_{ℂ} ⟶ ℂ

defined by

I(f̆, ğ, h̆) := ⟨f̆, ğh̆⟩,

where ⟨ , ⟩ denotes the Petersson scalar product on S_{2}(N)_{ℂ} × M_{2}(N)_{ℂ}. More precisely, for any integer N, let 𝕋_{N} denote the Hecke algebra generated by the good Hecke operators T_{n} with n ∤ N. If M is any 𝕋_{N}-module and ϕ ∈ M is a simultaneous eigenvector for the action of 𝕋_{N}, satisfing T_{ℓ}ϕ = a_{ℓ}(ϕ)ϕ for all ℓ ∤ N, let I_{ϕ} ⊂ 𝕋_{N} be the ideal associated to this system of Hecke eigenvalues and denote by

M[ϕ] = Ker(I_{ϕ}) = {m ∈ M : T_{ℓ}m = a_{ℓ}(ϕ)m for all ℓ ∤ N}  (2)

the corresponding eigenspace. By the work [HaKu] of Harris-Kudla, the restriction

I_{fgh} : S_{2}(N)[f] × M_{1}(N, χ^{−1})[g] × M_{1}(N, χ)[h] ⟶ ℂ  (3)

of I to the product of isotypic components for f, g and h is identically zero if and only if the central critical value L(f ⊗ g ⊗ h, 1) vanishes. Since this article is solely interested in the setting where ϱ_{gh} admits irreducible constituents ϱ for which L(E, ϱ, 1) = 0, the following hypothesis is also made throughout:

Hypothesis B. (Global vanishing hypothesis). The L-function L(E, ϱ_{gh}, s) vanishes at s = 1, and hence the trilinear form I_{fgh} in (3) is identically zero.

Under Hypotheses A and B, the L-function L(E, ϱ_{gh}, s) vanishes to order at least 2 at s = 1. When L′′(E, ϱ_{gh}, 1) ≠ 0, the elliptic Stark conjecture of this article expresses the values of certain p-adic iterated integrals attached to the triple (f, g, h) in terms of the formal group logarithms of global points in the ϱ_{gh}-isotypic subspace E(H)_{L}^{ϱ_{gh}} of E(H) ⊗ L.

The definition of these p-adic iterated integrals is based on the theory of p-adic and overconvergent modular forms. Let p be a prime number and fix an embedding of ℚ̄ in ℚ̄_{p}. Let ℂ_{p} denote the completion of ℚ̄_{p}, and let H_{p} and L_{p} denote the p-adic closure of H and L in ℂ_{p} with respect to the fixed embedding. Assume that p is chosen in such a way that

ord_{p}(N_{f}) ≤ 1 and p ∤ N_{g}N_{h}, and set N := the prime-to-p part of N.  (4)

(This hypothesis is made primarily for ease of exposition, and would eventually be interesting to relax.)

The symbols S_{k}^{(p)}(N, χ) ⊂ M_{k}^{(p)}(N, χ) refer to the spaces of p-adic cusp forms (resp. modular forms) of weight k, level N and character χ with coefficients in ℂ_{p}, while S_{k}^{oc}(N, χ) ⊂ M_{k}^{oc}(N, χ) stand for the corresponding subspaces of overconvergent modular forms. The relations between these various spaces are summarised in the following sequence of natural inclusions of ℂ_{p}-vector spaces

\xymatrix{S_{k}(Np, χ)_{ℂ_{p}} \ar@{^{(}->}[r] \ar@{^{(}->}[d] & S_{k}^{oc}(N, χ) \ar@{^{(}->}[r] \ar@{^{(}->}[d] & S_{k}^{(p)}(N, χ) \ar@{^{(}->}[d] \\ M_{k}(Np, χ)_{ℂ_{p}} \ar@{^{(}->}[r] & M_{k}^{oc}(N, χ) \ar@{^{(}->}[r] & M_{k}^{(p)}(N, χ).}

These spaces are all endowed with an action of the Hecke algebra 𝕋_{N}^{(p)} generated by the Hecke operators T_{ℓ} with ℓ ∤ Np and by the Hecke operator U_{p}. Let e_{ord} := lim_{n} U_{p}^{n!} be Hida’s ordinary idempotent and write M^{ord} = e_{ord} · M, for any L_{p}[U_{p}]-module M. While the spaces S_{k}^{oc}(N, χ) and M_{k}^{oc}(N, χ) are infinite-dimensional over ℂ_{p}, their ordinary parts S_{k}^{oc,ord}(N, χ) and M_{k}^{oc,ord}(N, χ) are known to be finite-dimensional. In fact, Hida’s classicality theorem asserts that, for all k ≥ 2, the natural inclusion

M_{k}^{ord}(Np, χ)_{ℂ_{p}} ↪ M_{k}^{oc,ord}(N, χ)

is an isomorphism.

Serre’s differential operator

d = q \frac{d}{dq} : M_{k}^{(p)}(N, χ)⟶M_{k+2}^{(p)}(N, χ)

on p-adic modular forms sends overconvergent forms of weight 0 to overconvergent forms of weight 2. For any f̆ ∈ S_{2}^{oc}(N), the overconvergent modular function (or modular form of weight 0)

F̆ := d^{−1}f̆:= lim_{t→−1} d^{t}f̆ ∈ S_{0}^{oc}(N)

is called the overconvergent primitive of f̆. Here the limit is taken over positive integers t tending to −1 in weight space ℤ/(p−1)ℤ×ℤ_{p}. The Fourier expansions of f̆ and F̆ are related by

f̆(q) = ∑_{n≥1} a_{n}(f̆)q^{n},  F̆(q) = ∑_{p∤n} \frac{a_{n}(f̆)}{n}q^{n}.

For any k ≥ 2, the p-adic iterated integral attached to the data

(f̆, γ̆, h̆) ∈ S_{2}(Np)_{L} × M_{k}^{ord}(Np, χ)_{L}^{∨} × M_{k}(Np, χ)_{L}

(in which the superscript ∨ denotes the L-linear dual) is defined to be

∫γ̆f̆· h̆ := γ̆(e_{ord}(F̆ · h̆)) ∈ ℂ_{p}.  (5)

Expressions of this sort with k ≥ 2 play an important role in [DR1] and [DR2] where they are related (when f̆, γ̆ and h̆ are cuspidal eigenvectors) to the special values of Garrett-Hida triple product p-adic L-functions and to the images of certain generalised Gross-Schoen diagonal cycles under the p-adic Abel-Jacobi map. When γ̆ and/or h̆ arise from weight two Eisenstein series, these expressions are also related to p-adic regulators of Beilinson elements in the K-groups K_{2}(X_{1}(N)) or in K_{1}(X_{1}(N)^{2}) (cf. [BD2] and [BDR1], [BDR2] respectively).

On the computational side, using a method for computing with overconvergent modular forms using Katz expansions developed in [La1], the article [La2] describes an algorithm for the efficient numerical evaluation of these p-adic iterated integrals, and uses it to calculate certain Chow-Heegner points on the elliptic curve E which were first defined and studied by Shouwu Zhang. These global points, which arise when k ≥ 2 and γ̆ and h̆ are eigenvectors for 𝕋_{N} with the same eigenvalues, have a well-understood geometric provenance, and can also be calculated by complex analytic means following the strategy described in [DRS] and [DDLR].

The setting k = 1 which is germane to the present work is significantly more mysterious: the p-adic iterated integrals (and indeed, the modular forms g and h themselves) cease to enjoy geometric interpretations in terms of the cohomology of Kuga-Sato varieties. On a more technical level, the space M_{1}^{oc,ord}(N, χ), while still finite-dimensional, may contain the space M_{1}^{ord}(Np, χ)_{ℂ_{p}} as a proper subspace. Fix a root α_{g} of the Hecke polynomial

x^{2} − a_{p}(g)x + χ^{−1}(p) = (x − α_{g})(x − β_{g})

attached to g at p, and let g_{α} ∈ M_{1}(Np, χ^{−1})_{L} denote the corresponding p-stabilisation. It is an eigenform for the Hecke algebra 𝕋_{N}^{(p)} satisfying

U_{p}g_{α} = α_{g} · g_{α}.

The weight one form g is said to be regular at p if α_{g} ≠ β_{g}, i.e., if g admits two distinct p-stabilisations. Let g_{α}^{∗} = g_{α} ⊗ χ denote the newform of level N_{g} corresponding to the twist of g_{α} by χ, the inverse of its nebentype character.

If ϕ is a simultaneous eigenvector for either the Hecke algebra 𝕋_{N}^{(p)} or 𝕋_{Np}, then, extending the definitions in (2), let

M[ϕ] := Ker(I_{ϕ}), M[[ϕ]] = ∪_{n≥1}Ker(I_{ϕ}^{n})

denote the eigenspace and generalized eigenspace attached to its system of eigenvalues. The following assumption is a crucial substitute for the classicality theorem when k = 1:

Hypothesis C (Classicality property for g_{α}). The generalised eigenspace S_{1}^{oc,ord}(N, χ)_{ℂ_{p}}[[g_{α}^{∗}]] consists entirely of classical forms, and g satisfies one of the following explicit conditions:

(1) It is a cusp form which is regular at the prime p, and is not the theta series of a character of a real quadratic field in which p splits. (So that in particular, ϱ_{g} is irreducible and the frobenius element at p acts on it with two distinct eigenvalues.)

(2) It is an Eisenstein series which is irregular at p. (In other words, ϱ_{g} is the direct sum of two characters χ_{1} and χ_{2} for which χ_{1}(p) = χ_{2}(p).)

In the case where g satisfies condition (1) of Hypothesis C, Proposition 1.1 of Section 1.1 proves, as an immediate consequence of the work of Cho-Vatsal [CV] and Bellaïche-Dimitrov [BeDi], that all the forms in S_{1}^{oc,ord}(N, χ)_{ℂ_{p}}[[g_{α}^{∗}]] are classical, i.e., that the natural inclusion

i_{g_{α}} : S_{1}(Np, χ)_{ℂ_{p}}[[g_{α}^{∗}]] ↪ S_{1}^{oc,ord}(N, χ)[[g_{α}^{∗}]]  (6)

is an isomorphism of ℂ_{p}-vector spaces. Let us mention in passing that i_{g_{α}} is not an isomorphism when g is cuspidal and irregular at p, or is the theta series of a real quadratic field in which p splits. (The extension of Conjecture ES to these settings, which is far from routine and involves significant new features, will be taken up in future work [DLR2].)

In the case were g satisfies condition (2) of Hypothesis C, Proposition 1.2 of Section 1.1 below shows that g_{α} is a p-adic cusp form, i.e., the generalised eigenspace S_{1}^{oc,ord}(N, χ)[[g_{α}^{∗}]] is non-zero. While the larger space M_{1}^{oc,ord}(N, χ)[[g_{α}^{∗}]] always contains non-classical modular forms in this setting (whose fourier expansions can be written down explicitly), it nonetheless seems plausible that S_{1}(N, χ)[[g_{α}^{∗}]] should consist solely of classical forms, i.e., that the natural map

i_{g_{α}} : M_{1}(Np, χ)_{ℂ_{p}}[[g_{α}^{∗}]] ↪ S_{1}^{oc,ord}(N, χ)[[g_{α}^{∗}]]  (7)

should be an isomorphism. The analogue of the theorem of Bellaiche-Dimitrov for irregular weight one Eisenstein points is not available in the literature, and the assumption that i_{g_{α}} is an isomorphism has therefore been built into the statement of Hypothesis C.

Thanks to the assumption that the inclusions in (6) and (7) are isomorphisms, it becomes possible to adapt the definition of the p-adic iterated integrals to define a ℂ_{p}-valued trilinear form

I′_{p} : S_{2}(Np)_{L}[f] × M_{1}(Np, χ)_{L}^{∨}[g_{α}] × M_{1}(Np, χ)_{L}[h] ⟶ ℂ_{p}  (8)

by the simple rule

I′_{p}(f̆, γ̆, h̆) = ∫γ̆f̆· h̆ := γ̆(e_{g_{α}^{∗}}e_{ord}(F̆ · h̆)),

where

e_{g_{α}^{∗}} : S_{1}^{oc,ord}(N, χ)⟶S_{1}^{oc,ord}(N, χ)[[g_{α}^{∗}]] = M_{1}(Np, χ)_{ℂ_{p}}[[g_{α}^{∗}]]  (9)

is the projector onto the generalized eigenspace attached to g_{α}^{∗} for the action of the Hecke algebra 𝕋_{N}^{(p)}. Note the crucial role played by Hypothesis C in the definition of this trilinear form via the equality in (9).

We are now almost ready to state Conjecture ES relating the trilinear form in (8) to arithmetic data associated to (E, ϱ_{g}, ϱ_{h}). Since the prime p is unramified in H, there is an embedding H⟶ℚ_{p}^{ur} into the maximal unramified extension of ℚ_{p}. Fix such an embedding and denote by σ_{p} ∈ Gal (H/ℚ) the associated frobenius element at p. If F is any finite extension of ℚ_{p}, with ring of integers O_{F}, let

log_{p} : O_{F}^{×} ⟶ F

be the usual p-adic logarithm, and let

log_{E,p} : E(F)⟶F

denote the canonical extension to E(F) of the p-adic formal group logarithm associated to a Néron differential on E. Via the chosen embedding of H into ℂ_{p} the maps log_{p} and log_{E,p} give rise to homomorphisms

log_{p} : (O_{H})_{L}^{×}⟶H_{p} ⊗ L, log_{E,p} : E(H)_{L}⟶H_{p} ⊗ L.

Hypotheses A and B together imply that r_{an}(E, ϱ_{gh}) ≥ 2, and of course the same is expected of the algebraic rank. When r(E, ϱ_{gh}) = 2, let (Φ_{1}, Φ_{2}) be an L-vector space basis for Hom_{G_{ℚ}}(V_{gh}, E(H)_{L}).

Using Hypothesis C, we make a choice of a one dimensional G_{ℚ_{p}}-stable subspace of V_{g}, denoted V_{g}^{g_{α}}, according to the following recipe:

(1) When condition (1) in Hypothesis C is satisfied, the frobenius element σ_{p} acts on the Artin representations V_{g} with distinct eigenvalues α_{g} and β_{g}, and thus determines a decomposition V_{g} = V_{g}^{g_{α}} ⊕ V_{g}^{g_{β}} into one-dimensional σ_{p}-eigenspaces with eigenvalue α_{g} and β_{g} respectively.

(2) When condition (2) in Hypothesis C is satisfied, we simply let V_{g}^{g_{α}} be any onedimensional subspace of the (reducible) representation V_{g} which is not stable under G_{ℚ}.

The choice of V_{g}^{g_{α}} then determines a canonical two-dimensional subspace of V_{gh} by the rule

V_{gh}^{g_{α}} := V_{g}^{g_{α}} ⊗ V_{h}.

Fix an L-vector space basis (v_{1}, v_{2}) for this space.

The bases (Φ_{1}, Φ_{2}) and (v_{1}, v_{2}) give rise to a 2 × 2 regulator matrix

R_{g_{α}}(E, ϱ_{gh}) := ( \begin{matrix}log_{E,p}(Φ_{1}(v_{1})) & log_{E,p}(Φ_{1}(v_{2})) \\ log_{E,p}(Φ_{2}(v_{1})) & log_{E,p}(Φ_{2}(v_{2})) \end{matrix} ) ,

which is independent of the choice of basis (Φ_{1}, Φ_{2}) (resp., of the basis (v_{1}, v_{2})) up to left (resp., right) multiplication by a matrix in GL_{2}(L). The determinant of R_{g_{α}}(E, ϱ_{gh}) is therefore well defined, up to multiplication by L^{×}. It can also be shown that this determinant does not depend on the choice of V_{g}^{g_{α}} when Condition (2) of Hypothesis C is satisfied. In both cases (1) and (2), it belongs to ℚ_{p}^{ur} ⊗ L, and σ_{p} acts on it with the same eigenvalue as on ∧^{2}V_{gh}^{g_{α}}, namely, α_{g}α_{h}α_{g}β_{h} = α_{g}/β_{g}.

Let Ad_{g} := Hom^{0}(V_{g}, V_{g}) denote the three-dimensional adjoint representation attached to ϱ_{g}, on which the frobenius element σ_{p} acts with eigenvalues 1, α_{g}/β_{g}, and β_{g}/α_{g}. Section 1.2 attaches to g_{α} a Stark unit u_{g_{α}} ∈ (O_{H_{g}}[1/p]^{×})_{L}^{Ad_{g}}, where H_{g} is the number field cut out by Ad_{g}.

(1) When g satisfies condition (1) in Hypothesis C and α_{g}/β_{g} ≠ −1, this unit is determined up to scaling by the twin requirements that

u_{g_{α}} belongs to (O_{H_{g}}^{×})_{L}^{Ad_{g}}, σ_{p}(u_{g_{α}}) = \frac{α_{g}}{β_{g}} u_{g_{α}}.

A more detailed discussion of u_{g_{α}} is given in Section 1.2. The unit u_{g_{α}} is analogous to the Stark units of [St2] and [DuLi] whose logarithms encode the Petersson scalar product of g_{α} with itself. Section 1.2 explains the striking connection between the existence of such a unit and Hypothesis C.

(2) When g satisfies condition (2) in Hypothesis C, then

u_{g_{α}} belongs to (O_{H_{g}}[1/p]^{×})_{L}^{Ad_{g}}.

Note that in this case H_{g} is the abelian extension cut out by the (odd) Dirichlet character χ_{1}/χ_{2}, in which the prime p splits completely.

The main conjecture of the present work is

Conjecture ES: (Elliptic Stark Conjecture). Assume that hypotheses A,B, and C are satisfied, so that in particular r_{an}(E, ϱ_{gh}) ≥ 2. If r(E, ϱ_{gh}) > 2, the trilinear form I′_{p} of (8) is identically zero. Otherwise, there exist test vectors

(f̆, γ̆, h̆) ∈ S_{2}(Np)_{L}[f] × M_{1}(Np, χ)_{L}^{∨}[g_{α}] × M_{1}(Np, χ)_{L}[h]

for which

∫γ̆f̆· h̆ = \frac{det(R_{g_{α}}(E, ϱ_{gh}))}{log_{p}(u_{g_{α}})}.  (10)

Note that both the numerator and denominator on the right-hand side of (10) belong to the same eigenspace for σ_{p}, with eigenvalue α_{g}/β_{g}, and hence that the ratio belongs to L_{p} ⊂ H_{p} ⊗ L, consistent with the fact that this is clearly true of the left-hand side.

The condition r(E, ϱ_{gh}) = 2 implies that either:

(1) the Artin representation ϱ_{gh} admits a decomposition

ϱ_{gh} = ϱ_{1} ⊕ ϱ_{2},  with r(E, ϱ_{1}) = 2, r(E, ϱ_{2}) = 0,  (11)

where ϱ_{1} is an irreducible constituent of ϱ_{gh} and ϱ_{2} is possibly trivial, or reducible;

(2) the Artin representation ϱ_{gh} admits a decomposition

ϱ_{gh} = ϱ_{1} ⊕ ϱ_{2} ⊕ ϱ_{3},  with r(E, ϱ_{1}) = r(E, ϱ_{2}) = 1, r(E, ϱ_{3}) = 0,  (12)

where ϱ_{1} and ϱ_{2} are irreducible constituents of ϱ_{gh} and ϱ_{3} is possibly trivial or reducible.

These scenarios will be referred to respectively as the rank 2 and rank (1,1) settings for the elliptic Stark conjecture. The apparent rarity of double zeroes for primitive L-functions at central critical points suggests that rank 2 settings may only occur sporadically, while a systematic supply of rank (1, 1) settings arises when the constituents ϱ_{1} and ϱ_{2} in (12) are self-dual and the signs in the functional equations for L(E, ϱ_{1}, s) and L(E, ϱ_{2}, s) are both equal to −1. This explains why more of the experiments reported on below have tested Conjecture ES in the rank (1, 1) setting (although enough data has been gathered to lend convincing evidence for Conjecture ES in the rank 2 scenario as well).

Another attractive feature of the rank (1, 1) setting is that it often leads to explicit formulae for the p-adic logarithms of global points in E(H)_{L}^{ϱ_{1}} and E(H)_{L}^{ϱ_{2}}. More precisely, the regulator R_{g_{α}}(E, ϱ_{gh}) is said to be factorisable if, after eventually interchanging α_{h} and β_{h} and/or ϱ_{1} and ϱ_{2}, the frobenius σ_{p} acting on V_{ϱ_{1}} admits α_{g}α_{h} but not α_{g}β_{h} as an eigenvalue. Relative to a basis (Φ_{1}, Φ_{2}) ∈ Hom_{G_{ℚ}}(V_{ϱ_{1}}, E(H)_{L}) × Hom_{G_{ℚ}}(V_{ϱ_{2}}, E(H)_{L}) and to a σ_{p}-eigenbasis (v_{1}, v_{2}) for V_{gh}^{g_{α}} satisfying

σ_{p}v_{1} = α_{g}α_{h}v_{1}, σ_{p}v_{2} = α_{g}β_{h}v_{2},

the regulator matrix R_{g_{α}} takes on the simple form

R_{g_{α}}(E, ϱ_{gh}) = ( \begin{matrix}log_{E,p} P_{1} & 0 \\ log_{E,p} Q & log_{E,p} P_{2} \end{matrix} ) ,

where the points P_{j} belong to E(H)_{L}^{ϱ_{j}} and are uniquely determined (up to L^{×}) by the requirement that

σ_{p}P_{1} = α_{g}α_{h}P_{1},  σ_{p}P_{2} = α_{g}β_{h}P_{2}.  (13)

Conjecture ES then admits the simpler formulation

∫γ̆f̆· h̆ = \frac{log_{E,p}(P_{1}) · log_{E,p}(P_{2})}{log_{p}(u_{g_{α}})}.

The prototype for such a formula, discovered experimentally and subsequently proved, is equation (45) in Remark 3.4.

One of the important motivations for studying Conjecture ES is the connection between the p-adic Rankin L-values it considers and the generalised Kato classes

κ(f, g_{α}, h_{α}), κ(f, g_{α}, h_{β}), κ(f, g_{β}, h_{α}), κ(f, g_{β}, h_{β}) ∈ H^{1}(ℚ, V_{p}(E) ⊗ V_{gh})

constructed in [DR2] (resp. in [BDR2]) as p-adic limits of étale Abel-Jacobi images of Gross-Kudla-Schoen diagonal cycles (resp. of étale regulators of Beilinson-Kato and/or Beilinson-Flach elements) when g and h are cusp forms (resp. when g and/or h are Eisenstein series). When L(E, ϱ_{gh}, 1) = 0, these generalised Kato classes can be viewed as playing the role of Heegner points in settings of (analytic) rank two; the reader is referred to Theorems B and D and §6.3. of [DR2] for the relationship between their p-adic logarithms and the p-adic iterated integrals attached to (f, g, h). The calculations performed in this work have also played a crucial role in the more recent preprint [DR3], which conjectures a precise expression for the generalised Kato classes as linear combinations of points in the relevant Mordell-Weil group.

Turning to a more detailed overview of the contents of the paper, Chapter 1 discusses Hypothesis C and defines the Gross-Stark unit u_{g_{α}}, showing that Hypothesis C bears an intimate connection with the existence of this unit.

Building on the results of Harris-Kudla and Ichino on trilinear periods and Jacquet’s conjecture, Chapter 2 recasts the trilinear form I′_{p} as the value of a Garrett-Hida triple product p-adic L-function at a point lying outside its region of classical interpolation. This connection with p-adic L-functions justifies viewing Conjecture ES as an elliptic curve analogue of the (p-adic) Stark conjectures. It also suggests that Conjecture ES might be regarded as a p-adic limit of Jacquet’s conjecture on trilinear periods, prompting a possible strategy for proving it in some cases and suggesting that it might be viewed as a deep manifestation of the “Artin formalism” for certain p-adic L-series.

The rest of the paper is devoted to giving both theoretical and experimental evidence in support of Conjecture ES.

Chapters 3 and 4 treat the case where ϱ_{g} and ϱ_{h} are induced from characters ψ_{g} and ψ_{h} of the same quadratic field K, and with inverse central characters. The representation

V_{gh} = V_{ψ_{1}} ⊕ V_{ψ_{2}}

then decomposes as the direct sum of a pair of representations induced from ring class characters ψ_{1} and ψ_{2} of K.

Chapter 3 focusses on the setting where K is imaginary quadratic. Building on the results of Chapter 2 and the precise formulae of Watson [Wa] and Woodbury [Wo1], Conjecture ES is proved in Section 3.2 under the assumption that the prime p splits in K and a Heegner hypothesis is satisfied. The main ingredients in the proof are the p-adic Gross-Zagier formula of [BDP1] expressing Heegner points in terms of values of a Katz-style p-adic Rankin L-series, the relation established in [DR1] between p-adic iterated integrals and Garrett-Hida p-adic L-series, and a subtle relation between these two types of L-series. It is in the non-trivial comparison between periods used in their definitions that a value of Katz’s two variable p-adic L-function makes an appearance. The unit u_{g_{α}} can be expressed in terms of elliptic units in this case, and emerges from Katz’s p-adic analogue of the Kronecker limit formula.

Katz’s approach to p-adic L-functions (as well as the closely related approach of [BDP1]) is based on the evaluation of p-adic modular forms at CM points when they lie on the ordinary locus of the modular curve, i.e., when p is split in K. This approach breaks down entirely when p is inert in the quadratic imaginary field. Section 3.3 gives numerical evidence in support of Conjecture ES in the inert case, both in the rank (1, 1) setting—where the theory of elliptic units and Heegner points can still be used to produce the global objects that should enter into the right-hand side of (10)—and in the more mysterious rank 2 setting.

Chapter 4 attempts a parallel treatment when ϱ_{g} and ϱ_{h} are induced from characters (of mixed signature) of the same real quadratic field K. In some cases, the global points P_{1} and P_{2} can then be expressed, at least conjecturally, in terms of the Stark-Heegner points of [Dar] and [Gre] attached to “real quadratic cycles on H_{p} × H”. The resulting relation between the Stark points of the present work and these Stark-Heegner points might be amenable to proof via the methods of Section 3.2. The prospect of tying the real quadratic cycles on H_{p} × H to global (but not necessarily motivic) cohomology classes in the Selmer groups of E over ring class fields of real quadratic fields represents an encouraging first step towards better understanding their global nature.

Chapters 5, 6, and 7 consider the remaining interesting cases, all of which appear to lie beyond the reach of previously available theory or conjectural construction.

Chapter 5 reports on a number of experiments in the setting where h is an exotic weight one form, whose associated Galois representation has projective image A_{4}, S_{4} or A_{5}, and g = h^{∗} is its dual. The representation V_{gh} then breaks up as

V_{gh} = L ⊕ Ad_{g}.

In the rank (1, 1) setting, Conjecture ES leads to a p-adic analytic construction of a global point on E over the field cut out by Ad_{g}. The setting where the Hasse-Weil L-series L(E, s) has a double zero at the center and L(E, Ad_{g}, 1) ≠ 0 is particularly interesting: Conjecture ES then predicts the vanishing of the p-adic iterated integrals, even though the generalised Kato class κ(f, g_{α}, \overline{g}ᾱ) is expected to be a canonical, non-trivial element of the pro-p Selmer group of E/ℚ: we refer to [DR3] for a fuller discussion of this setting.

Chapter 6 takes up the setting where g ∈ S_{1}(N, χ̄) is a cusp form and h := E_{1}(1, χ) ∈ M_{1}(N, χ) is the weight one Eisenstein series attached to the pair (1, χ) of Dirichlet characters. A novelty of this setting is that

V_{gh} = V_{g} ⊕ V_{g^{∗}}

decomposes into a direct sum of two representations which are not self-dual in general, and the regulator matrix R_{g_{α}}(E, ϱ_{gh}) involves points in the isotypic parts for these non self-dual representations.

Chapter 7 concludes with the setting in which g := E_{1}(1, χ^{−1}) is an (irregular) Eisenstein series. The case where h := E_{1}(χ_{1}, χ_{2}) is also an Eisenstein series (with χ_{1}χ_{2} = χ) is particularly appealing, since

V_{gh} = χ_{1} ⊕ χ̄_{1} ⊕ χ_{2} ⊕ χ̄_{2}

is then a direct sum of one-dimensional representations. In this setting—the most “degenerate instance” of the one described in Ch. 6—the relevant “generalised Kato classes” coincide with the classes constructed by Kato from p-adic families of Beilinson elements, whose behaviour has a bearing on the arithmetic of elliptic curves over cyclotomic fields. Chapter 7 reports on experimental evidence in support of Conjecture ES, where the regulators attached to (f, g_{α}, h_{α}) involve logarithms of points on E defined over the abelian extension of ℚ cut out by either χ_{1} or χ_{2}. The scenario in which these Dirichlet characters are non-quadratic is particularly intriguing. According to predictions of David, Fearnley and Kisilevsky [DFK], the occurence of global points over cyclic extensions of degree greater than 2 is quite sporadic; the L-series that control them have non-real coefficients, and their functional equation gives no indication as to the parity of their order of vanishing at the center. Although ostensibly rare, global points over cyclic cubic extensions do occur in practice. Fairly extensive tables of pairs (E, χ) with χ cubic for which r_{an}(E, χ) > 0 have been drawn up by the authors of [DFK], and the computationally more challenging task of verifying that r(E, χ) > 0 for such pairs is carried out—albeit in more limited ranges—in [Co]. Section 7.2 describes a few experimental verifications of Conjecture ES involving the p-adic logarithms of such cyclic cubic points.

The article concludes with a table summarizing the main features of the numerical experiments described in this work.

Acknowledgements. The authors thank Kevin Buzzard for illuminating exchanges on weight one modular forms, Mladen Dimitrov for helpful discussions around Section 1.1, Neil Dummigan for his guidance on some classical formulae relating adjoint L-functions and the Petersson norm, Haruzo Hida for a suggestion which helped shape the statement of the conjectures, and Mike Woodbury for sharing his expertise on trilinear periods. The experimental work in this paper was undertaken using the Magma computer algebra package and the authors are very grateful and deeply indebted to Steve Donnelly for his help and advice on a number of computational problems.

The authors gratefully acknowledge the following institutions for their support and the pleasant locales in which their collaboration on this project was pursued: the Mathematics Center Heidelberg (MATCH), McGill’s Bellairs Research Station in Barbados, the CRM and Università Politecnica de Catalunya in Barcelona, the Mathematics Institute at the University of Oxford, the Pohang Mathematics Institute in Pohang, South Korea, and the Centro de Ciencias de Benasque Pedro Pascual in the Spanish Pyrénées.

The first author was supported by an NSERC Discovery grant. The work of the middle author was supported in part by a grant from the European Research Council (204083), and largely undertaken during a sabbatical year at the University of Kyushu—he thanks Shin Hattori, Masanobu Kaneko, and Yuichiro Taguchi for their hospitality. The third author was supported by Grant MTM2012-34611.

