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

HENRI DARMON, ALAN LAUDER AND VICTOR ROTGER

To Glenn Stevens on his 60th birthday

Abstract. This article can be read as a companion and sequel to [DLR], which proposes a conjectural expression for the so-called p-adic iterated integrals attached to a triple (f, g, h) of classical eigenforms of weights (2, 1, 1). When f is a cusp form, this expression involves the p-adic logarithms of so-called Stark points: distinguished points on the modular abelian variety attached to f, defined over the number field cut out by the Artin representations attached to g and h. The goal of this paper is to formulate an analogous conjecture when f is a weight two Eisenstein series rather than a cusp form. The resulting formula involves the p-adic logarithms of units and p-units in suitable number fields, and can be seen as a new variant of Gross’s p-adic analogue of Stark’s conjecture on Artin L-series at s = 0.

Contents

Introduction 1 1. The conjecture 2 2. Group-theoretic preliminaries 6 3. Circular units and Eisenstein series of weight one 10 4. Elliptic units and theta series of imaginary quadratic fields 16 5. Gross-Stark units and theta series of real quadratic fields 20 6. Exotic forms 23 References 26

Introduction

This article can (and probably should) be read as a companion and sequel to [DLR], which describes an arithmetic formula for the so-called p-adic iterated integrals attached to a triple (f, g, h) of classical modular eigenforms of weights (2, 1, 1). In the setting considered in [DLR], where f is assumed to be a cusp form, this formula involves the p-adic formal group logarithms of global points on the modular abelian variety A_{f} attached to f by the Eichler-Shimura construction. The points that arise are defined over the number field cut out by the tensor product ϱ_{g} ⊗ ϱ_{h} of the two Artin representations associated to g and h. The p-adic iterated integral of [DLR] can be understood as a p-adic avatar of the second derivative of the Hasse-Weil-Artin L-series L(A_{f}, ϱ_{g} ⊗ ϱ_{h}, s) at its central point s = 1, and the main conjecture of loc.cit. can therefore be viewed as an analogue (in the p-adic setting) of Stark’s conjecture on units [Sta], with unit groups replaced by Mordell-Weil groups of elliptic curves or more general modular abelian varieties.

The goal of the present work is to bring the analogy with Stark’s conjecture into sharper focus by studying the iterated integrals of [DLR] when f is an Eisenstein series of weight 2. The main conjecture of Section 1 below transposes the “Elliptic Stark conjecture” of loc.cit. to this setting, essentially by replacing points on elliptic curves with units or p-units in number fields having similar Galois-theoretic properties. When g and h are Eisenstein series or theta series attached to a common imaginary quadratic field in which the prime p splits, Sections 3 and 4 explain how the p-adic Stark conjecture can frequently be reduced to classical relations between the Kubota-Leopoldt or Katz p-adic L-functions and p-adic logarithms of circular or elliptic units. To make up for the lack of theoretical evidence in other settings, several numerical verifications of the remaining cases are described in Sections 5 and 6.

This article owes a lot to the vision and insights of Glenn Stevens, who in several works (notably, [Ste1], [Ste2], and [Ste3]) exploits the rich interplay between the arithmetic properties of cusp forms and Eisenstein series via their connection with irreducible and reducible two-dimensional Galois representations. Around thirteen years ago, Glenn encouraged the first author to revisit his construction of Stark-Heegner points attached to “real quadratic cycles on H_{p} × H” [Dar] by replacing the cusp form that arises in it with an Eisenstein series. This led to a refinement [DD] of the p-adic Gross-Stark units in ring class fields of real quadratic fields^{1}. (These p-units appear in a new guise in Section 5, where they play the same role as the Stark-Heegner points of [Dar] in Section 4 of [DLR].) Likewise, but in a purely archimedean setting, the construction [DL] of Stark-Heegner points attached to ATR cycles on Hilbert modular varieties gives rise to the geometric construction [CD] of Stark units in abelian extensions of ATR extensions of a totally real field from periods of Hilbert modular Eisenstein series. To the extent that the present work bears the same relation to [DLR] as do [DD] and [CD] to [Dar] and [DL] respectively, Glenn’s suggestion continues to resonate through the years. In the reverse direction, the strategy of substituting elliptic cusp forms for the weight two Eisenstein series that arise in the seminal work of Kato on the Birch and Swinnerton-Dyer conjecture has inspired the general program of studying the arithmetic of elliptic curves based on replacing Beilinson elements attached to a pair of modular units with Beilinson-Flach elements [BDR1], [BDR2] and generalised Gross-Schoen diagonal cycles [DR1], [DR2]. (See [BCDDPR] for an overview of this circle of ideas.) With profound gratitude for the influence he has exerted on the authors’ mathematical outlook, it is a great pleasure to dedicate this article to Glenn Stevens on the occasion of his 60th birthday.

Acknowledgements. The first author was supported by an NSERC Discovery grant, and the third by Grant MTM2012-34611.

1. The conjecture

Following the notations of [DLR] which will be adopted throughout, let M_{k}(N, χ)_{L} and S_{k}(N, χ)_{L} denote the spaces of modular forms and cusp forms respectively of weight k, level N, and character χ with Fourier coefficients in a field L. When L = ℂ, it shall be suppressed from the notation.

Let

g ∈ M_{1}(N_{g}, χ_{g}), h ∈ M_{1}(N_{h}, χ_{h})

be classical eigenforms of weight one. A crucial self-duality assumption made in [DLR] required the even Dirichlet character χ_{gh} := χ_{g}χ_{h} to be trivial, and thus the tensor product

ϱ_{gh} := ϱ_{g} ⊗ ϱ_{h}

of the Artin representations attached to g and h to be isomorphic to its contragredient.

GROSS-STARK UNITS AND p-ADIC ITERATED INTEGRALS 3

One of the novelties of the Eisenstein series setting is that this self-duality assumption is not necessary anymore, and can therefore be relaxed: the character χ_{gh} can now be an arbitrary Dirichlet character, of conductor dividing N_{gh} := lcm(N_{g}, N_{h}). Fix a prime p ∤ N_{gh} and let

1As another illustration of Glenn’s influence, let us mention the decisive impact, described in [DP], of his theory of overconvergent modular symbols for the efficient polynomial time calculation of the Stark-Heegner points of [Dar].

\begin{aligned}X^{2} − a_{p}(g)X + χ_{g}(p) & = & (X − α_{g})(X − β_{g}), \\ X^{2} − a_{p}(h)X + χ_{h}(p) & = & (X − α_{h})(X − β_{h}) \end{aligned}

be the p-th Hecke polynomials of g and h in ℂ[X].

It will be convenient to fix a finite extension L ⊂ ℂ of ℚ over which the representations ϱ_{g} and ϱ_{h} (and hence also ϱ_{gh}) can be realised. Such an L contains the field generated by the traces of these representations, and can always be chosen to be contained in a cyclotomic field. By enlarging L if necessary, we also assume it contains the roots of unity α_{g}, β_{g}, α_{h} and β_{h}.

Let g_{α}(z) = g(z) − β_{g}g(pz) and g_{β}(z) = g(z) − α_{g}g(pz) ∈ S_{1}(N_{g}p, χ_{g})_{L} denote the pstabilizations of g on which the Hecke operator U_{p} acts with eigenvalue α_{g} (resp. β_{g}).

Let

f := E_{2}(1, χ_{gh}^{−1}) ∈ M_{2}(N_{gh}, χ_{gh}^{−1})

be the weight two Eisenstein series with fourier expansion given by

f(q) := c_{0} + ∑_{n=1}^{∞} (∑_{d|n} χ_{gh}^{−1}(d)d) q^{n}.

The product of the three nebentypus characters attached to the triple (f, g, h) of modular forms of weights (2, 1, 1) is trivial, and one can define the p-adic iterated integral attached to this triple by letting

F := d^{−1}f = E_{0}^{[p]}(χ_{gh}^{−1}, 1)

be the overconvergent Eisenstein series of weight zero attached to the pair (χ_{gh}^{−1}, 1) of Dirichlet characters, with Fourier expansion given by

F (q) := ∑_{p∤n} (∑_{d|n} χ_{gh}^{−1}(n/d)d^{−1}) q^{n},

and considering the modular form

Ξ(g_{α}, h) = e_{g_{α}^{∗}}e_{ord}(F h),

where

e_{ord} : M_{1}^{oc}(N_{gh}, χ_{g}^{−1})⟶M_{1}^{oc}(N_{gh}, χ_{g}^{−1})

denotes Hida’s ordinary projection on the space M_{1}(N_{gh}, χ_{g}^{−1}) of overconvergent modular forms of weight one, and

e_{g_{α}^{∗}} : M_{1}^{oc}(N_{gh}, χ_{g}^{−1})⟶M_{1}^{oc}(N_{gh}, χ_{g}^{−1})[[g_{α}^{∗}]]

denotes the Hecke-equivariant projection to the generalised eigenspace attached to the system of Hecke eigenvalues for the dual form g_{α}^{∗} of g_{α}. The modular form Ξ(g_{α}, h) is overconvergent and ordinary, but need not be classical, since the conclusion of Coleman’s classicality theorem fails in general in weight one. For this reason, it is convenient to assume that Hypothesis C of [DLR], referred to there as the “classicality property”, holds for g_{α}. This hypothesis stipulates that the natural inclusion

M_{1}(Np, χ_{g}^{−1})_{ℂ_{p}}[g_{α}^{∗}] ↪ M_{1}^{oc,ord}(N, χ_{g}^{−1})[[g_{α}^{∗}]]

of the eigenspace attached to g_{α} (consisting entirely of classical forms) into the associated generalised eigenspace in the space of p-adic modular forms, maps M_{1}(Np, χ_{g}^{−1})_{ℂ_{p}}[g_{α}^{∗}] into the generalized eigenspace S_{1}^{oc,ord}(N, χ_{g}^{−1})[[g_{α}^{∗}]] of p-adic cusp forms and gives rise to an isomorphism

M_{1}(Np, χ_{g}^{−1})_{ℂ_{p}}[g_{α}^{∗}] ≃ S_{1}^{oc,ord}(N, χ_{g}^{−1})[[g_{α}^{∗}]]

between these two ℂ_{p}-vector spaces.

When g is cuspidal, it shall be assumed throughout that α_{g} ≠ β_{g}, i.e., that g is regular at p, and that g is not a theta series of a real quadratic field in which the prime p is split. As explained in [DLR, Proposition 1.2], the main result of [BeDi] then implies the classicality property for g_{α}.

On the other hand, when g = E_{1}(χ^{+}, χ^{−}) is the Eisenstein series associated to a pair of Dirichlet characters χ^{+} and χ^{−}, the space M_{1}(Np, χ_{g}^{−1})_{ℂ_{p}}[g_{α}^{∗}] maps into S_{1}^{oc,ord}(N, χ_{g}^{−1})[[g_{α}^{∗}]] if and only if χ^{+}(p) = χ^{−}(p), that is to say, if and only if g is not regular at p (cf. [DLR, Proposition 1.3]), and non-regularity is therefore a necessary condition for the classicality property to be satisfied.

Given an element γ in the L-linear dual space M_{1}(N_{gh}, χ_{g}^{−1})_{L}[g_{α}^{∗}]^{∨}, the p-adic iterated integral attached to the pair (γ, h) is defined to be

(1)  ∫_{γ} f · h := γ(e_{g_{α}^{∗}}e_{ord}(F h)) = γ(Ξ(g_{α}, h)).

Our goal is to study this p-adic iterated integral: more precisely, to understand when the classical modular form Ξ(g_{α}, h) with ℂ_{p} coefficients is non-trivial, and, when it is, give an arithmetic interpretation for its position in M_{1}(N_{gh}, χ_{g}^{−1})_{ℂ_{p}}[g_{α}^{∗}] relative to the natural L-rational structure on this ℂ_{p}-vector space given by M_{1}(N_{gh}, χ_{g}^{−1})_{L}[g_{α}^{∗}].

As in [DLR], the four-dimensional tensor product representation

ϱ_{gh} := ϱ_{g} ⊗ ϱ_{h}

plays a key role in this arithmetic interpretation. Let H be the smallest number field through which ϱ_{gh} factors, and set

U_{gh} := L ⊗ O_{H}^{×}, U_{gh}^{(p)} := L ⊗ (O_{H}[1/p]^{×}/p^{ℤ}),

viewed as finite-dimensional L-linear representations of G_{ℚ}.

Lemma 1.1. Let s denote the multiplicity of the trivial representation in ϱ_{gh}. Set

d_{gh} := dim_{L} hom_{G_{ℚ}}(ϱ_{gh}, U_{gh}), d_{gh}^{(p)} := dim_{L} hom_{G_{ℚ}}(ϱ_{gh}, U_{gh}^{(p)}).

Then (1) d_{gh} = 2 − s,

(2) d_{gh}^{(p)} = 2 + dim_{L}(ϱ_{gh})^{G_{ℚp}} − 2s.

