ELLIPTIC CURVES OF RANK TWO AND GENERALIZED KATO CLASSES

HENRI DARMON AND VICTOR ROTGER

In memory of Robert Coleman

Abstract. Heegner points play an outstanding role in the study of the Birch and Swinnerton-Dyer conjecture, providing canonical Mordell-Weil generators whose heights encode first derivatives of the associated Hasse-Weil L-series. Yet the fruitful connection between Heegner points and L-series also accounts for their main limitation, namely, that they are torsion in (analytic) rank > 1. This partly expository article discusses the generalised Kato classes introduced in [BDR2] and [DR2], stressing their analogy with Heegner points but explaining why they are expected to give non-trivial, canonical elements of the idoneous Selmer group in settings where the classical L-function (of Hasse-Weil-Artin type) that governs their behaviour has a double zero at the center.

The generalized Kato class denoted κ(f, g, h) is associated to a triple (f, g, h) consisting of an eigenform f of weight two and classical p-stabilised eigenforms g and h of weight one, corresponding to odd two-dimensional Artin representations V_{g} and V_{h} of Gal (H/ℚ) with p-adic coefficients for a suitable number field H. This class is germane to the Birch and Swinnerton-Dyer conjecture over H for the modular abelian variety E over ℚ attached to f. One of the main results of [BDR2] and [DR2] is that κ(f, g, h) lies in the pro-p Selmer group of E over H precisely when L(E, V_{gh}, 1) = 0, where L(E, V_{gh}, s) is the L-function of E twisted by V_{gh} := V_{g} ⊗ V_{h}. In the setting of interest, parity considerations imply that L(E, V_{gh}, s) vanishes to even order at s = 1, and the Selmer class κ(f, g, h) is expected to be trivial when ord_{s=1}L(E, V_{gh}, s) > 2. The main new contribution of this article is a conjecture expressing κ(f, g, h) as a canonical point in (E(H)⊗V_{gh})^{G_{ℚ}} when ord_{s=1}L(E, V_{gh}, s) = 2. This conjecture strengthens and refines the main conjecture of [DLR1], and supplies a framework for understanding the results of [DLR1], [BDR2] and [DR2].

Contents

1. Background and motivation 1 2. Hida families and periods for weight one forms 6 3. Generalized Kato classes 9 3.1. Definition 9 3.2. Basic properties 11 3.3. Enhanced regulators 14 3.4. The conjecture 15 4. Special cases 16 4.1. Beilinson-Kato classes 16 4.2. Belinson-Flach classes 17 4.3. Complex multiplication classes and Heegner points 18 4.4. Real multiplication classes and Stark-Heegner points 21 4.5. Adjoint classes 22 References 24

1. Background and motivation

The theme of modularity of p-adic Galois representations has occupied center stage in number theory for the last several decades, and Robert Coleman has been a major figure in many of its key developments, notably through the theory of Coleman families of p-adic modular forms and of the

1991 Mathematics Subject Classification. 11G18, 14G35.

Coleman-Mazur eigencurve parametrising these families and their associated Galois representations. By way of background and motivation, this section explains how much of the progress achieved on the Birch and Swinnerton-Dyer conjecture, including the results of [DLR1], [BDR2] and [DR2], can be viewed as part of the larger program of understanding the modularity of (non-semisimple) p-adic Galois representations.

One of the most celebrated modularity results is the statement that all elliptic curves over ℚ arise as quotients of suitable modular curves: more precisely, that an elliptic curve E over ℚ of conductor N is equipped with a surjective parameterization

(1)  π_{E} : X_{0}(N) ⟶ E,

where X_{0}(N) is the modular curve attached to Hecke’s congruence subgroup Γ_{0}(N). This was proved in [Wi95], [TW], and [BCDT] by showing that the p-adic representation

H^{1}(E) := H_{et}^{1}(Eℚ̄, ℚ_{p})(1) = (lim_{←,n} E[p^{n}]) ⊗_{ℤ_{p}} ℚ_{p}

of G_{ℚ} := Gal (ℚ̄/ℚ) arises as a quotient of the étale cohomology group^{1}

H^{1}(X_{0}(N)) := H_{et}^{1}(X_{0}(N)ℚ̄, ℚ_{p}(1)).

The existence of a Galois-equivariant projection

(2)  π_{E} : H^{1}(X_{0}(N)) ⟶ H^{1}(E)

is the real content of the breakthrough in [Wi95] and [TW], the ostensibly stronger geometric version (1) being deduced from it by invoking the Tate conjecture for curves^{2}.