1. Overconvergent weight one forms and Gross-Stark units

1.1. Classical and overconvergent modular forms of weight one. One of the main results of this section is the following concrete criterion, following directly from a result of Cho-Vatsal and Bellaiche-Dimitrov, which shows that Hypothesis C is frequently satisfied in practice.

Proposition 1.1. Let g ∈ S_{1}(N, χ^{−1}) be a cusp form of weight one which is regular at p, and let g_{α} denote one of its p-stabilisations. Then the natural inclusion

S_{1}(Np, χ)_{ℂ_{p}}[g_{α}^{∗}] ↪ S_{1}^{oc,ord}(N, χ)[[g_{α}^{∗}]]

is an isomorphism of ℂ_{p}-vector spaces if and only if

ϱ_{g} is not induced from a character of a real quadratic field in which p splits. (14)

Proof. Let 𝕋_{k}(N) denote the ℤ_{p}-algebra generated by the Hecke operators T_{ℓ} (with ℓ ∤ N) and U_{q} (with q|N), together with the diamond operators, acting on the space M_{k}(N) of classical modular forms on Γ_{1}(N). When k ≥ 2, there is a natural perfect duality

𝕋_{k}(N) × M_{k}(N)⟶ℤ_{p}  (15)

defined by ⟨T, f⟩ := a_{1}(T f).

Hida’s Λ-adic Hecke algebra is defined to be

\underline{𝕋}(N) = lim_{←,t}(e_{ord}𝕋_{2}(Np^{t})) ⊗_{ℤ_{p}} ℚ_{p},

where the inverse limit is taken with respect to the natural projection maps. The algebra \underline{𝕋}(N) has a natural structure of a Λ⊗_{ℤ_{p}}ℚ_{p}-module arising from the action of the diamond operators, and it can also be viewed as a module over the Λ ⊗ ℚ_{p}-subalgebra \underline{𝕋}^{0}(N) of \underline{𝕋}(N) generated by the “good” Hecke operators T_{n} with (n, pN) = 1 and by U_{p}. Let φ_{k} : Λ ⊗ ℚ_{p}⟶ℚ_{p} denote the “weight k specialisation map” sending a group-like element ⟨a⟩ ∈ 1 + pℤ_{p} ⊂ Λ^{×} to a^{k−1}, and let ℘_{k} denote its kernel. The pairing of (15) induces a perfect duality

((\underline{𝕋}(N)/℘_{k}) ⊗_{ℚ_{p}} ℂ_{p}) × M_{k}^{oc,ord}(N)⟶ℂ_{p}.

In particular, the space M_{1}^{oc,ord}(N) is naturally identified with the vector space of ℂ_{p}-linear homomorphisms Hom(\underline{𝕋}(N)/℘_{1}, ℂ_{p}). Under this identification, the classical weight one form g_{α} ∈ M_{1}^{oc,ord}(N), which is a normalised eigenvector for the good Hecke operators T_{n} with (n, Np) = 1 and for U_{p}, corresponds to an algebra homomorphism

φ_{g_{α}} ∈ Hom_{alg}(\underline{𝕋}^{0}(N)/℘_{1}, ℂ_{p}).

Let ℘_{g_{α}} ⊂ \underline{𝕋}^{0}(N) be the kernel of φ_{g_{α}}. Its image under the “projection map to weight space”

Hom_{alg}(\underline{𝕋}^{0}(N), ℂ_{p})⟶ Hom_{alg}(Λ, ℂ_{p})

induced by restriction is equal to φ_{1}, i.e., ℘_{g_{α}} contains the ideal ℘_{1}\underline{𝕋}^{0}(N).

The eigenspace M_{1}^{oc,ord}(N)[g_{α}^{∗}] that arises in the statement of Proposition 1.1 can be identified with the dual of \underline{𝕋}(N)/℘_{g_{α}}:

M_{1}^{oc,ord}(N)[g_{α}^{∗}] = Hom(\underline{𝕋}(N)/℘_{g_{α}}, ℂ_{p}).

Furthermore,

M_{1}^{oc,ord}(N)[[g_{α}^{∗}]] = Hom(\underline{𝕋}(N)_{℘_{gα}} /℘_{1}, ℂ_{p}),

where

\underline{𝕋}^{0}(N)_{℘_{gα}} := lim_{←,n} \underline{𝕋}^{0}(N)/℘_{g_{α}}^{n}, \underline{𝕋}(N)_{℘_{gα}} := lim_{←,n} \underline{𝕋}(N)/℘_{g_{α}}^{n}

denote the completions at the maximal ideal ℘_{g_{α}} of \underline{𝕋}^{0}(N) ⊗ ℂ_{p}.

Hence the dual of the natural inclusion M_{1}^{oc,ord}(N)[g_{α}^{∗}] ↪ M_{1}^{oc,ord}(N)[[g_{α}^{∗}]] is the natural surjection

\underline{𝕋}(N)_{℘_{gα}} /℘_{1}⟶\underline{𝕋}(N)/℘_{g_{α}}.  (16)

The ideal ℘_{1}\underline{𝕋}^{0}(N)_{℘_{gα}} is equal to a positive power ℘_{g_{α}}^{t} (with t ≥ 1) of the maximal ideal ℘_{g_{α}} in the local ring \underline{𝕋}^{0}(N)_{℘_{gα}} ⊗ ℂ_{p}. The map (16) is an isomorphism if and only if t = 1, i.e., if and only if the point of Hom(\underline{𝕋}^{0}(N)_{℘_{g}}, ℂ_{p}) attached to φ_{g_{α}} is smooth and étale over weight space. Under the regularity assumption on g at p, Theorem 1.1. of [BeDi] implies that this étaleness condition is satisfied precisely when (14) holds. (The case where g an Eisenstein series does not appear to be included in the statement of this theorem, but the result is well-known in this case and can be proved by similar techniques.) The proposition follows. □

We next turn to the case where g = E_{1}(χ_{1}, χ_{2}) is a weight one Eisenstein series with α_{g} := χ_{1}(p) and β_{g} := χ_{2}(p), and let

g_{α}(q) := g(q) − χ_{2}(p)g(q^{p}), g_{β}(q) := g(q) − χ_{1}(p)g(q^{p})

denote its (not necessarily distinct) p-stabilisations.

Proposition 1.2. The eigenform g satisfies condition (2) of Hypothesis C if and only if g_{α} = g_{β} is a p-adic cusp form, i.e., S_{1}^{oc,ord}(N, χ)_{ℂ_{p}}[[g_{α}^{∗}]] is non-zero.

Proof. Assume for simplicity that χ_{1} and χ_{2} are primitive with relatively prime conductors N_{1} and N_{2}, so that χ := χ_{1}χ_{2} is primitive and N = N_{1}N_{2}. We show that S_{1}^{oc}(N, χ)[g_{α}^{∗}] ≠ 0 by proving that g_{α} ∈ M_{1}(Np, χ) is a p-adic cusp form, i.e, has zero constant terms at the cusps in the ordinary locus of the modular curve X(N; p) = X_{1}(N) ×_{X(1)} X_{0}(p) associated to the congruence group Γ_{1}(N) ∩ Γ_{0}(p). More precisely, let X_{1}(N)_{ℂ_{p}}^{ord} denote the complement of the supersingular residue discs in X_{1}(N)(ℂ_{p}), regarded as an affinoid domain. For a suitable wide open neighbourhood W of X_{1}(N)_{ℂ_{p}}^{ord}, let

s : W⟶X(N; p)_{ℂ_{p}}

be the rigid analytic section of the natural projection X(N; p)⟶X_{1}(N), defined by sending the point attached to an ordinary pair (E, P_{N})_{/ℂ_{p}} ∈ W to the (moduli of the) triple (E, P_{N}, C_{can}), where P_{N} is a point of order N on E and C_{can} denotes the canonical subgroup of order p. The inclusion i_{g_{α}} of (7) in the Introduction is described geometrically in terms of s, as follows: letting \underline{ω} denote the sheaf over X(N; p) with logarithmic poles at the cusps discussed in [BDP1, §1.1], a classical weight one form ϕ in the source of (7) may be viewed as a section of \underline{ω} over X(N; p), and the map of (7) sends it to its pull-back s^{∗}(ϕ) ∈ H^{0}(W, \underline{ω}), naturally identified with an overconvergent modular form of level N.

The non-zero constant terms of the classical form g = E_{1}(χ_{1}, χ_{2}) are concentrated at the cusps of X_{1}(N) attached to the “Katz test objects”

c_{ab} := (𝔾_{m}/q^{ℤ}, ζ_{N_{1}}^{a}q^{b/N_{2}}), d_{ab} := (𝔾_{m}/q^{ℤ}, q^{a/N_{1}}ζ_{N_{2}}^{b}),

where 𝔾_{m}/q^{ℤ} is the Tate generalised elliptic curve over Spec(ℂ_{p}[[q]]), and ζ_{N_{1}} and ζ_{N_{2}} are fixed primitive N_{1} and N_{2}-th roots of unity; the indices (a, b) range over (ℤ/N_{1}ℤ)^{×} × (ℤ/N_{2}ℤ)^{×}. If g is any weight one modular form on Γ_{0}(N) with character χ, then the modular form g_{p} on X(N; p) whose q-expansion is equal to g(q^{p}) at the cusp ∞ = (𝔾_{m}/q^{ℤ}, ζ_{N}, μ_{p}) is defined on test objects of the form (E, P_{N}, C_{p}), with C_{p} ⊂ E of order p, by the rule

g_{p}((E, P_{N}, C_{p})) := χ^{−1}(p)p^{−1}φ_{C_{p}}^{∗}(g(E/C_{p}, P_{N} + C_{p})),

where φ_{C_{p}} : E⟶E/C_{p} is the natural quotient morphism. This can be seen by checking this identity on the test object (𝔾_{m}/q^{ℤ}, ζ_{N}, μ_{p}) and invoking the q-expansion principle. Let g_{ab}(q) and g′_{ab}(q) ∈ ℂ_{p}[[q^{1/N}]] denote the q-expansions of g at the cusps c_{ab} and d_{ab} respectively, and let x_{ab} and y_{ab} denote the associated leading terms. The q-expansion of g_{p}, viewed as a p-adic modular form, at the cusps c_{ab} and d_{ab} is then given by the rule

g_{p}(𝔾_{m}/q^{ℤ}, ζ_{N_{1}}^{a}q^{b/N_{2}}) = χ^{−1}(p)p^{−1}φ^{∗}(g(𝔾_{m}/q^{pℤ}, ζ_{N_{1}}^{pa}q^{pb/N_{2}})) = χ_{2}^{−1}(p)g_{ab}(q^{p})\frac{dt}{t},  (17)
g_{p}(𝔾_{m}/q^{ℤ}, q^{a/N_{1}}ζ_{N_{2}}^{b}) = χ^{−1}(p)p^{−1}φ^{∗}(g(𝔾_{m}/q^{pℤ}, q^{pa/N_{1}}ζ_{N_{2}}^{pb})) = χ_{1}^{−1}(p)g′_{ab}(q^{p})\frac{dt}{t}.  (18)

Here we have used the fact the canonical subgroup μ_{p} ⊂ 𝔾_{m}/q^{ℤ} of the Tate elliptic curve is the kernel of the map φ : 𝔾_{m}/q^{ℤ}⟶𝔾_{m}/q^{pℤ} sending t to t^{p}. Equations (17) and (18) yield the constant terms of the forms g, g_{p}, g_{α}, and g_{β} at the cusps c_{ab} and d_{ab}, as summarised in the table below:

c_{ab}  d_{ab}
g  x_{ab}  y_{ab}
g_{p}  χ_{2}^{−1}(p)x_{ab}  χ_{1}^{−1}(p)y_{ab}
g_{α}  0  (1 − χ_{2}/χ_{1}(p))y_{ab}
g_{β} (1 − χ_{1}/χ_{2}(p))x_{ab}  0

The proposition follows from the fact that χ_{1}(p) = χ_{2}(p). □

Remark 1.3. If g is regular, then the fact that S_{1}^{oc,ord}(N, χ)[[g_{α}^{∗}]] = 0 shows that the p-adic iterated integrals attached to (f, g_{α}, h_{α}) vanish identically, since the overconvergent weight one forms e_{ord}(d^{−1}f̆× h̆) belong to S_{1}^{oc,ord}(N, χ). It is not hard to check that the regulator term det(R_{g_{α}}(E, ϱ_{gh})) attached to the subspace V_{g}^{g_{α}} = ℂ_{p}(χ_{1}) always vanishes as well, in harmony with Conjecture ES, which thus amounts to statement 0 = 0 when g is a regular Eisenstein series.

1.2. The Gross-Stark unit attached to a weight one form. The aim of this section is to associate to the p-stabilised eigenform g_{α} a global element u_{g_{α}} ∈ H^{×} ⊗ L (often well-defined up to scaling by L^{×}), whose p-adic logarithm arises in Conjecture ES. This element belongs to the Ad_{g}-isotypic subspace of H^{×} ⊗ L, where

Ad_{g} := End_{0}(V_{g})

is the adjoint representation attached to V_{g}, the 3-dimensional Artin representation consisting of trace zero L-linear endomorphisms of V_{g} on which G_{ℚ} acts by conjugation via ϱ_{g}.

In defining u_{g_{α}}, it shall be convenient to replace the field H of the introduction by the smaller field H_{g} cut out by Ad_{g}, and L by any field over which the representation Ad_{g} can be defined. To give a rough feeling for the nature of u_{g_{α}}, let us mention at the outset that

L · u_{g_{α}} ⊂ \{ \begin{matrix}(O_{H_{g}}^{×})_{L}^{Ad_{g}} & if g is cuspidal, \\ (O_{H_{g}}[1/p]^{×})_{L}^{Ad_{g}} & if g is Eisenstein, \end{matrix}  (19)

and

the frobenius element σ_{p} acts on u_{g_{α}} with eigenvalue α_{g}/β_{g}.

As explained below, these requirements often characterize u_{g_{α}} uniquely up to scaling by L^{×}.

Write σ_{∞} ∈ G := Gal (H_{g}/ℚ) for the involution given by complex conjugation induced by the fixed choice of complex embedding of H_{g} ⊂ ℚ̄. The representation Ad_{g} decomposes as a direct sum of + and − eigenspaces for the action of σ_{∞}, denoted Ad_{g}^{+} and Ad_{g}^{−}. These eigenspaces are of dimensions 1 and 2 respectively:

dim_{L} Ad_{g}^{+} = 1,  dim_{L} Ad_{g}^{−} = 2.  (20)

As suggested by (19), the nature of the invariant u_{g_{α}} depends on whether ϱ_{g} is irreducible or not, and each case will be dealt with in turn.

1.2.1. The case where ϱ_{g} is irreducible. If ϱ_{g} is irreducible, then g is a cuspidal eigenform of weight one. The definition of u_{g_{α}} rests on the following proposition:

Proposition 1.4. (1) The L-vector space Hom_{G_{ℚ}}(Ad_{g}, O_{H_{g}}^{×} ⊗ L) is one-dimensional. (2) The Artin L-function L(Ad_{g}, s) has a simple zero at s = 0.

Proof. The finite group G = Gal (H_{g}/ℚ) acts faithfully on Ad_{g} and is identified with the image of the projective representation attached to V_{g}. Note that G_{∞} = ⟨σ_{∞}⟩ is of order two since Ad_{g} is not totally even. Dirichlet’s unit theorem determines the structure of O_{H_{g}}^{×} ⊗ L as an L[G]-module:

O_{H_{g}}^{×} ⊗ L = (Ind_{G_{∞}}^{G} L) − L,

where the two occurrences of L on the right refer to trivial representations (of G_{∞} and G respectively). The irreducibility of V_{g} implies that the representation Ad_{g} contains no Ginvariant vectors, by Schur’s lemma, and hence

Hom_{G}(Ad_{g}, O_{H_{g}}^{×} ⊗ L) = Hom_{G}(Ad_{g}, Ind_{G_{∞}}^{G} L).

By Frobenius reciprocity,

Hom_{G}(Ad_{g}, Ind_{G_{∞}}^{G} L) = Hom_{G_{∞}}(Ad_{g}, L) = Hom(Ad_{g}^{+}, L).

The first part of the proposition now follows from (20). The second statement follows from the shape of the Γ-factors in the functional equation of L(Ad_{g}, s) determined by (20). □

Choose a basis φ for the one-dimensional vector space Hom_{G}(Ad_{g}, O_{H_{g}}^{×} ⊗ L), and let

(O_{H_{g}}^{×})_{L}^{Ad_{g}} := φ(Ad_{g}) ⊂ O_{H_{g}}^{×} ⊗ L.

The L[G]-module (O_{H_{g}}^{×})_{L}^{Ad_{g}} is of dimension ≤ 3 over L and is abstractly isomorphic to the unique irreducible subrepresentation of Ad_{g} containing Ad_{g}^{+}.

Recall the prime number p chosen in the introduction. The frobenius element σ_{p} acts on Ad_{g} with the (not necessarily different) eigenvalues 1, α_{g}/β_{g} and β_{g}/α_{g}. Define

U_{g_{α}} := \{u ∈ (O_{H_{g}}^{×})_{L}^{Ad_{g}} such that σ_{p}(u) = \frac{α_{g}}{β_{g}} u\} .

Lemma 1.5. Assume that V_{g} is regular. If ϱ_{g} is induced from a character of a real quadratic field in which p splits, then U_{g_{α}} = 0. Otherwise, dim_{L}(U_{g_{α}}) ≥ 1, with equality if either

(1) α_{g} ≠ −β_{g}, i.e., Ad_{g} is regular, or,

(2) ϱ_{g} is induced from a character of a quadratic field in which p is inert.

Proof. This lemma is proved by a case by case analysis, according to the following possibilities for V_{g}:

a) V_{g} has imaginary dihedral projective image;

b) V_{g} has real dihedral projective image;

c) g is an exotic weight one form with projective image A_{4}, S_{4} or A_{5}.

a) Imaginary dihedral forms. Assume that the representation V_{g} is induced from a character ψ_{g} of an imaginary quadratic field K. Then

Ad_{g} = L(χ_{K}) ⊕ V_{ψ},  (21)

where χ_{K} is the quadratic Dirichlet character associated to K and V_{ψ} := Ind_{K}^{ℚ} ψ is the twodimensional representation induced from the ring class character ψ := ψ_{g}/ψ′_{g} of K. Since σ_{∞} acts with eigenvalue −1 on the first factor L(χ_{K}), it follows that (O_{H_{g}}^{×})_{L}^{Ad_{g}} is a subrepresentation of V_{ψ}.

If V_{ψ} is irreducible, then

(O_{H_{g}}^{×})_{L}^{Ad_{g}} ≃ V_{ψ}.

The eigenvalues of σ_{p} on V_{ψ} are α_{g}/β_{g} and β_{g}/α_{g} if χ_{K}(p) = 1, and 1 and −1 = α_{g}/β_{g} if χ_{K}(p) = −1. In either case, dim_{L}(U_{g_{α}}) ≥ 1, with equality if either α_{g}/β_{g} ≠ −1 or if p is inert in K.

If V_{ψ} is reducible, the character ψ is necessarily quadratic, and V_{ψ} further breaks up as a sum of one-dimensional representations attached to odd and even Dirichlet quadratic characters χ^{−} and χ^{+}, respectively, satisfying χ^{+}χ^{−} = χ_{K}. Because of this,

(O_{H_{g}}^{×})_{L}^{Ad_{g}} = (O_{H_{g}}^{×})_{L}^{χ^{+}} ≃ L(χ^{+})

is isomorphic to the one-dimensional representation associated to χ^{+}. The space U_{g_{α}} ⊂ (O_{H_{g}}^{×})_{L}^{χ^{+}} is therefore of dimension at most one, with equality if and only if α_{g}/β_{g} = χ^{+}(p). This equality always holds when p = ℘ ¯℘ splits in K, since in that case

α_{g}/β_{g} = ψ(℘) = ψ( ¯℘) = χ^{+}(p) = χ^{−}(p).

(Note that the regularity assumption α_{g} ≠ β_{g} that is made on g forces all these expressions to be equal to −1 rather than 1.) If p is inert in K, then σ_{p} acts on V_{ψ} with eigenvalues 1 and −1, and hence α_{g}/β_{g} = β_{g}/α_{g} = −1. In that case, either χ^{+}(p) = −1 and dim(U_{g_{α}}) = 1, or p splits in the real quadratic field K^{+} cut out by χ^{+}, and U_{g_{α}} = 0. In this last scenario, the representations V_{ψ} and V_{g} are also induced from a character of the real quadratic field K^{+}, in which p splits.

b) Real dihedral forms. If the representation V_{g} is induced from a character ψ_{g} of signature (+, −) of a real quadratic field K, then the character ψ = ψ_{g}/ψ′_{g} is a totally odd ring class character of K, and cuts out an abelian extension of K of dihedral type. The decomposition (21) continues to hold, but now, since σ_{∞} acts with eigenvalue 1 on L(χ_{K}), it follows that

(O_{H_{g}}^{×})_{L}^{Ad_{g}} ≃ L(χ_{K})  (22)

in all cases. If p splits in K, then χ_{K}(p) = 1 ≠ α_{g}/β_{g}, and hence U_{g_{α}} = 0. If p is inert in K, then χ_{K}(p) = α_{g}/β_{g} = −1, and hence U_{g_{α}} is also one-dimensional, and generated over L by a fundamental unit of the real quadratic field K.

c) Exotic forms. Assume to conclude that the projective image G of the representation V_{g} is isomorphic to A_{4}, S_{4} or A_{5}. Since these groups admit no faithful irreducible representations of dimension < 3, it follows that Ad_{g} is irreducible, and hence

(O_{H_{g}}^{×})_{L}^{Ad_{g}} ≃ Ad_{g}.

The lemma follows directly in this case, since σ_{p} acts on Ad_{g} with eigenvalues 1, α_{g}/β_{g}, and β_{g}/α_{g}. □

Remark 1.6. Note that condition (14) which was shown in Ch. 1.1 to imply Hypothesis C for g_{α} is precisely what guarantees the non-triviality of the group U_{g_{α}}.

Remark 1.7. The scenarios considered in (a) and (b) in the above proof are not mutually exclusive, and can arise simultaneously in certain cases where ψ_{g} is quadratic and ϱ_{g} has image isomorphic to the dihedral group of order 8: some of the numerical examples below focus on such settings.

Definition 1.8. A non-zero element u_{g_{α}} of U_{g_{α}} is called a Stark unit attached to g_{α}.

Remark 1.9. Lemma 1.5 furnishes us with many concrete instances where the unit u_{g_{α}} is well-defined up to multiplication by L^{×}. This happens, for instance, when ϱ_{g} is induced from a character of a real quadratic field K in which p is inert, when the vector space U_{g_{α}} is always generated by the fundamental unit of K. The case where p splits in K is more mysterious, since U_{g_{α}} = 0, and a natural definition of u_{g_{α}} does not seem to be available, consistent with the failure of Hypothesis C.

More generally, dim_{L}(U_{g_{α}}) > 1 occurs precisely when α_{g}/β_{g} = ±1. In the special case where V_{g} is induced from a character of an imaginary quadratic field in which p splits, there is nonetheless a natural candidate for u_{g_{α}} which is well-defined up to scaling by L^{×}: cf. §3.2 for more details.

Remark 1.10. Several results in the literature relate the Petersson norm of g to L′(Ad_{g}, 0), which in turn is expected to admit an elementary expression as the logarithm of units in the subspace of (O_{H_{g}}^{×})_{L}^{Ad_{g}} on which σ_{∞} = 1. See for example §6 of [St1], or Prop. 4.1 of [DuLi]. It is in this sense that the p-adic logarithm of u_{g_{α}} might be viewed as a p-adic avatar of ⟨g_{α}, g_{α}⟩.

1.2.2. The case where ϱ_{g} is reducible. Consider now the case where g corresponds to the Eisenstein series E_{1}(χ_{1}, χ_{2}) ∈ M_{1}(N, χ), where χ_{1} and χ_{2} are odd and even Dirichlet characters satisfying (as motivated in Section 1.1) the condition

χ_{1}(p) = χ_{2}(p).  (23)

In that case,

V_{g} = L(χ_{1}) ⊕ L(χ_{2}), Ad_{g} = L ⊕ L(η) ⊕ L(η^{−1}), where η := χ_{1}/χ_{2}.

Condition (23) implies that η(p) = 1 and hence p splits completely in the abelian extension H_{η}/ℚ cut out by η.

To define u_{g_{α}}, choose a prime ideal ℘ of O_{H_{η}} over p inducing the embedding H_{η} ↪ ℂ_{p} chosen at the outset, and let u_{℘} ∈ O_{H_{η}}[1/p]^{×} be a generator of the principal ideal ℘^{h}, where h is the class number of H_{η}:

℘^{h} = (u_{℘}), u_{℘} ∈ (O_{H_{η}}[1/℘])^{×} (mod O_{H_{η}}^{×}).

The unit u_{g_{α}} is defined as follows (in additive notation):

Definition 1.11. Let

e_{η} := \frac{1}{#G} ∑_{σ∈G} η^{−1}(σ)σ, e_{η^{−1}} := \frac{1}{#G} ∑_{σ∈G} η(σ)σ,

be the usual idempotents in L[G] giving the projection onto the η and η^{−1}-isotypic component for the action of G. Set

u_{g_{α}} = e_{η}u_{℘} + e_{η^{−1}}u_{℘}.

The unit u_{g_{α}} is a linear combination of Gross-Stark units attached to the characters η and η^{−1}, whose p-adic logarithms

log_{℘}(u_{g_{α}}) = \frac{1}{#G} ∑_{σ∈Gal (H_{η}/ℚ)}(η(σ) + η^{−1}(σ)) log_{℘}(σ(u_{℘}))

account for the derivative at 0 of the Kubota-Leopoldt p-adic L-function of η. In this formula, log_{℘} denotes Iwasawa’s logarithm satisfying log_{℘}(p) = 0, composed with the p-adic embedding of H_{η} attached to ℘. It is not hard to see that

(1) Although the unit u_{℘} is only well-defined modulo O_{H_{η}}^{×}, there is no resulting ambiguity in the unit u_{g_{α}}, since η and η^{−1} are odd characters and hence the idempotents e_{η} and e_{η^{−1}} annihilate O_{H_{η}}^{×} ⊗ L.

(2) The unit u_{℘}, and hence also the unit u_{g_{α}}, depends crucially (and not just by scaling by L^{×}!) on the choice of the ideal ℘ above p, but the expression log_{℘} u_{g_{α}} is independent of this choice, up to multiplication by a root of unity.

2. Iterated integrals and the Garrett-Hida p-adic L-function

The main goal of this chapter is to recast the p-adic iterated integral that arose in the introduction as the value of a p-adic L-function at a point which lies outside its region of classical interpolation. This interpretation is crucial for the proof of Theorem 3.3 of Section 3.2, which provides the main piece of theoretical evidence for Conjecture ES in a setting where the p-adic trilinear form I′_{p} does not vanish.

We begin by briefly recalling a few key facts about Garrett’s triple product classical Lfunction and the work of Harris-Kudla, Ichino, Watson and Woodbury on Jacquet’s conjecture on trilinear periods.

Let (k, ℓ, m) be a triple of positive integers and let

g_{k} ∈ M_{k}(N_{g}, χ_{g}), f_{ℓ} ∈ M_{ℓ}(N_{f}, χ_{f}), h_{m} ∈ M_{m}(N_{h}, χ_{h})

be three newforms (assumed to be cuspidal if the weight is at least 2) satisfying χ_{g}χ_{f}χ_{h} = 1. Note that this forces k + ℓ + m to be even. Set N = lcm(N_{f}, N_{g}, N_{h}) and write ℚ(g_{k}, f_{ℓ}, h_{m}) for the number field generated by the Fourier coefficients of the three forms.

Let L(g_{k} ⊗ f_{ℓ} ⊗ h_{m}, s) denote Garrett’s triple-product L-function associated to (g_{k}, f_{ℓ}, h_{m}). It is a self-dual L-function defined by an Euler product of degree 8 and satisfying a functional equation with central critical point c = \frac{k+ℓ+m−2}{2} and root number

ε = ∏_{v|N∞} ε_{v}(g_{k} ⊗ f_{ℓ} ⊗ h_{m}),  with  ε_{v}(g_{k} ⊗ f_{ℓ} ⊗ h_{m}) ∈ {±1}.

The archimedean sign ε_{∞} is equal to 1 if and only if the triple (k, ℓ, m) of weights is unbalanced, i.e., one of the weights is greater or equal to the sum of the other two. If v | N is a finite place, the main result of [Pr] asserts that ε_{v}(g_{k} ⊗ f_{ℓ} ⊗ h_{m}) = +1 if and only if there exists a GL_{2}(ℚ_{v})-invariant linear form in the tensor product π_{f,v} ⊗ π_{g,v} ⊗ π_{h,v} of local admissible representations of GL_{2}(ℚ_{v}) associated to the three eigenforms. This is the case if v ∤ gcd(N_{f}, N_{g}, N_{h}), or if one of π_{f,v}, π_{g,v} or π_{h,v} does not belong to the discrete series.

Motivated by the Ph. D thesis of D. Prasad, Jacquet formulated a conjecture suggesting a necessary and sufficient criterion for the non-vanishing of the central critical value L(g_{k} ⊗f_{ℓ} ⊗ h_{m}, c) in terms of certain trilinear periods. This conjecture was proved by Harris and Kudla in [HaKu], and several authors have subsequently refined this result by providing explicit formulae relating the L-value to the periods.

In order to describe the precise relationships between L-values and trilinear periods, we assume for the rest of this chapter that

ε_{v}(g_{k} ⊗ f_{ℓ} ⊗ h_{m}) = +1 for all v | N∞.  (24)

When (k, ℓ, m) = (1, 2, 1) and (g_{1}, f_{2}, h_{1}) = (g, f, h) are the modular forms considered in the introduction, then L(g_{k} ⊗f_{ℓ} ⊗h_{m}, s) = L(E/ℚ, ϱ_{gh}, s) is the Hasse-Weil-Artin L-function considered there, and (24) is just Hypothesis A. The above discussion yields sufficient conditions for Hypothesis A to hold, and shows that this is not a very stringent assumption, in practice. Condition (24) at v = ∞ implies that the triple (k, ℓ, m) is unbalanced; assume for the sake of definiteness that k ≥ ℓ + m from now on.

As in the introduction, let

S_{k}(N, χ_{g})[g_{k}], S_{ℓ}(N, χ_{f})[f_{ℓ}], S_{m}(N, χ_{h})[h_{m}]

denote the eigenspaces corresponding to the three forms in level N with respect to the Hecke operators {T_{n}, n ∤ N}.