Proof. This follows from Herbrand’s Galois-equivariant strengthening of Dirichlet’s S-unit theorem, combined with Frobenius reciprocity. See for instance [Ta, Ch.I, §3, §4] or [Das, Prop. 3.2.2]. □

In particular, when s = 0, Lemma 1.1 shows that d_{gh} = 2 and that the same is true for d_{gh}^{(p)}

when ϱ_{gh} admits no G_{ℚ_{p}}-invariants. Furthermore, it is not hard to see that d_{gh}^{(p)} ≥ 2 when at least one of g or h is a cusp form. This is in marked contrast with the setting explored in [DLR] where U_{gh} and U_{gh}^{(p)} were replaced by the Mordell-Weil group E(H) of an elliptic curve E, tensored with L: in this case no simple elementary formula was available for the analogues of d_{gh} and d_{gh}^{(p)}, and these quantities appear to be 0 quite often. This is why the necessity of the “vanishing hypothesis” (Hypothesis B) arose in [DLR], whereas here it is unecessary.

Fix an embedding of H into ℂ_{p}. This equips H, and hence also U_{gh} and U_{gh}^{(p)}, with natural actions of G_{ℚ_{p}}. Since p is assumed to be prime to N_{gh}, the action of G_{ℚ_{p}} on H is unramified; let Fr_{p} denote the arithmetic Frobenius operator, inducing the automorphism x ↦ x^{p} on the residue fields.

The galois element Fr_{p} acts on ϱ_{g} (resp. on ϱ_{h}) with eigenvalues α_{g} and β_{g} (resp. α_{h} and β_{h}). Write

ϱ_{g} := ϱ_{g}^{α} ⊕ ϱ_{g}^{β}, ϱ_{h} := ϱ_{h}^{α} ⊕ ϱ_{h}^{β}

for the decomposition of ϱ_{g} and ϱ_{h} into one-dimensional frobenius eigenspaces, when these are defined, i.e., when Fr_{p} acts with distinct eigenvalues.

When g is cuspidal, so that α_{g} ≠ β_{g} by our running assumption, we attach to g_{α} a twodimensional subspace of the representation ϱ_{gh} = ϱ_{g} ⊗ ϱ_{h} by setting

(2)  ϱ_{gh,α} := ϱ_{g}^{β} ⊗ ϱ_{h}.

When g is Eisenstein, we have α_{g} = β_{g} and we simply define

(3)  ϱ_{gh,α} := W ⊗ ϱ_{h},

where W is any line in ϱ_{g} which is not stable under G_{ℚ}.

When d_{gh}^{(p)} = 2, it is possible to associate to the pair (g_{α}, h) a p-adic regulator by choosing

an L-basis (Φ_{1}, Φ_{2}) for hom_{G_{ℚ}}(ϱ_{gh}, U_{gh}^{(p)}) and an L-basis (v_{1}, v_{2}) for ϱ_{gh,α} and setting

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

This matrix is well-defined up to left and right multiplication by invertible matrices with entries in L, and hence its determinant is well-defined modulo L^{×}. This determinant belongs to ℚ_{p}^{u-r} ⊗ L, and Fr_{p} acts on it with the eigenvalue β_{g}α_{h}β_{g}β_{h} = \frac{β_{g}}{α_{g}} · χ_{gh}(p).

Recall the unit u_{g_{α}} in U_{gh} (when g is a cusp form) or in U_{gh}^{(p)} (when g is an Eisenstein series) attached to the p-stabilised eigenform g_{α} in [DLR, §1.2]. It belongs to the Ad^{0}(g)-isotypic part of U_{gh} (resp. of U_{gh}^{(p)}), and is an eigenvector for^{2} Fr_{p} with eigenvalue β_{g}/α_{g}.

Finally, if χ is a Dirichlet character of conductor m, let

𝔤(χ) := ∑_{a=1}^{m} χ^{−1}(a)e^{2πia/m}

be the usual Gauss sum, on which G_{ℚ} acts through χ and thus Fr_{p} acts with eigenvalue χ(p). Our main conjecture is:

Conjecture 1.2. If d_{gh}^{(p)} > 2, then the modular form Ξ(g_{α}, h) is 0. Otherwise,

Ξ(g_{α}, h) belongs to Ω_{gh} · M_{1}(N_{gh}, χ_{g}^{−1})[g_{α}^{∗}]_{L},

where the period Ω_{gh} is given by the formula

Ω_{gh} := \frac{det(R_{g_{α}}(ϱ_{gh}))}{𝔤(χ_{gh}) log_{p}(u_{g_{α}})}.

As discussed in the introduction, this conjecture is directly inspired by the “elliptic Stark conjectures” of [DLR] which apply to the setting where f is replaced with a weight two cusp form attached to an elliptic curve. Conjecture 1.2 is closer in spirit to the p-adic Gross-Stark conjectures, since its regulator term involved units and p-units of number fields rather than global points on elliptic curves, but it does not appear to follow from any of the variants of Stark’s conjecture previously available in the literature.

The remainder of this paper shall be devoted to providing both theoretical and experimental support for Conjecture 1.2.

2This is indeed the same unit as in [DLR], where we rather employed the geometric frobenius σ_{p} with acts on the unit with eigenvalue α_{g}/β_{g}.

2. Group-theoretic preliminaries

This section collects some general representation-theoretic results concerning the structure of the representation ϱ_{gh} = ϱ_{g} ⊗ ϱ_{h} appearing in Conjecture 1.2. We refer to [DS] and [Se] for generalities on Artin representations and their connection to modular forms of weight one.

Let V_{g} and V_{h} denote two-dimensional L-vector spaces which realise ϱ_{g} and ϱ_{h} respectively, and denote by V_{gh} = V_{g} ⊗_{L} V_{h} the L-vector space realising ϱ_{gh}.

The image of ϱ_{g} is a finite subgroup G ⊂ GL_{2}(ℂ). The well-known classification of such groups implies that the natural image Γ of G in PGL_{2}(ℂ) is isomorphic to

(1) a cyclic group, if g is an Eisenstein series;

(2) a dihedral group, if g is the theta series attached to a finite order character of a (real or imaginary) quadratic field;

(3) the group A_{4}, S_{4} or A_{5}. In this last case, the form g is said to be exotic.

The group G is then isomorphic to

(5)  G = (Z × Γ̃)/⟨±1⟩,

where Z = G ∩ ℂ^{×} is the center of G, and Γ̃ = G ∩ SL_{2}(ℂ) is the universal central extension of Γ. It is a non-split central extension of Γ by the group ⟨±1⟩ ⊂ Z, and is called the binary tetrahedral, octahedral, or icosahedral group (of order 24, 48, and 120) when Γ = A_{4}, S_{4} and A_{5} respectively. The irreducible representations of G are obtained in a simple way from those of Γ̃: more precisely, if ϱ_{1}, … ϱ_{t} is a complete list of the irreducible representations of Γ̃, the irreducible representations of G are simply of the form χ_{Z} ⊗ ϱ_{j}, where χ_{Z} is a character of Z satisfying χ_{Z}(−1) = ϱ_{j}(−1).

When ϱ_{g} has cyclic projective image, it decomposes as a direct sum of two one-dimensional representations attached to Dirichlet characters χ_{1} and χ_{2}, and g = E_{1}(χ_{1}, χ_{2}) is the weight one Eisenstein series attached to this pair of characters.

When ϱ_{g} (or ϱ_{h}) has dihedral projective image, it is induced from a one-dimensional character ψ_{g} : G_{K}⟶ℂ^{×} of a quadratic field K, and g is the theta series attached to this character. The representation

V_{ψ_{g}} := Ind_{K}^{ℚ}(ψ_{g}) : G_{ℚ}⟶GL_{2}(ℂ)

becomes reducible when restricted to K, where it is isomorphic to L(ψ_{g})⊕L(ψ′_{g}), where ψ′_{g} denotes the character obtained by composing ψ_{g} with the automorphism of K/ℚ. Furthermore, V_{ψ_{g}} is odd if and only if either

(a) K is an imaginary quadratic field, or

(b) K is a real quadratic field, and ψ_{g} is of mixed signature, i.e., it is even at one of the

archimedean places of K, and odd at the other. Note that V_{ψ_{g}} is irreducible (and hence, g is a cusp form) precisely when ψ_{g} ≠ ψ′_{g}.

The most basic invariant of the tensor product V_{gh} is its decomposition

V_{gh} = V_{1} ⊕ ⋯ ⊕ V_{t}

into irreducible representations. Write \underline{d} := (d_{1}, …, d_{t}) for the dimensions of the V_{j} that arise in this decomposition, arranged by convention in non-increasing order. There are five possibilities for \underline{d}, corresponding to the possible partitions of 4.

If g and h are both reducible, i.e., correspond to Eisenstein series, then clearly \underline{d} = (1, 1, 1, 1). Assume from now on that at least one of g or h is a cusp form, hence is associated to an irreducible two-dimensional representation.

Lemma 2.1. The representation V_{gh} contains a subrepresentation of dimension 1 if and only if h is a twist of ḡ, i.e., V_{h} = V_{g}^{∨}(ι), for some Dirichlet character ι. In that case,

(1) \underline{d} = (3, 1) if and only if g is exotic; (2) \underline{d} = (2, 1, 1) if and only if g is dihedral and ψ_{g}^{−} := ψ_{g}/ψ′_{g} is not a quadratic (genus)

character;

(3) \underline{d} = (1, 1, 1, 1) if and only if g is dihedral and ψ_{g}^{−} is a genus character.

Proof. The one-dimensional subrepresentation of V_{gh} is isomorphic to L(ι) for some Dirichlet character ι (viewed as a Galois character). This implies that

V_{gh} ⊗ L(ι^{−1}) = hom(V_{g}^{∨}(ι), V_{h})

has a G_{ℚ}-invariant vector. Since at least one of V_{g} or V_{h} is irreducible, Schur’s Lemma implies that V_{h} is isomorphic to V_{g}^{∨}(ι). It follows that

V_{gh} = W ⊕ L(ι), where W := Ad^{0}(V_{g})(ι)

is the twist by ι of the adjoint representation attached to V_{g}. Lemma 2.1 now follows by noting that:

(i) if ϱ_{g} has projective image isomorphic to Γ = A_{4}, S_{4} or A_{5}, then Ad^{0}(V_{g}) factors through Γ and is a three-dimensional irreducible representation of this group, as can be seen, for instance, by consulting the character tables of Γ and Γ̃;

(ii) if ϱ_{g} has dihedral projective image, and thus is induced from a character ψ_{g} of a quadratic field K, then

Ad^{0}(V_{g}) = L(χ_{K}) ⊕ V_{ψ_{g}^{−}},

where χ_{K} is the quadratic Dirichlet character attached to K.

□

The next lemma takes up the case where \underline{d} = (2, 2).

Lemma 2.2. The representation V_{gh} decomposes as a sum of two irreducible representations of dimension two if and only if:

(1) Exactly one of g or h is a weight one Eisenstein series, say h = E_{1}(χ_{1}, χ_{2}), in which

case

V_{gh} = V_{g}(χ_{1}) ⊕ V_{g}(χ_{2}).

(2) Both g and h are theta series attached to characters ψ_{g} and ψ_{h} of the same quadratic field K for which neither of the characters

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

is equal to its conjugate over K. In this case,

V_{gh} = V_{ψ_{1}} ⊕ V_{ψ_{2}}.

The representations V_{ψ_{1}} and V_{ψ_{2}} are non-isomorphic, and if K is real quadratic, they are of opposite (pure) signatures.

Proof. It is clear that \underline{d} = (2, 2) when exactly one of g or h is an Eisenstein series. Assume for the rest of the proof that both g and h are cuspidal. Let H_{g}/ℚ and H_{h}/ℚ be the extensions of ℚ cut out by the representations ϱ_{g} and ϱ_{h}, let H := H_{g} ∩H_{h}, and let H_{gh} be the compositum of the extensions H_{g} and H_{h}, which are linearly disjoint over H. Write

G_{gh} := Gal (H_{gh}/ℚ), G_{g} := Gal (H_{g}/ℚ), G_{h} := Gal (H_{h}/ℚ),
A_{g} := Gal (H_{g}/H), A_{h} := Gal (H_{h}/H). G_{0} = Gal (H/ℚ).

The resulting field diagram is described below

[figure: H_{hg}; H_{g}; A_{h}; H_{h}; A_{g}; H; A_{g}; A_{h}; ℚ; G_{g}; G_{0}; G_{h}]

By hypothesis, the group G_{g} (resp. G_{h}) acts faithfully on V_{g} (resp. V_{h}).