Let E′ be an open subvariety of E, i.e., the complement of a zero-dimensional subvariety Σ of E over ℚ. The p-adic Galois representation H^{1}(E′) sits in the middle of the short exact excision sequence

0 ⟶ H^{1}(E) ⟶ H^{1}(E′) ⟶ H^{0}(Σ)_{0} ⟶ 0

of étale cohomology groups, where the subscript of 0 denotes the degree 0 elements of H^{0}(Σ). By analogy with (2), the curve E′ is (provisionally) said to be modular if H^{1}(E′) arises as a subquotient of H^{1}(Y), where Y is an open sub-Shimura variety of X_{0}(N)—the latter being defined, in the style of La Palice, as the complement of a closed sub-Shimura variety.

To completely describe the open sub-Shimura varieties of the modular curve X_{0}(N) over ℚ, note that the latter is the coarse moduli space of elliptic curves A with a marked subgroup scheme of order N, and that its closed sub-Shimura varieties are obtained by imposing additional endomorphism rings, which can only be equal to orders in quadratic imaginary fields. Given such an order O ⊂ K, the associated closed sub-Shimura variety Σ_{O} ⊂ X_{0}(N) consists of CM points for O, and is the coarse moduli space of elliptic curves A with level N structure equipped with an optimal embedding ι : O ⟶ End(A) (respecting the level structure) and acting in a prescribed way on the cotangent space of A. By the theory of complex multiplication, the 0-dimensional variety Σ_{O} is isomorphic over K (at least, when the discriminant of O is prime to N) to ϕ_{K}(N) copies of spec(H_{O}), where ϕ_{K}(N) is the number of primitive ideals of K of norm N and H_{O} is the ring class field of K attached to O, whose Galois group over K is canonically identified with the Picard group of O via global class field theory.

The complements

Y_{O}(N) := X_{0}(N) − Σ_{O}

thus provide an exhaustive list of the open sub-Shimura varieties of X_{0}(N). Given the modularity of E, the modularity of E′ amounts to the existence of a Galois-equivariant inclusion

i : H^{0}(Σ)_{0} ⟶ H^{0}(Σ_{O})_{0}

1The systematic shorthand H^{i}(X) := H_{et}^{i}(Xℚ̄, ℚ_{p}(i)) for any variety X over ℚ is adopted henceforth to lighten the notations.

2Subsequently, (2) has been generalised to a host of other p-adic Galois representations, while analogues of (1) remain unavailable in all but the simplest geometric settings.

for suitable O, realising H^{1}(E′) as a subquotient of H^{1}(Y_{O}(N)) via the pushforward under π_{E} and the pullback under ι of the first row in the following diagram with exact rows:

(3)  [figure: 0; H^{1}(X_{0}(N)); π_{E}; H^{1}(Y_{O}(N)); ?; H^{0}(Σ_{O})_{0}; 0; 0; H^{1}(E); H^{1}(E′); H^{0}(Σ)_{0}; i; 0.]

Consider the simplest non-trivial setting where Σ = {P_{1}, P_{2}} ⊂ E(ℚ) consists of two points defined over ℚ, so that H^{0}(Σ)_{0} = ℚ_{p} with trivial Galois action. The resulting extension

(4)  \xymatrix{0 \ar[r] & H^{1}(E) \ar[r] & H^{1}(E′) \ar[r] & ℚ_{p} \ar[r] & 0}

encodes the image of the point P_{2} − P_{1} ∈ E(ℚ) under the connecting homomorphism

δ : E(ℚ) ⟶ H^{1}(ℚ, H^{1}(E)) := Ext_{G_{ℚ}}^{1}(ℚ_{p}, H^{1}(E))

of Kummer theory, where the Ext group is taken in the category of continuous p-adic representations of G_{ℚ}. The following statement, which gives a “modularity criterion” for E′ and encapsulates many of the deepest theorems on the Birch and Swinnerton-Dyer conjecture obtained in the last decades, is of course expected to hold for all elliptic curves E, but the reader is cautioned that the proof of the implication (d) ⇒ (a) currently requires that E be a semistable elliptic curve having at least one odd prime of nonsplit multiplicative reduction or at least two odd primes of split multiplicative reduction.