For any w ≥ 1 and any Dirichlet character χ of conductor N, let S_{w}^{nh}(N, χ) denote the space of nearly holomorphic modular forms of weight w, level N and nebentype χ, and let

δ = \frac{1}{2πi} (\frac{d}{dz} + \frac{w}{z − z̄}) : S_{w}^{nh}(N, χ)⟶S_{w+2}^{nh}(N, χ)

denote the Shimura-Maass operator, as described for instance in [Hi2, Ch. X]. Let also

⟨ , ⟩ : S_{w}^{nh}(N, χ) × S_{w}^{nh}(N, χ)⟶ℂ, ⟨f_{1}, f_{2}⟩ = ∫_{Γ_{1}(N)\ℌ} \overline{f_{1}(z)} f_{2}(z) y^{w} \frac{dxdy}{y^{2}} ,

denote the Petersson scalar product on these spaces.

Set t = (k − ℓ − m)/2 ≥ 0. A test vector for the triple (g_{k}, f_{ℓ}, h_{m}) is a triplet

(ğ_{k}, f̆_{ℓ}, h̆_{m}) ∈ S_{k}(N, χ_{g})[g_{k}] × S_{ℓ}(N, χ_{f})[f_{ℓ}] × S_{m}(N, χ_{h})[h_{m}].

For such a test vector, recall that we denote g_{k}^{∗} = g_{k} ⊗ χ_{g}^{−1}, and define the quantities

I(ğ_{k}, f̆_{ℓ}, h̆_{m}) := ⟨ğ_{k}^{∗}, δ^{t}(f̆_{ℓ}) · h̆_{m}⟩

as well as

J(ğ_{k}, f̆_{ℓ}, h̆_{m}) := I(ğ_{k}, f̆_{ℓ}, h̆_{m})/⟨g_{k}^{∗}, g_{k}^{∗}⟩,  ω(ğ_{k}, f̆_{ℓ}, h̆_{m}) := I(ğ_{k}^{∗}, f̆_{ℓ}^{∗}, h̆_{m}^{∗})/I(ğ_{k}, f̆_{ℓ}, h̆_{m}). (25)

The following result is a direct consequence of the fundamental work of Ichino [Ich]. The precise formulation given here is only valid under assumption (24) and follows by combining loc. cit. with a formula due to Hida for a special value of the L-function of the adjoint of a modular form and the precise formulae of Watson [Wa] and Woodbury [Wo1], [Wo2].

Proposition 2.1. a) For any test vector (ğ_{k}, f̆_{ℓ}, h̆_{m}),

|I(ğ_{k}, f̆_{ℓ}, h̆_{m})|^{2} = 𝔞(k, ℓ, m) · 𝔣(k, ℓ, m) · ∏_{v|N∞} C_{v}(ğ_{k}, f̆_{ℓ}, h̆_{m}) · L (g_{k} ⊗ f_{ℓ} ⊗ h_{m}, c)  (26)

where i) 𝔞(k, ℓ, m) = (\frac{k+ℓ+m−4}{2})! · (\frac{k+ℓ−m−2}{2})! · (\frac{k−ℓ+m−2}{2})! · (\frac{k−ℓ−m}{2})! · π^{−2k}. ii) 𝔣(k, ℓ, m) = A · 2^{B} where A ∈ ℚ^{×} depends only on the levels of the eigenforms and of their characters, B = B(k, ℓ, m) ∈ ℤ is a linear function on the weights k,ℓ,m. iii) for all v | N∞, the local term C_{v}(ğ_{k}, f̆_{ℓ}, h̆_{m}) ∈ ℚ(g_{k}, g_{ℓ}, h_{m}) depends only on the admissible representations of GL_{2}(ℚ_{v}) associated to the three cuspforms and on the local components at v of the choice of (ğ_{k}, f̆_{ℓ}, h̆_{m}).

b) There exists a test vector (ğ_{k}, f̆_{ℓ}, h̆_{m}) for which C_{v}(ğ_{k}, f̆_{ℓ}, h̆_{m}) ≠ 0 for all v | N∞.

c) Assume N is square-free. Then there exists a particular test vector (ğ_{k}, f̆_{ℓ}, h̆_{m}) such that W_{v} := C_{v}(ğ_{k}, f̆_{ℓ}, h̆_{m}) ∈ ℚ^{×} is a rational constant for all v | N.

Proof. Let Γ_{ℝ}(s) := π^{−s/2}Γ(s/2) and Γ_{ℂ}(s) := 2(2π)^{−s}Γ(s), where Γ(s) is the usual gamma function. For any eigenform φ ∈ S_{w}(N, χ), let L(Ad(φ), s) = L(sym^{2}(φ), χ^{−1}, s) denote the L-function of the adjoint square of φ and let

L^{∗}(Ad(φ), s) = L_{∞}(Ad(φ), s)L(Ad(φ), s), with L_{∞}(Ad(φ), s) = Γ_{ℝ}(s − w + 2)Γ_{ℂ}(s)

denote the completed L-function. The Rankin-Selberg method relates the special value of L(Ad(φ), s) at s = w to the Petersson norm of φ. A completely explicit formula is given in [Hi1, Theorem 5.1], which, combined with the value of L_{∞}(Ad(φ), s) at s = w, yields

L^{∗}(Ad(φ), w) = 2^{w}A_{φ} × ⟨φ, φ⟩  (27)

where A_{φ} ∈ ℚ^{×} is a constant which only depends on the level of φ and conductor of χ. Likewise, let

L^{∗}(g_{k} ⊗ f_{ℓ} ⊗ h_{m}, s) := L_{∞}(g_{k} ⊗ f_{ℓ} ⊗ h_{m}, s)L(g_{k} ⊗ f_{ℓ} ⊗ h_{m}, s)

denote the completed Garrett L-function, where

L_{∞}(g_{k} ⊗ f_{ℓ} ⊗ h_{m}, s) := Γ_{ℂ}(s)Γ_{ℂ}(s + 1 − m)Γ_{ℂ}(s + 1 − ℓ)Γ_{ℂ}(s + 2 − ℓ − m). (28)

The main result of [Ich] (see also [Wo1, Theorem 2.1] for a reformulation of the same statement) asserts that for any test vector (ğ_{k}, f̆_{ℓ}, h̆_{m}),

\frac{|I(ğ_{k}, f̆_{ℓ}, h̆_{m})|^{2}}{⟨g_{k}, g_{k}⟩ · ⟨f_{ℓ}, f_{ℓ}⟩ · ⟨h_{m}, h_{m}⟩} = C · \frac{∏_{v|N∞} C_{v}(ğ_{k}, f̆_{ℓ}, h̆_{m}) · L^{∗} (g_{k} ⊗ f_{ℓ} ⊗ h_{m}, c)}{L^{∗}(Ad(g_{k}), k)L^{∗}(Ad(f_{ℓ}), ℓ)L^{∗}(Ad(h_{m}), m)}, (29)

where

• C ∈ ℚ^{×} is an absolute constant which only depends on the normalisation taken for the Petersson scalar product;

• for any v | N∞, C_{v}(ğ_{k}, f̆_{ℓ}, h̆_{m}) ∈ ℚ(g_{k}, f_{ℓ}, h_{m}) is as described in iii) above. Combining (29) with (27), and evaluating (28) at s = c it follows that

|I(ğ_{k}, f̆_{ℓ}, h̆_{m})|^{2} = 𝔞(k, ℓ, m) · 𝔣(k, ℓ, m) · ∏_{v|N∞} C_{v} · L (g_{k} ⊗ f_{ℓ} ⊗ h_{m}, c) ,

where we have set 𝔣(k, ℓ, m) = 2^{−3k−ℓ−m+4} ·C ·(A_{g_{k}}A_{f_{ℓ}}A_{h_{m}})^{−1}. This proves a). Statement b) was originally known as Jacquet’s conjecture, and was proved by Harris and Kudla in [HaKu]. As for c), we refer to [Wo1, Theorem 5.1]. □

Remark 2.2. The choice of test vector in c) is a completely explicit one, given by Watson [Wa] in the case N_{g} = N_{f} = N_{h} and more recently by Woodbury [Wo1] for arbitrary squarefree levels. We shall refer to it as the Watson-Woodbury test vector. The resulting value of W_{v} ∈ ℚ^{×} for v | N is also explicit (see [Wa, Theorem 3] and [Wo1, Theorem 5.1] for the recipes), and we shall refer to it as the Watson-Woodbury local constant at v. Note we say nothing about C_{∞}(ğ_{k}, f̆_{ℓ}, h̆_{m}), although having a tight control of it would be desirable. See [Wo2] for the latest developments, which assert that in favorable scenarios (such as the one considered in §3.2 below) the Watson-Woodbury test vector yields W_{∞} = 1.

Up to this point we have summarized the basic material we need about Garrett tripleproduct classical L-functions. Fix as in the introduction a prime number p ∤ N and let us discuss now the p-adic interpolation properties of their central critical values. The vehicle that allows to do so is the notion of Hida families, which recall are families of p-adic overconvergent ordinary modular forms parametrized by a rigid analytic space: we refer to [DR1, §2.6] and [DR2, §1.8] for a brief introduction to this subject, presented in a language suitable for our purposes. Let

Ω := Hom(ℤ_{p}^{×}, ℤ_{p}^{×}) ≃ ℤ/(p − 1)ℤ × ℤ_{p}  (30)

denote the weight space, which contains the integers as a dense subset by means of the identification k ↔ φ_{k}, where φ_{k} ∈ Ω is the homomorphism defined by the rule φ_{k}(n) = n^{k−1}.

Let g, f, and h be three Hida families of new tame level N_{g}, N_{f}, and N_{h} and tame nebentype characters χ_{g}, χ_{f} and χ_{h}, respectively. Assume χ_{g}χ_{f}χ_{h} = 1 as above.

Let U_{g} denote the parameter space of the Hida family g, and let k_{g} : U_{g} → Ω denote the finite weight map associated to it. If confusion is not likely to arise, in order to simplify the notation we may freely identify a point x ∈ U_{g} with its image k = k_{g}(x) ∈ Ω.

For any point x ∈ U_{g}, let x(g) denote the specialisation of g at x. In general x(g) is an overconvergent ordinary modular form. Define U_{g}^{cl} ⊆ U_{g} to be the subset of points x ∈ U_{g} such that x(g) is a classical ordinary eigenform. By the work of Hida, U_{g}^{cl} contains all points of weight k_{g}(x) = k ∈ ℤ_{≥2}. Points of weight 1 may belong to U_{g}^{cl} or not. Define further U_{g}^{∘} to be the subset of classical points x ∈ U_{g}^{cl} such that the eigenform x(g) is old at p. For such points, there exists a classical newform that we denote g_{k} ∈ M_{k}(N_{g}, χ_{g}) whose ordinary p-stabilisation is the specialisation of g at x. Phrased in terms of q-expansions, this amounts to saying that

x(g)(q) = g_{k}(q) − βg_{k}(q^{p}) ∈ M_{k}(N_{g}p, χ_{g}),

where β ∈ ℂ_{p} is a root of the p-th Hecke polynomial x^{2} − a_{p}(g_{k})x + χ_{g}(p)p^{k−1} whose p-adic valuation is k − 1. Adopt throughout similar notations as above for f and h.

Note that for any triple of classical points (x, y, z) ∈ U_{g}^{∘} × U_{f}^{∘} × U_{h}^{∘} of weights (k, ℓ, m), the subfield ℚ(g_{k}, f_{ℓ}, h_{m}) of ℂ_{p} generated by the fourier coefficients of g_{k}, f_{ℓ} and h_{m} is a finite extension of ℚ, whose degree varies in general as a function of the weights (k, ℓ, m).

Let ğ, f̆, h̆ be Λ-adic test vectors of tame level N associated to g, f and h as described in [DR1, Definition 2.16]. Such test vectors may be written down explicitly as ğ = ∑_{d|N/N_{g}} λ_{d}g(q^{d}), where d ranges over the positive divisors of N/N_{g} and λ_{d} ∈ Λ_{g} = O(U_{g}) are Iwasawa functions on the domain U_{g} (and similarly for f̆ and h̆).

Assume that gcd(N_{f}, N_{g}, N_{h}) = 1 and that the three Hida families are cuspidal, meaning that all their specialisations at classical points of weight greater than 1 (though not necessarily at points of weight 1) are cuspidal eigenforms. The first assumption implies that the local signs hypothesis (24) holds for all triples (g_{k}, f_{ℓ}, h_{m}) at points (k, ℓ, m) ∈ U_{g}^{∘} × U_{f}^{∘} × U_{h}^{∘}.

Define a dense subset U_{gfh}^{∘} ⊂ U_{g} × U_{f} × U_{h} by

U_{gfh}^{∘} = {(k, ℓ, m) ∈ U_{g}^{∘} × U_{f}^{∘} × U_{h}^{∘} such that k ≥ ℓ + m}.

Under the running assumptions, a three-variable p-adic L-function

ℒ_{p}^{g}(ğ, f̆, h̆) : U_{g} × U_{f} × U_{h} ⟶ ℂ_{p}  (31)

was constructed in [DR1, §4.2], which is characterized by the following interpolation property at triplets (k, ℓ, m) ∈ U_{fgh}^{∘}:

ℒ_{p}^{g}(ğ, f̆, h̆)(k, ℓ, m) = 𝔢(k, ℓ, m) · J(ğ_{k}, f̆_{ℓ}, h̆_{m}),  (32)

where

𝔢(k, ℓ, m) = E(g_{k}, f_{ℓ}, h_{m})/E_{0}(g_{k})E_{1}(g_{k}) = E(g_{k}^{∗}, f_{ℓ}^{∗}, h_{m}^{∗})/E_{0}(g_{k}^{∗})E_{1}(g_{k}^{∗})

denotes the Euler factor introduced in [DR1, Theorem 1.3].

The next proposition spells out the interpolation property tying this p-adic L-function to central critical L-values.

Proposition 2.3. For any Λ-adic test vectors ğ, f̆, h̆ and every (k, ℓ, m) ∈ U_{fgh}^{∘}, the following formula is valid:

ℒ_{p}^{g}(ğ, f̆, h̆) · ℒ_{p}^{g}(ğ^{∗}, f̆^{∗}, h̆^{∗})(k, ℓ, m) = 𝔢(k, ℓ, m)^{2} · 𝔞(k, ℓ, m) · 𝔣(k, ℓ, m) · ∏_{v|N∞} C_{v}(ğ_{k}, f̆_{ℓ}, h̆_{m})

×\frac{L(g_{k} ⊗ f_{ℓ} ⊗ h_{m}, \frac{k+ℓ+m−2}{2})}{⟨g_{k}^{∗}, g_{k}^{∗}⟩^{2}}.  (33)

Proof. Note that

|I(ğ_{k}, f̆_{ℓ}, h̆_{m})|^{2} = I(ğ_{k}, f̆_{ℓ}, h̆_{m}) · \overline{I(ğ_{k}, f̆_{ℓ}, h̆_{m})} = I(ğ_{k}, f̆_{ℓ}, h̆_{m}) · I(ğ_{k}^{∗}, f̆_{ℓ}^{∗}, h̆_{m}^{∗}),

and that

⟨g_{k}, g_{k}⟩ = ⟨g_{k}^{∗}, g_{k}^{∗}⟩.

It thus follows from (32) and (26) that

ℒ_{p}^{g}(ğ, f̆, h̆) · ℒ_{p}^{g}(ğ^{∗}, f̆^{∗}, h̆^{∗})(k, ℓ, m) = 𝔢(k, ℓ, m)^{2} \frac{|I(ğ_{k}, f̆_{ℓ}, h̆_{m})|^{2}}{⟨g_{k}^{∗}, g_{k}^{∗}⟩^{2}}
= 𝔢(k, ℓ, m)^{2} · 𝔞(k, ℓ, m) · 𝔣(k, ℓ, m)

× ∏_{v|N∞} C_{v}(ğ_{k}, f̆_{ℓ}, h̆_{m}) · \frac{L(g_{k} ⊗ f_{ℓ} ⊗ h_{m}, \frac{k+ℓ+m−2}{2})}{⟨g_{k}^{∗}, g_{k}^{∗}⟩^{2}}.

□

Remark 2.4. An inspection of the definition of the Watson-Woodbury test vector shows that, when we let it vary along the Hida families g, f and h, it gives rise to a triplet (ğ, f̆, h̆) of Λ-adic test vectors of tame level N, whose specialization at any (k, ℓ, m) ∈ U_{g}^{∘} × U_{f}^{∘} × U_{h}^{∘} is an eigenform with coefficients in ℚ(g_{k}, f_{ℓ}, h_{m}), a finite extension of ℚ; and moreover, the constants W_{v} = C_{v}(ğ_{k}, f̆_{ℓ}, h̆_{m}) ∈ ℚ^{×} in Proposition 2.1 c) depend on v and N but not on the triplet (k, l, m) of weights.

For the remainder of this section (and throughout the rest of the paper, as we shall recall again in later sections) we impose the following hypothesis:

Assumption 2.5. There exists a point x ∈ U_{g}^{∘} of weight k_{g}(x) = 1 such that the specialisation g_{α} := x(g) ∈ M_{1}(N_{g}p, χ_{g}) satisfies Hypothesis C.

To make sure that the notations introduced in the Introduction and in this section are compatible, let us emphasize that g_{α} is an eigenform of weight 1 and level N_{g}p that is new at N_{g} and old at p. The eigenform g_{α} is the p-stabilisation of a newform g := g_{1} ∈ M_{1}(N_{g}, χ_{g}).

The next proposition relates this restriction of the Garrett-Hida p-adic L-function to p-adic iterated integrals:

Proposition 2.6. There exists a linear form

γ̆_{α} : S_{1}(Np, χ_{g}^{−1})_{L}[g_{α}^{∗}] ⟶ L

with L = ℚ(g_{1}, f_{2}, h_{1}) such that

ℒ_{p}^{g}(ğ, f̆, h̆)(1, 2, 1) = ∫γ̆_{α} f̆_{2} · h̆_{1}.

Proof. Fix a finite flat extension Λ^{†} of Λ, sufficiently large so that it contains the coefficients of all the Λ-adic modular forms considered below, and write S^{ord}(N; Λ^{†}) for the space of Λ-adic modular forms with coefficients in Λ^{†}. The Hida family g gives rise to a subspace

S^{ord}(N; Λ^{†})[g] := {ğ ∈ S^{ord}(N; Λ^{†}) such that T_{n}ğ = a_{n}(g)ğ, for all (n, N) = 1} .

Letting L^{†} denote the fraction field of Λ^{†}, the vector space S^{ord}(N; L^{†})[g] is finite-dimensional over L^{†} and has for basis the set

{g(q^{d})}_{d|(N/N_{g})}  (34)

of Λ-adic forms. By [DR1, Lemma 2.19], there exists a linear operator

J(ğ^{∗}) : S^{ord}(N, Λ^{†})⟶L^{†} ⊗_{Λ} Λ^{†}, ϕ ↦ J(ğ^{∗}, ϕ)

that is characterized uniquely by the following interpolation property: at every point in U_{g}^{∘}∩U_{ϕ}^{∘} of weight k ≥ 2, the specialisation of J(ğ^{∗}) is regular (meaning that there is no pole at this point) and is described by the following rule:

ν_{k}(J(ğ^{∗})) : S_{k}^{ord}(N, χ_{g}^{−1})_{ℚ_{p}(g_{k})}⟶ℚ_{p}(g_{k}), ϕ ↦ \frac{⟨ğ_{k}^{∗}, ϕ⟩}{⟨g_{k}^{∗}, g_{k}^{∗}⟩}.

We turn now to analyze the specialisation of J(ğ^{∗}) in weight 1. For any d|(N/N_{g}) let c_{d} : S^{ord}(N, L^{†})⟶L^{†} denote the functional which associates to ϕ its coefficient in g(q^{d}) with respect to the basis in (34). It follows from the proofs of [DR1, Lemmas 2.12 and 2.19] that

J(ğ^{∗}) = ∑_{d|(N/N_{g})} λ_{d} · c_{d}

where λ_{d} ∈ L^{†} are elements which, viewed as functions of ℤ_{≥2} by means of the rule k ↦ ν_{k}(λ_{d}), can be expressed as polynomials in the expressions q^{k}, a_{q}(g_{k}), \frac{1}{q} and \frac{1}{q+1} as q ranges over the divisors of N. This implies that the specialisation of J(ğ^{∗}) in weight 1 is also regular and hence gives rise to a linear operator

γ̆_{α} := ν_{1}(J(ğ^{∗})) : S_{1}^{oc,ord}(N, χ_{g}^{−1})_{L_{p}}⟶L_{p}.

It factors through S_{1}^{oc,ord}(N, χ_{g}^{−1})_{L_{p}}[[g_{α}^{∗}]], which by the classicality hypothesis imposed on g_{α}^{∗} is isomorphic to the corresponding space S_{1}(Np, χ_{g}^{−1})_{L_{p}}[g_{α}^{∗}] of classical modular forms.

As explained in the introduction, Hypothesis C equips S_{1}^{oc,ord}(N, χ_{g}^{−1})_{L_{p}}[[g_{α}^{∗}]] with an Lrational structure, denoted S_{1}^{oc,ord}(N, χ_{g}^{−1})_{L}[[g_{α}^{∗}]]. In view of the description given above for λ_{d} as functions of k, it follows that ν_{1}(λ_{d}) ∈ L and hence γ̆_{α} is L-rational, that is to say, it belongs to S_{1}(Np, χ_{g}^{−1})_{L}[g_{α}^{∗}]^{∨} = S_{1}^{oc,ord}(N, χ_{g}^{−1})_{L}[[g_{α}^{∗}]]^{∨}.

Let e_{ord}(d^{•}f̆_{2}^{[p]} × h̆_{1}) denote the Λ-adic modular form whose specialisation in weight k is e_{ord}(d^{\frac{k−3}{2}} f̆_{2}^{[p]} ×h̆_{1}) for all k ∈ ℤ_{≥2}. By [DR1, Definition 4.4], ℒ_{p}^{g}(ğ, f̆_{2}, h̆_{1}) = J(ğ, e_{ord}(d^{•}f̆_{2}^{[p]} × h̆_{1})) and, by construction,

ℒ_{p}^{g}(ğ, f̆_{2}, h̆_{1})(1) = γ̆_{α}(e_{ord}(d^{−1}f̆_{2}^{[p]} × h̆_{1})) = ∫γ̆_{α} f̆_{2} · h̆_{1},

as was to be shown. □

Proposition 2.6 recasts the p-adic iterated integral of Conjecture ES with (γ̆, f̆, h̆) = (γ̆_{α}, f̆_{2}, h̆_{1}) as a p-adic avatar of the classical special value L(E, ϱ_{gh}, 1). This suggests a possible strategy for proving Conjecture ES in the the rank (1, 1) setting where ϱ_{gh} admits a decomposition of the form ϱ_{gh} = ϱ_{1} ⊕ϱ_{2} ⊕ϱ_{3} as in (12). Assuming the regulator R_{g_{α}}(E, ϱ_{gh}) is factorisable, Conjecture ES might then be proved by constructing appropriate “p-adic Artin L-series” L_{p}(Ad_{g_{α}}) associated to (the irreducible constituent containing Ad_{g}^{+} in) Ad_{g} and “p-adic Hasse-Weil-Artin L-series” L_{p}(E, ϱ_{i}) attached to the irreducible constituents of ϱ_{gh}, and proving that

ℒ_{p}^{g}(ğ, f̆_{2}, h̆_{1}) · L_{p}(Ad_{g_{α}}) =^{?} L_{p}(E, ϱ_{1}) · L_{p}(E, ϱ_{2}) · L_{p}(E, ϱ_{3})  (35)

and

L_{p}(Ad_{g_{α}})(1) =^{?} log_{p}(u_{g_{α}}),  L_{p}(E, ϱ_{1})(1) =^{?} log_{E,p}(P_{1}),  (36)
L_{p}(E, ϱ_{3})(1) ∈ L^{×},  L_{p}(E, ϱ_{2})(1) =^{?} log_{E,p}(P_{2}).

This is the strategy carried out in §3.2: see Theorem 3.9 for the variant of (35) and (36) in the special setting considered therein. In general, identities like (35) appear to lie quite deep, and we are not even able to provide an independent definition of the p-adic L-values L_{p}(E, ϱ_{i}) that figure in its putative right-hand side; they are perhaps somewhat akin to the “genuine” p-adic L-functions sought for in [Hi3].

3. Heegner points and theta series of imaginary quadratic fields

Let K be an imaginary quadratic field of discriminant −D_{K}, and let N_{K/ℚ} denote the norm map on fractional ideals of K. The aim of this chapter is to study Conjecture ES in the case where both g and h are theta series associated to K. After setting up the basic notations and assumptions, the cases in which the prime p splits or remains inert in K are treated separately in Sections 3.2 and 3.3.

Just as in the introduction, let E be an elliptic curve over ℚ and f ∈ S_{2}(N_{f}) denote its associated newform. Given a finite order character

ψ : G_{K}⟶ℂ^{×}

of conductor 𝔠_{ψ} ⊆ O_{K} and central character χ (viewed as a Dirichlet character), let

θ_{ψ} ∈ M_{1}(D_{K} · N_{K/ℚ}(𝔠_{ψ}), χ)

denote the theta series associated to ψ and let V_{ψ} := Ind_{K}^{ℚ} ψ denote the two-dimensional induced representation of ψ from G_{K} to G_{ℚ}. It is known that θ_{ψ} is Eisenstein if and only if V_{ψ} is reducible, which occurs precisely when ψ = ψ′, where ψ′ denotes the character of G_{K} defined by the rule ψ′(σ) = ψ(σ_{0}σσ_{0}^{−1}). Here σ_{0} is any choice of element in G_{ℚ} \ G_{K}.

Fix two finite order characters ψ_{g} and ψ_{h} of K of conductors 𝔠_{g} and 𝔠_{h}. Assume that the central character χ of ψ_{h} is inverse to that of ψ_{g}, and set

g := θ_{ψ_{g}} ∈ M_{1}(N_{g}, χ̄), h := θ_{ψ_{h}} ∈ M_{1}(N_{h}, χ).

Define ψ_{1} = ψ_{g}ψ_{h} and ψ_{2} = ψ_{g}ψ′_{h}. The characters ψ_{1} and ψ_{2} are ring class characters of K, associated to orders O_{c_{1}} and O_{c_{2}} in O_{K} of conductors c_{1} and c_{2} respectively. Such characters are sometimes also called, synonymously, dihedral or anticyclotomic, characters of K. Note that ψ_{g} and ψ_{h} were not assumed to be necessarily anticyclotomic.

One checks that there is a decomposition of Artin representations and, in parallel to that, a factorisation of L-series as follows:

V_{gh} = V_{ψ_{1}} ⊕ V_{ψ_{2}}, L(E, ϱ_{gh}, s) = L(E/K, ψ_{1}, s) · L(E/K, ψ_{2}, s).  (37)

Assume also that gcd(N_{f}, 𝔠_{g}𝔠_{h}) = 1. This implies that for all places of K above any place v | N∞ of ℚ, the local signs of L(E/K, ψ_{1}, s) and of L(E/K, ψ_{2}, s) are all equal, and hence ε_{v}(E, ϱ_{gh}) = (±1)^{2} = 1. This automatically implies that Hypothesis A holds.

Let H denote the ring class field of conductor c := lcm(c_{1}, c_{2}) of K, which contains the fields cut out by ψ_{1} and ψ_{2}. The field L of coefficients that is implicit in (37) can be taken to be any finite extension of the field generated by the traces of V_{ψ_{1}} and V_{ψ_{2}}.

3.1. Heegner points and elliptic units. In the rank (1, 1) setting of Conjecture ES in which r(E, V_{ψ_{1}}) = r(E, V_{ψ_{2}}) = 1, the global points and units that arise on the right hand side are expected to be expressible in terms of Heegner points and elliptic units. Not surprisingly, these global constructions are the key to the theoretical evidence that has been amassed in support of Conjecture ES in the rank (1, 1) setting when p is split in K.

The Heegner point construction is available under the following

Heegner hypothesis: The level of f may be factorised as N_{f} = N_{f}^{+}N_{f}^{−} where

• there exists an integral ideal 𝔑 of K such that O_{K}/𝔑 ≃ ℤ/N_{f}^{+}ℤ,

• N_{f}^{−} is the square-free product of an even number of primes which are all inert in K. This hypothesis implies that the signs in the functional equations for L(E, V_{ψ_{1}}, s) and

L(E, V_{ψ_{2}}, s) are −1, and hence that the analytic ranks r_{an}(E, V_{ψ_{1}}) and r_{an}(E, V_{ψ_{2}}) are both odd. In particular, Hypothesis B of the introduction is automatically satisfied. It is further expected that, in “generic” instances,

r_{an}(E, V_{ψ_{1}}) = r_{an}(E, V_{ψ_{2}}) = 1,  (38)

and that these equalities certainly hold when the p-adic iterated integrals attached to the triple (f, g, h) are not identically zero.

The decomposition in (37) can be further refined to

ϱ_{gh} = ϱ_{1} ⊕ ϱ_{2} ⊕ ϱ_{3},  (39)

where • for i = 1, 2, ϱ_{i} is the unique irreducible constituent of V_{ψ_{i}} for which r_{an}(E, ϱ_{i}) is odd;

• ϱ_{3} is the G_{ℚ}-stable complement of ϱ_{1} ⊕ ϱ_{2} in ϱ_{gh}.

This decomposition is uniquely determined by (f, g, h), up to possibly switching ϱ_{1} and ϱ_{2}. When (38) is satisfied, the decomposition (39) satisfies the conditions

r_{an}(E, ϱ_{1}) = r_{an}(E, ϱ_{2}) = 1,  r_{an}(E, ϱ_{3}) = 0.  (40)

As will be explained shortly, the work of Gross, Zagier, Kolyvagin and Zhang further shows that the same equations as in (40) hold for the corresponding algebraic ranks:

r(E, ϱ_{1}) = r(E, ϱ_{2}) = 1,  r(E, ϱ_{3}) = 0,  (41)

consistent with the predictions made by the Birch and Swinnerton-Dyer conjecture, and so we are in the rank (1, 1) setting of Conjecture ES. It also leads to the construction of explicit non-trivial elements in E(H)_{L}^{ϱ_{1}} and E(H)_{L}^{ϱ_{2}} from Heegner points, which we now describe.

Let X be the Shimura curve associated to an Eichler order of level N_{f}^{+} in the (indefinite, thanks to the Heegner hypothesis) quaternion algebra of discriminant N_{f}^{−} over ℚ. Let

π_{E} : Jac(X) → E

denote a modular parametrisation of E arising from the fact that E is an abelian variety quotient of the jacobian of X.

Let X(H)_{O_{c}} ⊂ X(H) denote the set of Heegner points attached to the order of conductor c in K. It is a finite set, and any two points in it are related by the action of an Atkin-Lehner involution and an element of G := Gal (H/K). A Heegner divisor of conductor c on X is any degree 0 divisor supported on X(H)_{O_{c}}, and a Heegner point on E of conductor c is the image of such a divisor.