Observe first that the groups A_{g} and A_{h} are necessarily abelian. For if A_{g} (say) were nonabelian, the two-dimensional vector space V_{g} on which A_{g} acts faithfully would be irreducible as an A_{g}-module. Since A_{g} acts trivially on V_{h}, the irreducible A_{g}-submodules of V_{gh} would have to all be of the form V_{g} ⊗ L_{h} for some one-dimensional L-vector subspace L_{h} ⊂ V_{h}. This would force any two-dimensional irreducible sub-representation of V_{gh} to also be of this form. But V_{h} contains no G_{h}-stable lines and the vector space V_{g} ⊗ L_{h} therefore cannot be stable under the full action of G_{gh}.

One can now distinguish two cases: Case A: At least one of G_{g} or G_{h} has a projective image isomorphic to A_{4}, S_{4} or A_{5}. Assume without loss of generality that this is the case for G_{g}. The only abelian normal subgroups of Γ̃_{g} are contained in its center, and hence, the same is true for the abelian normal subgroups of G_{g}. It follows that A_{g} acts on V_{g} as scalar multiplications, and that G_{0} has a quotient isomorphic to Γ. Hence the same assertions hold for G_{h}: in other words, both G_{g} and G_{h} are central extensions of a common Γ ⊂ PGL_{2}(ℂ) which arises as a quotient of G_{0}. A study of the character table of Γ̃ reveals that the tensor products of two irreducible two-dimensional representations of this group never decomposes as a sum of two-dimensional representations. This rules out the scenario where \underline{d} = (2, 2) in Case A.

Case B: The groups G_{g} and G_{h} have dihedral projective image. The representation ϱ_{g} is then induced from a character ψ_{g} of a quadratic field K, which satisfies ψ_{g} ≠ ψ′_{g}, since V_{g} is irreducible. Hence V_{g} is isomorphic to L(ψ_{g}) ⊕ L(ψ′_{g}) as a G_{K}-module, and any element of G_{ℚ} − G_{K} interchanges L(ψ_{g}) and L(ψ′_{g}). In particular, the representation V_{gh} is isomorphic to V_{h}(ψ_{g}) ⊕ V_{h}(ψ′_{g}) as a G_{K}-module, and the following cases arise:

• V_{h} is reducible as a G_{K}-module, and hence ϱ_{h} is induced from a character of K.

• V_{h} remains irreducible as a G_{K}-module. The reducibility of V_{gh} then forces

(6)  V_{h}(ψ_{g}) ≃ V_{h}(ψ′_{g}), hence V_{h} ≃ V_{h}(ψ_{g}/ψ′_{g}) as G_{K} − modules.

For otherwise, the only non-trivial G_{K}-stable subspaces of V_{gh} would be V_{h}(ψ_{g}) and V_{h}(ψ′_{g}), which are not preserved by elements of G_{ℚ}−G_{K}. Taking determinants on both sides of (6) reveals that ψ_{g}/ψ′_{g} is a quadratic (and hence a genus) character of K, whose induced representation is a direct sum χ_{1} ⊕ χ_{2} of two quadratic Dirichlet characters attached to quadratic extensions K_{1} and K_{2} respectively, satisfying χ_{1}χ_{2} = χ_{K}, where χ_{K} is the quadratic character attached to K. Inducing (6) from K to ℚ shows that

V_{h} ⊕ V_{h}(χ_{K}) ≃ V_{h}(χ_{1}) ⊕ V_{h}(χ_{2}) as G_{ℚ}-modules.

Since V_{h} and V_{h}(χ_{K}) are assumed to be non-isomorphic, it may be assumed, after eventually interchanging χ_{1} and χ_{2}, that V_{h} is isomorphic to V_{h}(χ_{1}), and therefore, that V_{h} is induced from a character of K_{1}. But the same is also true of V_{g}, which is a twist of a representation of the dihedral group of order 8, and as such is simultaneously induced from characters of all three of the quadratic fields K, K_{1} and K_{2}.

After replacing K by K_{1} in the second scenario, it follows that V_{g} and V_{h} are induced from characters of the same quadratic field K. Writing ψ_{h} for the character of K satisfying V_{h} = V_{ψ_{h}}, the representation V_{gh} breaks up as

V_{gh} = V_{ψ_{1}} ⊕ V_{ψ_{2}}, ψ_{1} := ψ_{g}ψ_{h}, ψ_{2} := ψ_{g}ψ′_{h}.

It follows from this that ψ_{1} and ψ_{2} have the same central character, and that \underline{d} = (2, 2) precisely when ψ_{1} ≠ ψ′_{1} and ψ_{2} ≠ ψ′_{2}. (Recall that the central character of a character ψ of G_{K} is the Dirichlet characters arising from the restriction to the idéles of ℚ of the Hecke character of K associated by class field theory to ψ.) To see that V_{ψ_{1}} and V_{ψ_{2}} are never isomorphic, note that the irreducibility of V_{h} implies that ψ_{h} ≠ ψ′_{h}, and hence that ψ_{1} ≠ ψ_{2}; and that likewise, the irreducibility of V_{g} implies that ψ_{1} ≠ ψ′_{2}. When K is real quadratic, it is readily checked that, since ψ_{g} and ψ_{h} are both of mixed signature, the ring class characters ψ_{1} = ψ_{g}ψ_{h} and ψ_{2} = ψ_{g}ψ′_{h} are of opposite pure signatures. The lemma follows. □

The “generic” case where \underline{d} = (4) encompasses all scenarios where the representations ϱ_{g} and ϱ_{h} cut out linearly disjoint extensions of ℚ. When it is irreducible, the Artin representation V_{gh} occurs with mutiplicity two in L ⊗ O_{H}^{×} and is undistinguished enough to probably never have garnered explicit attention in connection with the Stark or Gross-Stark conjectures. Nonetheless the special cases described in Lemmas 2.3 and 2.4 below are worth singling out.

Suppose that

ϱ_{h} : Gal (H_{h}/ℚ)⟶Γ̃ ⊂ GL_{2}(ℂ)

has projective image Γ = Γ̃/Z ≃ S_{4}, and that ϱ_{g} is the (non-faithful) two-dimensional representation of Gal (H_{h}/ℚ) obtained from ϱ_{h} by composing it with the natural sequence of homomorphisms Γ̃⟶S_{4}⟶S_{3} ⊂ GL_{2}(ℂ). Then ϱ_{gh} can be described concretely in terms of the induced representation of a quadratic character of the quartic subfield K of H_{h} fixed by S_{3} ⊂ S_{4}. More precisely, the natural exact sequence

1⟶Z⟶Gal (H_{h}/K)⟶S_{3}⟶1

splits. Since Z is a cyclic group of even cardinality, it follows that Gal (H_{h}/K) has three distinct surjective homomorphisms to ±1, of which only one is trivial on Z. Let η be a quadratic character of Gal (H_{h}/K) which is non-trivial on Z, and let M be the octic extension of ℚ cut out by η, as summarised in the following field diagram:

[figure: H_{h}; S_{3}×Z; Z; M; ±1; H̄_{h}; ℤ/2ℤ×ℤ/2ℤ; S_{3}; S_{4}; K; 4; H_{g}; S_{3}; ℚ]

Lemma 2.3. For ϱ_{g} and ϱ_{h} as above,

ϱ_{gh} = Ind_{K}^{ℚ} η.

In particular,

hom(V_{gh}, ℂ ⊗ O_{H}^{×}) = ℂ ⊗ O_{M,1}^{×},

where O_{M,1}^{×} denotes the group of units of M whose norm to K is equal to 1.

Proof. The first assertion follows from a direct calculation using the character table of S̃_{4}, and the second from Frobenius reciprocity. □

Another simple, concrete instance of an irreducible V_{gh} arises when K is a non-totally real quintic extension of ℚ whose Galois closure K̃ has Galois group A_{5} over ℚ. There are exactly two conjugacy classes of embeddings of Gal (K̃/ℚ) = A_{5} into PGL_{2}(ℂ); choosing one gives rise to a projective representation

ϱ̄_{g} : G_{ℚ} ↠ Gal (K̃/ℚ) ≃ A_{5} ⊂ PGL_{2}(ℂ).

Tate’s lifting theorem produces a linear lift ϱ_{g} : G_{ℚ}⟶GL_{2}(ℂ) of ϱ̄_{g}, which is odd since K is not totally real, and hence corresponds to a modular form g. It can further be assumed that the field H cut out by ϱ_{g} is a cyclic extension of K̃ of 2-power degree, so that ϱ_{g}(Gal(H/K̃)) consists of scalar matrices whose diagonal entries are 2-power roots of unity. Let σ be an automorphism of ℂ which agrees with complex conjugation on the 2-power roots of unity and sends \sqrt{5} to −\sqrt{5}, and let h be the modular form obtained by applying σ to g.

Lemma 2.4. For ϱ_{g} and ϱ_{h} as above,

(7)  V_{gh} ⊕ ℂ = V_{5},

where V_{5} is the natural permutation representation of A_{5} acting on 5 letters. In particular,

hom(V_{gh}, ℂ ⊗ O_{H}^{×}) = ℂ ⊗ O_{K}^{×}, hom(V_{gh}, ℂ ⊗ O_{H}[1/p]^{×}) = ℂ ⊗ (O_{K}[1/p]^{×}/p^{ℤ}).

Proof. See for instance the proof of Thm. 6.9 of [DR2]. □

3. Circular units and Eisenstein series of weight one

In this section we begin by considering the simplest case where

g = E_{1}(χ_{g}^{+}, χ_{g}^{−}), h = E_{1}(χ_{h}^{+}, χ_{h}^{−})

are both Eisenstein series attached to pairs of Dirichlet characters. It is assumed that χ_{g}^{+} is even and χ_{g}^{−} is odd, and likewise for χ_{h}^{+} and χ_{h}^{−}. The nebentype characters of g and h are equal to χ_{g} := χ_{g}^{+}χ_{g}^{−} and χ_{h} := χ_{h}^{+}χ_{h}^{−} respectively.

Fix a prime p not dividing the conductors of χ_{g} and χ_{h}, and recall we assume the classicality hypothesis on g, which in particular implies that χ_{g}^{+}(p) = χ_{g}^{−}(p).

The representation ϱ_{gh} decomposes as a direct sum of four one-dimensional characters

ϱ_{gh} = χ++_{gh}⊕ χ--_{gh} ⊕ χ{+ \atop -}_{gh} ⊕ χ-+_{gh}:= χ_{1} ⊕ χ_{2} ⊕ χ_{3} ⊕ χ_{4},

where

χ++_{gh}:= χ_{g}^{+}χ_{h}^{+},  χ--_{gh} := χ_{g}^{−}χ_{h}^{−},  χ{+ \atop -}_{gh} := χ_{g}^{+}χ_{h}^{−},  χ-+_{gh}:= χ_{g}^{−}χ_{h}^{+}.

Note that the characters χ++_{gh}and χ--_{gh} are even, while the characters χ{+ \atop -}_{gh} and χ-+_{gh}are odd (and in particular, non-trivial).

Given an even character χ factoring through a finite abelian extension H_{χ} of ℚ and taking values in L, denote by u(χ) ∈ L ⊗ (O_{H_{χ}}^{×}) the fundamental unit in the χ-eigenspace for the natural G_{ℚ}-action. Given an odd character χ for which χ(p) = 1, let u_{p}(χ) ∈ L⊗(O_{H_{χ}}[1/p])^{×} denote the fundamental p-unit in the χ-eigenspace for the Galois action.

In light of Lemma 1.1, the form taken on by Conjecture 1.2 depends on the number of even characters among the χ_{j} which are non-trivial, and hence it is convenient to break up our discussion into three cases:

Case 1: The even characters χ++_{gh}and χ--_{gh} are both non-trivial. In that case, Conjecture 1.2 predicts that Ξ(g_{α}, h) = 0 if χ_{j}(p) = 1 for some 1 ≤ j ≤ 4, and that otherwise

(8) Ξ(g_{α}, h) ∈ Ω_{gh} · M_{1}(N_{gh}, χ_{g}^{−1})[g_{α}^{∗}]_{L} where Ω_{gh} = \frac{log_{p}(u(χ++_{gh} )) log_{p}(u(χ--_{gh}))}{𝔤(χ_{gh}) log_{p}(u_{g_{α}})}.

Case 2: Exactly one of the even characters χ++_{gh}and χ--_{gh} is trivial. Our running assumption that χ_{g}^{+}(p) = χ_{g}^{−}(p) implies that χ-+_{gh}(p) = 1 if χ++_{gh}= 1 and that χ{+ \atop -}_{gh}(p) = 1 if χ--_{gh} = 1. Conjecture 1.2 predicts that Ξ(g_{α}, h) = 0 if χ_{j}(p) = 1 for at least two non-trivial characters among the χ_{j}. Assuming that this is not the case, let χ_{gh}^{+} denote the unique non-trivial character in \{χ++_{gh} , χ--_{gh}\} and let χ_{gh}^{−} denote the unique character in \{χ{+ \atop -}_{gh}, χ-+_{gh}\} satisfying χ_{gh}^{−}(p) = 1. Conjecture 1.2 then predicts that

(9) Ξ(g_{α}, h) ∈ Ω_{gh} · M_{1}(N_{gh}, χ_{g}^{−1})[g_{α}^{∗}]_{L} where Ω_{gh} = \frac{log_{p}(u(χ_{gh}^{+})) log_{p}(u_{p}(χ_{gh}^{−}))}{𝔤(χ_{gh}) log_{p}(u_{g_{α}})}.