Theorem 1.1. Assume that the point P_{2} − P_{1} is of infinite order in E(ℚ). Then the following are equivalent:

(a) The curve E′ = E \ {P_{1}, P_{2}} is modular;

(b) the Hasse-Weil L-series L(E, s) has a simple zero at s = 1;

(c) the point P_{2} − P_{1} generates E(ℚ) ⊗ ℚ and LLI(E/ℚ) is finite;

(d) for all primes p, the group Ext_{fin}^{1}(ℚ_{p}, H^{1}(E)) of extensions of p-adic representations of the Galois group of ℚ that are cristalline at p is one-dimensional over ℚ_{p}.

Sketch of proof. The modularity of E′ amounts to the statement that there exists an order O in an imaginary quadratic field K such that the extension (4) can be obtained as the pullback of (3) via an inclusion i : ℚ_{p} ⟶ H^{0}(Σ_{O})^{G_{ℚ}}, whose image contains a degree 0 divisor

D_{K} ∈ Div^{0}(Σ_{O})^{G_{ℚ}} ⊂ Div^{0}(X_{0}(N))(ℚ).

This means that the point P_{1}−P_{2} ∈ E(ℚ) is a non-zero multiple of the Heegner point P_{E,K} := π_{E}(D_{K}).

The implication (a)⇒(b) therefore follows from the Gross-Zagier formula [GZ86] expressing the height of P_{E,K} as a non-zero multiple of

L′(E/K, 1) = L′(E, 1) · L(E^{K}, 1),

where E^{K} is the quadratic twist of E by K. The existence of a suitable K for which L(E^{K}, 1) ≠ 0 follows from a non-vanishing result of Waldspurger or can be deduced from analytic number theory techniques (cf. [MM]).

The implication (b) ⇒ (c) was subsequently proved by Kolyvagin [Ko89], who parlayed the nontriviality of P_{E,K} into a bound on the Mordell-Weil rank and the Selmer group of E over K.

The implication (c) ⇒ (d) is a direct consequence of the definitions: in fact (d) is ostensibly weaker than (c), Selmer groups being less subtle to control than Mordell-Weil and Shafarevich-Tate groups.

The striking implication (d) ⇒ (a) follows from Skinner’s “converse of the Gross-Zagier-Kolyvagin Theorem” [Sk15]. This last step is the most recent, and combines several new ingredients: the powerful techniques developed by Skinner and Urban to prove the Iwasawa-Greenberg Main Conjecture for elliptic curves over ℚ [SU14], an important variant explored by Xin Wan in his PhD thesis [Wa], and the p-adic analogue of [GZ86] formulated and proved in [BDP13].

More precisely, choose a prime p ≥ 5 of good ordinary reduction for E such that E[p] is an irreducible G_{ℚ}-representation and the image of the restriction map Sel_{p}(E) ⟶ E(ℚ_{p})/pE(ℚ_{p}) does not lie in the image of E(ℚ_{p})[p]. A result of Waldspurger ensures the existence of an odd quadratic character χ such that L(E, χ, 1) ≠ 0, which can be chosen so that χ(2) = χ(p) = 1. Let K denote the imaginary quadratic field associated to χ. The p-adic Selmer group Ext_{K,fin}^{1}(ℚ_{p}, H^{1}(E)) of E over K (defined as an Ext group in the category of cristalline representations of G_{K}) decomposes as a direct sum of eigenspaces

Ext_{K,fin}^{1}(ℚ_{p}, H^{1}(E)) ≃ Ext_{fin}^{1}(ℚ_{p}, H^{1}(E)) ⊕ Ext_{K,fin}^{1}(ℚ_{p}, H^{1}(E))^{−}

with respect to the action of complex conjugation. Because L(E, χ, 1) ≠ 0, the results of Kolyvagin (or of Kato) imply the triviality of Ext_{K,fin}^{1}(ℚ_{p}, H^{1}(E))^{−}. Assumption (d) therefore implies that Ext_{K,fin}^{1}(ℚ_{p}, H^{1}(E)) is one-dimensional over ℚ_{p}. One can then argue as in [Sk15]. Namely, the running hypotheses ensure that both Lemma 2.3.2 and Proposition 2.7.3 of loc.cit. apply, and hence, that a p-adic L-function of the type that occurs in [Wa] and [BDP13] (which interpolates critical values of the L-series of the Rankin convolution of the modular form f associated to E with suitable Hecke characters of K of higher infinity-type) does not vanish at the trivial point, which lies outside its region of classical interpolation. This in turn implies, in light of [Sk15, Corollary 2.6.2] resting on the variant of the Gross-Zagier formula of [BDP13], that the Heegner point P_{E,K} has non-trivial p-adic formal group logarithm, and is therefore non-torsion. As already explained, the non-triviality of P_{E,K} is equivalent to (a), and the implication (d) ⇒ (a) follows. □