After fixing a choice of P ∈ X(H)_{O_{c}}, any ring class character ψ of conductor c gives rise to a degree zero divisor on X by setting

D_{ψ} := \{ \begin{matrix}∑_{σ∈G} ψ^{−1}(σ)P^{σ}, & if ψ ≠ 1, \\ ∑_{σ∈G} P^{σ} − #G · ∞ & if ψ = 1, \end{matrix}  (42)

where ∞ is the degree one divisor in Div(X) ⊗ ℚ defined by ∞ = \frac{1}{2g−2}K_{X}, and K_{X} denotes the canonical class on X. The Heegner point attached to ψ is then defined to be

P_{ψ} := π_{E}(D_{ψ}) ∈ E(H)_{L}.

Note that the point P_{ψ} actually belongs to E(H)_{L}^{V_{ψ}}, and that the same is true of Pψ̄.

Lemma 3.1. Assume that (40) is satisfied. Then (41) holds as well, and, for i = 1 and 2, the Mordell-Weil group E(H)_{L}^{ϱ_{i}} is generated by the vectors P_{ψ_{i}} and Pψ̄_{i}. If ψ_{i} ≠ ψ̄_{i}, these points are linearly independent and form a basis for E(H)_{L}^{ϱ_{i}}. If ψ_{i} = ψ̄_{i} (i.e., if ψ_{i} is a genus character) then P_{ψ_{i}} = Pψ̄_{i} and this vector is a basis for E(H)_{L}^{ϱ_{i}}.

Proof. Assumption (40) asserts that the central critical derivative L′(E, ϱ_{i}, 1) is non-zero. The Gross-Zagier formula and its extension to ring class characters given in [Zh] shows that the vectors P_{ψ_{i}} and Pψ̄_{i} have non-zero canonical height and hence are non-zero in E(H)_{L}^{V_{ψ_{i}}}. The main theorem of [BD0] (and a suitable extension to Heegner points arising from Shimura curves as given in [YZZ]) implies that r(E, V_{ψ_{i}}) = 1 and that E(H)_{L}^{ϱ_{i}} is generated by the Heegner vectors P_{ψ_{i}} and Pψ̄_{i}. The result follows. □

Note that the points P_{ψ_{i}} and Pψ̄_{i} depend on the choice of a point P ∈ X(H)_{O_{c}} that was made in (42), but only up to multiplication by a root of unity. More precisely, replacing P by wP^{σ_{0}} for σ_{0} ∈ G and w a Fricke involution on X has the effect of replacing P_{ψ_{i}} by ±ψ̄_{i}(σ_{0})P_{ψ_{i}} and Pψ̄_{i} by ±ψ_{i}(σ_{0})Pψ̄_{i}. In particular, the L-vector space in E(H)_{L} spanned by P_{ψ_{i}} is completely canonical and does not depend on the choice of P.

It will also be convenient to introduce a related basis of E(H)_{L}^{ϱ_{i}} attached to a choice of reflection τ in the dihedral Galois group Gal (H/ℚ), by setting

P_{ψ_{i}}^{+} = P_{ψ_{i}} + P_{ψ_{i}}^{τ},  P_{ψ_{i}}^{−} = P_{ψ_{i}} − P_{ψ_{i}}^{τ}.  (43)

Sometimes, the basic Heegner point P used to define the points P_{ψ_{i}}^{±} can be chosen to be fixed (up to sign) by the reflection τ, i.e., in such a way that τP = ±P. When this is the case,

P_{ψ_{i}}^{+} = P_{ψ_{i}} ± Pψ̄_{i}, P_{ψ_{i}}^{−} = P_{ψ_{i}} ∓ Pψ̄_{i}.

But this simpler expression for the basis (P_{ψ_{i}}^{+}, P_{ψ_{i}}^{−}) need not be invoked in general.

Let

ψ_{0} := ψ_{g}/ψ′_{g}

be the non-trivial ring class character attached to the adjoint representation of g. In addition to the Heegner points P_{ψ_{i}} and P_{ψ_{i}}^{±}, Conjecture ES also involves the elliptic unit u ∈ O_{H_{g}}^{×} and its ψ_{0}-component

u_{ψ_{0}} = ∑_{σ∈G} ψ_{0}^{−1}(σ)u^{σ} ∈ (O_{H_{g}}^{×})_{L}^{V_{ψ_{0}}} ,

where the group law in the unit group O_{H_{G}}^{×} is written additively, and the units u_{ψ_{0}}^{±} relative to any choice of a reflection τ ∈ Gal (H/ℚ) are defined just as in (43):

u_{ψ_{0}}^{+} = u_{ψ_{0}} + u_{ψ_{0}}^{τ},  u_{ψ_{0}}^{−} = u_{ψ_{0}} − u_{ψ_{0}}^{τ}.  (44)

Note that, just as for the points P_{ψ_{i}} and P_{ψ_{i}}^{±}, the group Gal (H/K) acts on the element u_{ψ_{0}} through the character ψ_{0} but does not preserve the line spanned by either u_{ψ_{0}}^{+} or u_{ψ_{0}}^{−}.

3.2. Conjecture ES when p splits in K. When p is split in K, the frobenius element σ_{p} belongs to Gal (H/K), and the choice of a prime ℘ of K above p is sufficient to determine σ_{p} as an element of this group. The choices of α_{g}, β_{g}, α_{h} and β_{h} can then be made so that

α_{g} = ψ_{g}(℘),  β_{g} = ψ_{g}( ¯℘),  α_{h} = ψ_{h}(℘),  β_{h} = ψ_{h}( ¯℘).

With the above choices,

α_{g}α_{h} = ψ_{g}ψ_{h}(℘) = ψ_{1}(σ_{p}), α_{g}β_{h} = ψ_{g}ψ′_{h}(℘) = ψ_{2}(σ_{p}).

The following lemma describes the p-adic regulator det R_{g_{α}}(E, ϱ_{gh}) that enters into Conjecture ES.

Lemma 3.2. The regulator det(R_{g_{α}}(E, ϱ_{gh})) is given by the following expressions: (1) If r(E, V_{ψ_{1}}) = r(E, V_{ψ_{2}}) = 1,

det R_{g_{α}}(E, ϱ_{gh}) = log_{E,p}(P_{ψ_{1}}) · log_{E,p}(P_{ψ_{2}}).

(2) If r(E, V_{ψ_{1}}) ≥ 2 or r(E, V_{ψ_{2}}) ≥ 2,

det R_{g_{α}}(E, ϱ_{gh}) = 0;

Proof. The first part follows from the fact that, after choosing a σ_{p}-eigenbasis for V_{gh}^{g_{α}}, the matrix R_{g_{α}}(E, ϱ_{gh}) takes the form

R_{g_{α}}(E, ϱ_{gh}) = ( \begin{matrix}log_{E,p}(P_{ψ_{1}}) & 0 \\ 0 & log_{E,p}(P_{ψ_{2}}) \end{matrix} ) .

In the rank 2 setting, assume without loss of generality that r(E, V_{ψ_{1}}) = 2 and that r(E, V_{ψ_{2}}) = 0. The two dimensional subspace V_{gh}^{g_{α}} then intersects V_{ψ_{1}} and V_{ψ_{2}} in one-dimensional subspaces which determine a decomposition of V_{gh}^{g_{α}}. Choosing a basis (v_{1}, v_{2}) for V_{gh}^{g_{α}} which is compatible with this decomposition, and a basis Φ_{1}, Φ_{2} of Hom_{G_{ℚ}}(V_{ψ_{1}}, E(H)_{L}), produces a matrix R_{g_{α}}(E, ϱ_{gh}) whose second column is zero. □

The lemma above motivates focussing our attention on the rank (1, 1) setting of Conjecture ES when p is split, which is what shall be done for the remainder of Section 3.2. Assume as in §3.1 that the Heegner hypothesis is satisfied and let

P_{1} := P_{ψ_{1}}, P_{2} := P_{ψ_{2}}

be the Heegner points constructed in the previous section. The main result of this section is:

Theorem 3.3. Assume N = lcm(N_{f}, N_{g}, N_{h}) is square-free and p ∤ 2N. Assume also that Hypothesis C holds. If r_{an}(E, V_{ψ_{1}}) > 1 or r_{an}(E, V_{ψ_{2}}) > 1, the iterated integrals attached to the triple (f, g, h) vanish identically. Otherwise, there exists a finite extension L of ℚ(ψ_{g}, ψ_{h}), a scalar λ ∈ L^{×} and test vectors

(f̆, γ̆_{g_{α}}, h̆) ∈ S_{2}(N)_{L}[f] × M_{1}(Np, χ)_{L}^{∨}[g_{α}] × M_{1}(N, χ)_{L}[h]

for which

∫γ̆_{gα} f̆· h̆ = λ · \frac{log_{E,p}(P_{1}) log_{E,p}(P_{2})}{log_{p}(u_{g_{α}})}.

Remark 3.4. We believe that in fact L can always be taken to be ℚ(ψ_{g}, ψ_{h}). It would be interesting to be able to pin down a concrete choice of test vectors (f̆, γ̆_{g_{α}}, h̆) for which the precise value of λ ∈ L^{×} could be computed explicitly. For example, let χ_{K} denote a quadratic Dirichlet character of prime conductor D_{K} associated to an imaginary quadratic field K. Assume N = D_{K} and that ψ_{g} = ψ_{h} = 1, so that

g = h = θ_{1} ∈ M_{1}(D_{K}, χ_{K})

is the Eisenstein theta series associated to the trivial character of K. Since N_{f} = N_{g} = N_{h} = N, the only possible choice of test vectors (up to scaling by elements in L^{×}) is (f, γ_{g_{α}}, h). Then one observes that the following striking formula holds in a number of examples:

∫_{γ_{gα}} f · h = \frac{|E(𝔽_{p})|^{2} log_{E,p}(P_{K})^{2}}{p(p − 1)h_{K} log_{p}(u_{℘})} .  (45)

Here |E(𝔽_{p})| is the number of points on E over the finite field 𝔽_{p}, h_{K} is the class number of K, P_{K} is the Heegner point in E(K), and u_{℘} ∈ K^{×} is any p-unit satisfying (u_{℘}) = ℘^{h_{K}}.

Formula (45) was first discovered experimentally by the authors (on Christmas day), after a long theoretical and experimental investigation trying to uncover the arithmetic meaning of these p-adic iterated integrals. It is the prototype for all the formulae in this paper.

The remainder of this section is devoted to proving Theorem 3.3. The proof rests on a comparison between the Garrett-Hida p-adic L-function of the previous section and two different types of “Katz-style” p-adic L-functions whose definitions are recalled below.

1. Katz’s two-variable p-adic L-function associated to the imaginary quadratic field K. Let 𝔠 ⊆ O_{K} be an integral ideal of K and denote Σ the set of Hecke characters of K of

conductor dividing 𝔠. Define Σ_{K} = Σ_{K}^{(1)} ∪ Σ_{K}^{(2)} ⊂ Σ to be the disjoint union of the sets

Σ_{K}^{(1)} = {ψ ∈ Σ of infinity type (κ_{1}, κ_{2}), with κ_{1} ≤ 0, κ_{2} ≥ 1},
Σ_{K}^{(2)} = {ψ ∈ Σ of infinity type (κ_{1}, κ_{2}), with κ_{1} ≥ 1, κ_{2} ≤ 0}.

For all ψ ∈ Σ_{K}, the complex argument s = 0 is a critical point for the Hecke L-function L(ψ^{−1}, s), and Katz’s p-adic L-function is constructed by interpolating the (suitably normalized) values L(ψ^{−1}, 0) as ψ ranges over Σ_{K}^{(2)}.

More precisely, let Σ̂_{K} denote the completion of Σ_{K}^{(2)} with respect to the compact open topology on the space of O_{L_{p}}-valued functions on a certain subset of A_{K}^{×}, as described in [BDP1, §5.2]. By the work of Katz [Katz1], there exists a p-adic analytic function

L_{p}(K) : Σ̂_{K}⟶ℂ_{p}

which is uniquely characterized by the following interpolation property: for all ψ ∈ Σ_{K}^{(2)} of infinity type (κ_{1}, κ_{2}),

L_{p}(K)(ψ) = 𝔞(ψ) × 𝔢(ψ) × 𝔣(ψ) × \frac{Ω_{p}^{κ_{1}−κ_{2}}}{Ω^{κ_{1}−κ_{2}}} × L_{c}(ψ^{−1}, 0),  (46)

where • 𝔞(ψ) = (κ_{1} − 1)!π^{−κ_{2}}, 𝔢(ψ) = (1 − ψ(℘)p^{−1})(1 − ψ^{−1}( ¯℘)), and 𝔣(ψ) = D_{K}^{κ_{2}/2}2^{−κ_{2}};

• Ω_{p} ∈ ℂ_{p}^{×} is a p-adic period attached to K, as defined in [BDP2, (2-17)];

• Ω ∈ ℂ^{×} is the complex period associated to K defined in [BDP2, (2-15)];

• L_{c}(ψ^{−1}, s) is Hecke’s L-function associated to ψ^{−1} with the Euler factors at primes

dividing c removed. Formula (46) is [BDP2, Proposition 3.1], which in turn follows from [Katz2, §5.3.0] (cf. also

[Gro, Theorem 2.3] and [deS, II, Theorem 4.14]; the reader is cautioned that the latter reference needs to be slightly modified, for the power of 2π is not correct there.)

Letting N_{K} denote the norm of K/ℚ regarded as a Hecke character of K of infinity type (1, 1), Katz’s p-adic L-function satisfies the functional equation L_{p}(K)(ψ) = L_{p}(K)((ψ′)^{−1}N_{K}) (cf. [Gro, p. 90-91]).

The values of p-adic L-functions at classical points lying outside their region of interpolation are frequently of great arithmetic interest, since they may encode further invariants of the associated motives. An instance of this philosophy is Katz’s p-adic analogue of the Kronecker limit formula (cf. [Katz1, §10.4, 10.5], [Gro, p. 90], [deS, Ch. II, §5.2]), expressing the value of L_{p}(K) at a finite order character ψ of G_{K} in terms of a suitable linear combination of p-adic logarithms of elliptic units:

L_{p}(K)(ψ) = \begin{cases}\frac{1}{2}(\frac{1}{p} − 1) · log_{p}(u_{℘}) & if ψ = 1 \\ \frac{−1}{24c}(1 − ψ^{−1}( ¯℘))(1 − ψ(℘)p^{−1}) · ∑_{σ∈G} ψ^{−1}(σ) log_{p}(σ(u)) & if ψ ≠ 1. \end{cases} (47)

Here, as above, u_{℘} ∈ K^{×} is any element satisfying (u_{℘}) = ℘^{h_{K}} and u ∈ U_{CM} is an elliptic unit; the integer c > 0 is the smallest positive integer in the conductor ideal of ψ.

2. The p-adic Rankin L-function attached to a cusp form and an imaginary quadratic field. For any Hecke character ψ of K of infinity type (κ_{1}, κ_{2}), let

L(f, ψ, s) := L (π_{f} × π_{ψ}, s − \frac{κ_{1} + κ_{2} + 1}{2})

denote the L-series associated with the product of the global automorphic representations attached to the weight two cuspform f and the Hecke character ψ, normalised conveniently. As usual, L(f, ψ, s) = ∏_{q} L_{(q)}(q^{−s}) is defined as a product of Euler factors ranging over the set of prime numbers.

Fix a positive integer c ≥ 1 which is relatively prime to pN_{f}. Let Σ_{f,c} be the set of Hecke characters ψ ∈ Σ of conductor c and trivial central character for which L(f, ψ^{−1}, s) is self-dual and has s = 0 as its central critical point. This set can be expressed as the disjoint union of the three subsets

Σ_{f,c}^{(1)} = {ψ ∈ Σ_{f,c} of infinity type (1, 1)},
Σ_{f,c}^{(2)} = {ψ ∈ Σ_{f,c} of infinity type (κ + 2, −κ), κ ≥ 0},

and

Σ_{f,c}^{(2′)} = {ψ ∈ Σ_{f,c} of infinity type (−κ, κ + 2), κ ≥ 0}.

The three sets Σ_{f,c}^{(1)}, Σ_{f,c}^{(2)} and Σ_{f,c}^{(2′)} are each dense in the completion Σ̂_{f,c} of Σ_{f,c} with respect to the p-adic compact open topology as explained in [BDP1, §5.2]. As shown in loc. cit., there exists a unique p-adic analytic function

L_{p}(f, K) : Σ̂_{f,c}⟶ℂ_{p}

interpolating the critical values L(f, ψ^{−1}, 0) for ψ ∈ Σ_{f,c}^{(2)}, suitably normalized. The function L_{p}(f, K) is referred to as the p-adic Rankin L-function attached to the pair (f, K). By [BDP2, Proposition 3.11], the interpolation formula reads precisely as follows: for any character ψ ∈ Σ_{f,c}^{(2)} of infinity type (κ + 2, −κ),

L_{p}(f, K)(ψ) = 𝔞(f, ψ) × 𝔢^{2}(f, ψ) × 𝔣(f, ψ) × \frac{Ω_{p}^{4κ+4}}{Ω^{4κ+4}} × L(f, ψ^{−1}, 0)  (48)

where • 𝔞(f, ψ) = κ!(κ + 1)! π^{2κ+1}, 𝔢(f, ψ) = 1 − a_{p}(f)ψ^{−1}( ¯℘) + ψ^{−2}( ¯℘)p and

• 𝔣(f, ψ) = (2/c\sqrt{D_{K}})^{2κ+1} · ∏_{q|c} \frac{q−χ_{K}(q)}{q−1} · ω(f, ψ)^{−1}.

Here ω(f, ψ) denotes the complex number defined in [BDP1, (5.1.11)]. It is an algebraic scalar of absolute value 1 relative to any complex embedding of ℚ̄.

Consider now a Hecke character ψN_{K} ∈ Σ_{f,c}^{(1)}, where ψ has finite order. The character ψN_{K} lies outside the region of interpolation of L_{p}(f, K) and the main theorem of [BDP1]—cf. [Br] for a proof of it under the more relaxed Heegner hypothesis imposed here—asserts that

L_{p}(f, K)(ψN_{K}) = (1 − \frac{a_{p}(f)}{ψ( ¯℘)p} + \frac{1}{ψ^{2}( ¯℘)p})^{2} × log_{ω_{E}}(P_{ψ})^{2}.  (49)

We now turn to the comparison of the Rankin p-adic L-function with the Garrett-Hida p-adic L-function of the previous section: the Katz p-adic L-function then arises indirectly from the ratio of periods (51) used in constructing these two p-adic L-functions.

Recall the theta series g of weight 1, level N_{g} = D_{K}N_{K/ℚ}(𝔠(ψ_{g})) and character χ^{−1} appearing in the statement of Theorem 3.3.

Let g be the primitive cuspidal Hida family of theta series of tame level N_{g} and tame character χ_{g} = χ^{−1} constructed in e.g. [Hi2, p. 235-236] and [Gh, §5], specialising in weight 1 to the p-stabilisation g_{α} of g satisfying U_{p}(g_{α}) = α_{g} · g_{α}. It can be described explicitly by fixing a Hecke character λ of infinity type (1, 0) and conductor ℘. Let ℚ(λ) denote the number field generated by the values of λ and let ℚ_{p}(λ) be the completion of ℚ(λ) at the prime determined by an embedding ℚ(λ) ↪ ℚ̄_{p} which is fixed at the outset. Writing O_{ℚ_{p}(λ)} for its ring of integers, decompose its group of units as O_{ℚ_{p}(λ)}^{×} = μ × W, where μ is finite and W is free over ℤ_{p}. Let ⟨·⟩ : O_{ℚ_{p}(λ)}^{×} → W denote the natural projection.

For every integer k ≥ 1 define

ψ_{g,k−1}^{(p)} = ψ_{g}⟨λ⟩^{k−1}.

This definition does not depend on the choice of λ, for any other such choice λ′ differs from λ by a finite character and hence ⟨λ′⟩ = ⟨λ⟩. Let now ψ_{g,k−1} be the Hecke character defined by

ψ_{g,k−1}(𝔮) = \begin{cases}ψ_{g,k−1}^{(p)}(𝔮) & if 𝔮 ≠ ℘ \\ χ^{−1}(p)p^{k−1}/ψ_{g,k−1}^{(p)}( ¯℘) & if 𝔮 = ℘. \end{cases}

The Hecke character ψ_{g,k−1} has conductor 𝔠(ψ_{g}) and infinity type (k − 1, 0). For any k ∈ ℤ_{≥1} ∩ U_{g}, the weight k specialisation of g is the ordinary p-stabilisation of the theta series g_{k} = θ_{ψ_{g,k−1}} = θ_{ψ′_{g,k−1}} associated with ψ_{g,k−1} (or with its Gal (K/ℚ)-conjugate ψ′_{g,k−1}). In particular ψ_{g,0} = ψ_{g} is the character we considered at the outset and g_{1} = g.

Together with g, recall also the eigenforms f ∈ S_{2}(N_{f}) and h ∈ M_{1}(N_{h}, χ) of the statement. Let (ğ, f̆, h̆) be Watson-Woodbury’s test vector of level N associated to these three modular forms as described in Remark 2.2, and let also ğ denote Watson-Woodbury’s Λ-adic test vector discussed in Remark 2.4. Recall that, as discussed in Chapter 2, W_{v} = C_{v}(ğ_{k}, f̆, h̆) ∈ ℚ^{×} are constants independent of k for all v | N. Moreover, the results of [Wo2] guarantee that in this setting C_{∞}(ğ_{k}, f̆, h̆) = 1 for all k.

Recall the triple-product p-adic L-functions introduced in the previous chapter. Denote by k ↦ ℒ_{p}^{g}(ğ, f̆_{2}, h̆_{1})(k) the restriction of ℒ_{p}^{g}(ğ, f̆, h̆) to U_{g} × {2} × {1}. Because it does not involve the variation of f_{2} in a Hida family, this restriction has the virtue of being defined a bit more generally, without any ordinariness assumptions on f_{2}.

Proposition 2.3 implies, for all k ∈ U_{g}^{∘}, that

ℒ_{p}^{g}(ğ, f̆, h̆)(k) · ℒ_{p}^{g}(ğ^{∗}, f̆, h̆^{∗})(k) = 𝔢(k, 2, 1)^{2} · 𝔣(k, 2, 1) · 𝔞(k, 2, 1) · ∏_{v|N} W_{v}

× \frac{L(g_{k} ⊗ f ⊗ h, \frac{k+1}{2})}{⟨g_{k}^{∗}, g_{k}^{∗}⟩^{2}}.

Here the factors on the right are described in Proposition 2.3. In particular,

𝔣(k, 2, 1) = A · 2^{B} and 𝔞(k, 2, 1) = ((\frac{k − 3}{2})! (\frac{k − 1}{2})!)^{2} · π^{−2k}.

For every integer k ≥ 1 define the Hecke characters

Ψ_{gh}(k) = (ψ′_{g,k−1})^{−1} · ψ_{h}^{−1} · NK^{\frac{k+1}{2}} and Ψ_{gh′}(k) = (ψ′_{g,k−1})^{−1} · (ψ′_{h})^{−1} · NK^{\frac{k+1}{2}} .

While for k = 1 the characters Ψ_{gh}(1) = N_{K} · (ψ′_{g}ψ_{h})^{−1} and Ψ_{gh′}(1) = N_{K} · (ψ′_{g}ψ′_{h})^{−1} lie in Σ_{f,c}^{(1)}, for weights k ∈ U_{g}^{∘}, k > 1, the characters Ψ_{gh}(k), Ψ_{gh′}(k) belong to Σ_{f,c}^{(2)} with c = lcm(c_{1}, c_{2}), because their infinity type is (κ + 2, −κ) with κ = \frac{k−3}{2} ≥ 0.

It follows that for weights k ∈ U_{g}^{∘}, k > 1, the values L_{p}(f, K)(Ψ_{gh}(k)), L_{p}(f, K)(Ψ_{gh′}(k)) and ℒ_{p}^{g}(ğ, f̆, h̆)(k) all lie in the region of interpolation. An inspection of the Euler factors defining their classical counterparts yields the following relationship between critical values: for every k ∈ U_{g} there is a fudge factor 𝔣_{0}(k) ≠ 0 such that

L (g_{k} ⊗ f ⊗ h, \frac{k + 1}{2}) = 𝔣_{0}(k) · L(f, Ψ_{gh}^{−1}(k), 0) · L(f, Ψ_{gh′}^{−1}(k), 0),  (50)

and we shall prove the theorem by letting k ∈ U_{g}^{∘} ∩ ℤ_{>1} in (50) tend p-adically to 1 within U_{g}, so that the triplets (k, 2, 1) tend to (1, 2, 1).

Several of the fudge factors that arise in comparing different L-values, viewed as functions of weight space, satisfy the property described in the following definition:

Definition 3.5. Let F be a number field. A function

𝔣 : U_{g}^{∘} ∩ ℤ_{>1} ⟶ ℚ̄

is F -admissible if it extends to an element of the fraction field L_{g} of the algebra Λ_{g} = O(U_{g}) of Iwasawa functions on U_{g}, and if 𝔣(1) is well-defined and belongs to F^{×}.

As in the statement of the definition, an F -admissible function 𝔣 will customarily be identified with the function on U_{g} ⊆ ℤ_{p} which it induces. In order to make the calculations described below easier to follow, vowels will generally be used to denote non-admissible functions (such as rational functions of p^{k}, π^{k} or k!), and consonants to denote admissible ones (such as q^{k} or a_{q}(g_{k}) for some prime number q ≠ p).

Instances of admissible functions were already encountered in the proof of Proposition 2.6: using the notation employed there, the functions λ_{d} were shown in loc. cit. to be ℚ(g_{1}, h_{1})- admissible. Recall also that the fudge factor 𝔣_{0}(k) appearing in (50) accounts for the discrepancy of the Euler factors at the primes q | cD_{K}N_{f} in the definitions of the above L-series. It may be computed explicitly, but we content to note here that it is ℚ(ψ_{g}, ψ_{h})-admissible because 𝔣_{0}(k) is a rational function on q^{k} for q | cD_{K}N_{f} with coefficients in ℚ(ψ_{g}, ψ_{h}), with 𝔣_{0}(k) ≠ 0 for all k ∈ ℤ.

For every k ∈ U_{g}^{∘}, k > 1, define

σ(k) := ⟨g_{k}^{∗}, g_{k}^{∗}⟩ × (\frac{πΩ_{p}}{Ω} )^{2k−2}  (51)

and

𝔢_{1}(k) = \frac{𝔢(k, 2, 1)^{2}}{𝔢^{2}(f, Ψ_{gh}(k))𝔢^{2}(f, Ψ_{gh′}(k))}, 𝔣_{1}(k) = \frac{∏_{v|N} W_{v} · 𝔣_{0}(k) · 𝔣(k, 2, 1)}{𝔣(f, Ψ_{gh}(k))𝔣(f, Ψ_{gh′}(k))}.

Lemma 3.6. The function 𝔣_{1}(k) is K(ψ_{g}, ψ_{h})-admissible and

ℒ_{p}^{g}(ğ, f̆, h̆)(k) · ℒ_{p}^{g}(ğ^{∗}, f̆, h̆^{∗})(k) · σ(k)^{2} = 𝔢_{1}(k)𝔣_{1}(k) · L_{p}(f, K)(Ψ_{gh}(k)) · L_{p}(f, K)(Ψ_{gh′}(k)).

Proof. We already saw that 𝔣_{0}(k) is ℚ(ψ_{g}, ψ_{h})-admissible, and Proposition 2.1 (b) (ii) implies that 𝔣(k, 2, 1) = A · 2^{B(k)} is ℚ-admissible because p ≠ 2 and B(k) is a linear function of k. It remains to prove that the denominator of 𝔣_{1}(k) is K(ψ_{g}, ψ_{h})-admissible. Recall from (48) that 𝔣(f, ψ) = (2/c\sqrt{D_{K}})^{k−2} · ∏_{q|c} \frac{q−χ_{K}(q)}{q−1} · ω(f, ψ)^{−1} for any Hecke character ψ ∈ Σ_{f,c}^{(2)} of infinity type (\frac{k−3}{2} + 2, −\frac{k−3}{2}). The first two factors are K-admissible, because p is assumed not to divide 2N. Hence, in order to conclude, we need to show that the functions ω(f, Ψ_{gh}(k)) and ω(f, Ψ_{gh′}(k)), or at least their product, is admissible. By [BDP1, (5.1.11)]:

ω(f, Ψ_{gh}(k)) · ω(f, Ψ_{gh′}(k)) = \frac{N_{K}^{2k−2}(𝔟)N^{k−1}}{(ψ′_{g,k−1})^{2}ψ_{h}ψ′_{h}(𝔟) · b^{2k−2}} , (𝔟, pNc) = 1, 𝔟 · 𝔑 = (b). (52)

This shows that the function in (52) is K(ψ_{g}, ψ_{h})-admissible, because p ∤ 𝔟𝔑 and its value at k = 1 is the non-zero algebraic number ((ψ′_{g})^{2}ψ_{h}ψ′_{h}(𝔟))^{−1} = (ψ′_{g})^{2}ψ_{h}ψ′_{h}(𝔑), which does not depend on the choice of 𝔟.

Finally, the factorisation formula of Lemma 3.6 is a direct consequence of the interpolating formulae (33) and (48), combined with (50), once one observes that the factorial terms and powers of π on both sides cancel out. □

At this point it is convenient to invoke the following classical formula:

Lemma 3.7. There is a K-admissible function 𝔣_{2}(k) such that

⟨g_{k}^{∗}, g_{k}^{∗}⟩ = \frac{(k − 1)!}{π^{k}} · 𝔣_{2}(k) · L(ψ_{g,k−1}^{2} · χ, k).

Proof. Since the L-series of the adjoint of a CM-form factors as

L(Ad(g_{k}^{∗}), s) = L(Ad(g_{k}), s) = L(χ_{K}, s − k + 1)L(ψ_{g,k−1}^{2} · χ_{g}^{−1}, s)

and χ_{g} = χ^{−1}, Lemma 3.7 follows directly from (27) after evaluating at s = k and applying Dirichlet’s class number formula L(χ_{K}, 1) = 2πh_{K}(|O_{K}^{×}|\sqrt{D})^{−1}. □

It now becomes clear how Katz’s p-adic L-function enters in the calculation: note that

L(ψ_{g,k−1}^{2} · χ, k) = 𝔣_{3}(k) · L_{c}(ψ_{g,k−1}^{2} · χ · N_{K}^{−k}, 0),  (53)

where again 𝔣_{3}(k) accounts for the discrepancy between the Euler factors at primes q | cD_{K} defining both L-series above, and hence is ℚ(ψ_{g}, ψ_{h})-admissible.

Define

Ψ_{g}(k) := (ψ′_{g,k−1})^{−2} χ^{−1} N_{K}^{k}.