A notable special case arises when χ_{g}^{+} = χ_{h}^{+} = 1 and the odd characters χ_{g}^{−} and χ_{h}^{−} are both quadratic, with χ_{g}^{−}(p) = −χ_{h}^{−}(p) = 1. In that scenario, the p-units u_{p}(χ_{gh}^{−}) and u_{g_{α}} that occur in the numerator and denominator of (9) cancel out, and the conjectural formula for the period simplifies to

(10)  Ω_{gh} = \frac{log_{p}(u(χ--_{gh}))}{𝔤(χ_{gh})}.

One obtains an appealing expression for the p-adic logarithm of the fundamental unit of a real quadratic field in terms of p-adic iterated integrals. This leads to a (somewhat) new method for solving Pell’s equation, albeit one which is hardly an improvement over the classical method of continued fractions!

Case 3: Both of the even characters χ++_{gh}and χ--_{gh} are trivial. This implies that

χ_{h}^{+} = (χ_{g}^{+})^{−1}, χ_{h}^{−} = (χ_{g}^{−})^{−1},

so that, after setting χ_{g}^{±} := χ_{g}^{+}/χ_{g}^{−} and χ_{g}^{∓} := χ_{g}^{−}/χ_{g}^{+},

ϱ_{gh} = 1 ⊕ 1 ⊕ χ_{g}^{±} ⊕ χ_{g}^{∓}.

Both of the odd characters that arise in this decomposition are trivial at p. Conjecture 1.2 therefore predicts that Ξ(g_{α}, h) ≠ 0 and that

(11) Ξ(g_{α}, h) ∈ Ω_{gh} · M_{1}(N_{gh}, χ_{g}^{−1})[g_{α}^{∗}]_{L} where Ω_{gh} = \frac{log_{p}(u_{p}(χ_{g}^{±})) log_{p}(u_{p}(χ_{g}^{∓}))}{log_{p}(u_{p}(χ_{g}^{±})) + log_{p}(u_{p}(χ_{g}^{∓}))}.

A notable special case arises when the odd character χ_{g}^{±} is quadratic, and hence equal to χ_{g}^{∓}. The expression in (11) then simplifies to yield (modulo L^{×}, as always)

(12)  Ω_{gh} = log_{p}(u_{p}(χ_{g}^{±})).

One thus obtains, somewhat like in (10), an appealing expression for the p-adic logarithm of the fundamental p-unit of an imaginary quadratic field in terms of p-adic iterated integrals of Eisenstein series.

The main result of this section is the unconditional proof of certain special cases of Conjecture 1.2 when g and h are both Eisenstein series:

Theorem 3.1. Assume that g and h are Eisenstein series, and that χ_{g}^{+} and χ_{g}^{−} are both quadratic characters of relatively prime conductors. Then Conjecture 1.2 holds, i.e., equations (8), (9) and (11) hold in cases 1, 2 and 3 respectively.

Proof. Recall that the classicality hypothesis imposed on g implies that

(13)  χ_{g}^{+}(p) = χ_{g}^{−}(p).

The pair (χ_{g}^{+}, χ_{g}^{−}) corresponds to a genus character ψ_{g} of the imaginary quadratic field K cut out by the odd Dirichlet character χ_{g} := χ_{g}^{+}χ_{g}^{−}. Condition (13) implies that χ_{g}(p) = 1, and hence that the prime p =: 𝔭 𝔭̄ splits in K/ℚ. The weight one Eisenstein series g = E_{1}(χ_{g}^{+}, χ_{g}^{−}) is equal to the theta series θ_{K}(ψ_{g}) attached to ψ_{g}, and the modular form g admits three natural ordinary deformations, arising from the fact that E_{1}(χ_{g}^{+}, χ_{g}^{−}) = E_{1}(χ_{g}^{−}, χ_{g}^{+}) = θ_{K}(ψ_{g}):

• The Hida families \underline{E}(χ_{g}^{+}, χ_{g}^{−}) and \underline{E}(χ_{g}^{−}, χ_{g}^{+}) of Eisenstein series, whose weight k specialisations are described by

\underline{E}_{k}(χ_{1}, χ_{2}) := ∑_{n=1}^{∞} σ_{k−1,χ_{1},χ_{2}}^{(p)}(n)q^{n}, σ_{k−1,χ_{1},χ_{2}}^{(p)}(n) := ∑_{d|n, p∤d}χ_{1}(n/d)χ_{2}(d)d^{k−1};

• The cuspidal family \underline{θ}(ψ_{g}) of ordinary theta series whose weight k specialisation is

given by

\underline{θ}_{k}(ψ_{g}) = θ(ψ_{g}ψ_{p}^{k−1}) = ∑_{𝔞⊂O_{K}, 𝔭̄∤𝔞} ψ_{g}(𝔞)ψ_{p}^{k−1}(𝔞)q^{𝔞𝔞̄}.

In this equation, the character ψ_{p} : I(K)⟶ℚ̄_{p}^{×} on the group I(K) of fractional ideals of K is obtained from a canonical Hecke character ψ_{∞} : I(K)⟶ℂ^{×} in the sense of Gross and Rohrlich, sending a principal ideal (α) to ±ᾱ for all α which are relatively prime to the conductor of ψ_{∞}. The character ψ_{∞} has infinity type (0, 1) and takes its values in a finite extension L of K, and ψ_{p} is obtained by composing it with a p-adic embedding of L sending K to its completion at 𝔭.

Let θ_{3+2t}(ψ_{g}) denote the weight 3 + 2t specialisation of the Hida family of theta-series arising from the genus character ψ_{g}. The classicality assumption on θ_{1}(ψ_{g}) implies that \underline{θ}(ψ_{g}) is the unique cuspidal Hida family specialising to θ_{1}(ψ_{g}).

Because of this, the p-adic iterated integral (1) attached to the pair (θ_{K}(ψ_{g}), E_{1}(χ_{h}^{+}, χ_{h}^{−})) agrees, up to non-zero algebraic factors, with the p-adic limit of the algebraic numbers (depending on t ≥ 0)

Ω_{gh}^{[p]}(t) := \frac{⟨θ_{3+2t}(ψ_{g}), d^{t}E_{2}^{[p]}(1, χ_{gh}^{−1}) × E_{1}(χ_{h}^{+}, χ_{h}^{−})⟩}{⟨θ_{3+2t}(ψ_{g}), θ_{3+2t}(ψ_{g})⟩}

as t tends to −1 in weight space. (Note that this ratio does not depend on any choice of normalisation for the Petersson scalar product.) Applying Cor. 4.13 of [DR1] after making the substitutions

(k, ℓ, m) ← (3 + 2t, 2, 1), f_{k} ← θ_{3+2t}(ψ_{g}), g ← E_{2}(1, χ_{gh}^{−1}), h ← E_{1}(χ_{h}^{+}, χ_{h}^{−}),

one obtains

Ω_{gh}^{[p]}(t) = \frac{E_{gh}(t)}{E_{g}(t)} × Ω_{gh}(t),

where

E_{gh}(t) := (1 − ψ_{p}(𝔭̄)^{2+2t}χ_{h}^{+}(p)p^{−2−t}) × (1 − ψ_{p}(𝔭̄)^{2+2t}χ_{h}^{−}(p)p^{−2−t})
×(1 − ψ_{p}(𝔭̄)^{2+2t}χ̄_{h}^{−}(p)p^{−1−t}) × (1 − ψ_{p}(𝔭̄)^{2+2t}χ̄_{h}^{+}(p)p^{−1−t}),
E_{g}(t) := (1 − ψ_{p}(𝔭̄)^{4+4t}p^{−3−2t}),
Ω_{gh}(t) := \frac{⟨θ_{3+2t}(ψ_{g}), d^{t}E_{2}(1, χ_{gh}^{−1}) × E_{1}(χ_{h}^{+}, χ_{h}^{−})⟩}{⟨θ_{3+2t}(ψ_{g}), θ_{3+2t}(ψ_{g})⟩}.

It will be convenient to express Ω_{gh}^{[p]}(t) as a ratio

Ω_{gh}^{[p]}(t) := \frac{Ω_{gh}^{num}(t)}{Ω_{g}^{den}(t)}

of two quantities that can (as will be shown) be individually interpolated p-adically, namely

Ω_{gh}^{num}(t) := \frac{E_{gh}(t) × Ω_{K,p}^{4+4t} × ⟨θ_{3+2t}(ψ_{g}), d^{t}E_{2}(1, χ_{gh}^{−1}) × E_{1}(χ_{h}^{+}, χ_{h}^{−})⟩}{Ω_{K}^{4+4t}},

Ω_{g}^{den}(t) := \frac{E_{g}(t) × Ω_{K,p}^{4+4t} × ⟨θ_{3+2t}(ψ_{g}), θ_{3+2t}(ψ_{g})⟩}{Ω_{K}^{4+4t}},

where Ω_{K} ∈ ℂ^{×} is Shimura’s canonical period attached to the imaginary quadratic field K and Ω_{K,p} ∈ ℂ_{p}^{×} is the corresponding p-adic period which appears in the interpolation formula for the Katz p-adic L-function. The strategy of proof is to show that both of these functions of t ∈ ℤ^{≥0}, viewed as taking values in ℂ_{p}, extend to an analytic function on weight space by relating them to the Katz p-adic L-function, and then to evaluate them at the point t = −1 by invoking Katz’s p-adic analogue of the Kronecker limit formula.

We begin with the expression Ω_{gh}^{num}(t). To evaluate it, we invoke formula (18) of [BD2], after making the substitutions

(k, ℓ, m) ← (3 + 2t, 1, 2), c ← 2 + t, f ← θ_{3+2t}(ψ_{g}), g ← E_{1}(χ_{h}^{+}, χ_{h}^{−}),

to obtain

⟨θ_{3+2t}(ψ_{g}), d^{t}E_{2}(1, χ_{gh}^{−1}) × E_{1}(χ_{h}^{+}, χ_{h}^{−})⟩ = C(t) × L(θ_{3+2t}(ψ_{g}) ⊗ E_{1}(χ_{h}^{+}, χ_{h}^{−}), 2 + t),

where

C(t) := \frac{(1 + t)!^{2} × Ω_{K,p}^{4+4t}}{(−1)^{t−1}2^{2+2t}(2π)^{4+2t}(−iN)^{−2}𝔤(χ_{gh})}.

Formula (19) of loc.cit. implies that

Ω_{gh}^{num}(t) = C(t)E_{gh}(t) × \frac{L(θ_{3+2t}(ψ_{g}), χ_{h}^{+}, 2 + t)}{Ω_{K}^{2+2t}} × \frac{L(θ_{3+2t}(ψ_{g}), χ_{h}^{−}, 2 + t)}{Ω_{K}^{2+2t}}

= C(t)E_{gh}(t) × \frac{L(ψ_{g}χ_{h}^{+}ψ_{t}^{−1}, 0)}{Ω_{K}^{2+2t}} × \frac{L(ψ_{g}χ_{h}^{−}ψ_{t}^{−1}, 0)}{Ω_{K}^{2+2t}},

where the L-functions in the second line are those attached to Hecke characters of K, and ψ_{t} is the Hecke character

ψ_{t} := \frac{(ψ′_{p})^{2+t}}{ψ_{p}^{t}} .

In view of our definition of ψ_{∞}, the character ψ_{t} has infinity type (2 + t, −t) and therefore lies in the range of classical interpolation for the Katz p-adic L-function precisely when t ≥ 0.

Let L_{p}^{Katz} denote the two-variable Katz p-adic L-function attached to Hecke characters of K, and Ω_{K,p} denote the p-adic counterpart of the complex period Ω_{K} that appears in the interpolation property of L_{p}^{Katz}.

The interpolation formula for the Katz p-adic L-function, as it is described for instance in [DLR, §3, (45)], shows that

Ω_{gh}^{num}(t) ∼_{t} 𝔤(χ_{gh})^{−1} × L_{p}^{Katz}(ψ_{g}χ̄_{h}^{+}ψ_{t}) × L_{p}^{Katz}(ψ_{g}χ̄_{h}^{−}ψ_{t}),

where the symbol ∼_{t} denotes an equality up to an elementary fudge factor which (as a function of the variable t) extends to an analytic function on weight space and takes L-rational, nonzero values at t = −1.