Lemma 3.8. The following equality holds for every k ∈ U_{g}^{∘}, k > 1:

L_{p}(K)(Ψ_{g}(k)) = 𝔢(Ψ_{g}(k)) · \frac{𝔣(Ψ_{g}(k))}{𝔣_{2}(k)𝔣_{3}(k)}σ(k) := \frac{𝔢(Ψ_{g}(k)) · 𝔣(Ψ_{g}(k))}{𝔣_{2}(k) · 𝔣_{3}(k)}⟨g_{k}^{∗}, g_{k}^{∗}⟩ × (\frac{πΩ_{p}}{Ω} )^{2k−2}

Proof. In light of Katz’s interpolation formula (46), since the infinity type of Ψ_{g}(k) is (κ_{1}, κ_{2}) = (k, −k + 2), it follows that

L(ψ_{g,k−1}^{2} · χ, k) = L((ψ′_{g,k−1})^{2} · χ, k) = \frac{π^{−k+2}}{(k − 1)!} · \frac{1}{𝔢(Ψ_{g}(k))} · \frac{𝔣_{3}(k)}{𝔣(Ψ_{g}(k))} · \frac{Ω^{2k−2}}{Ω_{p}^{2k−2}} · L_{p}(K)(Ψ_{g}(k)).

Comparing Lemma 3.7 and (53) with the definition of σ(k), one finds that the powers of π and the factorials (which, as functions of k, are not admissible) cancel out and the lemma follows. □

For all integers k ∈ U_{g}^{∘}, k > 1, define

ω(k) = I(ğ_{k}^{∗}, f̆, h̆^{∗}) / I(ğ_{k}, f̆, h̆)

as in (25) and

𝔣(k) = \frac{𝔣_{1}(k)}{ω(k)} (\frac{𝔣(Ψ_{g}(k))}{𝔣_{2}(k)𝔣_{3}(k)})^{2} .

Let 𝔤(χ) ∈ ℚ̄ ⊂ ℂ_{p} denote the Gauss sum associated to the character χ and define the number field L_{0} = K(\sqrt{N_{g}}, \sqrt{N_{h}}, 𝔤(χ), ψ_{g}, ψ_{h}). Theorem 3.3 will follow from the next result:

Theorem 3.9. The function 𝔣(k) is L_{0}-admissible and the following factorisation of p-adic L-series is valid

ℒ_{p}^{g}(ğ, f̆, h̆)(k)^{2} · L_{p}(K)(Ψ_{g}(k))^{2} = 𝔣(k) · L_{p}(f, K)(Ψ_{gh}(k)) · L_{p}(f, K)(Ψ_{gh′}(k)).  (54)

Moreover, there exist explicit elements λ_{1}, λ_{2}, λ_{3} ∈ ℚ(ψ_{g}, ψ_{h})^{×} such that

L_{p}(f, K)(Ψ_{gh′}(1)) = λ_{1}^{2} · log_{E,p}(P_{1})^{2}, L_{p}(f, K)(Ψ_{gh}(1)) = λ_{2}^{2} · log_{E,p}(P_{2})^{2}, (55)
and L_{p}(K)(Ψ_{g}(1)) = λ_{3} · log_{p}(u_{g_{α}}).

Proof. We first prove (55): note that Ψ_{g}(1) = (ψ′_{g})^{−2} · χ^{−1} · N_{K} = \frac{ψ_{g}}{ψ′_{g}} N_{K}. By the func-

tional equation of Katz’s p-adic L-function quoted above, since \frac{ψ_{g}}{ψ′_{g}} is anticyclotomic we have

L_{p}(K)(\frac{ψ_{g}}{ψ′_{g}} N_{K}) = L_{p}(K)(\frac{ψ_{g}}{ψ′_{g}}) and hence (47) applies, yielding L_{p}(K)(Ψ_{g}(k))(1) = λ_{3}^{2} log_{p}(u_{g_{α}}) for some explicit value of λ_{3} ∈ ℚ(ψ_{g}, ψ_{h})^{×}. Moreover,

Ψ_{gh}(1) = (ψ′_{g})^{−1} · ψ_{h}^{−1} · N_{K} = ψ_{2} · N_{K}, Ψ_{gh′}(1) = (ψ′_{g})^{−1} · (ψ′_{h})^{−1} · N_{K} = ψ_{1} · N_{K},

and hence (49) can be applied, thereby proving (55).

In order to prove the admissibility of 𝔣 and the factorisation formula, recall we already saw in Lemma 3.6 that 𝔣_{1} is K(ψ_{g}, ψ_{h})-admissible, and likewise for 𝔣_{2} and 𝔣_{3}. An inspection of the fudge factor 𝔣(Ψ_{g}(k)) shows that it also is. Hence it remains to prove that ω(k) is L_{0}-admissible. Let us first show that ω interpolates to a meromorphic Iwasawa function on U_{g} such that (54) holds and ω(1) ∈ ℂ_{p}^{×} is a non-zero p-adic number, leaving for later that ω(1) actually lies in L_{0}^{×}.

Note that Lemmas 3.6 and 3.8 give rise to the following identity for all integers k ≥ 0:

ℒ_{p}^{g}(ğ, f̆, h̆)(k) · ℒ_{p}^{g}(ğ^{∗}, f̆, h̆^{∗})(k) · L_{p}(K)(Ψ_{g}(k))^{2} =  (56)

𝔢(k) · 𝔣_{1}(k) (\frac{𝔣(Ψ_{g}(k))}{𝔣_{2}(k)𝔣_{3}(k)})^{2} · L_{p}(f, K)(Ψ_{gh}(k)) · L_{p}(f, K)(Ψ_{gh′}(k))

where

𝔢(k) := 𝔢(Ψ_{g}(k))^{2}𝔢_{1}(k) = ( \frac{𝔢(Ψ_{g}(k)) · 𝔢(k, 2, 1)}{𝔢(f, Ψ_{gh}(k))𝔢(f, Ψ_{gh′}(k))} )^{2} .

An inspection of the factors appearing above shows that

𝔢(k) = 1 for all k ≥ 0.  (57)

More precisely, one checks that the four factors in the numerator E(g_{k}, f, h) of 𝔢(k, 2, 1) are equal to the four factors appearing in 𝔢(f, Ψ_{gh}(k))𝔢(f, Ψ_{gh′}(k)), and hence cancel out. In addition, the two factors showing up in 𝔢(Ψ_{g}(k)) are identical to those in the denominator of 𝔢(k, 2, 1), and thus also cancel out.

Since the unit u_{g_{α}} and the points P_{1} and P_{2} are non-torsion, it also follows from (55) and (56) that neither ℒ_{p}^{g}(ğ, f̆, h̆)(k) nor ℒ_{p}^{g}(ğ^{∗}, f̆, h̆^{∗})(k) vanish at k = 1. Note that (32) implies that

ω(k) = ℒ_{p}^{g}(ğ^{∗}, f̆, h̆^{∗})(k) / ℒ_{p}^{g}(ğ, f̆, h̆)(k),  (58)

from which it follows that ω ∈ L_{g}, with no poles at k = 1 and satisfying ω(1) ≠ 0. The factorisation formula claimed in the theorem is also a consequence of (56), (57) and (58).

To finish the proof, we need to show that ω(1) ∈ L_{0}. To do that, let W_{N} denote the Atkin-

Lehner operator induced by the matrix (\begin{matrix}0 & −1 \\ N & 0 \end{matrix}). As shown in [AL78] and [Hi2, Ch. X], it

commutes with the Shimura-Maass operator, and its adjoint with respect to the Petersson scalar product is −W_{N}. This implies that

I(ğ_{k}, f̆, h̆) = I(W_{N}(ğ_{k}), W_{N}(f̆), W_{N}(h̆)).

By the formulae in [AL78], W_{N}(h̆(q)) is a linear combination with coefficients in ℚ(h) of the modular forms λ_{h} · h^{∗}(q^{d}) where d ranges over the positive divisors of N/N_{h} and λ_{h} is the pseudo-eigenvalue of the newform h of level N_{h} in the sense of [AL78], characterized by the equation W_{N_{h}}(h) = λ_{h}h^{∗}. The analogous claim, with similar notations, holds for g_{k} and f. It thus follows from e.g. [DR1, Lemma 2.12] that the ratio

ω(k) = I(ğ_{k}^{∗}, f̆, h̆^{∗}) / I(W_{N}(ğ_{k}), W_{N}(f̆), W_{N}(h̆))

lies in ℚ(g_{k}, f, h, λ_{f}, λ_{g}, λ_{h}).

By [AL78, Theorem 2.1], since a_{N_{g}}(g_{k}), a_{N_{f}} (f) and a_{N_{h}}(h) are all non-zero, the pseudoeigenvalues of f, g_{k} and h are

λ_{f} = −a_{N_{f}} (f), λ_{g} = Ng^{\frac{k−2}{2}} 𝔤(χ^{−1})/a_{N_{g}}(g_{k}), λ_{h} = 𝔤(χ)/(\sqrt{N_{h}} · a_{N_{h}}(h)).

This implies that ω(1) lies in L_{0}, as claimed. Note that, since p ∤ N and the coefficients a_{N_{g}}(g_{k}) are Iwasawa functions on U_{g}, the above formulae give rise to a rather explicit expression of ω(k) as a p-adic continuous function. In any case, Theorem 3.9 follows. □

To conclude the proof of Theorem 3.3, recall that Proposition 2.6 recasts ℒ_{p}^{g}(ğ, f̆, h̆)(1) as the p-adic iterated integral. Combined with Theorem 3.9 we deduce that

(∫γ̆_{α} f̆· h̆)^{2} · log_{p}(u_{g_{α}})^{2} = 𝔣(1) log_{E,p}(P_{1})^{2} log_{E,p}(P_{2})^{2}  (59)

up to squares in ℚ(ψ_{g}, ψ_{h})^{×}. Set L = L_{0}(\sqrt{𝔣(1)}). Taking square-roots in (59) shows that

∫γ̆_{α} f̆· h̆ = λ · \frac{log_{E,p}(P_{1}) log_{E,p}(P_{2})}{log_{p}(u_{g_{α}})}

for some λ ∈ L^{×}, thereby proving Theorem 3.3.

3.3. Numerical evidence for inert primes. The proof of Conjecture ES based on Katzstyle p-adic L-series and their factorisations breaks down when the prime p is inert in K. Yet the units and points that arise can still be expressed in terms of elliptic units and Heegner points, at least in the rank (1, 1) setting, making this a natural first testing ground for Conjecture ES.

3.3.1. General remarks. Retain all the notations and assumptions of Section 3.2, but assume now that the prime p is inert in K.

Let K_{p} denote the completion of K at p, a quadratic unramified extension of ℚ_{p}. Since the frobenius element σ_{p} is a reflection in the dihedral group Gal (H/ℚ), the prime p splits completely in H/K and the choice of σ_{p} amounts to choosing an embedding H ↪ K_{p}, which shall be fixed from now on.

Because ϱ_{g} and ϱ_{h} have dihedral projective image, it follows that σ_{p} acts on V_{g} (resp. on V_{h}) with eigenvalues λ and −λ (resp. λ^{−1} and −λ^{−1}), for a suitable root of unity λ. Label the four values

α_{g} = λ, β_{g} = −λ, α_{h} = λ^{−1}, β_{h} = −λ^{−1},

so that

α_{g}α_{h} = 1, α_{g}β_{h} = −1,

as shall be assumed from now on. The decomposition

V_{gh} ≃ V_{ψ_{1}} ⊕ V_{ψ_{2}}

of Gal (H/ℚ)-modules continues to hold, but now σ_{p} acts on both V_{ψ_{1}} and V_{ψ_{2}} as reflections, with eigenvalues 1 and −1.

The asymmetry in the roles of g and h in the definition of the p-adic iterated integral continues to determine a pair (ψ_{1}, ψ_{2}) of global characters of G_{K} by requiring that ψ_{1} and ψ_{2} “each be divisible by ψ_{g}”. This pair is uniquely defined up to the action of Gal (K/ℚ), since replacing ψ_{g} by ψ′_{g} has the effect of replacing (ψ_{1}, ψ_{2}) by (ψ′_{2}, ψ′_{1}). Fix v_{ψ_{j}} ∈ V_{ψ_{j}} to be a non-zero vector in the one-dimensional ψ_{j}-eigenspace for the action of G_{K} on V_{ψ_{j}}. With this choice in hand, set v_{ψ′_{j}} := σ_{p}v_{ψ_{j}}, as in the previous section. Given any P ∈ Hom(V_{ψ_{j}}, E(H)_{L}), we shall write

P_{ψ_{j}} := P (v_{ψ_{j}}), P_{ψ′_{j}} := P (v_{ψ′_{j}}).

Lemma 3.10. The regulator det(R_{g_{α}}(E, ϱ_{gh})) is given by (1) If r(E, V_{ψ_{1}}) = r(E, V_{ψ_{2}}) = 1,

det R_{g_{α}}(E, ϱ_{gh}) = log_{E,p}(P_{ψ_{1}}) · log_{E,p}(Q_{ψ_{2}}) − log_{E,p}(P_{ψ′_{1}}) · log_{E,p}(Q_{ψ′_{2}}).

In particular, if ψ_{1} (say) is a quadratic (genus) character, so that the point P_{ψ_{1}} is a σ_{p}-eigenvector with eigenvalue ι = ±1, then

det R_{g_{α}}(E, ϱ_{gh}) = log_{E,p}(P_{ψ_{1}}) · log_{E,p}(Q_{ψ_{2}} − ιQ_{ψ′_{2}}).

(2) If r(E, V_{ψ_{1}}) = 2 and r(E, V_{ψ_{2}}) = 0,

det R_{g_{α}}(E, ϱ_{gh}) = log_{E,p}(P_{ψ_{1}}) · log_{E,p}(Q_{ψ′_{1}}) − log_{E,p}(P_{ψ′_{1}}) · log_{E,p}(Q_{ψ_{1}}),

where (P, Q) is a basis for Hom_{G_{ℚ}}(V_{ψ_{1}}, E(H)_{L}).

Observe in particular that the regulator vanishes in the special case where there are no global points (P_{1}, P_{2}) ∈ E(H)_{L}^{ϱ_{1}} × E(H)_{L}^{ϱ_{2}} belonging to distinct eigenspaces for the action of σ_{p}; this exceptional scenario in which the regulator is degenerate occurs precisely when

dim(ϱ_{1}) = dim(ϱ_{2}) = 1 (so that ψ_{1}^{2} = ψ_{2}^{2} = 1) and ϱ_{1}(p) = ϱ_{2}(p).  (60)

If (60) holds, Conjecture ES predicts that the iterated integrals attached to (f, g, h) vanish, in spite of the running hypothesis r_{an}(E, ϱ_{1}) = r_{an}(E, ϱ_{2}) = 1. See Example 4.3 below for a numerical verification of this interesting phenomenon.

Suppose for the rest of this section that (60) is not satisfied, so that, switching ϱ_{1} and ϱ_{2} if necessary, it may be assumed that 1 and −1 are eigenvalues of ϱ_{1}(σ_{p}) and ϱ_{2}(σ_{p}) respectively. Recall furthermore the points P_{ψ_{i}}^{±} defined from Heegner points in Lemma 3.1 and (43), and the unit u_{ψ_{0}}^{−} defined in (44) from elliptic units. Assuming (38) or (40), Conjecture ES predicts that there is some choice of test vectors f̆, γ̆ and h̆ for which

∫γ̆f̆· h̆ = \frac{1}{log_{p}(u_{ψ_{0}}^{−})} · \begin{cases}log_{E,p}(P_{ψ_{1}}^{+}) log_{E,p}(P_{ψ_{2}}^{−}) − log_{E,p}(P_{ψ_{1}}^{−}) log_{E,p}(P_{ψ_{2}}^{+}) & if ψ_{1}^{2}, ψ_{2}^{2} ≠ 1, \\ & and V_{ψ_{1}} ≠ V_{ψ_{2}}, \\ log_{E,p}(P_{ψ_{1}}^{+}) log_{E,p}(P_{ψ_{2}}^{−}) & otherwise. \end{cases}

3.3.2. Numerical evidence. The following experiment in support of (61) gave the authors their first evidence that the p-adic iterated integrals continue to have geometric meaning beyond the setting treated in Section 3.2.

Example 3.11. Consider the elliptic curve

E : y^{2} + xy + y = x^{3} + x^{2} + x

of conductor 83 labelled 83a in Cremona’s tables and write f for the associated weight 2 newform. Let χ be the odd quadratic character of conductor 83. The space S_{1}(83, χ) has dimension 1 and is spanned by the theta series g := θ_{ψ_{g}} attached to a cubic character ψ_{g} of G = Gal (H/K), where H is the Hilbert class field of K = ℚ(\sqrt{−83}), which has class number 3. The representation ρ_{g} is equal to the induced representation V_{ψ_{g}}. Let h = E(1, χ) ∈ M_{1}(83, χ) be the Eisenstein series attached to the pair (1, χ), whose Galois representation ϱ_{h} is induced from the trivial character ψ_{h} = 1 of G. Since g and h have rational Fourier coefficients it will be enough to take L = ℚ (although the representations are in fact defined over ℚ(ζ_{3})). The representations V_{gg} and V_{gh} decompose into irreducible constituents as follows:

\begin{aligned}V_{gg} & = & V_{ψ_{g}^{2}} ⊕ V_{ψ_{g}·ψ_{g}^{−1}} & = & V_{ψ_{g}} ⊕ ℚ ⊕ ℚ(χ) & =: & ϱ_{1} ⊕ ϱ_{2} ⊕ ϱ_{3}, \\ V_{gh} & = & V_{ψ_{g}·ψ_{h}} ⊕ V_{ψ_{g}·ψ_{h}^{−1}} & = & V_{ψ_{g}} ⊕ V_{ψ_{g}} & = & ϱ_{1} ⊕ ϱ_{1}. \end{aligned}

A direct calculation shows that

r(E, ϱ_{1}) = r(E, ϱ_{2}) = 1, r(E, ϱ_{3}) = 0,

and similarly for the analytic ranks. More precisely, the Mordell-Weil group

E(H)_{ℚ} = E(H)_{ℚ}^{V_{ψg}} ⊕ E(ℚ)_{ℚ}

is of rank 3, and is generated by the Heegner point

P = (t, −t^{2} − 2) ∈ E(H),  where t^{3} − t^{2} + t − 2 = 0,  (62)

along with its Galois translates P^{σ} and P^{σ^{2}}, where σ is a generator of Gal (H/K). The vector spaces E(ℚ)_{ℚ} and E(H)_{ℚ}^{V_{ψg}} are generated by

P_{ℚ} := P + P^{σ} + P^{σ^{2}} = (1, −3),  and  \{ \begin{matrix}P_{ψ_{g}}^{+} := 2P − P^{σ} − P^{σ^{2}}, \\ P_{ψ_{g}}^{−} := P^{σ} − P^{σ^{2}}. \end{matrix}

Let p = 5, which is in inert in K. The cubic polynomial in (62) has exactly one root over ℚ_{5}. There is therefore a unique embedding of H into ℂ_{5} for which the Heegner point P maps to E(ℚ_{5}); the images of the points P^{σ} and P^{σ^{2}} are then interchanged by the frobenius element σ_{5}.

The Stark unit that enters into the denominator of the conjectural expression for the p-adic iterated integral is the elliptic unit defined by u_{ψ_{0}}^{−} = u^{σ}/u^{σ^{2}}, so that

log_{p}(u_{ψ_{0}}^{−}) = log_{p}(u^{σ}) − log_{p}(u^{σ^{2}}),

where u ∈ H is the unique root of the polynomial

x^{3} − 2x^{2} − 2x − 1

whose image under the previously chosen embedding of H belongs to ℚ_{5}.

Let γ_{±1} ∈ S_{1}^{oc,ord}(83, χ)^{∨} be the dual basis to g_{±1} = g(q) ∓ g(q^{5}) in the space of 5-adic overconvergent modular forms, which maps g_{±1} to 1 and all other generalised eigenspaces in S_{1}^{oc,ord}(83, χ) to zero. A numerical calculation of 5-adic iterated integrals shows that

\begin{aligned}∫_{γ_{±1}} f · g & = & −1688341751390720654446425615337716320464561776401 mod 5^{70} \\ ∫_{γ_{±1}} f · h & = & −2981481189608571355501040629889815717824676581453 mod 5^{70}. \end{aligned}

Let E(g_{±1}, f, g) denote the algebraic factor defined at the start of Section 5.2. In all the examples where p is inert, this factor is always equal to

E(g_{±1}, f, g) := (1 − \frac{a_{p}(f)}{p} + \frac{1}{p}) (1 + \frac{a_{p}(f)}{p} + \frac{1}{p}) ∈ ℚ.

The following identities were checked to 70 digits of 5-adic precision:

∫_{γ_{±1}} f · g = \frac{5 · E(g_{±1}, f, g) · log_{E,5}(P_{ℚ}) log_{E,5}(P_{ψ_{g}}^{−})}{24 · log_{5}(u_{ψ_{0}}^{−})}  (63)

∫_{γ_{±1}} f · h = −\frac{5 · E(g_{±1}, f, h) · log_{E,5}(P_{ψ_{g}}^{+}) log_{E,5}(P_{ψ_{g}}^{−})}{24 · log_{5}(u_{ψ_{0}}^{−})}  (64)

in perfect agreement with the prediction (61) for both γ_{+1} and γ_{−1}. Note that u_{ψ_{0}}^{−} is a unit in the (−1)-eigenspace for σ_{5} while the numerators involve points in both eigenspaces, so that the resulting ratio belongs to ℚ_{5}.

Although the Heegner point P is only defined up to sign, it is worth noting that the righthand side of (63) and (64) are unchanged if P is replaced by −P.

Formulae similar to (63) and (64) (with the terms 5 and 24 in the numerator and denominator replaced by p and 4(p + 1) respectively) were verified to 15 digits of 13-adic precision (respectively, 10 digits of 19-adic precision) replacing p = 5 by the inert primes p = 13 and p = 19. Similar experiments were also performed with the class number 3 field K = ℚ(\sqrt{−331}) and the inert prime p = 7, where the same formulae (mutatis mutandis) were verified to 10 significant 7-adic digits.

The next example also involves Heegner points but is one of the most illuminating in this paper, insofar as it concerns a rank (1, 1) setting of Conjecture ES in which nonetheless the regulator R_{g_{α}}(E, ϱ_{gh}) is not factorisable. It illustrates a feature of Conjecture ES which initially surprised the authors, namely, that the numerator on the right hand side of Conjecture ES needs not, in contrast with Theorem 3.3, be the product of logarithms of two points but may involve a linear combination of such products. This ran counter to the intuition the authors had first gained in their proof of Conjecture ES in the setting of Section 3.2.

Example 3.12. Consider the curve

E : y^{2} + xy + y = x^{3} + x^{2} − 2x

of conductor 79 labelled 79a in Cremona’s tables and write f for the associated weight 2 newform.

Let χ be the odd Dirichlet character of order 2 and conductor 79 associated to the imaginary quadratic field K = ℚ(\sqrt{−79}), of class number 5. As in the previous example, denote by H the Hilbert class field of K and set G = Gal (H/K). The space S_{1}(79, χ) has dimension 2 and is spanned by a pair of dihedral eigenforms with coefficients in the field L = ℚ(\sqrt{5}) generated by the golden ratio ω = (1 + \sqrt{5})/2, namely

g = q + (−1 + ω) q^{2} + (1 − ω) q^{4} − ωq^{5} − q^{8} + q^{9} − q^{10} + ⋯

and its Galois conjugate g^{†}. Let h = E(1, χ) ∈ M_{1}(79, χ) be the Eisenstein series attached to the pair (1, χ). After fixing a quintic character ψ of G, it can be assumed that

V_{g} = V_{ψ^{2}} = V_{ψ^{3}},  V_{g^{†}} = V_{ψ} = V_{ψ^{4}},  V_{h} = V_{ψ_{h}} = L ⊕ L(χ_{K}),

where ψ_{h} is the trivial character of G_{K}. The representations V_{g^{†}g} and V_{g^{†}h} decompose into irreducible summands as follows:

\begin{aligned}V_{g^{†}g} & = & V_{ψ·ψ^{−2}} ⊕ V_{ψ·ψ^{2}} & = & V_{ψ} ⊕ V_{ψ^{2}} & =: & ϱ_{1} ⊕ ϱ_{2}, \\ V_{g^{†}h} & = & V_{ψ·ψ_{h}} ⊕ V_{ψ·ψ_{h}^{−1}} & = & V_{ψ} ⊕ V_{ψ} & = & ϱ_{1} ⊕ ϱ_{1}. \end{aligned}

A Heegner point calculation reveals that the Mordell-Weil group E(H) has rank 5, and is generated by the Heegner point

P = (t, \frac{−t^{4} − 3t^{3} + 2t^{2} + 7t − 6}{3}) , where t^{5} + t^{4} − 5t^{3} − 3t^{2} + 8t − 3 = 0,

and its Galois translates P^{σ}, P^{σ^{2}}, P^{σ^{3}} and P^{σ^{4}}, where σ is a generator of Gal (H/K), chosen in such a way that ψ(σ) = ζ_{5} with ζ_{5} a primitive fifth root of unity satisfying ζ_{5} + ζ_{5}^{4} = −ω. Let ω′ = (1 − \sqrt{5})/2 denote the conjugate of the golden ratio. The L-vector space E(H)_{L}^{ϱ_{1}} is then spanned by the points

P_{1}^{+} = 2P − ωP^{σ} − ω′P^{σ^{2}} − ω′P^{σ^{3}} − ωP^{σ^{4}},
P_{1}^{−} = (ζ_{5} − ζ_{5}^{4}) (P^{σ} − ωP^{σ^{2}} + ωP^{σ^{3}} − P^{σ^{4}}) ,

and E(H)_{L}^{ϱ_{2}} has a similar basis (P_{2}^{+}, P_{2}^{−}) obtained by replacing ζ_{5} by ζ_{5}^{3}, and thus ω by its conjugate ω′, in the above expressions for P_{1}^{+} and P_{1}^{−}.

Let p = 29 be the smallest prime which is inert in K but split in L/ℚ. (The latter property is not essential to the conjectures, but makes it easier to calculate the p-adic iterated integrals.) Fix the unique embeding of H into ℂ_{p} for which the image of P belongs to E(ℚ_{p}), and for which σ_{p} interchanges P^{σ} with P^{σ^{4}} and P^{σ^{2}} with P^{σ^{3}}.

For the elliptic unit, one may start with u as the unique root of x^{5} − 1014x^{4} + 299243x^{3} − 21035x^{2} +195x−1 in H ∩ℚ_{29} and define u_{g_{±1}^{†}} in exactly the manner as P_{2}^{−} was defined above

(only replacing P by u and using additive notation for (O_{H}^{×})_{L}), i.e.,

u_{g_{±1}^{†}} := (ζ_{5}^{3} − ζ_{5}^{2}) (u^{σ} − ωu^{σ^{2}} + ωu^{σ^{3}} − u^{σ^{4}}) .

Note that u_{g_{±1}^{†}} may be alternatively denoted u_{ψ_{0}}^{−}.

Let γ_{±1}^{†} ∈ S_{1}^{oc,ord}(79 · 23, χ)^{∨} be the dual element to g_{±1}^{†}, defined as before. A numerical calculation of p-adic iterated integrals reveals that

∫_{γ_{±1}^{†}} f · h = 371806430046362 mod 29^{10}
∫_{γ_{±1}^{†}} f · g = 259114157339642 mod 29^{10}.

The following identity was then verified to 10 digits of 29-adic precision:

∫_{γ_{±1}^{†}} f · g = \frac{p · E(g_{±1}^{†}, f, g) · (log_{E,29}(P_{1}^{+}) log_{E,29}(P_{2}^{−}) − log_{E,29}(P_{1}^{−}) log_{E,29}(P_{2}^{+}))}{4 · 5 · \sqrt{5} log_{29}(u_{g_{±1}^{†}})}. (65)

where here \sqrt{5} denotes the unique 29-adic square root of 5 which is congruent to 18 modulo 29.

Turning to ∫_{γ_{±1}^{†}} f · h, notice that E(g_{±1}^{†}, f, h) = E(g_{±1}^{†}, f, g) (= 864/841) and the transcendental factor log_{29}(u_{g^{†}}) expected on the righthand side of the formula is the same as for ∫_{γ_{±1}^{†}} f · g. Thus these factors should cancel upon taking a ratio and indeed one observes

(modulo 29^{10}) the identity

\frac{∫_{γ_{±1}^{†}} f · g}{∫_{γ_{±1}^{†}} f · h} = \frac{1}{2} · \frac{log_{E,29}(P_{1}^{+}) log_{E,29}(P_{2}^{−}) − log_{E,29}(P_{1}^{−}) log_{E,29}(P_{2}^{+})}{log_{E,29}(P_{1}^{+}) log_{E,29}(P_{1}^{−})}  (66)

= \frac{1}{2} (\frac{log_{E,29}(P_{2}^{−})}{log_{E,29}(P_{1}^{−})} − \frac{log_{E,29}(P_{2}^{+})}{log_{E,29}(P_{1}^{+})}) ,

which suggests that

∫_{γ_{±1}^{†}} f · h = \frac{p · E(g_{±1}^{†}, f, h) · log_{E,29}(P_{1}^{+}) log_{E,29}(P_{1}^{−})}{2 · 5 · \sqrt{5} log_{29}(u_{g_{±1}^{†}})}.  (67)

Both equations (65) and (67) are in perfect agreement with Conjecture ES.

Remark 3.13. It was a ratio identity like the one in (66) (for a simpler ratio in Example 3.11) which first convinced the authors that the p-adic iterated integral had geometric meaning in the setting where p is inert, even prior to their full understanding of the transcendental factor occurring in the denominator of the identity of Conjecture ES.

We now turn to our first numerical example in the rank 2 setting, illustrating a notable feature of Conjecture ES, namely that its right hand side can be a non-trivial p-adic combination of logarithms of global points in a “genuine rank two” setting involving a primitive L-function with a double zero at the center.

Example 3.14. We take the curve

E : y^{2} + xy + y = x^{3} − 7x + 5

of conductor 57 labelled 57b in Cremona’s tables and write f for the attached newform of weight 2.

Let χ be the odd Dirichlet character of order 2 and conductor 23 associated to the imaginary quadratic field K = ℚ(\sqrt{−23}), of class number 3. The space S_{1}(23, χ) has dimension 1 and is spanned by the theta series g := θ_{ψ_{g}} attached to a cubic character ψ_{g} of G = Gal (H/K), where H is the class field of K. As in Example 3.11 we find

V_{gg} = V_{ψ_{g}} ⊕ L ⊕ L(χ) with L = ℚ(ζ_{3})

but a calculation reveals that here

r(E, V_{ψ_{g}}) = 2, r(E, L) = r(E, L(χ)) = 0

and likewise for the analytic ranks. Here the Heegner hypothesis is not satisfied, but the descent code in Magma can be used to construct the required global points in E(H)_{L}^{V_{ψg}}. Over the cubic subfield

ℚ(a), a^{3} − a^{2} + 2a − 1 = 0

of H, the curve E has rank 2 with generators for the Mordell-Weil lattice

P := (9a^{2} − 4a + 17, −45a^{2} + 20a − 80), Q := (a^{2} + 3, 2a^{2} + 2).

We embed ℚ(a) into H = ℚ(b), where b^{6} − 6b^{4} + 9b^{2} + 23 = 0, by taking a = (b^{4} − 2b^{2} + 1)/9. Taking σ as a generator for G we compute four points

\begin{matrix}P_{ψ_{g}}^{+} := 2P − P^{σ} − P^{σ^{2}}, & P_{ψ_{g}}^{−} := P^{σ} − P^{σ^{2}} \\ Q_{ψ_{g}}^{+} := 2Q − Q^{σ} − Q^{σ^{2}}, & Q_{ψ_{g}}^{−} := Q^{σ} − Q^{σ^{2}} \end{matrix}

and similarly a unit u_{ψ_{0}}^{−} starting from the root (b^{4} − 5b^{2} + 4)/9 of x^{3} − x − 1 in H. We take p = 19 and find we may embed H into ℂ_{19} so that P, Q and u are defined over ℚ_{19}.

To compute the relevant 19-adic iterated integral we must work in 19-adic level 3 · 23 and so choose a 3-stabilisation ğ of g. We find x^{2} − a_{3}(g)x + χ(3) = (x − ζ_{3})(x − ζ_{3}^{2}) and take ğ to be the 3-stabilisation with U_{3}-eigenvalue ζ_{3}. (With the other choice the formula is exactly the same, except the algebraic factor is replaced by its complex conjugate.) We embed ℚ(ζ_{3}) into ℚ_{19} so that ζ_{3} ≡ 7 mod 19, and find

∫γ̆_{±1} f · g = −25103076413984358720047537708218 mod 19^{25}.

One checks to 25 digits of 19-adic precision that

∫γ̆_{±1} f · g = \frac{63ζ_{3} + 18}{26 · 19} · \frac{log_{E,19}(P_{ψ_{g}}^{+}) log_{E,19}(Q_{ψ_{g}}^{−}) − log_{E,19}(P_{ψ_{g}}^{−}) log_{E,19}(Q_{ψ_{g}}^{+})}{log_{p} u_{ψ_{0}}^{−}},

as predicted by Conjecture ES.

In the next example, the relevant Heegner point arises from a Shimura curve rather than a classical modular curve parametrisation. Note that Conjecture ES in principle allows the efficient computation of these Heegner points in terms of the Fourier expansions of classical modular forms rather than working with their Jacquet-Langlands lifts. For the purpose of verifying Conjecture ES, the authors did not actually calculate the Heegner points arising from the Shimura curve parametrisation, which raises significant computational challenges, but rather found appropriate global points by general descent methods.

Example 3.15. Consider the curve

E : y^{2} + y = x^{3} + x^{2} + 9x + 1

of conductor 35 labelled 35a in Cremona’s tables and write f for the associated weight 2 newform. Let K = ℚ(\sqrt{−23}) be the smallest imaginary quadratic field of class number 3, and note that both primes dividing 35 are inert in K. This is the simplest setting where there is no Heegner point on E arising from a modular curve, and in which one must rather work with the Shimura curve attached to the indefinite quaternion algebra ramified at 5 and 7.

Let χ be the quadratic Dirichlet character of conductor 23. Then S_{1}(23, χ) is one-dimen-sional and is spanned by the single dihedral form

g = q − q^{2} − q^{3} + q^{6} + q^{8} − q^{13} − q^{16} + · · · .

Let H be the Hilbert class field of K, set G = Gal (H/K), and let ϱ_{g} be the representation induced from a cubic character ψ_{g} of G with image contained in L = ℚ(ζ_{3}).

The representation V_{gg} decomposes as

V_{gg} = V_{ψ_{g}·ψ_{g}} ⊕ V_{ψ_{g}·ψ_{g}^{−1}} = V_{ψ_{g}} ⊕ L(χ) ⊕ L.

A direct calculation reveals that

r(E, V_{ψ_{g}}) = r(E, L(χ)) = 1, r(E, L) = rank(E(ℚ)) = 0,

and the same is true of the analytic ranks. Let P_{K} be a generator of E(K) up to torsion, which lies in the −1 eigenspace for the action of the involution in Gal (K/ℚ), and let P be a ℚ[Gal (H/K)]-module generator of E(H) ⊗ ℚ. The points P_{K} and P were not constructed using Lemma 3.1, since the Heegner point evoked in this lemma is difficult to compute directly in practice. Instead, one just takes

P_{K} := (−5, \frac{−5\sqrt{−23} − 1}{2}) , P := (\frac{−7t^{4} + 14t^{2} + 29}{36}, \frac{7t^{4} − 14t^{2} − 68}{24}) ∈ E(H),

where t is a normal basis generator for H over ℚ satisfying the equation

t^{6} − 6t^{4} + 9t^{2} + 23 = 0.

The point P just turned out to be the first point in a basis for the rank 3 Mordell-Weil group E(H) computed by Magma.

Let p = 5 be the smallest prime which is inert in K. After a suitable choice of embedding of H into K_{5} for which P maps to E(ℚ_{5})^{−}, the frobenius element σ_{5} interchanges P^{σ} and P^{σ^{2}}. With these conventions, we may set

P_{ψ_{g}}^{+} := 2P − (P^{σ} + P^{σ^{2}}).

Both f and g are 5-adic overconvergent modular forms of level 7 and 23 respectively. Stabilise g to an eigenform ğ = g(q) − g(q^{7}) of level 161 = 7 · 23 with U_{7} eigenvalue 1. Then ğ has two stabilisations

ğ_{±1} = ğ(q) ∓ ğ(q^{5})

at 5, which can be viewed either as classical modular forms of level 161 · 5 or as 5-adic overconvergent forms of level 161. Write γ̆_{±1} ∈ S_{1}^{oc,ord}(7 · 23, χ)^{∨} for the usual dual element to ğ_{±1}. A direct evaluation of the 5-adic iterated integral reveals that

∫γ̆_{+1} f · g = − ∫γ̆_{−1} f · g = 376357248757242028241 mod 5^{30}.

Starting with the elliptic unit u which is the unique root of x^{3} + 6x^{2} + 29x − 1 in H ∩ ℚ_{5}, the unit u^{−} is defined by u^{−} = (u^{τ} /u^{τ^{2}}). The following identity was then checked to 30 digits of 5-adic precision:

∫γ̆_{+1} f · g = \frac{3 · 7 · log_{E,5}(P_{K}) · log_{E,5}(P_{ψ_{g}}^{+})}{2 · 5 · log_{5}(u^{−})}

as predicted by (61). Note that once the point P_{K} is known, this identity leads to an explicit analytic formula for a global point on E over the Hilbert class field H, namely P_{ψ_{g}}^{+}, without passing through the calculation of the Shimura curve parametrisation of E, as would be necessary if one were to resort to Heegner points instead of p-adic iterated integrals.

4. Stark-Heegner points and theta series of real quadratic fields

4.1 Theoretical preliminaries. This chapter focusses on the case where g and h are theta series attached to characters—denoted ψ_{g} and ψ_{h} respectively—of mixed signature (+, −) of the same real quadratic field K. The self-duality assumption on V_{gh} implies that

ψ_{1} := ψ_{g}ψ_{h}, ψ_{2} := ψ_{g}ψ′_{h}

are dihedral characters of K, which are totally even and totally odd respectively and that

V_{g} = V_{ψ_{g}},  V_{h} = V_{ψ_{h}},  V_{gh} = V_{ψ_{1}} ⊕ V_{ψ_{2}}.

As in the previous chapter, the Hasse-Weil-Artin L-function factors as L(E, ϱ_{gh}, s) = L(E/K, ψ_{1}, s) · L(E/K, ψ_{2}, s), and Hypothesis A is fulfilled as soon as the conductors of ψ_{g} and ψ_{h} are coprime to N_{f}. In all examples where Conjecture ES is tested below, Hypothesis B holds because we have there L(E/K, ψ_{1}, 1) = L(E/K, ψ_{2}, 1) = 0, and Hypothesis C holds too by Proposition 1.1.

Let p be a rational prime, at which the representations V_{g} and V_{h} are unramified, and let α_{g} and β_{g} (resp. α_{h} and β_{h}) denote as before the frobenius eigenvalues for V_{g} (resp. V_{h}) attached to this prime. In order for g to satisfy the classicality condition of Hypothesis C, it is necessary to assume that

p is inert in K.  (68)

In that case, by the remarks surrounding (22), the Stark unit u_{g_{α}} =: u_{K} of Conjecture ES can be taken to be equal to the fundamental unit of K, and in particular it depends only on K and not on ψ_{g}.

Assumption (68) also implies that the frobenius element σ_{p} corresponds to reflections in the dihedral groups cut out by the representations V_{ψ_{1}} and V_{ψ_{2}}. As in Section 3.3, it is assumed that

α_{g}α_{h} = 1, α_{g}β_{h} = −1, β_{g}α_{h} = −1, β_{g}β_{h} = 1.

The eigenvalues of σ_{p} on V_{ψ_{1}} are (1, −1), and likewise for V_{ψ_{2}}. Note in particular that in this setting, the representation V_{gh} is never regular at p, even though the representations V_{g}, V_{h}, V_{ψ_{1}} and V_{ψ_{2}} are all regular.

Consider first the “generic” setting where neither of the characters ψ_{1} or ψ_{2} is a quadratic (genus) character, so that the representations V_{ψ_{1}} and V_{ψ_{2}} are both irreducible. Conjecture ES then predicts the existence of points P_{1} ∈ E(H)_{L}^{V_{ψ_{1}}} and P_{2} ∈ E(H)_{L}^{V_{ψ_{2}}} whose projections onto the different σ_{p}-eigenspaces

P_{1}^{+} ∈ (E(H)_{L}^{V_{ψ_{1}}} )^{σ_{p}=+1},  P_{2}^{−} ∈ (E(H)_{L}^{V_{ψ_{2}}} )^{σ_{p}=−1},
P_{1}^{−} ∈ (E(H)_{L}^{V_{ψ_{1}}} )^{σ_{p}=−1},  P_{2}^{+} ∈ (E(H)_{L}^{V_{ψ_{2}}} )^{σ_{p}=+1},

are related to a suitable p-adic iterated integral by

∫γ̆_{α} f̆· h̆ =^{?} \frac{log_{E,p}(P_{1}^{+}) log_{E,p}(P_{2}^{−}) − log_{E,p}(P_{1}^{−}) log_{E,p}(P_{2}^{+})}{log_{p}(u_{K})}.  (69)

This conjecture becomes slightly easier to verify when one of the characters ψ_{1} or ψ_{2} is a quadratic (genus) character. Assume without loss of generality that this is the case for ψ_{1}, so that

V_{ψ_{1}} = L(χ_{1}) ⊕ L(χ_{2})

becomes reducible as a G_{ℚ}-module, breaking up into two one-dimensional representations attached to quadratic Dirichlet characters χ_{1} and χ_{2} satisfying χ_{1}χ_{2} = ϵ_{K}, so that in particular χ_{1}(p) = −χ_{2}(p). When the iterated integral of the left-hand side of (69) is non-zero, Conjecture ES predicts that precisely one of the characters χ_{1} or χ_{2} occurs in the Mordell-Weil group of E. Assume that it is χ_{1} which arises, and set ι := χ_{1}(p) = ±1.

One then has

P_{1} ∈ E(H)_{L}^{χ_{1}}.

Conjecture ES predicts that, for a suitable choice of test vectors,

∫γ̆_{α} f̆· h̆ =^{?} \frac{log_{E,p}(P_{1}) log_{E,p}(P_{2}^{−ι})}{log_{p}(u_{K})}.  (70)

Finally consider the case in which ψ_{1}^{2} = ψ_{2}^{2} = 1. The representation V_{gh} then decomposes as a sum of one dimensional summands, and Conjecture ES takes the form

∫γ̆_{α} f̆· h̆ =^{?} \frac{log_{E,p}(P_{1}) log_{E,p}(P_{2})}{log_{p}(u_{K})},  (71)

provided the points P_{1} and P_{2}, which both belong to quadratic extensions of ℚ, belong to distinct eigenspaces for σ_{p}; in the case where they are in the same eigenspace, Conjecture ES predicts a subtle vanishing of the p-adic iterated integral arising from a degeneracy in the regulator term. This case is illustrated in Example 4.3 below.

4.2. Numerical evidence. Weight one modular forms whose Artin representations can be induced only from characters of class groups of real quadratic fields are rather rare in small level. The first such occurrence is in level 145, where one finds such a form coming from a quartic character of ℚ(\sqrt{5}) with conductor a prime of norm 29. The next example is concerned with this modular form.

Example 4.1. Let K = ℚ(\sqrt{5}) and let

ψ_{g} = ψ_{h}^{−1}

be the quartic character of K which is ramified at the prime λ = (29, \frac{11+\sqrt{5}}{2}) of norm 29. Let χ_{5} and χ_{29} be the quadratic and quartic Dirichlet characters of conductors 5 and 29 respectively, and let χ := χ_{5}χ_{29} be the odd quartic character of conductor 5 · 29. Both ψ_{g} and χ take values in the field L := ℚ(i) and are unique up to conjugation by Gal (L/ℚ). Fix a choice of these characters in such a way that the theta series

g ∈ S_{1}(5 · 29, χ^{−1}), h ∈ S_{1}(5 · 29, χ)

associated to ψ_{g} and ψ_{h} belong to spaces of weight 1 modular forms with those characters (and Fourier coefficients in L). The Artin representation V_{gh} is given by

V_{gh} = L ⊕ L(χ_{5}) ⊕ V_{ψ},

where ψ is the unique (up to conjugation) quartic ring class character of K of conductor 29. Let H be the abelian quartic extension of K cut out by ψ, and write Gal (H/K) =: ⟨σ⟩. The field H is Galois over ℚ, with Galois group G = D_{8}, the dihedral group of order 8. The subfield of H fixed by the center ⟨σ^{2}⟩ of G is the totally real biquadratic field M := ℚ(\sqrt{5}, \sqrt{29}), and H is the quadratic extension of M given by

H = M(\sqrt{δ}) = M(\sqrt{δ′}), δ = (\frac{−29 + 3\sqrt{29}}{2}) , δ′ = (\frac{−29 − 3\sqrt{29}}{2}) .

Note that δ and δ′ are conjugate totally negative elements of M and that δδ′ = 5 · 29. Let f be the weight two cusp form of level 17 attached to the unique elliptic curve

E : y^{2} + xy + y = x^{3} − x^{2} − x − 14

of conductor 17 denoted 17a in the tables of Cremona. The conductor of E is inert in the quadratic field K. An explicit calculation shows that

E(ℚ)_{L} = 0, E(K)_{L}^{χ_{5}} = L · P_{K}, where P_{K} = (\frac{379}{20}, \frac{−1995 + 7218\sqrt{5}}{200}) .

The group G admits four distinct one-dimensional representations, on which σ^{2} acts trivially, and a unique irreducible representation of dimension two with rational traces, which is isomorphic to V_{ψ} and on which σ^{2} acts as multiplication by −1. Hence

E(H)_{ℚ}^{V_{ψ}} = E(H)_{ℚ}^{σ^{2}=−1}.

Considerations involving signs in the functional equations of L-series suggest that the representation V_{ψ} occurs with odd multiplicity in E(H) (cf. for example the introduction of [Dar]). Note that the field H, a totally odd cyclic quartic extension of K, contains no imaginary quadratic subfields. The classical Heegner point construction is therefore unavailable to produce the global points in E(H)_{ℚ}^{V_{ψ}} whose existence is predicted by the Birch and Swinnerton-Dyer conjecture. The Stark-Heegner point algorithms described in [DP] can be used as a substitute for the method based on Heegner points, and yield four non-trivial points in E(H) given by

P = ( \frac{−220777 − 17703\sqrt{145}}{5800} , \frac{214977 + 17703\sqrt{145}}{11600} + \frac{28584525 + 3803103\sqrt{5} + 1645605\sqrt{29} + 2364771\sqrt{145}}{290000} \sqrt{δ})
P^{σ} = ( \frac{−220777 + 17703\sqrt{145}}{5800} , \frac{214977 − 17703\sqrt{145}}{11600} + \frac{28584525 + 3803103\sqrt{5} − 1645605\sqrt{29} − 2364771\sqrt{145}}{290000} \sqrt{δ′})
P^{σ2} = ( \frac{−220777 − 17703\sqrt{145}}{5800} , \frac{214977 + 17703\sqrt{145}}{11600} − \frac{28584525 + 3803103\sqrt{5} + 1645605\sqrt{29} + 2364771\sqrt{145}}{290000} \sqrt{δ})
P^{σ3} = ( \frac{−220777 + 17703\sqrt{145}}{5800} , \frac{214977 − 17703\sqrt{145}}{11600} − \frac{28584525 + 3803103\sqrt{5} − 1645605\sqrt{29} − 2364771\sqrt{145}}{290000} \sqrt{δ′}) .

These four Galois conjugate points are constructed from periods arising from a p-adic integration on H_{17} × H based on the minus modular symbol of f, as described in [Dar]. The conjectures of loc.cit. predict that these points should lie in the minus eigenspace for complex conjugation, which coincides in this case with the central involution σ^{2} of Gal (H/ℚ). It can indeed be observed that σ^{2}P = −P and that σ^{3}P = −σP. In particular, the Stark-Heegner point P and its conjugates generate a rank two subgroup of E(H) which is isomorphic to V_{ψ}.

This shows that r(E, V_{ψ}) ≥ 1, and the belief that Stark-Heegner points ought to behave like their quadratic imaginary counterparts strongly suggests that in fact r_{an}(E, V_{ψ}) = 1. If such is the case, Conjecture ES predicts the non-vanishing of some p-adic iterated integral attached to (f, g, h).

The global point P and its conjugates were obtained from a 17-adic calculation. This is an essential limitation of [Dar], where one is forced to work p-adically at a prime p dividing the conductor of E. On the other hand, the p-adic iterated integral appearing on the left of (70) can be calculated using any prime p which is inert in K. It turns out to be expedient (albeit, not essential, as in the calculations relying on the algorithms of [DP]) to work at p = 17. This is because, with this choice of p, the classical modular form f of level 17 is p-adically of level 1, and the p-adic iterated integral can therefore be calculated by applying the ordinary projection algorithms of [La2] to a space of ordinary overconvergent 17-adic modular forms of weight one and relatively modest level 5 · 29.

It can be checked that, for a suitable embedding of H into the quadratic extension ℚ_{17^{2}} of ℚ_{17}, the points P and σ^{2}P are defined over ℚ_{17} while the points σP and σ^{3}P are defined over ℚ_{17^{2}} and are interchanged by the frobenius element σ_{17}. Fix such an embedding from now on, and let γ_{±} ∈ S_{1}(Γ_{1}(5 · 29), χ)_{L}^{∨}[g_{±}] be the unique element satisfying

γ_{±}(g_{±}^{∗}) = 1.

Here g_{±} = g_{±ζ_{8}^{3}} denotes the 17-stabilisations of g. A numerical evaluation of the 17-adic iterated integrals, to 16 significant digits shows that

∫_{γ_{±}} f · h = 1259389260500681328 × 17 mod 17^{16},

and that

∫_{γ_{±}} f · h = \frac{log_{E,17}(P_{K}) log_{E,17}(P)}{3 · 17 · log_{17}(u_{K})}  (72)

to an accuracy of 16 significant 17-adic digits. Since σ_{17}P_{K} = −P_{K}, while P generates E(H)_{L}^{V_{ψ}} ∩ E(ℚ_{17}), this is in perfect agreement with the prediction of (70).

Remark 4.2. It is worth emphasising that the Stark-Heegner point P is a priori quite mysterious and not known to correspond to a global point, while the 17-adic iterated integral appearing on the left-hand side of (72) can be related to the 17-adic logarithm of the global cohomology class κ(f̆, ğ, h̆) ∈ H^{1}(H, V_{17}(E)) constructed in [DR2] as a limit of 17-adic étale Abel-Jacobi images of suitable “generalised Gross-Schoen diagonal cycles”. Since both the 17-adic iterated integral appearing on the left hand side of (72) and the “integrals on H_{17}×H” used to compute the Stark-Heegner point P are given by explicit 17-adic analytic expressions, a proof of formula (72) does not seem entirely out of reach. Such a proof would represent substantial progress in understanding the global properties of Stark-Heegner points, perhaps making it possible to use them in Euler system arguments of the kind exploited originally by Kolyvagin. This would extend the approach described in Theorems A and C of [DR2] from “rank zero” to certain “rank one” settings.

The next example involves forms g and h with representation having projective image D_{4}. Such odd representations are induced from class characters of both real and imaginary quadratic fields, so this example would be equally at home in Chapter 3. Its principal novelty is that it illustrates situations where the p-adic iterated integrals vanish in rank (1, 1) settings because the regulator term of Conjecture ES is 0.

Example 4.3. Consider the curves

E_{a} : y^{2} + y = x^{3} − x^{2} + 10x + 6
E_{b} : y^{2} + xy + y = x^{3} + x^{2} − x − 2
E_{c} : y^{2} + y = x^{3} − x^{2} − x + 1

of conductor 155 labelled 155a, 155b, and 155c in Cremona’s tables and write f_{a}, f_{b} and f_{c} for the associated weight two newforms.

Let χ_{1} denote the trivial character, let χ_{5} and χ_{31} denote the (even and odd, respectively) characters of conductor 5 and 31, and set χ := χ_{155} := χ_{5} · χ_{31}, which takes values in L = ℚ. The space S_{1}(155, χ) has dimension 3 and contains three dihedral forms: one form g with integer Fourier coefficients

g = q + q^{4} − q^{5} − q^{9} + q^{16} − 2q^{19} − ⋯

and a second pair of complex conjugate forms each defined over the cyclotomic field ℚ(ζ_{3}). The form g has associated linear representation ρ_{g} with projective image D_{4}, and ρ_{g} can be described in the following equivalent ways:

(1) It is induced from a quartic character of the Hilbert class field of ℚ(\sqrt{−155});

(2) it is induced from a quadratic character of the imaginary quadratic field ℚ(\sqrt{−31}), whose conductor is a prime ideal of norm 5;

(3) it is induced from a quadratic character of the real quadratic field ℚ(\sqrt{5}), whose

conductor is a prime ideal of norm 31. (The remaining pair of complex conjugate forms are associated to representations which arise from quartic characters of ℚ(\sqrt{−31}) with conductor a prime ideal of norm 5—they are not considered further in this example.)

The representation V_{gg} decomposes as

V_{gg} = L ⊕ L(χ_{5}) ⊕ L(χ_{31}) ⊕ L(χ_{155}),

and the corresponding ranks are given by

\begin{matrix}r(E_{a}/ℚ) = 1, & r(E_{a}, χ_{5}) = 0, & r(E_{a}, χ_{31}) = 1, & r(E_{a}, χ_{155}) = 0 \\ r(E_{b}/ℚ) = 0, & r(E_{b}, χ_{5}) = 0, & r(E_{b}, χ_{31}) = 0, & r(E_{b}, χ_{155}) = 0 \\ r(E_{c}/ℚ) = 1, & r(E_{c}, χ_{5}) = 1, & r(E_{c}, χ_{31}) = 0, & r(E_{c}, χ_{155}) = 0, \end{matrix}

and likewise for the analytic ranks. In particular, the central critical value L(E_{b}, ϱ_{gg}, 1) is non-zero, and hence the curve E_{b} fails to satisfy Hypothesis B of the introduction. It will therefore be ignored from now on.

As predicted by the BSD conjecture, the points on E_{a} and E_{c} over ℚ and the fields ℚ(\sqrt{−31}) and ℚ(\sqrt{5}) are given by

P_{a} = (2, 5) ∈ E_{a}(ℚ), P_{a,31} = (\frac{−1630}{729}, \frac{40175\sqrt{−31} − 19683}{39366}) ∈ E_{a}(ℚ(\sqrt{−31}))^{χ_{31}},

P_{c} = (1, 0) ∈ E_{c}(ℚ), P_{c,5} = (0, \frac{−1 + \sqrt{5}}{2}) ∈ E_{c}(ℚ(\sqrt{5}))^{χ_{5}}.

Consider both the primes p = 7 and p = 13, which have different splitting behaviours in the three quadratic fields, as shown in the table below.

\begin{matrix}p & χ_{5}(p) & χ_{31}(p) & χ_{155}(p) \\ 7 & −1 & +1 & −1 \\ 13 & −1 & −1 & +1 \end{matrix}

For p = 7 the eigenvalues of σ_{p} on V_{g} are α_{g} = 1 and β_{g} = −1, and so

(α_{g} · α_{g}, α_{g} · β_{g}) = (1, −1), (β_{g} · α_{g}, β_{g} · β_{g}) = (−1, 1).

The 7-adic iterated integrals attached to (f_{a}, γ_{±1}, g) and (f_{c}, γ_{±1}, g) are

∫_{γ_{1}} f_{a} · g = ∫_{γ_{−1}} f_{a} · g = 0 mod 7^{30}
∫_{γ_{1}} f_{c} · g = ∫_{γ_{−1}} f_{c} · g = 1416540068067896281523634 × 7 mod 7^{30}.

For p = 13 the eigenvalues of σ_{p} on V_{g} are α_{g} = i and β_{g} = −i, and so

(α_{g} · α_{g}, α_{g} · β_{g}) = (−1, 1), (β_{g} · α_{g}, β_{g} · β_{g}) = (1, −1).

The 13-adic iterated integrals attached to (f_{a}, γ_{±i}, g) and (f_{c}, γ_{±i}, g) are

∫_{γ_{i}} f_{a} · g = − ∫_{γ_{−i}} f_{a} · g = 10546307292618612 mod 13^{15}
∫_{γ_{i}} f_{c} · g = ∫_{γ_{−i}} f_{c} · g = 21913581880269709 mod 13^{15}.

Following the notations in the rank (1, 1) setting of Conjecture ES, the pair (ϱ_{1}, ϱ_{2}) of irreducible subrepresentations of V_{gg} attached to the curves E_{a} and E_{c} can be taken to be (χ_{1}, χ_{31}) and (χ_{1}, χ_{5}), respectively, where χ_{1} is the trivial character. Observe that the eigenvalues of σ_{13} are (1, −1) in both cases. One checks to 15-digits of 13-adic precision that

∫_{γ_{i}} f_{a} · g = \frac{80 · log_{E_{a},13}(P_{a}) log_{E_{a},13}(P_{a,31})}{i · 63 · 13 · log_{13}(u^{−})}

where i is the unique square root of −1 in ℤ_{13} which is ≡ 5 mod 13, and u^{−} = (1 + \sqrt{5})/2 is a fundamental unit for ℚ(\sqrt{5}). Note that the same formula holds if one replaced i by −i on both sides. Similarly one checks to the same amount of 13-adic precision that

∫_{γ_{i}} f_{c} · g = −\frac{80 · log_{E_{c},13}(P_{c}) log_{E_{c},13}(P_{c,5})}{21 · 13 · log_{13}(u^{−})}.

Turning now to p = 7 and the curve E_{a}, observe that (χ_{1}(7), χ_{31}(7)) = (1, 1), and hence the regulator term in Conjecture ES vanishes. This explains why the iterated integrals attached to (f_{a}, γ_{±1}, g) were observed to vanish, at least numerically to 30 digits of 7-adic precision. The situation for the curve E_{c} is different, since (χ_{1}(7), χ_{5}(7)) = (1, −1). As predicted by Conjecture ES, the authors were able to verify, to 30 digits of 7-adic precision, that

∫_{γ_{±1}} f_{c} · g = −\frac{8 · log_{E_{c},7}(P_{c}) log_{E_{c},7}(P_{c,5})}{3 · 7 · log_{7}(u^{−})}.

The final two chapters summarize some of the numerical evidence that was obtained in support of Conjecture ES in settings where g and h are not both theta series. It is an apt moment to pause momentarily and thank again Steve Donnelly: most of the interesting global points in both chapters were found by him upon our request using his descent code. We might also stress here that although the overconvergent projection algorithms used to compute p-adic iterated integrals are efficient, such computations still remain something of a technological tour de force: many of our examples pushed the available hardware and software (the algorithms of [La2] in particular) to their current limits.

5. Stark points over A_{4}, S_{4} or A_{5} extensions and exotic weight one forms

This chapter focusses on the case where V_{gh} is reducible and at least one of g or h is an exotic cuspform whose associated projective Galois representation cuts out a non-totally real extension of ℚ with Galois group A_{4}, S_{4} or A_{5}. (The case in which V_{gh} is irreducible is very difficult to test experimentally, due to the apparent scarcity of such V_{gh} of manageable size occurring with multiplicity two in the Mordell-Weil group of elliptic curves of reasonable conductor. See [DLR1, Example 6.2] though for a successful experiment in the analogous setting in which Mordell-Weil groups are replaced by groups of units.)

Lemma 5.1. Assume that one of g or h is an exotic weight one form and that g and h satisfy the self-duality condition. Then the representation V_{gh} is irreducible, unless g is the twist of h^{∗} by a quadratic character ι, in which case

V_{gh} = L(ι) ⊕ Ad_{g}(ι).

Proof. The discussion in Section 2 of [DLR1] rules out the case where V_{gh} contains an irreducible subrepresentation of dimension two. Hence V_{gh} contains an irreducible onedimensional subrepresentation, and the result follows from Schur’s Lemma (cf. [DLR1, §2]. □

So if V_{gh} is reducible, Lemma 5.1 implies that g is a quadratic twist of h^{∗}. Since it is always possible to replace the form f by one of its quadratic twists, there is no real loss in generality in assuming that g = h^{∗} which is what shall be done in the rest of this chapter. In this setting, Conjecture ES suggests a new, efficiently computable (conjectural) analytic formula for the p-adic logarithm of global points defined over the A_{4}, S_{4} or A_{5}-extension cut out by the adjoint Ad_{g} (or, equivalently, by the projective representation attached to g).

5.1. Theoretical preliminaries. Let h be an exotic weight one form, let g = h^{∗} be its conjugate form, and let L be as before the field generated by the Fourier coefficients of h, or of g = h^{∗}. The representation V_{gh} = L ⊕ Ad_{g}, where L refers to the trivial representation of G_{ℚ} with coefficients in L, factors through Gal (H/ℚ), where H is the field cut out by the projective representation attached to g, so that

Gal (H/ℚ) ≃ A_{4}, S_{4} or A_{5}.