Evaluating this relation at t = −1, one finds

Ω_{gh}^{num}(−1) ∼ 𝔤(χ_{gh})^{−1} × L_{p}^{Katz}(ψ_{g}χ̄_{h}^{+}N)L_{p}^{Katz}(ψ_{g}χ̄_{h}^{−}N),

where N denotes the norm character (of infinity type (1, 1)) and we have (in many of the equations above) identified the Dirichlet characters χ_{h}^{+} and χ_{h}^{−} with their restrictions to G_{K}, viewed as (finite order) Hecke characters of K. Note that the smallest extension H of K cut out by the characters ψ_{g}χ_{h}^{+} and ψ_{g}χ_{h}^{−} is Galois and abelian over ℚ as well.

If ψ is any non-trivial character of Gal (H/K), denote by u_{K}(ψ) any element of L⊗O_{H}^{×} which generates the ψ-eigenspace for the action of Gal (H/K), and when χ is trivial, write u_{K}(1) for the fundamental p-unit of the imaginary quadratic field K. Katz’s p-adic analogue of the Kronecker limit formula (Cf. for instance [Katz1, §10.4, 10.5], [Gro, p. 90], [deS, Ch. II,§5.2]) shows that

(14)  L_{p}^{Katz}(ψ_{g}χ̄_{h}^{+}N) = \{ \begin{matrix}0 & if ψ_{g}χ_{h}^{+} ≠ 1, ψ_{g}χ_{h}^{+}(p) = 1; \\ log_{p}(u_{K}(ψ_{g}χ_{h}^{+})) & otherwise. \end{matrix}

Because the character ψ_{g}χ_{h}^{+} arises from the restriction of a character of ℚ and ψ_{g} is the genus character attached to the pair (χ_{g}^{+}, χ_{g}^{−}) of quadratic Dirichlet characters, it can be seen that:

u_{K}(ψ_{g}χ_{h}^{+}) = \{ \begin{matrix}u(χ++_{gh} ) & if ψ_{g}χ_{h}^{+} ≠ 1; \\ u_{p}(χ-+_{gh}) & if ψ_{g}χ_{h}^{+} = 1. \end{matrix}

Therefore, it follows that

(15)  L_{p}^{Katz}(ψ_{g}χ̄_{h}^{+}N) ∼ \{ \begin{matrix}0 & if χ++_{gh}≠ 1, and χ_{h}^{+}(p) = χ_{g}^{+}(p) = χ_{g}^{−}(p); \\ log_{p}(u(χ++_{gh} )) & if χ++_{gh}≠ 1, and χ_{h}^{+}(p) ≠ χ_{g}^{+}(p); \\ log_{p}(u_{p}(χ-+_{gh})) & if χ++_{gh}= 1. \end{matrix}

Similar considerations applied to the character ψ_{g}χ_{h}^{−} show that

(16)  L_{p}^{Katz}(ψ_{g}χ̄_{h}^{−}N) ∼ \{ \begin{matrix}0 & if χ--_{gh} ≠ 1, and χ_{h}^{−}(p) = χ_{g}^{+}(p) = χ_{g}^{−}(p); \\ log_{p}(u(χ--_{gh})) & if χ--_{gh} ≠ 1, and χ_{h}^{−}(p) ≠ χ_{g}^{−}(p); \\ log_{p}(u_{p}(χ{+ \atop -}_{gh})) & if χ--_{gh} = 1. \end{matrix}

It follows that

(17)  Ω_{gh}^{num}(−1) ∼ \{ \begin{matrix}\frac{log_{p}(χ++_{gh} ) log_{p}(χ--_{gh})}{𝔤(χ_{gh})} & in case 1; \\ \frac{log_{p}(u(χ_{gh}^{+})) log_{p}(u_{p}(χ_{gh}^{−}))}{𝔤(χ_{gh})}, & in case 2; \\ log_{p}(u_{p}(χ_{g}^{±})) log_{p}(u_{p}(χ_{g}^{∓})) & in case 3. \end{matrix}

In addition, it follows from [DLR, Lemma 3.8], where t = k − 2, that Ω_{g}^{den}(t) likewise extends to a p-adic analytic function of t, and

(18)  Ω_{g}^{den}(−1) ∼ log_{p}(u_{p}(χ_{g}))

by [DLR, Theorem 3.9 (54)].

The theorem follows directly from equations (17) and (18) in light of the fact that

Ω_{gh} ∼ \frac{Ω_{gh}^{num}(−1)}{Ω_{g}^{den}(−1)}.

□

Remark 3.2. The analogue of Theorem 3.1 in the setting explored in [DLR, §7], where the weight two Eisenstein series E_{2}(1, χ_{gh}^{−1}) is replaced by a cusp form f attached to an elliptic curve E over ℚ, poses significant difficulties and remains unproved. This analogue would involve points on E defined over the cyclotomic fields cut out by the characters χ++_{gh} , χ{+ \atop -}_{gh}, χ-+_{gh}, and χ--_{gh}, and no appropriate variant of the Heegner construction leading to such points is available, at least when the characters involved are non quadratic. Such a variant would occupy the same position vis-à-vis Heegner points as circular units vis-à-vis elliptic units.

Numerical examples.

Example 3.3. The following experiments illustrate Theorem 3.1 in the special case where g = E_{1}(1, χ_{g}^{−}) and h = E_{1}(1, χ_{h}^{−}), where χ_{g}^{−} and χ_{h}^{−} are odd quadratic Dirichlet characters satisfying (χ_{g}^{−}(p), χ_{h}^{−}(p)) = (1, −1). This corresponds to the special case of Conjecture 1.2 described in equation (10) of Case 2 treated above, involving the p-adic logarithm of the fundamental unit of the real quadratic field cut out by the even character χ--_{gh}.

3.3a. For this experiment, let χ_{g}^{−} and χ_{h}^{−} be the odd quadratic Dirichlet characters of conductors 3 and 11 respectively, and choose p := 7. The 7-stabilised eigenform g_{α} := E_{1}^{(7)}(1, χ_{g}^{−}) (of level 3·7) has two distinct 11-stabilisations, denoted g_{1} and g_{−1}, of level 3·7·11, with U_{11}-eigenvalues 1 and −1, respectively. Denote by γ_{1} and γ_{−1} the linear functionals on S_{1}(3 · 7 · 11)[g_{α}] = ℂ_{p}g_{1} ⊕ ℂ_{p}g_{−1} sending (g_{1}, g_{−1}) to (1, 0) and (0, 1) respectively. A computer calculation using the algorithms based on [La] shows that

\begin{aligned}∫_{γ_{1}} E_{2}(1, χ--_{gh}) · E_{1}(1, χ_{h}^{−}) & = & −9955228630224805563482516 (mod 7^{30}) \\ ∫_{γ_{−1}} E_{2}(1, χ--_{gh}) · E_{1}(1, χ_{h}^{−}) & = & 0 (mod 7^{30}). \end{aligned}

A careful examination of the non-vanishing 7-adic iterated integral above suggests that

∫_{γ_{1}} E_{2}(1, χ--_{gh}) · E_{1}(1, χ_{h}^{−}) = −\frac{2 · \sqrt{33} · log_{7} (u(χ--_{gh}))}{7}

to 30 digits of 7-adic precision, where u(χ--_{gh}) is the fundamental unit in the real quadratic field cut out by the even character χ--_{gh}, satisfying the polynomial x^{2} − 46x + 1.

3.3b. We place ourselves in exactly the same settting as in 3.3a, but taking this time χ_{h}^{−} to be the odd quadratic character of conductor 23 rather than 11. Adopting the same notations as for the previous example, we find:

\begin{aligned}∫_{γ_{1}} E_{2}(1, χ--_{gh}) · E_{1}(1, χ_{h}^{−}) & = & 10649075716435865817575346 (mod 7^{30}) \\ ∫_{γ_{−1}} E_{2}(1, χ--_{gh}) · E_{1}(1, χ_{h}^{−}) & = & 0 (mod 7^{30}) \end{aligned}

and

∫_{γ_{1}} E_{2}(1, χ--_{gh}) · E_{1}(1, χ_{h}^{−}) = −\frac{3 · \sqrt{69} · log_{7}(u(χ--_{gh}))}{7}

to 30 digits of 7-adic precision.

3.3c. In the third experiment of this kind, we take χ_{g}^{−} to have conductor 7 and χ_{h}^{−} to have conductor 3, with p := 11. Then

\begin{aligned}∫_{γ_{1}} E_{2}(1, χ--_{gh}) · E_{1}(1, χ_{h}^{−}) & = & −11038147038719013039751993 (mod 11^{25}) \\ ∫_{γ_{−1}} E_{2}(1, χ--_{gh}) · E_{1}(1, χ_{h}^{−}) & = & 0 (mod 11^{25}) \end{aligned}

and we find again that

∫_{γ_{1}} E_{2}(1, χ--_{gh}) · E_{1}(1, χ_{h}^{−}) = −\frac{\sqrt{21} · log_{11}(u(χ--_{gh}))}{11}

to 25 digits of 11-adic precision.

Example 3.4. Let χ_{3} and χ_{4} be the odd quadratic characters of conductors 3 and 4, respectively, and let χ_{5} be the even quadratic character of conductor 5. Let g := E_{1}(1, χ_{3}) and h := E_{1}(χ_{5}, χ_{4}). We have found experimentally that the 7-adic iterated integrals attached to g and h, for all the stabilisations of g in level 3 · 7 · 4 · 5, are all zero, except for

∫_{γ} E_{2}(1, χ_{3} · χ_{4} · χ_{5}) · E_{1}(χ_{5}, χ_{4}) = −1510481263266793866295817 (mod 7^{30}),

where γ is the unique element of the dual space of S_{1}(7 · 60, χ_{3})[g_{α}] sending the stabilisation of g with U_{2} and U_{5} eigenvalues 1 and −1, respectively to 1, and all other stabilisations to 0. Experimentally, one finds

∫_{γ} E_{2}(1, χ_{3} · χ_{4} · χ_{5}) · E_{1}(χ_{5}, χ_{4}) = \frac{2^{4} · \sqrt{60} · log_{7}(u(χ_{5})) · log_{7}(u(χ_{12}))}{3 · 7 · log_{7}(u_{7}(χ_{3}))}

to 30-digits of 7-adic precision, in perfect agreement with the most “generic” instance of Conjecture 1.2 treated in Case 1 above.

4. Elliptic units and theta series of imaginary quadratic fields

4.1. Theoretical evidence. Let K be an imaginary quadratic field of discriminant −D_{K} and let χ_{K} denote the quadratic Dirichlet character associated to it. The aim of this chapter is to study Conjecture 1.2 in the case where both g and h are theta series associated to K.

Fix two arbitrary finite order characters ψ_{g}, ψ_{h} of K of conductors 𝔠_{g}, 𝔠_{h}. Let χ_{g}, χ_{h} denote the central characters of ψ_{g} and ψ_{h}. Let

g := θ_{ψ_{g}} ∈ M_{1}(N_{g}, χ_{K}χ_{g}), h := θ_{ψ_{h}} ∈ M_{1}(N_{h}, χ_{K}χ_{h})

be the theta series associated to them. Here N_{g} = D_{K} · N_{K/ℚ}(𝔠_{g}) and N_{h} = D_{K} · N_{K/ℚ}(𝔠_{h}).

Define ψ_{1} = ψ_{g}ψ_{h} and ψ_{2} = ψ_{g}ψ′_{h} where, for any character ψ of G_{K}, we denote ψ′ the character given by the rule ψ′(σ) = ψ(σ_{0}σσ_{0}^{−1}) for any choice of σ_{0} ∈ G_{ℚ} − G_{K}. Unlike in [DLR, §3], here we are not imposing the restriction χ_{g} = χ_{h}^{−1}, and thus ψ_{1} and ψ_{2} are not necessarily ring class characters.

Let V_{g} = Ind_{K}^{ℚ}(ψ_{g}) and V_{h} = Ind_{K}^{ℚ}(ψ_{h}) denote the two-dimensional induced representations of ψ_{g} and ψ_{h} from G_{K} to G_{ℚ}, with coefficients in a finite extension L of ℚ which we may take to contain the fourier coefficients of g and h and the Gauss sum 𝔤(χ_{gh}). By Lemma 2.2, there is a decomposition of Artin representations

(19)  V_{gh} = V_{ψ_{g}} ⊗ V_{ψ_{h}} ≃ V_{ψ_{1}} ⊕ V_{ψ_{2}},

where for i = 1, 2, V_{ψ_{i}} is irreducible if and only if ψ_{i} ≠ ψ′_{i}. If ψ_{i} = ψ′_{i}, let χ_{i} denote the central character of ψ_{i}. Then V_{ψ_{i}} ≃ χ_{i} ⊕ χ_{K}χ_{i} decomposes as the sum of an odd and even Dirichlet character. The odd one can never be trivial, while the even one is trivial precisely when ψ_{i} = 1.

Let N = gcd(N_{g}, N_{h}) and let p ∤ N be a prime that splits in K. By Lemma 1.1, it follows that

(20)  d_{gh}^{(p)} = 2 if and only if ψ_{g} ≠ ψ_{h}^{−1}, (ψ′_{h})^{−1},

which we assume for the remainder of this chapter. In that case we in fact have d_{gh} = d_{gh}^{(p)} = 2 and, more precisely,

dim_{L}(V_{ψ_{i}}, U_{gh}) = 1 for i = 1, 2.