Conjecture ES predicts that the trilinear form I′_{p} in (8) arising from p-adic iterated integrals may be non-identically zero only when

r(E, L) = r(E, Ad_{g}) = 1,  (73)

or

r(E, L) = 0, r(E, Ad_{g}) = 2. (74)

In the rank (1, 1) setting, Conjecture ES asserts that, for a suitable choice of elements

P_{ℚ} ∈ E(ℚ),  P_{g_{α}} ∈ (E(H)_{L}^{Ad_{g}})^{σ_{p}=α_{g}/β_{g}},  u_{g_{α}} ∈ ((O_{H}^{×})_{L}^{Ad_{g}})^{σ_{p}=α_{g}/β_{g}},

which, since α_{g} ≠ ±β_{g}, are unique up to scaling by L^{×}, and for a suitable choice f̆, γ̆_{g_{α}} and h̆ of test vectors attached to (f, g_{α}, h),

∫γ̆_{gα} f̆· h̆ =^{?} \frac{log_{E,p}(P_{ℚ}) log_{E,p}(P_{g_{α}})}{log(u_{g_{α}})}.

In the rank 2 setting, Conjecture ES asserts that

∫γ̆_{gα} f̆· h̆ =^{?} \frac{log_{E,p}(P_{1}) log_{E,p}(Q_{g_{α}}) − log_{E,p}(Q_{1}) log_{E,p}(P_{g_{α}})}{log(u_{g_{α}})},

where (P, Q) is a basis of Hom_{G_{ℚ}}(Ad_{g}, E(H)_{L}), and P_{1}, Q_{1} (resp. P_{g_{α}}, Q_{g_{α}}) denote the images of a vector v_{1} ∈ Ad_{g}^{σ_{p}=1} (resp. v_{2} ∈ Ad_{g}^{σ_{p}=α_{g}/β_{g}}) under these homomorphisms.

In the rank (1, 1) setting, the conjectural formula above gives an explicit, albeit conjectural, p-adic analytic expression for a global point—namely P_{g_{α}}—defined over the field H cut out by Ad_{g}, depending on quantities which are a priori accessible computationally:

P_{g_{α}} =^{?} exp_{E,p} (\frac{log(u_{g_{α}})}{log_{E,p}(P_{ℚ})} · ∫γ̆_{gα} f̆· h̆) .  (75)

The point P_{g_{α}} can be viewed as an elliptic curve analogue of the Stark unit u_{g_{α}}. For the sake of numerical experiments, it is worth giving a more explicit description of P_{g_{α}}, depending on the nature of Gal (H/ℚ).

5.1.1. Cases where g is of tetrahedral type. The group G = Gal (H/ℚ) ≃ A_{4} has three representations of dimension one which factor through the quotient A_{4}/V_{4} ≃ A_{3}, and a unique faithful irreducible representation of dimension three, which is therefore isomorphic to the adjoint representation Ad_{g}. Let R(x) be any degree four polynomial whose splitting field is H, and let M be the quartic field which is generated by a single root of R. Then Ad_{g} is related to a permutation representation by the formula

Ad_{g} = Ind_{G_{M}}^{G_{ℚ}} L − L.

It follows that

r(E, Ad_{g}) = r(E, Ind_{M}^{ℚ} L) − r(E, L) = rank(E(M)) − rank(E(ℚ)).

(The Artin formalism for L-series implies similar equalities for the analytic ranks.) In particular when r(E, Ad_{g}) = 1, there is (up to scaling) a unique point P ∈ E(M) whose trace to E(ℚ) is zero, and E(H)_{L}^{Ad_{g}} is generated, as an L-vector space, by the four Galois translates of P, whose sum is zero. Let P =: P_{1}, P_{2} , P_{3} and P_{4} denote these Galois translates, on which Gal (H/ℚ) = A_{4} acts via the natural permutation action. The point P_{g_{α}} can be expressed as a linear combination of the P_{j}, depending on the conjugacy class of σ_{p}.

Case 1: σ_{p} = 1. In the case α_{g}/β_{g} = β_{g}/α_{g} = 1, the representations ϱ_{g} and Ad_{g} are not regular at p, and Hypothesis C is not satisfied.

Case 2: σ_{p} is of order 2. In that case it may be assumed that the points P_{1} and P_{2} are defined over ℚ_{p^{2}} and are interchanged by σ_{p}, and likewise for P_{3} and P_{4}. It follows that α_{g}/β_{g} = −1 and that

(E(H)_{L}^{Ad_{g}})^{σ_{p}=1} = L · (P_{1} + P_{2}) = L · (P_{3} + P_{4})  is one-dimensional,
(E(H)_{L}^{Ad_{g}})^{σ_{p}=−1} = L · (P_{1} − P_{2}) ⊕ L · (P_{3} − P_{4})  is two-dimensional.

In this scenario, in which V_{g} is regular but Ad_{g} is not, the point P_{g_{α}} is a specific L-linear combination of the points P_{1} − P_{2} and P_{3} − P_{4}, which lies in the well-defined one-dimensional subspace hom(V_{g}^{g_{β}}, V_{g}^{g_{α}}) of Ad_{g}.

Case 3: σ_{p} is of order 3. In that case, it may be assumed that σ_{p} fixes the point P_{1}, which is thus defined over ℚ_{p}, and cyclically permutes the points P_{2}, P_{3} and P_{4}. The eigenvalues of σ_{p} on Ad_{g} are then equal to 1, α_{g}/β_{g} = ζ_{3} and β_{g}/α_{g} = ζ_{3}^{−1}, for some primitive cube root ζ_{3} of 1, and P_{g_{α}} is determined, up to scaling, by the rule

P_{g_{α}} = P_{2} + ζ_{3}^{−1}P_{3} + ζ_{3}P_{4}.  (76)

In particular,

(E(H)_{L}^{Ad_{g}})^{σ_{p}=α_{g}/β_{g}} = L · P_{g_{α}}  is one-dimensional.

This is the only regular case, in which the point P_{g_{α}} is completely pinned down by its specified behaviour under the action of σ_{p}.

Remark 5.2. There are two distinct conjugacy classes of elements of order 3 in A_{4}, which are rational over ℚ(ζ_{3}) and interchanged by the action of Gal (ℚ(ζ_{3})/ℚ). Note that 8 of the 12 elements in A_{4} are of order 3, and hence, by the Chebotarev density theorem, the Frobenius element σ_{p} is regular for Ad_{g} for a set of primes p of Dirichlet density 2/3.

5.1.2. Cases where g is of octahedral type. The dimensions of the irreducible representations of the group G = Gal (H/ℚ) ≃ S_{4} are 1, 1, 2, 3, and 3. The first two representations are the trivial and sign representations, and the two-dimensional representation factors through the quotient S_{4}/V_{4} = S_{3}. In particular, the only faithful irreducible representations of G are three-dimensional, and there are two such, whose characters are given in the following table:

size 1  6  3  8  6
class 1 (12) (12)(34) (123) (1234)
ϱ_{1} 3  1  −1  0  −1
ϱ_{2} 3 −1  −1  0  1

The representation ϱ_{1} occurs in the natural permutation representation of G on 4 letters, while ϱ_{2} is just the twist of ϱ_{1} by the sign character. Among the two representations it is ϱ_{2} which is the adjoint representation. This can be seen by noting that the eigenvalues of ϱ_{1}((1234)) are visibly −1, i and −i, and that those of ϱ_{2}((1234)) are 1, i, and −i, and that only the latter set of eigenvalues is consistent with the property of the adjoint representation that all elements in G admit 1 as eigenvalue.

Let R(x) be any degree four polynomial whose splitting field is H, and let M be the quartic field which is generated by a single root of R. The field H is an S_{3}-extension of M and hence there is a unique quadratic extension M̃ of M contained in H. The field H also contains a unique quadratic field K/ℚ, which is cut out by the sign character of S_{4} and satisfies M̃ = MK. The representation Ad_{g} is described by the formula

Ad_{g} = Ind_{G_{M}}^{G_{ℚ}} sgn − sgn,

where sgn is viewed both as the quadratic character of G_{M} and G_{ℚ} induced by the sign characters on the finite quotients Gal (H/M) ≃ S_{3} and Gal (H/ℚ) ≃ S_{4}. It follows that

r(E, Ad_{g}) = r(E, Ind_{M}^{H} sgn) − r(E, sgn) = dim_{L} E(M̃)_{L}^{−} − dim_{L} E(K)^{−},

where the superscript − refers to the −1 eigenspace for the involutions in Gal (M̃/M) and Gal (K/ℚ) respectively. When r(E, Ad_{g}) = 1, there is, by Frobenius reciprocity, a unique point P ∈ E(M̃)^{−}, up to scaling. Let P_{1}, P_{2}, P_{3} and P_{4} denote the A_{4} translates of P = P_{1}, and observe that σ ∈ G = S_{4} acts on these points by the rule

σP_{j} = \{ \begin{matrix}P_{σj} & if σ ∈ A_{4}; \\ −P_{σj} & if σ ∈ S_{4} − A_{4}. \end{matrix}

The resulting explicit description of the action of G on these four points, which together generate E(H)_{L}^{Ad_{g}}, make it possible to describe the point P_{g_{α}} explicitly, depending on the conjugacy class of σ_{p} in G.

Case 1: σ_{p} = 1. In that case, α_{g} = β_{g}, and hence Hypothesis C is not satisfied.

Case 2: σ_{p} is of order 2 and odd, i.e., it is conjugate to the transposition (12). In that case, after relabelling the P_{j}, it can be assumed that

σ_{p}P_{1} = −P_{2}, σ_{p}P_{2} = −P_{1}, σ_{p}P_{3} = −P_{3}, σ_{p}P_{4} = −P_{4}.

In particular, the x-coordinates of the points P_{1} and P_{2} lie in ℚ_{p^{2}} and are permuted by Gal (ℚ_{p^{2}}/ℚ_{p}), while the x coordinates of P_{3} and P_{4} are defined over ℚ_{p}, even though all four points are defined over ℚ_{p^{2}}. The eigenvalues of σ_{p} are equal to 1 and −1 = α_{g}/β_{g} = β_{g}/α_{g}, and

(E(H)_{L}^{Ad_{g}})^{σ_{p}=1} = L · (P_{1} − P_{2})  is one-dimensional,
(E(H)_{L}^{Ad_{g}})^{σ_{p}=−1} = L · P_{3} ⊕ L · P_{4}  is two-dimensional.

This is another irregular case, in which the point P_{g_{α}} is a specific L-linear combination of P_{3} and P_{4} lying in the line hom(V_{g}^{g_{β}}, V_{g}^{g_{α}}) of Ad_{g}.

Case 3: σ_{p} is of order 2 and even, i.e., it is conjugate to the permutation (12)(34). After eventually relabelling the P_{j}, it can be assumed that

σ_{p}P_{1} = P_{2},  σ_{p}P_{2} = P_{1},  σ_{p}P_{3} = P_{4},  σ_{p}P_{4} = P_{3},

so that

(E(H)_{L}^{Ad_{g}})^{σ_{p}=1} = L · (P_{1} + P_{2}) = L · (P_{3} + P_{4})  is one-dimensional,
(E(H)_{L}^{Ad_{g}})^{σ_{p}=−1} = L · (P_{1} − P_{2}) ⊕ L · (P_{3} − P_{4})  is two-dimensional.

This is yet another irregular case, and P_{g_{α}} is some unspecified L-linear combination of P_{1} −P_{2} and P_{3} − P_{4}.

Case 4: σ_{p} is of order 3. In that case, it can be assumed that σ_{p} fixes the point P_{1}, which is defined over ℚ_{p}, and permutes the points P_{2}, P_{3} and P_{4} cyclically. The eigenvalues of σ_{p} on Ad_{g} are then equal to 1, α_{g}/β_{g} = ζ_{3} and β_{g}/α_{g} = ζ_{3}^{−1}, and P_{g_{α}} is determined, up to scaling, by the rule

P_{g_{α}} = P_{2} + ζ_{3}^{−1}P_{3} + ζ_{3}P_{4}.

In particular,

(E(H)_{L}^{Ad_{g}})^{σ_{p}=α_{g}/β_{g}} = L · P_{g_{α}}  is one-dimensional.

Case 5: σ_{p} is of order 4, and hence is conjugate to (1234). In that case, it can be assumed that

σ_{p}P_{1} = −P_{2}, σ_{p}P_{2} = −P_{3}, P_{3} = −P_{4}, σP_{4} = −P_{1}.

The eigenvalues of σ_{p} are equal to 1, i and −i, and

(E(H)_{L}^{Ad_{g}})^{σ_{p}=1} = L · (P_{1} − P_{2} + P_{3} − P_{4}),
(E(H)_{L}^{Ad_{g}})^{σ_{p}=i} = L · (P_{1} + iP_{2} − P_{3} − iP_{4}),  (77)
(E(H)_{L}^{Ad_{g}})^{σ_{p}=−i} = L · (P_{1} − iP_{2} − P_{3} + iP_{4}).  (78)

In particular, the α_{g}/β_{g}-eigenspace for σ_{p} is one-dimensional if and only if σ_{p} has order > 2, which occurs for 14 of the 24 permutations in S_{4}. This regularity condition is therefore satisfied for a set of primes p of density 7/12.

5.1.3. Cases where g is of icosahedral type. The dimensions of the irreducible representations of the group G = Gal (H/ℚ) ≃ A_{5} are 1, 3, 3, 4 and 5, and the characters of the two distinct three-dimensional representations are described in the following table:

size 1  15  20  12  12
class 1 (12)(34) (123) (12345) (12354)
ϱ_{1} 3  −1  0  \frac{1+\sqrt{5}}{2}  \frac{1−\sqrt{5}}{2}
ϱ_{2} 3  −1  0  \frac{1−\sqrt{5}}{2}  \frac{1+\sqrt{5}}{2}

In this case, both the representations ϱ_{1} and ϱ_{2} arise as the adjoint of the Artin representation ϱ_{g} and ϱ′_{g} respectively, where ϱ′_{g} denotes the Artin representation of the conjugate form of g, whose Fourier coefficients lie in the field ℚ(\sqrt{5}).

It turns out to be easier to describe the direct sum ϱ_{1} ⊕ ϱ_{2} in Galois-theoretic terms rather than each ϱ_{i} individually. Let D_{10} be the dihedral group of order 10, embedded naturally in A_{5}, and let M be the non-Galois sextic field fixed by D_{10}. The field M is generated by a root of a sextic polynomial R(x), the so-called Lagrange sextic resolvent attached to the A_{5} extension. Since Gal (H/M) = D_{10}, there is a unique quadratic extension M̃ of M contained in H, which is cut out by the unique non-trivial quadratic character χ^{−} of D_{10}. It is not hard to check the identity of virtual representations:

ϱ_{1} ⊕ ϱ_{2} = Ind_{G_{M}}^{G_{ℚ}} χ^{−},

which implies that the six-dimensional representation E(H)_{L}^{Ad_{g}} ⊕ E(H)_{L}^{W_{g′}} is generated as an L-vector space, by the unique point P ∈ E(M̃)^{−} together with its 12 Galois translates

P =: P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, P_{6}, −P_{1}, −P_{2}, −P_{3}, −P_{4}, −P_{5}, −P_{6}.

The Stark point P_{g_{α}} is a ℚ(\sqrt{5})-linear combination of the 6 linearly independent points P_{j} with 1 ≤ j ≤ 6. The coefficients that enter in this linear combination are completely determined by the requirement that σ_{p}P_{g_{α}} = α_{g}/β_{g}P_{g_{α}}, provided the frobenius element σ_{p} is not of order ≤ 2. Since A_{5} has precisely 44 elements of order > 2, this condition holds for a set of primes of Dirichlet density 11/15.

5.2. Numerical evidence. In contrast to the situation in Sections 4.2 and 7.2, numerical evidence is somewhat more bountiful in the setting of exotic modular forms. In particular, despite the appellation “exotic” such forms seem to be more plentiful in small level than those studied in Section 4.2. (Note that in all our exotic examples we checked that hypotheses A,B and C hold.)

As in Example 3.11 it is helpful to consider algebraic factors similar to those defined in [DR1, Theorem 3.1], given by

E(g_{α}, f, h) := (1 − α_{g}α_{f}α_{h}p^{−1}) × (1 − α_{g}α_{f}β_{h}p^{−1})
×(1 − α_{g}β_{f}α_{h}p^{−1}) × (1 − α_{g}β_{f}β_{h}p^{−1})
E_{0}(g_{α}) := (1 − α_{g}^{2}χ^{−1}(p))
E_{1}(g_{α}) := (1 − α_{g}^{2}χ^{−1}(p)p^{−1}).

Here when p divides the level of f we set α_{f} = a_{p}(f) and β_{f} = 0, and otherwise define these so that x^{2} − a_{p}(f)x + p = (x − α_{f})(x − β_{f}). Economising on notation, we define E_{α} as

E_{α} := \frac{E(g_{α}, f, h)}{E_{0}(g_{α})E_{1}(g_{α})} ∈ L.  (79)

We define E_{β} similarly, switching the roles of α_{g} and β_{g} in the above.

Our first exotic example involves the tetrahedral form of level 133 first constructed by Tate.

Example 5.3. Let χ be a Dirichlet character of level 133 = 7 · 19 of order 2 at 7 and order 3 at 19. (There are two such characters which are Galois conjugate to each other.) Denote by ζ a primitive 12-th root of unity and define ζ_{3} := ζ^{−4}, a primitive cube root of unity. Then S_{1}(133, χ) is defined over ℚ(ζ_{3}) and has dimension two. It contains two eigenforms defined over ℚ(ζ), namely

h = q + (ζ^{2} − 1)q^{2} − ζq^{3} + ζq^{5} + (−ζ^{3} + ζ)q^{6} + ζ^{3}q^{7} − q^{8} + (ζ^{3} − ζ)q^{10} + (−ζ^{3} + ζ)q^{13} − ⋯

and its Galois conjugate obtained by replacing ζ by −ζ. Let g = h^{∗}, the twist of h by χ^{−1} (this has Fourier coefficients complex conjugate to those of h). The A_{4}-extension H cut out by the projective representation attached to g is the normal closure of the quartic field

M := ℚ(w), w^{4} + 3w^{2} − 7w + 4 = 0,

which has discriminant 133^{2}.

Consider the elliptic curves

\begin{matrix}E_{a} : y^{2} + y = x^{3} + x \\ E_{b} : y^{2} + y = x^{3} + x^{2} − 7x + 5 \end{matrix}

of conductor 91 = 7 · 13 labelled 91a and 91b in Cremona’s tables and write f_{a} and f_{b} for the associated weight 2 newforms. These curves both satisfy the conditions in the rank (1, 1) setting of Conjecture ES, namely

r(E_{a}, 1) = 1,  r(E_{a}, Ad_{g}) = 1,  r(E_{b}, 1) = 1,  r(E_{b}, Ad_{g}) = 1.

Let P_{ℚ}^{(a)} and P_{ℚ}^{(b)} denote Mordell-Weil generators of E_{a}(ℚ) and E_{b}(ℚ) respectively (modulo torsion), and let P^{(a)} and P^{(b)} denote generators of the one-dimensional subspaces of E_{a}(M)_{ℚ} and E_{b}(M)_{ℚ} respectively, consisting of points with trace 0. A combination of naive search and two-descent shows that one can take

\begin{matrix}P_{ℚ}^{(a)} = (0, 0), & P^{(a)} = (\frac{−156w^{3}−64w^{2}−598w+654}{289}, \frac{823w^{3}+1464w^{2}+5515w−7663}{4913}) , \\ P_{ℚ}^{(b)} = (−1, 3), & P^{(b)} = (9w^{3} + 5w^{2} + 31w − 45, −w^{3} + 16w^{2} + 16w + 83). \end{matrix}

Letting p = 13, the Hecke polynomial t^{2} − a_{13}(h)t + χ(13) has roots β_{h} = ζ and α_{h} = −ζ^{3}. Hence the two 13-stabilisations g_{α} and g_{β} of g with U_{p}-eigenvalue α_{g} = \overline{β_{h}} = ζ^{−1} and β_{g} = \overline{α_{h}} = ζ^{3} are distinct. (Note we are twisting by χ^{−1} here and χ^{−1}(p)β_{h} = (α_{h}β_{h})^{−1}β_{h} = \overline{α_{h}}, and likewise χ^{−1}(p)α_{h} = \overline{β_{h}}.) Let γ_{g_{α}} and γ_{g_{β}} denote the corresponding dual elements.

Choose the embedding of ℚ(ζ) into ℚ_{13} which sends ζ to the unique 12th root of unity in ℤ_{13} which is congruent to 6 modulo 13. The values of the p-adic iterated integrals, evaluated to 20 digits of 13-adic accuracy, are then given by the four quantities

∫_{γ_{gα}} f_{a} · h = −6603156733219298602528 mod 13^{20}
∫_{γ_{g_{β}}} f_{a} · h = 2705941994573919260213 mod 13^{20}
∫_{γ_{gα}} f_{b} · h = −679336918469999467881 × 13 mod 13^{20}
∫_{γ_{g_{β}}} f_{b} · h = −2983964554989928769557 mod 13^{20}.

The ratio α_{g}/β_{g} = ζ^{−4} = ζ_{3} is a primitive cube root of unity, and hence the eigenvalues of σ_{13} on Ad_{g} are 1, ζ_{3} and ζ_{3}^{2}. The fact that σ_{13} has order 3 puts us in Case 3 in the analysis in Section 5.1.1. Let

P_{g_{α}}^{(a)}, P_{g_{β}}^{(a)} ∈ E_{a}(H)_{L}^{Ad_{g}}, P_{g_{α}}^{(b)}, P_{g_{β}}^{(b)} ∈ E_{b}(H)_{L}^{Ad_{g}}

be the points constructed from P^{(a)} and P^{(b)} using (76). The units u_{g_{α}} and u_{g_{β}} are constructed using the same formula with the point P replaced by the unit u_{1} := −w + 1 which is the root of x^{4} − 4x^{3} + 9x^{2} − 3x + 1 = 0 in M and its Galois conjugates u_{2}, u_{3} and u_{4}.

It then turns out that to 20 digits of 13-adic accuracy,

∫_{γ_{gα}} f_{a} · g = −ζ_{3}^{2} · E_{α} · \frac{log_{E_{a},13}(P_{ℚ}^{(a)}) log_{E_{a},13}(P_{g_{α}}^{(a)})}{log_{13}(u_{g_{α}})},

∫_{γ_{g_{β}}} f_{a} · g = −ζ_{3} · E_{β} · \frac{log_{E_{a},13}(P_{ℚ}^{(a)}) log_{E_{a},13}(P_{g_{β}}^{(a)})}{log_{13}(u_{g_{β}})},

where E_{α} is defined as in (79), and likewise for E_{β}.

A similar pattern recurs with the elliptic curve E_{b}, for which it was checked to 20 digits of 13-adic precision that

∫_{γ_{gα}} f_{b} · h = \frac{ζ}{3} · E_{α} · \frac{log_{E_{b},13}(P_{ℚ}^{(b)}) log_{E_{b},13}(P_{g_{α}}^{(b)})}{log_{13}(u_{g_{α}})},

∫_{γ_{g_{β}}} f_{b} · h = −\frac{ζ^{−1}}{3} · E_{β} · \frac{log_{E_{b},13}(P_{ℚ}^{(b)}) log_{E_{b},13}(P_{g_{β}}^{(b)})}{log_{13}(u_{g_{β}})},

with E_{α} as in (79), and likewise for E_{β}. (Here ζ · E_{α}/3 = ζ^{−1} · E_{β}/3 = 4/(13\sqrt{3}).)

The choice of the prime p = 13 in the above calculations was dictated largely by convenience. This prime has the virtue of splitting completely in L = ℚ(ζ), so that L embeds into ℚ_{13}, and it divides the level of f_{a} and f_{b}, which can thus be viewed as overconvergent 13-adic modular forms of tame level 133.

The weight one tetrahedral form g of level 133 was considered by Chinburg [Ch] in connection with Stark’s original conjectures, motivated by the final sentence in [St2],

A meaningful verification for N = 133 would be of some interest. Stark’s concluding comment follows the verification of his conjecture for the weight one modular form considered in Example 4.1.

The next example involves the tetrahedral modular form of minimal level 124 = 4 · 31 [Bu, §2.2] which is also considered in [Ch]. It illustrates our conjecture in both rank (1, 1) and rank 2 exotic settings.

Example 5.4. Let χ be a sextic Dirichlet character of conductor 124 = 4 · 31, having order 2 at 4 and order 3 at 31. Denote by ζ a primitive 12-th root of unity and define ζ_{3} := ζ^{2} − 1, a primitive cube root of unity. The space S_{1}(124, χ) is defined over ℚ(ζ_{3}) and has dimension two. Just as in the previous example, it contains two eigenforms, which are defined over ℚ(ζ) and conjugate over ℚ(ζ_{3}), namely

h = q+ζ^{3}q^{2}+ζq^{3}−q^{4}−ζ^{2}q^{5}+(ζ^{2}−1)q^{6}−ζq^{7}−ζ^{3}q^{8}+(−ζ^{3}+ζ)q^{10}+(ζ^{3}−ζ)q^{11}−ζq^{12}+ζ^{2}q^{13}+⋯

and its conjugate obtained by replacing ζ by −ζ. Let g = h^{∗}. The field cut out by the projective representation attached to g is the normal closure H of the quartic field

M = ℚ(w), w^{4} + 7w^{2} − 2w + 14 = 0,

which has discriminant 124^{2} and Galois group isomorphic to A_{4}.

Consider the elliptic curve

E : y^{2} = x^{3} + x^{2} − 2x + 1

of conductor 124 labelled 124a in Cremona’s tables and write f for the associated weight 2 newform. The curve E has rank one over ℚ (with a torsion subgroup of order 3) and rank two over M—this places us in the rank (1, 1) setting. Following the same notations of the previous example, generators for E(ℚ) and for the subgroup of points in E(M) of trace zero are given by

P_{ℚ} = (−2, 1), P = (\frac{w^{3} − w^{2} + 3w − 7}{2}, \frac{w^{3} + 4w^{2} + 9w + 10}{4}) .

These points were found using a naive search.

Choosing the prime p = 13 as in the previous example, the Hecke polynomial t^{2} − a_{13}(h)t + χ(13) has roots β_{h} = 1 and α_{h} = ζ_{3}. Hence α_{g} = 1 and β_{g} = ζ_{3}^{2} and therefore the 13stabilisations g_{α} and g_{β} of g are distinct.

Choose the embedding of ℚ(ζ) into ℚ_{13} which sends ζ to the unique 12-th root of unity in ℤ_{13} which is congruent to 7 modulo 13. The values of the 13-adic iterated integrals were computed to be

\begin{aligned}∫_{γ_{gα}} f · h & = & 684158979097768450057 × 13 mod 13^{20} \\ ∫_{γ_{g_{β}}} f · h & = & 5953034774844704381281 mod 13^{20}. \end{aligned}

Note that α_{g}/β_{g} and β_{g}/α_{g} are both primitive cube roots of unity, and hence, just as in the previous example, the setting is the one discussed in Case 3 in the analysis in Section 5.1.1. Just as before, the points P_{g_{α}} and P_{g_{β}} in E(H)_{L}^{Ad_{g}} can be defined as suitable linear combinations of the point P and its Galois conjugates, following the formula in (76). The units u_{g_{α}} and u_{g_{β}} are likewise constructed using the same formula with the point P replaced by the unit u_{1} := w^{2} − w + 3, the unique root of x^{4} + 2x^{3} + 18x^{2} − 6x + 1 in M.

With these conventions in place, it has been checked to 20 digits of 13-adic precision that

∫_{γ_{gα}} f · h = \frac{2}{\sqrt{3}} · E_{α} · \frac{log_{E,13}(P_{ℚ}) log_{E,13}(P_{g_{α}})}{log_{13}(u_{g_{α}})},

∫_{γ_{g_{β}}} f · h = \frac{2}{\sqrt{3}} · E_{β} · \frac{log_{E,13}(P_{ℚ}) log_{E,13}(P_{g_{β}})}{log_{13}(u_{g_{β}})},

where E_{α} is as in (79), and likewise for E_{β}, with \sqrt{3} = ζ^{3} − 2ζ.

We now turn to the curve

E : y^{2} + xy + y = x^{3} − x^{2} − 3x + 3

of conductor 26 labelled 26b. One checks

r(E) = 0, r(E, Ad_{g}) = 2,  (80)

and likewise for analytic ranks, and so here we are in the rank 2 setting. The required points in E(H)^{Ad^{g}} may be constructed from the generators

\begin{aligned}P & = & (\frac{283w^{3}+1295w^{2}+1442w+9528}{4802}, \frac{29371w^{3}−35875w^{2}+146552w−319814}{235298}) \\ Q & = & (\frac{−25w^{3}+98w^{2}+41w+775}{121}, \frac{67w^{3}+3832w^{2}+2465w+17404}{1331}) \end{aligned}

of the rank 2 Mordell-Weil lattice in E(M). We define P_{1} and P_{g_{α}} as in the Case 3 analysis for tetrahedral forms (and similarly P_{g_{β}} switching ζ_{3} for ζ_{3}^{−1}). Likewise we define the points Q_{1}, Q_{g_{α}} and Q_{g_{β}}.

Writing now f for the form attached to E we find, with the same embedding as before, that

\begin{aligned}∫_{γ_{gα}} f · h & = & 6721481652857547558786 mod 13^{20} \\ ∫_{γ_{g_{β}}} f · h & = & −2549577176522983012568 mod 13^{20}. \end{aligned}

Then to 20-digits of 13-adic precision

∫_{γ_{gα}} f · h = \frac{ζ_{3} · E_{α}}{\sqrt{3}} · \frac{log_{E,13}(P_{1}) log_{E,13}(Q_{g_{α}}) − log_{E,13}(Q_{1}) log_{E,13}(P_{g_{α}})}{log_{p} u_{g_{α}}}  (81)

∫_{γ_{g_{β}}} f · h = −\frac{ζ_{3}^{−1} · E_{β}}{\sqrt{3}} · \frac{log_{E,13}(P_{1}) log_{E,13}(Q_{g_{β}}) − log_{E,13}(Q_{1}) log_{E,13}(P_{g_{β}})}{log_{p} u_{g_{β}}} (82)

where E_{α} is as in (79), and likewise for E_{β}, exactly as predicted by Conjecture ES. Finally we consider the curve

E : y^{2} = x^{3} + x − 10

of conductor 52 labelled 52b for which again (80) holds and so we are in the rank 2 setting. Here we take

\begin{aligned}P & = & (\frac{−225w^{3}−1533w^{2}−1458w+1078}{2809}, \frac{−247041w^{3}+68295w^{2}−1265094w+317530}{297754}) \\ Q & = & (\frac{−63391w^{3}−51546w^{2}−698338w−942346}{390625}, \frac{84293978w^{3}−33352782w^{2}+603648754w−1348508882}{244140625}) \end{aligned}

as generators for the Mordell-Weil lattice in E(M) and construct the points P_{1}, P_{g_{α}}, P_{g_{β}} and Q_{1}, Q_{g_{α}}, Q_{g_{β}} from them. Letting f now be the newform attached to E we have

\begin{aligned}∫_{γ_{gα}} f · h & = & −1602512412162131808818 mod 13^{20} \\ ∫_{γ_{g_{β}}} f · h & = & 4989460414989865668627 mod 13^{20} \end{aligned}

and one checks to 20-digits of 13-adic precision that again (81) and (82) hold. (The algebraic constants which occur in equation (81) for 26b and 52a are in fact exactly the same, and likewise for (82): both curves have non-split multiplicative reduction at 13.)

The following example illustrates the vanishing of p-adic iterated integrals predicted by Conjecture ES in a rank 2 setting but where the regulator term vanishes for trivial reasons.

Example 5.5. Consider the curve

E : y^{2} + xy = x^{3} − 18x + 31

of conductor 4811 = 17 · 283 labelled 4811a in Cremona’s table and write f for the associated weight 2 newform. Letting h be the octahedral weight one form in level 283 with quadratic character and g = h^{∗}, it is readily checked that

r(E/ℚ) = r_{an}(E/ℚ) = 2, r_{an}(E, Ad_{g}) = 0.

(The third equality was obtained by calculating L(E, Ad_{g}, 1) numerically to a modest precision sufficient to verify its non-vanishing.) Taking p a regular prime for ρ_{g}, it is clear that hypotheses A,B,C are satisfied but that the matrix R_{g_{α}}(E, ϱ_{gh}) of Conjecture ES has determinant 0. The prediction that all associated p-adic iterated integrals should vanish was verified for p = 17 to 5 digits of 17-adic accuracy.

We note also here that we tested the vanishing prediction of Conjecture ES in several examples when r(E, ϱ_{gh}) = 4 and 6 (involving CM rather than exotic forms).

Our final exotic example involves the exotic octahedral modular form of minimal level 148 = 4 · 37 [Bu, §2.3] and rank (1, 1) settings of Conjecture ES.

Example 5.6. Let χ be the Dirichlet character of level 148 = 2^{2} · 37 which is trivial at 2 and has order 4 at 37 (there are two such characters and they are Galois conjugate). The space S_{1}(148, χ) is defined over L = ℚ(i) and has dimension 1. It contains the single eigenform

h = q − iq^{3} − q^{7} + iq^{11} + (i − 1)q^{17} + (−i + 1)q^{19} + · · · .

The field H cut out by the projective representation attached to g = h^{∗} is the normal closure of the quartic field

M := ℚ(a), a^{4} − a^{3} + 5a^{2} − 7a + 12

of discriminant 2^{2} · 37^{3} and having Galois group S_{4}. The unique quadratic subfield of H is K = ℚ(\sqrt{37}), and hence M̃ = MK is the unique quadratic extension of M contained in H.

Testing Conjecture ES in the rank (1, 1) setting requires elliptic curves E of rank 1 over ℚ for which in addition r(E, Ad_{g}) = 1. We choose p = 17 and consider elliptic curves E of rank one and conductor dividing 148p which satisfy the necessary hypotheses. (Restricting the conductor of E in this way makes it possible to confine the 17-adic iterated integral computations to spaces of overconvergent p-adic modular forms tame level 148.) Analytic rank computations suggest these are the curves 37a, 629a, 629c and 629d in Cremona’s tables.

Following the general notations introduced in Section 5.1.2, Conjecture ES was verified for each of E = 37a, 629a and 629d using points P ∈ E(M̃)^{−} found by Steve Donnelly. (For 629c the authors were unable to find the point P. This leaves opens possibility of constructing it using Conjecture ES, a challenge that has not yet been addressed.)

The elliptic curves of interest are

\begin{matrix}E_{37a} : & y^{2} + y = x^{3} − x \\ E_{629a} : & y^{2} + xy = x^{3} − x^{2} + 11x − 18 \\ E_{629d} : & y^{2} + xy + y = x^{3} − x^{2} − 171x + 1904 \end{matrix}

with associated modular forms f_{37a}, f_{629a} and f_{629d}.

With p = 17 the Hecke eigenvalues for the octahedral form h are β_{h} = i and α_{h} = −1, and therefore α_{g} = −i and β_{g} = −1. It follows that the Frobenius eigenvalues at 17 for the adjoint representation Ad_{g} are 1, α/β = i and β/α = −i, which places this experiment in Case 5 of the analysis in Section 5.1.2.

Choose the embedding of L = ℚ(i) into ℚ_{17} which sends i to the unique square root of −1 which is congruent to 4 modulo 17. A calculation of iterated integrals to 15-digits of 17-adic precision reveals that

∫_{γ_{gα}} f_{37a} · h = 795182232610993468 mod 17^{15}
∫_{γ_{g_{β}}} f_{37a} · h = 591309544450892129 mod 17^{15}
∫_{γ_{gα}} f_{629a} · h = 231921888434186915 mod 17^{15}
∫_{γ_{g_{β}}} f_{629a} · h = 776162921415439615 mod 17^{15}
∫_{γ_{gα}} f_{629d} · h = −612176557243786922 × 17 mod 17^{15}
∫_{γ_{g_{β}}} f_{629d} · h = 329220710071055453 mod 17^{15}.

Consider first the elliptic curve E = E_{37a}. The generator of E(ℚ) is chosen to be P_{ℚ} = (0, 0). The point

( \frac{−152551a^{3} + 327117a^{2} − 384541a + 1037221}{21316} , \frac{−110958819a^{3} + 69702635a^{2} + 95157821a + 314525497}{3112136} )

on the quadratic twist y^{2} + y = x^{3} − 1369x + 12663 of E by 37 gives rise to the point

P = ( \frac{−10817b^{7} − 15535b^{6} − 11412b^{5} − 43863b^{4} − 97162b^{3} + 30857b^{2} + 39063b − 42}{21316} ,
\frac{2553946b^{7} − 379967b^{6} − 1189613b^{5} + 10482748b^{4} + 12706439b^{3} − 35039416b^{2} + 19176309b + 6012477}{15560680} )

in E(M̃)^{−}, where

M̃ = ℚ(b), b^{8} + b^{7} + b^{6} + 4b^{5} + 8b^{4} − 4b^{3} + b^{2} − b + 1 = 0.

Writing P_{1}, P_{2}, P_{3} and P_{4} for the Galois translates of P so that they are permuted cyclically by σ_{17}, define

P_{g_{α}} = P_{1} − iP_{2} − P_{3} + iP_{4}, P_{g_{β}} = P_{1} + iP_{2} − P_{3} − iP_{4},

following (78) and (77). Likewise, taking u := −b^{2} as one of the two roots of x^{8} − x^{7} + 9x^{6} − 10x^{5} + 102x^{4} − 10x^{3} + 9x^{2} − x + 1 in M̃ leads to a definition of the Stark units u_{g_{α}} and u_{g_{β}} ∈ (O_{H}^{×})_{L}^{Ad_{g}} attached to the two stabilisations of g.

The following identities were verified to 15-digits of 17-adic precision:

∫_{γ_{gα}} f_{37a} · h = \frac{4}{3} · E_{α} · \frac{log_{E,17}(P_{ℚ}) log_{E,17}(P_{g_{α}})}{log_{17}(u_{g_{α}})},

∫_{γ_{g_{β}}} f_{37a} · h = \frac{4}{3} · E_{β} · \frac{log_{E,17}(P_{ℚ}) log_{E,17}(P_{g_{β}})}{log_{17}(u_{g_{β}})},

with E_{α} as in (79), and likewise for E_{β}.

Consider next the elliptic curve E = E_{629a}. Here the generator for E(ℚ) is chosen to be P_{ℚ} = (2, 2), and, just as above, the point

P = ( \frac{−5b^{7} − 11b^{6} − 16b^{5} − 31b^{4} − 74b^{3} − 47b^{2} − 25b + 57}{36} ,
\frac{1083b^{7} + 1784b^{6} + 2366b^{5} + 5569b^{4} + 11572b^{3} + 3647b^{2} + 1907b − 3464}{1080} )

belongs to E(M̃)^{−}. The points P_{g_{α}} and P_{g_{β}}, and the units u_{g_{α}} and u_{g_{β}}, are defined just as they were before. It was then checked to 15-digits of 17-adic precision that

∫_{γ_{gα}} f_{629a} · h = \frac{−2i − 2}{3} · E_{α} · \frac{log_{E,17}(P_{ℚ}) log_{E,17}(P_{g_{α}})}{log_{17}(u_{g_{α}})},

∫_{γ_{g_{β}}} f_{629a} · h = \frac{2i − 2}{3} · E_{β} · \frac{log_{E,17}(P_{ℚ}) log_{E,17}(P_{g_{β}})}{log_{17}(u_{g_{β}})},

with E_{α} as in (79), and likewise for E_{β}.

Finally, for the curve E = E_{629d}, let P_{ℚ} = (16, 47) be the Mordell-Weil generator for E(ℚ) and let

P = ( \frac{−1369b^{7} + 259b^{6} − 518b^{5} − 4625b^{4} − 5994b^{3} + 12987b^{2} − 17871b + 9656}{400} ,
\frac{−1258148b^{7} + 64491b^{6} − 649461b^{5} − 5104594b^{4} − 5377617b^{3} + 14354298b^{2} − 11310937b + 2516289}{40000} )

be the generator of E(M̃)_{L}^{−}.

With the same definitions for the points P_{g_{α}} and P_{g_{β}} in terms of P as before, the following remarkably simple relation was checked to 15 digits of 17-adic precision:

∫_{γ_{gα}} f_{629d} · h = (−10i + 10) · E_{α} · \frac{log_{E,17}(P_{ℚ}) log_{E,17}(P_{g_{α}})}{log_{17}(u_{g_{α}})},

∫_{γ_{g_{β}}} f_{629d} · h = (−10i − 10) · E_{β} · \frac{log_{E,17}(P_{ℚ}) log_{E,17}(P_{g_{β}})}{log_{17}(u_{g_{β}})},

with E_{α} as in (79), and likewise for E_{β}. Here (−10i + 10) · E_{α} = 160/17 and (−10i − 10) · E_{β} = −160/17.

6. Stark points attached to odd two-dimensional Artin representations

This chapter considers the case where g ∈ S_{1}(N, χ̄) is a cuspidal newform and h = E(1, χ) is the Eisenstein series attached to the pair (1, χ). In that case,

V_{gh} = V_{g} ⊕ V_{g^{∗}}, L(E, ϱ_{gh}, s) = L(E, ϱ_{g}, s)L(E, ϱ̄_{g}, s).

The factorisation of L-functions guarantees that Hypothesis A of the introduction is automatically satisfied in this setting but Hypothesis B is only satisfied when L(E, ϱ_{g}, 1) = 0. Furthermore, the conditions formulated in Conjecture ES are fulfilled precisely when L(E, ϱ_{gh}, s) has a double zero at s = 1, so that

r_{an}(E, V_{g}) = r_{an}(E, V̄_{g}) = 1.

The eigenvalues of σ_{p} acting on V_{g} and on V_{h} respectively are given by

α_{g},  β_{g},  α_{h} = 1,  β_{h} = χ(p) = (α_{g}β_{g})^{−1},

and therefore

α_{g}α_{h} = α_{g}, α_{g}β_{h} = β̄_{g}.

Let H be the Galois extension of ℚ through which ϱ_{g} (or, equivalently, ϱ̄_{g}) factors. Assuming that V_{g} is not isomorphic to its contragredient, its underlying field L of coefficients is a CM subfield of a cyclotomic field, and the spaces E(H)_{L}^{V_{g}} and E(H)_{L}^{V_{g^{∗}}} decompose as σ_{p}-modules as

E(H)_{L}^{V_{g}} = L · P_{g,α} ⊕ L · P_{g,β}, E(H)V̄_{g} ; L = L · P_{g^{∗},ᾱ} ⊕ L · P_{g^{∗},β̄},

where P_{g,α} is an eigenvector for σ_{p} with eigenvalue α_{g}, and similar conventions are adopted for the other points.

When g is regular at p and (14) is satisfied—and so Hypothesis C holds by Proposition 1.1—Conjecture ES of the introduction predicts that, for a suitable choice of test vectors

(f̆, γ̆_{g_{α}}, h) ∈ S_{2}(N)_{L} × S_{1}(N, χ)_{L}^{∨}[g_{α}] × M_{1}(N, χ)_{L},

(and likewise with α_{g} replaced by β_{g}) the following identities hold

∫γ̆_{gα} f̆· h̆ = \frac{log_{E,p}(P_{g,α}) · log_{E,p}(P_{g^{∗},β̄})}{log_{p}(u_{g_{α}})},  (83)

∫γ̆_{g_{β}} f̆· h̆ = \frac{log_{E,p}(P_{g,β}) · log_{E,p}(P_{g^{∗},ᾱ})}{log_{p}(u_{g_{β}})},  (84)

where u_{g_{α}} is precisely the same Stark unit as arose in Chapter 5.

The identities above turned out to be harder to check experimentally, because

• instances where L(E, ϱ_{g}, s) acquires a simple zero at s = 1 are ostensibly sporadic in nature. When ϱ_{g} is not isomorphic to its contragredient representation, this L-function does not admit a sign in its functional equation which might be used to systematically predict the vanishing of L(E, ϱ_{g}, 1), and it is expected that cases where L(E, ϱ_{g}, 1) = 0 are exceedingly rare (although very little data has been gathered to be able to make serious pronouncements in this direction);

• the field cut out by ϱ_{g} is larger than the field cut out by its adjoint Ad_{g}. More precisely, it is an abelian extension of it, of degree t ≥ 2, where t is the order of the non-trivial central character attached to ϱ_{g}. In particular, the point P_{g,α} that arises in this section is entirely independent of the point denoted P_{g_{α}} in Chapter 5, is defined over a larger degree extension, and is therefore harder to produce numerically in practice, in addition to not arising in as systematic a manner.

The problem of producing numerical evidence becomes more tractable when g = g^{∗}, which occurs precisely when g is induced from a ring class character of an imaginary quadratic field. Such examples are arguably less interesting since they also fall under the scope of Chapter 3. See the second half of Example 3.12 of Section 3.3 for a numerical verification of Conjecture ES in the case where g is further assumed to be self-dual. (See [DLR1, Example 5.2] though for numerical evidence when g ≠ g^{∗} in the analogous setting in which f is replaced by a weight two Eisenstein series, and Mordell-Weil groups are replaced by unit groups.)

7. Stark points over cyclotomic fields and Eisenstein series

7.1. Theoretical preliminaries. This section turns to the case where g is an Eisenstein series satisfying condition (2) of Hypothesis C. Recall that in this case, V_{gh}^{g_{α}} := W ⊗ V_{h} where W is any line in V_{g} which is not an eigenspace for the global Galois action; that is, not one of the two one-dimensional irreducible summands of V_{g}. Conjecture ES of the introduction predicts that, for a suitable choice of test vectors

(f̆, γ̆_{g_{α}}, h̆) ∈ S_{2}(N)_{L} × S_{1}(N, χ)_{L}^{∨}[g_{α}] × M_{1}(N, χ)_{L},

(and likewise with α_{g} replaced by β_{g}) the following identities hold

∫γ̆_{gα} f̆· h̆ = \frac{log_{E,p}(P_{g,α}) · log_{E,p}(P_{g^{∗},β̄})}{log_{p}(u_{g_{α}})},

∫γ̆_{g_{β}} f̆· h̆ = \frac{log_{E,p}(P_{g,β}) · log_{E,p}(P_{g^{∗},ᾱ})}{log_{p}(u_{g_{β}})},

where u_{g_{α}} and u_{g_{β}} are the Gross-Stark p-units rather than genuine units. Otherwise, the statement of Conjecture ES is identical to the case (83) where the roles of g and h are interchanged.

For the same reason, we have been unable to gather much experimental evidence for it, except in the simplest (and particularly appealing) special case where g and h are both weight one Eisenstein series of the form

g = E_{1}(1, χ^{−1}), h = E_{1}(χ_{1}, χ_{2}), where χ_{1}χ_{2} = χ.

We also require that at least one of χ_{1} or χ_{2} is not quadratic, to exclude the case (already treated in Chapter 3) where both g and h are induced from genus characters of the imaginary quadratic field attached to the quadratic Dirichlet character χ. Let L be the field in which the characters χ_{1} and χ_{2} take values. The representations V_{g}, V_{h}, and V_{gh} decompose as direct sums of one-dimensional representations:

V_{g} = L ⊕ L(χ^{−1}), V_{h} = L(χ_{1}) ⊕ L(χ_{2}), V_{gh} = L(χ_{1}) ⊕ L(χ_{2}) ⊕ L(χ̄_{1}) ⊕ L(χ̄_{2}).

It is worth noting that the representation V_{gh} is isomorphic to its dual, but that its irreducible constituents are not all self-dual when χ_{1} or χ_{2} is non-quadratic. This phenomenon, which is specific to the case where at least one of the forms g or h is an Eisenstein series, makes the occurence of situations where L-functions vanish somewhat more rare and sporadic, just as in the setting described in Chapter 6.

The adjoint representation Ad_{g} attached to V_{g} breaks up as a direct sum

Ad_{g} = L ⊕ L(χ) ⊕ L(χ^{−1}).

Let H_{χ} be the abelian extension of ℚ cut out by the character χ. In order for the p-adic iterated integral attached to (f, g, h) where f is a cusp form to have the possibility of not vanishing, it is necessary to make the assumption

χ(p) = 1, i.e., V_{g} is not regular and χ_{1}(p) = χ_{2}(p)^{−1}.

Since χ is non-quadratic the L-vector spaces

(O_{H_{χ}}[1/p]^{×})_{L}^{χ},  (O_{H_{χ}}[1/p]^{×})_{L}^{χ^{−1}},  (O_{H_{χ}}[1/p]^{×})_{L}^{χ+χ^{−1}}

have dimensions 1, 1 and 2 respectively. Let u_{g_{α}} be the element of O_{H_{χ}}[1/p]^{×})_{L}^{χ+χ^{−1}} associated to g_{α} following the recipe of Section 1.2.

Let f be a weight two cusp form attached to an elliptic curve. Note that Hypothesis A is automatically satisfied in this setting. When r_{an}(E, χ_{1} ⊕ χ_{2}) > 0 (and so Hypothesis B holds) Conjecture ES predicts that the iterated integrals attached to (f, g, h) is non-vanishing if and only if r_{an}(E, χ_{1} ⊕ χ_{2}) = 1. Assume (after possibly re-ordering χ_{1} and χ_{2}) that

r_{an}(E, χ_{1}) = 1, r_{an}(E, χ_{2}) = 0.

Under this assumption, Conjecture ES predicts the existence of points P_{χ_{1}} ∈ E(H_{χ_{1}})_{L}^{χ_{1}} and

Pχ̄_{1} ∈ E(H_{χ_{1}})χ̄_{1} ; L for which

∫_{γ} f · h =^{?} \frac{log_{E,p}(P_{χ_{1}}) log_{E,p}(Pχ̄_{1})}{log_{p}(u_{g_{α}})}.

A striking feature of this conjecture is that it involves global points over abelian extensions of ℚ, whose occurrence seems sporadic, in contrast with the self-dual setting that is germane to the theory of Heegner and Stark-Heegner points.

7.2. Numerical evidence. Although Eisenstein series are plentiful, it is the very scarcity of points over odd degree abelian extensions, and the necessity of working at a prime p which splits completely in this abelian extension, which makes the numerical verification of Conjecture ES so challenging in this situation. Frequently, when such a point does exist, the common level of the Eisenstein series and cusp form f, and size of the prime p, pushes the example beyond the current reach of experimentation.

The authors were rather fortunate to encounter the following two examples (and in addition another cubic field example in which r_{an}(E, χ_{1}⊕χ_{2}) = 2 and the integral vanished, as predicted by Conjecture ES). In the first one we give for variety numerical details of the logarithms computed — these were suppressed elsewhere to lighten the burden of unilluminating long numbers in the paper.

Example 7.1. Let

E : y^{2} + xy + y = x^{3} + x^{2} − 4x + 5

be the elliptic curve of conductor 2 · 3 · 7 denoted 42a in the tables of Cremona. Let χ_{1} be the (even) cubic character of conductor 19, and let χ_{2} be the (odd) quadratic character of conductor 3, so that χ = χ_{1}χ_{2} is an odd sextic character of conductor 57. As before, write L = ℚ(ζ_{3}) for the field of coefficients generated by the values of χ_{1}. A direct calculation reveals that

E(ℚ)_{L} = 0,  r(E, χ_{1}) = r(E, χ̄_{1}) = 1,  r(E, χ_{2}) = r(E, χ̄_{2}) = 0,

and likewise for the analytic ranks. More precisely, the cyclic cubic field cut out by χ_{1} is given concretely by

H_{χ_{1}} = ℚ(α), where α satisfies α^{3} − α^{2} − 6α + 7 = 0.

The Mordell-Weil group E(H_{χ_{1}}) has rank two, and is generated as a Gal (H_{χ_{1}}/ℚ)-module by the point

P := (2α^{2} + 3α − 8, 4α^{2} + 6α − 10)

with coordinates in H_{χ_{1}}.

Fix the prime p = 7 and let ζ_{3} be the unique cube root of unity in ℚ_{7} which is congruent to 2 modulo 7. Fix a primitive 6th root of unity ζ_{6} = −ζ̄_{3}, so ζ_{3} = ζ_{6}^{2}. After letting σ be a generator of the Galois group Gal (H_{χ_{1}}/ℚ), let

P_{χ_{1}} = P + χ̄_{1}(σ)P^{σ} + χ̄_{1}(σ^{2})P^{σ^{2}},
Pχ̄_{1} = P + χ_{1}(σ)P^{σ} + χ_{1}(σ^{2})P^{σ^{2}}.

A numerical evaluation of the p-adic logarithms shows that

log_{E,7}(P_{χ_{1}}) = −199213220217977173722637405653124 × 7 mod 7^{38},  (85)
log_{E,7}(Pχ̄_{1}) = −61436332416679587407622380051451 × 7^{2} mod 7^{38}.

Turning now to the Gross-Stark unit attached to g, the defining polynomial of the sextic field H_{χ} is chosen to be x^{6} + 2x^{5} − 2x^{4} − 14x^{3} + 55x^{2} + 19x + 343, whose roots generate the group of 7-units in H_{χ}. Let u be the unique 7-adic root of this polynomial satisfying ord_{7}(u) = 3, and set

u_{χ} := ∑_{σ∈Gal (H_{χ}/ℚ)} χ̄(σ) ⊗ u^{σ},

uχ̄ := ∑_{σ∈Gal (H_{χ}/ℚ)}χ(σ) ⊗ u^{σ}.

An evaluation of the 7-adic logarithms of these units yields

log_{7}(u_{χ}) = 793452521874901120340916121448 × 7 mod 7^{38},
log_{7}(uχ̄) = 2536773629823487601064250408052 × 7 mod 7^{38},

and

log_{7}(u_{χ}) + log_{7}(uχ̄) = 475746593099769817343595218500 × 7^{2} mod 7^{38}.  (86)

Let

g := E_{1}(1, χ^{−1}), h := E_{1}(χ_{1}, χ_{2})

and take the stabilisation

ğ := (g(q) − g(q^{7})) − (−ζ_{6}^{−1} + 1) (g(q^{2}) − g(q^{14}))

in (classical) level 2 · 7 · 57 with U_{2}-eigenvalue 1. These are also 7-adic modular forms of level 114 = 2 · 57. Letting γ̆ be the element in the dual space which sends ğ to 1 and the other stabilisation of E_{1}(1, χ^{−1}) to zero, a computation of the p-adic iterated integral (by far the most resource intensive part of the experiment) reveals that

∫γ̆ f · h = −1264003828062411821439581214808 mod 7^{36}.  (87)

A direct comparison of (87) with the invariants obtained in (85) and (86) shows

∫γ̆ f · h = \frac{64}{7 · 9} \frac{log_{E,7}(P_{χ_{1}}) log_{E,7}(Pχ̄_{1})}{(log_{7}(u_{χ}) + log_{7}(uχ̄))} mod 7^{36}.

Replacing ğ by the other stabilisation of E_{1}(1, χ^{−1}), with U_{2}-eigenvalue −ζ_{6}^{−1}+1, and defining γ̆ accordingly just multiplies the integral value by ζ_{3}^{2}; and likewise, taking ğ to be a stabilisation of E_{1}(χ_{1}^{−1}, χ_{2}^{−1}) and h := E_{1}(1, χ) in the integral changes its value by at most a cube root of unity. This is as predicted by Conjecture ES.

Example 7.2. Consider the case where

E : y^{2} + xy + y = x^{3} − 5x − 8

is the elliptic curve of conductor 2 · 13 denoted 26a in the tables of Cremona. Write f for the newform of weight 2 attached to E. Let η_{1} be an (even) cubic character of conductor 7, and let η_{2} be the (odd) quadratic character of conductor 3. Define χ_{1} := η_{1}η_{2}, an odd sextic character of conductor 21, χ_{2} := 1, and χ := χ_{1}χ_{2} = χ_{1}. (This rather roundabout notation is chosen just to align with that in Section 7.1.) Letting g := E_{1}(1, χ^{−1}) and h := E_{1}(χ_{1}, χ_{2}) = E_{1}(χ, 1) = E_{1}(1, χ), the representation V_{gh} is given by

V_{gh} = L ⊕ L ⊕ L(χ) ⊕ L(χ̄)

where L = ℚ(ζ_{3}) is the field generated by the values of χ. A descent calculation reveals that

E(ℚ)_{L} = 0, r(E, χ) = r(E, χ̄) = 1

and likewise for the analytic ranks. The cyclic sextic field cut out by χ is given by

H_{χ} = ℚ(α), where α satisfies α^{6} − α^{5} + 3α^{4} + 5α^{2} − 2α + 1 = 0.

The Mordell-Weil group E(H_{χ}) has rank two, and is generated as a Gal (H_{χ}/ℚ)-module by the point

P := (2α^{5} − 6α^{4} + 8α^{3} − 10α^{2} + 4α − 23, −50α^{5} + 38α^{4} − 134α^{3} − 36α^{2} − 242α + 50)

with coordinates in H_{χ}. (This point is the image of one on the quadratic twist of E by −3 over the cyclic cubic field H_{η_{1}} cut out by η_{1}.) The points P_{χ} and Pχ̄, and the units u_{χ} and uχ̄, are now constructed in an analogous manner to Example 7.1.

We take p = 13. Noting here a_{2}(g) = 1 + ζ_{6}^{−1} with ζ_{6} a primitive sixth root of unity, we replace g by its stabilisation

ğ := (g(q) − g(q^{13})) − (g(q^{2}) − g(q^{26}))

in level 2 · 13 · 21, and let γ̆ be the element in the dual space which sends ğ to 1 and the other stabilisation to zero. Embedding ℚ(ζ_{6}) in ℚ_{13} so that ζ_{6} ≡ 4 mod 13 we find

∫γ̆ f · h = −1183909055963560023118938339892601 × 13 mod 13^{30}.

It was then checked to 30 digits of 13-adic precision that

∫γ̆ f · h = \frac{4}{13 · ζ_{6}} \frac{log_{E,13}(P_{χ}) log_{E,13}(Pχ̄)}{(log_{13}(u_{χ}) + log_{13}(uχ̄))}

as predicted by Conjecture ES.

Remark 7.3. The experimental work in this paper fell into two phases—one of discovery and one of verification—between which Conjecture ES was gradually formulated and refined. Formula (45) and Examples 3.11, 4.1 and 7.1 belong to the former phase, and the remaining examples to the latter. One notable feature which eluded the authors at the outset was the possibilitly of non-vanishing of the p-adic iterated integral in “genuine rank two” settings. This discovery was made experimentally in the companion paper [DLR1, Example 6.1], and it was this experiment which led to the definitive formulation of the conjecture.

Appendix A. Digest of numerical examples

The following table summarises some of the salient features of the numerical examples presented in support of Conjecture ES. It is intended as a navigational guide for the reader. Each numbered example involves a choice of weight one modular forms g and h of some common level N. This level, the dimensions of the irreducible summands of the tensor product representation V_{gh} := V_{g} ⊗ V_{h}, and the image of each of those irreducible representations as an abstract group are given in the second, third and fourth columns of the table, respectively. (Here D_{n} is the dihedral group of order n.) Each numbered example then involves a further choice of one or more elliptic curve E and one or more prime p with gcd(N, p) = 1. The curve E always has conductor dividing Np (and thus the attached form f has p-adic level dividing N). The Cremona labels of the curves E are shown in the fifth column, and the ranks with which the irreducible summands of V_{gh} occur in the Mordell-Weil group of E are given in the sixth. For reasons of space we do not show the different primes p considered; note however that whether p was chosen to divide the conductor of E or not can be readily deduced by comparing columns “Level” and “Curves”. The examples in the table are grouped under headings indicating the nature of the points constructed.

Ex. Level Dimension of V_{gh}  Galois groups  Curves  Ranks

Classical Heegner points over ring class fields of imaginary quadratic fields

3.11 83  2 + 1 + 1  D_{6} × 1 × C_{2}  83a  1 + 1 + 0
3.11 83  2 + 2  D_{6} × D_{6}  83a  1 + 1
3.12 79  2 + 2  D_{10} × D_{10}  79a  1 + 1
3.14 57  2 + 1 + 1  D_{6} × 1 × C_{2}  57b  2 + 0 + 0

Heegner points arising from a Shimura curve parametrisation

3.15 35  2 + 1 + 1  D_{6} × C_{2} × 1  35a  1 + 1 + 0

Stark-Heegner points over ring class fields of real quadratic fields

4.1 145  2 + 1 + 1  D_{8} × C_{2} × 1  17a  1 + 1 + 0

Points arising both as Heegner and Stark-Heegner points

4.3 155  1 + 1 + 1 + 1  1 × C_{2} × C_{2} × C_{2}  155a  1 + 0 + 1 + 0
4.3 155  1 + 1 + 1 + 1  1 × C_{2} × C_{2} × C_{2}  155c  1 + 1 + 0 + 0

Points over fields cut out by the projective representations attached to exotic forms

5.3 133  1 + 3  1 × A_{4}  91a, 91b  1 + 1
5.4 124  1 + 3  1 × A_{4}  124a  1 + 1
5.4 124  1 + 3  1 × A_{4}  26b, 52b  0 + 2
5.5 283  1 + 3  1 × S_{4}  4811a  2 + 0
5.6 148  1 + 3  1 × S_{4}  37a, 629a, 629d  1 + 1

Points over abelian extensions of ℚ

7.1 57  1 + 1 + 1 + 1 C_{3} × C_{2} × C_{3} × C_{2}  42a  1 + 0 + 1 + 0
7.2 21  1 + 1 + 1 + 1  1 × 1 × C_{6} × C_{6}  26a  0 + 0 + 1 + 1

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H. D.: McGill University, Montreal, Canada

   E-mail address: darmon@math.mcgill.ca

A. L.: University of Oxford, U. K.

   E-mail address: lauder@maths.ox.ac.uk

V. R.: Universitat Politècnica de Catalunya, Barcelona, Spain

   E-mail address: victor.rotger@upc.edu