Write U_{gh}^{ψ_{i}} for the image of V_{ψ_{i}} in U_{gh}. The theory of elliptic units allows us to pin down distinguished choices of elements in U_{gh}^{ψ_{i}} as follows.

Let H_{g}/K and H_{h}/K denote the abelian extensions cut out by the characters ψ_{g} and ψ_{h}, respectively, and let H_{gh} = H_{g}H_{h} denote their composition. Since p splits in K, the frobenius element Fr_{p} belongs to Gal (H_{gh}/K), and the choice of a prime ℘ of K above p is sufficient to determine Fr_{p} as an element of this group, satisfying ψ(℘) = ψ(Fr_{p}) for all characters ψ of Gal (H_{gh}/K).

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}( ¯℘).

Let K_{𝔣} denote the smallest ray class field containing H_{gh} and let u ∈ K_{𝔣}^{×} be an elliptic unit constructed as in [deS, §2.6]. For any character ψ of G = Gal (K_{𝔣}/K), define

(21)  u_{ψ} = ∑_{σ∈G} ψ^{−1}(σ)u^{σ}.

If ψ ≠ 1, the element ∑ ψ^{−1}(σ) · σ belongs to the augmentation ideal of L[G] and [deS, §2.6] shows that u_{ψ} lies in L ⊗ O_{K_{𝔣}}^{×}. Since, by (20), the characters ψ_{i}, i = 1, 2, are not trivial,

the four units u_{ψ_{1}}, u_{ψ′_{1}}, u_{ψ_{2}}, u_{ψ′_{2}} lie in U_{gh}. In fact, by construction u_{ψ_{1}} belongs to U_{gh}^{ψ_{1}} and Fr_{p} acts on it with eigenvalue ψ_{1}(Fr_{p}) = α_{g}α_{h}. Analogous properties hold for u_{ψ′_{1}} ∈ U_{gh}^{ψ_{1}} and u_{ψ_{2}}, u_{ψ′_{2}} ∈ U_{gh}^{ψ_{2}}.

Both when g is cuspidal or Eisenstein, the matrix in (4) is

R_{g_{α}}(ϱ_{gh}) = ( \begin{matrix}log_{p}(u_{ψ′_{2}}) & 0 \\ 0 & log_{p}(u_{ψ′_{1}}) \end{matrix} ) .

In addition to the above units, let us introduce also a canonical choice for the unit u_{g_{α}} entering in Conjecture 1.2.

If g is cuspidal, let ψ_{ad} := ψ′_{g}/ψ_{g} denote the character associated to the adjoint of g and define u_{g_{α}} = u_{ψ_{ad}}. Since ψ_{g} ≠ ψ′_{g}, it follows as above that u_{g_{α}} belongs to L ⊗ O_{H_{g}}^{×} and that Fr_{p} acts on it with eigenvalue β_{g}/α_{g}.

If g is Eisenstein, i.e. ψ_{g} = ψ′_{g}, define

(22)  u_{g_{α}} := u_{℘} ∈ L ⊗ O_{H_{g}}[1/p]^{×}

to be any element such that ℘^{h_{K}} = (u_{℘}), where h_{K} denotes the class number of K. Conjecture 1.2 then predicts that

(23)  Ξ(g_{α}, h) ∈ Ω_{gh} · M_{1}(N_{gh}, χ_{g}^{−1})[g_{α}^{∗}]_{L} where Ω_{gh} = \frac{log_{p}(u_{ψ′_{1}}) log_{p}(u_{ψ′_{2}})}{𝔤(χ_{gh}) log_{p}(u_{g_{α}})}.

The main result of this section is:

Theorem 4.1. Conjecture 1.2 holds in the setting described above, i.e., (23) holds true.

The remainder of this section is devoted to the proof of Theorem 4.1. Similarly as in the proof of the theorem of the previous section, fix a Hecke character λ of infinity type (0, 1) and conductor ¯℘, with values in a number field E. Write O for the ring of integers of the p-adic closure of E in ℚ̄_{p} and decompose its group of units as O^{×} = μ × W, where μ is finite and W is free over ℤ_{p}. Let ⟨·⟩ : O^{×} → W denote the natural projection.

For every integer k ≥ 1 define ψ_{g,k−1}^{(p)} = ψ_{g}⟨λ⟩^{k−1} and let ψ_{g,k−1} be the Hecke character of infinity type (k − 1, 0) given by

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

Let g_{k} = θ(ψ_{g,k−1}) denote the theta series associated to ψ_{g,k−1}, an eigenform of weight k. These modular forms vary p-adically, in the sense that there is a Hida family g of tame level N_{g} and character χ_{g} over an open subset U_{g} ⊂ 1 + pℤ_{p} of weight space such that its classical specializations are the ordinary p-stabilizations of g_{k}. Note that ψ_{g,1} = ψ_{g} and thus g = g_{1}; the specialization of g in weight one is g_{α}.

For the remainder of this section, we replace the Hida family g with a Λ-adic modular form ğ(q) = ∑_{d| \frac{N}{Ng}} λ_{d}g(q^{d}) of tame level N satisfying that all its classical specializations are not

only eigenvectors for the Hecke operators T_{ℓ} for ℓ ∤ Np but also for the operators U_{q} for all q | Np. Likewise, replace h with an oldform h̆ ∈ M_{1}(N, χ_{h}) that is an eigenvector for all good and bad Hecke operators.

Together with (24), define also the Hecke characters

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

where N is the Hecke character induced by the norm from K to ℚ.

It follows immediately from the definitions that, for all k ≥ 1, the Rankin-Selberg Lfunction L(ğ_{k}, h̆, s) factors as the product of the two L-series associated to the (inverse of the) Hecke characters appearing in (25), that is to say:

(26)  L (ğ_{k}, h̆, \frac{k + 1}{2}) = 𝔣_{0}(k) · L(Ψ_{gh}^{−1}(k), 0) · L(Ψ_{gh′}^{−1}(k), 0).

Here 𝔣_{0}(k) is a fudge factor which is a rational function on q^{k} as q ranges over the divisors of N, with coefficients in L. Moreover, 𝔣_{0}(k) ∈ L^{×} for all k ∈ ℤ_{≥1}. In the terminology coined in [DLR, Definition 4.3], functions satisfying these properties are called L-admissible, and we identify them with their unique analytic extension to U_{g}.

The factorisation formula (26) admits a p-adic analogue, of which Theorem 4.1 is a consequence, as we show later. Since the reader may find it of independent interest, we state it below as a separate result. For every k ≥ 1, define the Hecke character

Ψ_{g}(k) := ψ_{g,k−1}^{−2} χ_{g} N^{k}.

Let ℒ_{p}(ğ, h̆) denote Hida’s p-adic Rankin L-function on U_{g} associated to the pair (ğ, h̆). We refer e.g. to [BDR1] for its precise definition, and we shall use the normalizations adopted therein.

As in the previous chapter, let also L_{p}^{Katz} denote Katz’s p-adic L-function, defined and normalized as in [DLR].

Theorem 4.2. There is an L-admissible function 𝔣 such that the following factorisation of analytic functions on U_{g} is valid:

(27)  ℒ_{p}(ğ, h̆)(k) · L_{p}^{Katz}(Ψ_{g}(k)) = 𝔣(k) · L_{p}^{Katz}(Ψ_{gh}(k)) · L_{p}^{Katz}(Ψ_{gh′}(k)).

Proof. It suffices to show that (27) holds true for all k ∈ U_{g} := U_{g} ∩ ℤ_{≥2}, as this is a dense subset of U_{g} for the p-adic topology. Since points in U_{g} belong to the region of interpolation of the four p-adic L-functions appearing in (27), their values are non-zero multiples of critical classical L-values. Namely, there are explicit functions Ω_{i}(k), i = 1, …, 4, such that

ℒ_{p}(ğ, h̆)(k) = Ω_{1}(k) · L (ğ_{k}, h̆, \frac{k+1}{2}) , L_{p}^{Katz}(Ψ_{g}(k)) = Ω_{2}(k) · L(Ψ_{g}^{−1}(k), 0),
L_{p}^{Katz}(Ψ_{gh}(k)) = Ω_{3}(k) · L(Ψ_{gh}^{−1}(k), 0), L_{p}^{Katz}(Ψ_{gh′}(k)) = Ω_{4}(k) · L(Ψ_{gh′}^{−1}(k), 0).

Functions Ω_{i}(k) involve powers of suitable complex and p-adic periods and products of Euler-like factors at primes dividing pN. The reader may find completely explicit formulas for them in [BDR1, §2] and [DLR, §3.2], respectively. In particular, it is apparent from these expressions that each Ω_{i}(k) individually does not interpolate p-adically to a continuous function on U_{g}, thus a fortiori they are not L-admissible at all. However, [DLR, Lemma 3.8] together with a rather tedious but elementary computation show that the ratio

𝔣(k) := \frac{Ω_{1}(k)Ω_{2}(k)}{Ω_{3}(k)Ω_{4}(k)} · L(Ψ_{g}^{−1}(k), 0)

does vary p-adically continuously, and gives rise to an L-admissible function. We omit the details, since the proof of this fact proceeds along similar lines as in [DLR, Theorem 3.9], although the calculations are much simpler than in loc. cit. because here we avoid the subtleties arising in [DLR] from the choice of test vectors for Garrett’s triple-product L-function.

The theorem now follows from (27). □

In order to prove Theorem 4.1, we proceed by relating the values at k = 1 of the functions appearing in (27) to a p-adic iterated integral and the p-adic logarithm of suitable units. Note that

Ψ_{gh}(1) = ψ_{g}^{−1} · ψ_{h}^{−1} · N, Ψ_{gh′}(1) = ψ_{g}^{−1} · (ψ′_{h})^{−1} · N, Ψ_{g}(1) = ψ_{g}^{−2} χ_{g} N = \frac{ψ′_{g}}{ψ_{g}} N.

Since for any finite order character ψ of G_{K}, Katz’s p-adic L-function satisfies the functional equation

L_{p}^{Katz}(ψ) = L_{p}^{Katz}((ψ′)^{−1}N_{K}),

Theorem 4.2 specializes to the following formula at k = 1:

(28)  ℒ_{p}(ğ, h̆)(k) · L_{p}^{Katz}(\frac{ψ′_{g}}{ψ_{g}}) = 𝔣(1) · L_{p}^{Katz}(ψ′_{2}) · L_{p}^{Katz}(ψ′_{1}).

Lemma 4.3. Let g_{α}^{∗} = g_{α} ⊗ χ_{g}^{−1} denote the twist of g_{α} under the inverse of its nebentypus. There exists a linear functional

γ̆_{α} : M_{1}(N_{gh}p, χ_{g}^{−1})_{L}[g_{α}^{∗}] ⟶ L

such that

(29)  𝔤(χ_{gh}) × ℒ_{p}(ğ, h̆)(1) = ∫γ̆_{α} f · h̆.

Proof. Since the p-adic L-function ℒ_{p}(ğ, h̆) of [BDR1, §2.2.2] is constructed using the method of [DR1], the proof of this statement follows verbatim as in [DLR, Proposition 2.6]. □

In addition to the above lemma, recall Katz’s p-adic analogue of Kronecker’s limit formula, which asserts that for any finite order character ψ of G_{K}, the following equality holds up to scalars in L^{×}:

(30)  L_{p}^{Katz}(ψ) = \begin{cases}log_{p}(u_{ψ}) & if ψ ≠ 1 \\ log_{p}(u_{℘}) & if ψ = 1, \end{cases}

where u_{ψ} and u_{℘} are as in (21) and (22) respectively.

Since 𝔣 is L-admissible, 𝔣(1) is a non-zero scalar in L. Theorem 4.1 follows, as the combination of (28), (4.3) and (30) show the sought-after equality

γ̆_{α}(Ξ(g_{α}, h)) = λ · \frac{log_{p}(u_{ψ′_{1}}) log_{p}(u_{ψ′_{2}})}{𝔤(χ_{gh}) log_{p}(u_{g_{α}})}

up to some λ ∈ L^{×}.

Example 4.4. We revisit the setting treated in [DLR, Example 4.3]. Let χ_{5} and χ_{31} denote the (even and odd, respectively) quadratic Dirichlet characters of conductor 5 and 31, and let χ := χ_{5} · χ_{31} be the quadratic character attached to the quadratic field K := ℚ(\sqrt{−155}). The class group of K is cyclic of order 4, and the two-dimensional representation induced from a quartic character ψ of this class group is attached to the weight one cuspidal eigenform

g = θ_{ψ} = q + q^{4} − q^{5} − q^{9} + q^{16} − 2q^{19} − ⋯ ∈ S_{1}(155, χ).

The Galois representation ϱ_{g} attached to g has image the dihedral group D_{4} of order 8, and projective image isomorphic to the Klein 4-group. It is also induced from quadratic characters of ℚ(\sqrt{5}) and ℚ(\sqrt{−31}) (ramified at a prime above 31 and 5, respectively). In particular, the classicality hypothesis for g is satisfied for the prime p precisely when χ_{5}(p) = −1. Set h := g = g^{∗} and let f := E_{2}^{(p)}(1, 1) be the Eisenstein series of weight 2 and level p attached to the trivial character.

Although g = h = θ_{ψ} is cuspidal, the tensor product ϱ_{gh} decomposes as a direct sum

ϱ_{gh} = 1 ⊕ χ_{5} ⊕ χ_{31} ⊕ χ_{155},

while

Ad(ϱ_{g}) = χ_{5} ⊕ χ_{31} ⊕ χ_{155}.

The period Ω_{gh} of Conjecture 1.2 involves the p-adic logarithms of the fundamental unit u(χ_{5}) of the quadratic field ℚ(\sqrt{5}) in both its numerator and denominator. This quantity cancels out in the conjecture, which then predicts that the iterated integrals attached to (f, g, g) should agree with the logarithm of the fundamental p-unit of the imaginary quadratic field K = ℚ(\sqrt{−155}), if χ_{155}(p) = 1 (resp. of ℚ(\sqrt{−31}), if χ_{31}(p) = 1). The running assumption that χ_{5}(p) = −1 ensures that one is always in precisely one of these two scenarios.

The prime p := 7 is inert in K/ℚ, and the frobenius eigenvalues α_{g} and β_{g} are equal to 1 and −1 respectively. Letting γ_{1} and γ_{−1} denote the associated elements in the dual space of S_{1}(155, χ), we calculate:

∫_{γ_{±1}} E_{2}^{(7)}(1, 1) · θ_{ψ} = 4365724287399866821223816 (mod 7^{30}),

and check to 30 digits of 7-adic precision that

∫_{γ_{±1}} E_{2}^{(7)}(1, 1) · θ_{ψ} = −\frac{log_{7}(u_{7}(χ_{31}))}{7}.

The prime p := 13 splits in K/ℚ, and the associated frobenius eigenvalues are i and −i. A numerical calculation reveals that

∫_{γ_{±i}} E_{2}^{(13)}(1, 1) · θ_{ψ} = 11525893437665284 (mod 13^{15}).

One observes, to 15 digits of 13-adic precision, that

∫_{γ_{±i}} E_{2}^{(13)}(1, 1) · θ_{ψ} = −\frac{3 · log_{13}(u_{13}(χ_{155}))}{13},

as predicted by Conjecture 1.2.

5. Gross-Stark units and theta series of real quadratic fields

The setting in which

(31)  g := θ_{ψ_{g}},  h := θ_{ψ_{h}}

are induced from characters ψ_{g} and ψ_{h} of (the same) mixed signature of a common real quadratic field K is of special interest, since in that case ϱ_{gh} factors through the Galois group of an abelian extension of K. For instance, when (31) holds, the representation V_{gh} decomposes as a direct sum

V_{gh} = V_{ψ^{+}} ⊕ V_{ψ^{−}}, ψ^{+} := ψ_{g}ψ_{h}, ψ^{−} := ψ_{g}ψ′_{h}

of two induced representations by characters ψ^{+} and ψ^{−} of K which are totally even and totally odd respectively.

If V_{ψ^{+}} is irreducible, it occurs with multiplicity two in the unit group, and hence d_{gh} = 2. If p is any prime satisfying the regularity hypothesis for g, then necessarily p is inert in K and therefore α_{g} = −β_{g}, and likewise for α_{h} and β_{h}. The period Ω_{gh} of Conjecture 1.2 is predicted to be non-zero precisely when β_{g}α_{h} ≠ ±1 and β_{g}β_{h} ≠ ±1. This period then involves, in its numerator, a p-adic regulator formed from genuine units in the ψ^{+} and (ψ^{+})′ components of the unit group of an appropriate abelian extension of K. The unit u_{g_{α}} which appears in the denominator in the definition of Ω_{gh} is just the p-adic logarithm of a fundamental unit u_{K} of K, in light of the regularity hypothesis which implies that the prime p is inert in K. (Cf. case (b) in the proof of Lemma 1.5 in Sec. 1.2 of [DLR].) We have not tested any instance of this general conjecture numerically, and it might be interesting to do so.

In the special case where ψ_{h} = ψ_{g}^{−1}, the character ψ^{+} is trivial and ψ^{−} = ψ_{g}/ψ′_{g} is a ring class character of K. The abelian extension H of K cut out by ψ^{−} is Galois over ℚ and Gal (H/ℚ) is a generalised dihedral group. The regularity hypothesis for p, which forces p to be inert in K/ℚ, implies that the frobenius element at p acts on V_{ψ^{−}} with eigenvalues 1 and −1, and hence the Artin representation V_{ψ^{−}} occurs with multiplicity 1 in the space ℂ ⊗ O_{H}[1/p]^{×}. The p-units in this abelian extension are precisely the Gross-Stark units studied in [DD], and the regulator of Conjecture 1.2 involves the p-adic logarithms of such units. There is also the p-adic logarithm of a fundamental unit of K, which is in the minus eigenspace for the action of the frobenius automorphism at p, and which appears in both the numerator and the denominator and thus cancels out. The transcendental part of the period Ω_{gh} therefore involves only the p-adic logarithms of a p-unit u_{p}^{+}(ψ^{−}), which is in the V_{ψ^{−}}-component of O_{H}[1/p]^{×} and is fixed by the frobenius element at p. The next example illustrates this special instance of Conjecture 1.2:

Example 5.1. Let K = ℚ(\sqrt{5}) be the real quadratic field attached to the even Dirichlet character χ_{5} of conductor 5, and let ψ be a quartic character of K whose conductor is one of the primes of K above 29. The central character χ_{29} of ψ is an odd quartic Dirichlet character of conductor 29. Let χ := χ_{5}χ_{29}, and let

g := θ_{ψ} = q − iq^{4} + iq^{5} − iq^{9} + (i − 1)q^{11} − q^{16} + (i − 1)q^{19} + ⋯ ∈ S_{1}(5 · 29, χ)

be the weight one modular form considered in [DLR, Example 4.1], whose Galois representation agrees with the induced representation V_{ψ}. After setting h := g^{∗} = θ_{ψ^{−1}}, we have

V_{gh} = 1 ⊕ χ_{5} ⊕ V_{ψ^{−}},

where ψ^{−} = ψ/ψ′ is a quartic ring class character of ℚ(\sqrt{5}). The induced representation V_{ψ^{−}} has image the dihedral group D_{4} of order 8, and is notable in that it cannot be obtained as the induced representation of the character of any imaginary quadratic field.

Set p := 13 and choose the embedding of ℚ(i) into ℚ_{13} sending i to the unique 4th root of −1 which is congruent to 5 modulo 13. One finds that χ(13) = 1 and (α_{g}, β_{g}) = (i, −i). Letting f := E_{2}^{(13)}(1, 1) , one computes

∫_{γ_{±i}} E_{2}^{(13)}(1, 1) · θ_{ψ^{−1}} = −7724370085026350688358 (mod 13^{20}).

Let

H := ℚ(a) where a^{8} − 2a^{7} + 4a^{6} − 26a^{5} + 94a^{4} − 212a^{3} + 761a^{2} − 700a + 980.

This is the quartic subfield of the ring class field of K of conductor 29, which is cut out by ψ^{−}. Let

b := (−27a^{7} + 116a^{6} − 196a^{5} + 738a^{4} − 3602a^{3} + 11256a^{2} − 22295a + 53310)/5160,

denote the 13-unit of H satisfying the quartic polynomial x^{4}−11x^{3}+20x^{2}+89x+169. Writing Gal (H/K) = ⟨σ⟩, and defining u_{13}^{+}(ψ^{−}) := b/b^{σ^{2}}, we have verified that

∫_{γ_{±i}} E_{2}^{(13)}(1, 1) · θ_{ψ^{−1}} = \frac{3}{2 · 13} · log_{13}(u_{13}^{+}(ψ^{−})),

to 20-digits of 13-adic precision.

The setting in which

(32)  g := θ_{ψ_{g}},  h := E_{1}(1, χ_{g}^{−1})

is also of great relevance for explicit class field theory for K, since in that case

V_{gh} = V_{ψ_{g}} ⊕ V_{ψ_{g}^{−1}} = V_{g} ⊕ V_{g^{∗}}.

Let u(ψ_{g}), u(ψ′_{g}), u(ψ_{g}^{−1}), and u(ψ′_{g}) denote the generators in L ⊗ O_{H_{g}}^{×} for the relevant eigencomponent relative to the action of Gal (H_{g}/K). Since the prime p is inert in K, these four units can be chosen in such a way that

Fr_{p}u(ψ_{g}) = u(ψ′_{g}), Fr_{p}u(ψ_{g}^{−1}) = u(ψ_{g}^{′−1}).

Example 5.2. Let g be the cusp form in level 145 from Example 5.1 but now take

h := E_{1}(1, χ) = (i + 1) + q + ⋯ + 2q^{13} + ⋯

to be the Eisenstein series of weight 1 attached to the pair of characters (1, χ). The Artin representation ϱ_{gh} is then given by

V_{gh} = V_{ψ_{g}} ⊕ V_{ψ_{g}^{−1}} = V_{g} ⊕ V_{g^{∗}}.

The smallest abelian extension H_{g} of K cut out by the representation V_{g} (and/or V_{g^{∗}}) is larger than the ring class field H of degree 8 over ℚ that arose in Example 5.1: it is the ray class field of K of conductor

(29) = (29, 2\sqrt{5} + 17)(29, 2\sqrt{5} + 10) =: Q_{1}Q_{2},

and has degree 32 over ℚ. The field H_{g} = H_{g^{∗}} is the compositum of the ray class fields H_{1} and H_{2} of K of conductors Q_{1} and Q_{2}, these ray class fields being the abelian extensions of K cut out by the characters ψ_{g} and ψ′_{g} respectively.

A calculation with Magma reveals that H_{g} can be described as the splitting field of the polynomial

x^{8} − x^{7} − 3x^{6} + 3x^{5} + 3x^{4} − 6x^{3} − 2x^{2} + 3x + 1 = P_{1}(x)P_{2}(x),

where

P_{1}(x) = x^{4} + \frac{(−\sqrt{5} − 1)}{2}x^{3} − x^{2} + x + \frac{(−\sqrt{5} + 3)}{2}, P_{2}(x) = τP_{1}(x),

and τ denotes the generator of Gal (K/ℚ). The ray class fields H_{1} and H_{2} are just the splitting fields over K of the irreducible quartic factors P_{1}(x) and P_{2}(x) above with cyclic Galois group. Hence, H_{1} = K(u_{1}) and H_{2} = K(u_{2}), where u_{1} and u_{2} are roots of P_{1}(x) and P_{2}(x) respectively. We define

v_{1} := \frac{1}{2} (−\sqrt{5} − 1) u_{1}^{2}, v_{2} := \frac{1}{2} (\sqrt{5} − 1) u_{2}^{2}.

Letting σ_{1} denote a generator of Gal (H_{1}/K), we have

v_{1} = (σ_{1}^{2}(v_{1}))^{−1}, σ_{1}(v_{1}) = (σ_{1}^{3}(v_{1}))^{−1},

and the unit v_{1} and its Gal (H_{1}/K)-translates σ_{1}v_{1}, σ_{1}^{2}v_{1}, σ_{1}^{3}v_{1} generates the compositum of the ψ_{g} and ψ_{g}^{−1}-isotypic parts in the unit group O_{H_{1}}^{×}. More precisely, using additive notation in L ⊗ O_{H_{g}}^{×}, the elements

v(ψ_{g}) := v_{1} + i ⊗ v_{1}^{σ_{1}}, v(ψ′_{g}) := v_{2} + i ⊗ v_{2}^{σ_{2}}

generate the ψ_{g} and ψ′_{g} eigenspaces respectively for the action of Gal (H_{g}/K), while the elements

v(ψ_{g}^{−1}) := v_{1} − i ⊗ v_{1}^{σ_{1}}, v((ψ′_{g})^{−1}) := v_{2} − i ⊗ v_{2}^{σ_{2}}

generate the ψ_{g}^{−1} and (ψ′_{g})^{−1}-eigenspaces for this action.

Assume that p is inert in K and that α_{g} = −β_{g} = i. After fixing v_{1} and v_{2} in such a way that Fr_{p}(v_{1}) = v_{2}, a direct calculation reveals that the regulator matrix R_{g_{i}}(ϱ_{gh}) is a diagonal matrix of the form

R_{g_{i}}(ϱ_{gh}) = ( \begin{matrix}log_{p}(v(ψ_{g})) − i log_{p}(v(ψ′_{g})) & 0 \\ 0 & log_{p}(v(ψ_{g}^{−1})) − i log_{p}(v((ψ′_{g})^{−1})) \end{matrix} ) ,

while

R_{g_{−i}}(ϱ_{gh}) = ( \begin{matrix}log_{p}(v(ψ_{g})) + i log_{p}(v(ψ′_{g})) & 0 \\ 0 & log_{p}(v(ψ_{g}^{−1})) + i log_{p}(v((ψ′_{g})^{−1})) \end{matrix} ) .

Let p := 13 and embed ℚ(i) into ℚ_{p} so that i ≡ 5 mod p. A numerical calculation of the p-adic iterated integrals reveals that

\begin{aligned}∫_{γ_{+i}} E_{2}^{(13)}(1, 1) · E_{1}(1, χ) & = & −8501755713365462060121 (mod 13^{20}) \\ ∫_{γ_{−i}} E_{2}^{(13)}(1, 1) · E_{1}(1, χ) & = & 4274893846350140558653 (mod 13^{20}). \end{aligned}

After further embedding H_{1} and H_{2} into ℚ_{p^{4}} in such a way that σ_{p}(v_{1}) = v_{2}, one finds that

∫_{γ_{i}} E_{2}^{(13)}(1, 1) · h = \frac{(7i + 6)^{2} · det R_{g_{i}}(ϱ_{gh})}{3 · 8 · 7 · 13 · log_{13}(u_{K})}

∫_{γ_{−i}} E_{2}^{(13)}(1, 1) · h = \frac{(−7i + 6)^{2} · det R_{g_{−i}}(ϱ_{gh})}{3 · 8 · 7 · 13 · log_{13}(u_{K})},

to 20-digits of 13-adic precision.

The analogous setting to that in Example 5.2 with Mordell-Weil groups replacing groups of units was discussed in [DLR, Section 6]. However, this case of the “Elliptic Stark Conjecture” was difficult to check experimentally because of the apparent scarcity of elliptic curves E for which r(E, ϱ_{g}) > 0, and no evidence was given in [DLR] (beyond the case when g = g^{∗}). Example 5.2 can be seen as providing some indirect evidence for the main conjecture of [DLR], albeit with unit groups replacing Mordell-Weil groups.

6. Exotic forms

Cases where g and/or h are exotic forms often correspond to settings that have been little studied previously from the point of view of either the classical or the p-adic Stark conjectures.

One of the simplest settings occurs when g is exotic and h = g^{∗} is the dual form. The Artin representation ϱ_{gh} then factors through the extension H cut out by the projective representation attached to g, and it decomposes as

V_{gh} = 1 ⊕ Ad^{0}(V_{g}).

The adjoint representation Ad^{0}(V_{g}) always occurs with multiplicity one in ℂ ⊗ O_{H}^{×}, and with multiplicity at least two in ℂ ⊗ O_{H}[1/p]^{×}, since 1 always arises as an eigenvalue for any linear transformation on the adjoint. More precisely, the frobenius automorphism Fr_{p} acts on Ad^{0}(V_{g}) with eigenvalues 1, α_{g}/β_{g} and β_{g}/α_{g}. When Ad^{0}(V_{g}) is regular at p, there are therefore (unique up to scaling) elements

(33)  u_{1}, u_{g_{α}}, u_{g_{β}} ∈ L ⊗ O_{H}^{×}

generating a subrepresentation of L ⊗ O_{H}^{×} which is isomorphic to Ad^{0}(V_{g}), and such that

Fr_{p}u_{1} = u_{1},  Fr_{p}u_{g_{α}} = (β_{g}/α_{g}) ⊗ u_{g_{α}},  Fr_{p}u_{g_{β}} = (α_{g}/β_{g}) ⊗ u_{g_{β}}.

Likewise, the choice of a Galois-equivariant homomorphism v : V_{gh}⟶L ⊗ (O_{H}[1/p])^{×} whose image is not contained in L ⊗ O_{H}^{×} produces three independent p-units

(34)  v_{1}, v_{g_{α}}, v_{g_{β}} ∈ L ⊗ (O_{H}[1/p])^{×}

satisfying

Fr_{p}v_{1} = v_{1},  Fr_{p}v_{g_{α}} = (β_{g}/α_{g}) ⊗ v_{g_{α}},  Fr_{p}v_{g_{β}} = (α_{g}/β_{g}) ⊗ v_{g_{β}}.

These units are the counterparts of the Stark points considered in [DLR, Example 5.3]. A simple calculation reveals that

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

and therefore that the period Ω_{gh} of Conjecture 1.2 can be described in this case by

(35)  Ω_{gh} = \frac{log_{p} u_{1} log_{p} v_{g_{α}} − log_{p} u_{g_{α}} log_{p} v_{1}}{log_{p} u_{g_{α}}} .

Example 6.1. This example examines this special case of Conjecture 1.2 when g is Tate’s tetrahedral form in level 133. Recall the following from [DLR, Example 5.3]: Let χ be a Dirichlet character of level 133 = 7 · 19 of order 2 at 7 and order 3 at 19. Denote by ζ a primitive 12-th root of unity and define ζ_{3} := ζ^{−4}. 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}. The extension H of ℚ 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} and Galois group A_{4} over ℚ.

As in [DLR, Example 5.3], with p := 13 we find the two 13-stabilisations g_{α} and g_{β} of g have U_{p}-eigenvalues α_{g} = ζ^{−1} and β_{g} = ζ^{3}, and choose the embedding of ℚ(ζ) into ℚ_{13} which sends ζ to the 12th root of unity which is congruent to 6 modulo 13.

Letting f := E_{2}^{(13)}(1, 1) be the Eisenstein series of weight 2 and level 13, and letting γ_{α} be the element of the dual space of S_{2}(133 · p, χ)[g_{α}] sending g_{α} to 1, (and likewise for γ_{β}, with g_{α} replaced by g_{β}), we find

\begin{aligned}∫_{γ_{α}} E_{2}^{(13)}(1, 1) · g & = & 7956424094204877500900 (mod 13^{20}) \\ ∫_{γ_{β}} E_{2}^{(13)}(1, 1) · g & = & −5574982888910767132405 (mod 13^{20}). \end{aligned}

Define the unit and 13-unit

\begin{aligned}u & := & −b + 1 ∈ O_{M}^{×} \\ v & := & 6b^{3} + 3b^{2} + 22b − 21 ∈ O_{M}[1/p]^{×}, \end{aligned}

respectively. One may choose an embedding of H into ℚ_{13^{3}} so that u_{1} := u and v_{1} := v lie in ℚ_{13} and so are fixed by σ_{13}. Let

u_{g_{α}}, u_{g_{β}} ∈ (O_{H}^{×})_{L}^{Ad_{g}} , v_{g_{α}}, v_{g_{β}} ∈ (O_{H}[1/p]^{×})_{L}^{Ad_{g}}

be the units and p-units described in terms of u and v in (33) and (34). The unit u_{α_{g}} (resp. u_{β_{g}}) lies in the β_{g}/α_{g} = ζ_{3}^{−1} eigenspace (resp. the α_{g}/β_{g} = ζ_{3} eigenspace) for Fr_{p}, and likewise for the p-units v_{α_{g}} and v_{β_{g}}.

To 20-digits of 13-adic precision we have found

∫_{γ_{α}} E_{2}^{(13)}(1, χ^{−1}) · g = −\frac{log_{p}(u_{1}) log_{p}(v_{g_{α}}) − log_{p}(v_{1}) log_{p}(u_{g_{α}})}{13 · log_{p}(u_{g_{α}})}

and likewise replacing α by β throughout.

This example is of a similar flavour to [DLR, Example 5.4, curves 26b, 52a]. Indeed, it pre-dates the rank 2 examples in [DLR] and was the experiment which led to the first serious evidence for (and correct formulation of) the “Elliptic Stark conjecture” of [DLR] in the more striking, but harder to test scenarios of “genuine rank two”.

Our final example involves a scenario where V_{gh} is an irreducible 4-dimensional representation, occuring with multiplicity two in the group of p-units. It is worth noting that no instances of this sort were ever successfuly tested numerically in the elliptic curve setting considered in [DLR], owing to the apparent scarcity of irreducible V_{gh} of manageable size occuring with multiplicity two in the Mordell-Weil groups of elliptic curves of reasonable conductor. In the setting of units, such examples abound, since any irreducible V_{gh} has multiplicity two in the unit group. The experiment below can thus also be viewed as providing some indirect support for the elliptic Stark conjecture of [DLR] in the numerically less accessible setting where ϱ_{gh} is irreducible.

Example 6.2. Let χ be the quadratic Dirichlet character of level 283. The space S_{1}(283, χ) is 3-dimensional and spanned by the eigenforms

\begin{aligned}g & = & q + q^{4} − q^{7} + q^{9} − q^{11} − q^{13} + q^{16} − q^{23} + ⋯ \\ h_{±} & = & q ± \sqrt{−2}q^{2} ∓ \sqrt{−2}q^{3} − q^{4} ∓ \sqrt{−2}q^{5} + 2q^{6} − q^{7} − q^{9} + 2q^{10} + q^{11} ± \sqrt{−2}q^{12} + · · · . \end{aligned}

The pair h_{±} are exotic forms whose associated projective representations have image S_{4}, and g is the dihedral form who representation ϱ_{g} is induced from an unramified character of the imaginary quadratic field ℚ(\sqrt{−283}) (which has class number 3). The representation ϱ_{g} is obtained by composing the projective representation attached to ϱ_{h} with the exceptional homomorphism S_{4}⟶S_{3}. This is precisely the set-up of Lemma 2.3, which asserts that ϱ_{gh} is irreducible and induced from a suitable quadratic character η of a quartic extension K of ℚ. The quartic field K occuring in this lemma, whose Galois closure H̄_{h} is the S_{4}-extension cut out by the projective representation attached to the exotic forms h_{±}, is equal in this case to

K = ℚ(a), a^{4} − a − 1 = 0.

The quadratic extension M of K cut out by η is given by

M := K(b) where b^{2} = −6a^{3} + a^{2} − 3a + 5.

The elements

\begin{matrix}u := \frac{1}{2}(−a^{3} + a^{2} − a + 1)b + \frac{1}{2}(−4a^{3} + 3a^{2} − 2a + 4) \\ v := \frac{1}{2}(a^{3} − 2a)b + \frac{1}{2}(a^{2} + 2a − 3) \end{matrix}

are relative units in M/K, and hence their Galois translates generate the ϱ_{gh}-isotypic components of the unit group of H_{h}. In additive notation let

u^{ζ_{8}^{±1}} := ∑_{j=0}^{7} ζ_{8}^{±}ju^{Fr^{j}^{p}}

and likewise for v^{±ζ_{8}}. Here ζ_{8} is a primitive 8th root of unity such that \sqrt{−2} = ζ_{8}^{3} + ζ_{8}.

Let p := 5, so that χ(5) = −1 and α_{g} = −β_{g} = 1. Taking f = E_{2}^{(5)}(1, 1), one computes

∫_{γ_{+1}} E_{2}^{(5)}(1, 1) · h_{±} = (1365364418900524107116682 ± 982832041040545110371696\sqrt{−2}) /5

to 35-digits of 5-adic precision (and likewise for the −1-stabilisation, replacing ± on the righthand side by ∓), after choosing an embedding of ℚ(ζ_{8}) and H into \overline{ℚ_{p}}. A direct calculation shows that in this example,

det R_{g_{α}}(ϱ_{gh}) = log_{p}(u^{ζ_{8}}) log_{p}(v^{ζ_{8}^{−1}}) − log_{p}(u^{ζ_{8}^{−1}}) log_{p}(v^{ζ_{8}}).

As for the Stark unit u_{g_{α}} occuring in the denominator of the period Ω_{gh} of Conjecture 1.2 when β_{g}/α_{g} = −1, the Hilbert class field of ℚ(\sqrt{−283}) is generated by an algebraic number w satisfying the relation w^{3} − 16w^{2} − 8w − 1 = 0. The unit u_{g_{α}} is then described, using additive notation, by the formula

u_{g_{α}} := w^{Fr^{p}} − w^{Fr^{2}^{p}}

where an embedding of the Hilbert class field into ℚ_{p^{2}} has been fixed so that w is defined over ℚ_{p}. It has been verified experimentally that

∫_{γ_{+1}} E_{2}^{(5)}(1, 1) · h_{−} = \frac{19ζ_{8}^{3} + 34ζ_{8}^{2} − 19ζ_{8}}{2^{5} · 3^{2} · 5} · \frac{log_{5}(u^{ζ_{8}}) log_{5}(v^{ζ_{8}^{−1}}) − log_{5}(u^{ζ_{8}^{−1}}) log_{5}(v^{ζ_{8}})}{log_{5}(u_{g_{α}})}

to 35 digits of 5-adic precision, in excellent agreement with Conjecture 1.2.

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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