The Birch and Swinnerton-Dyer conjecture admits an extension to elliptic curves twisted by Artin representations which arises very naturally in the context of the modularity questions framed above. Let

ϱ : Gal (H/ℚ) ↪ Aut(V_{ϱ}) ≃ GL_{n}(ℚ̄_{p})

be an n-dimensional representation of the Galois group of a finite extension H/ℚ, a so-called Artin representation, viewed as having coefficients in ℚ̄_{p}. The pair (E, ϱ) gives rise to the Hasse-Weil-Artin L-series

L(E, ϱ, s) := ∏_{ℓ} det(1 − ℓ^{−s}(σ_{ℓ}^{−1})_{(H^{1}(E)⊗V_{ϱ})^{I_{ℓ}}})^{−1},

where the product is taken over the rational primes ℓ, the arithmetic frobenius element at ℓ is denoted by σ_{ℓ}, and I_{ℓ} denotes the inertia group at ℓ. The equivariant Birch and Swinnerton-Dyer conjecture for E and ϱ, denoted BSD(E, ϱ), asserts that

(5)  ord_{s=1}L(E, ϱ, s) = dimℚ̄_{p}(E(H) ⊗ V_{ϱ})^{G_{ℚ}}.

As a first step to understanding BSD(E, ϱ), it is natural to ask which κ ∈ Ext_{fin}^{1}(V_{ϱ}, H^{1}(E)) can be realised as a subquotient of a suitable H^{1}(Y_{O}(N)). The Artin representation H^{0}(Σ_{O})_{0} which appears in the upper rightmost term of the diagram (3) is readily analyzed using the theory of complex multiplication. Namely, the slightly larger Artin representation H^{0}(Σ_{O}) decomposes as a direct sum

H^{0}(Σ_{O}) ⊗ ℚ̄_{p} = ⊕_{j=1}^{ϕ_{K}(N)}W_{j}, where W_{j} = ⊕_{ψ}V_{j}(ψ), with V_{j}(ψ) ⊂ V_{ψ} := Ind_{K}^{ℚ}ψ.

In this equation, the second direct sum is taken over the non-trivial, ℚ̄_{p}-valued, finite order characters ψ of Gal (H_{O}/K) modulo the involution ψ ↦ ψ^{−1}, and V_{j}(ψ) is a non-trivial irreducible constituent of the two-dimensional representation V_{ψ} obtained by inducing the Galois character ψ from G_{K} to G_{ℚ}. The representation V_{ψ} is irreducible precisely when ψ ≠ ψ^{−1}, and in this case a non-trivial class κ ∈ Ext_{fin}^{1}(V_{ψ}, H^{1}(E)⊗ℚ̄_{p}) is expected to be modular if and only if (any of) the analogues of conditions (b)-(d) of Theorem 1.1 are satisfied, namely:

(b’) The Hasse-Weil-Artin L-series L(E, V_{ψ}, s) has a simple zero at s = 1;

(c’) the representation V_{ψ} occurs with multiplicity one in E(H) ⊗ ℚ̄_{p}, and the V_{ψ}-isotypic compo-

nent of the LLI(E/H) is finite; (d’) the group Ext_{fin}^{1}(V_{ψ}, H^{1}(E) ⊗ ℚ̄_{p}) is one-dimensional over ℚ̄_{p}, and generated by κ.

Although such a precise result does not seem to appear in the literature, all the ingredients needed to prove it seem to be available in principle.

The rather narrow notion of modularity described above has a few visible drawbacks:

(1) Very few Artin representations arise in the cohomology of the 0-dimensional Shimura varieties Σ_{O}, which are not even rich enough to capture all of the irreducible two-dimensional Artin representations of ℚ. The open Shimura varieties Y_{O}(N) thus appear to give no purchase on BSD(E, ϱ) when ϱ is not induced from a ring class character of an imaginary quadratic field.

(2) Theorem 1.1 suggests that the modularity of elements of Ext_{fin}^{1}(V_{ψ}, H^{1}(E)) is purely a “rank one phenomenon”: if this Ext group has dimension > 1, none of its elements are expected to be realised in subquotients of any H^{1}(Y_{O}(N)).

In order to relate a larger class of non-semisimple Galois representations to modular forms, it becomes desirable to relax the notion of modularity. One way in which one might try to do this is by replacing the curves Y_{O}(N) with more general “open Shimura varieties”. These should include all the varieties whose cohomology (at least, after semisimplification) is directly related to automorphic forms via a suitable generalisation of the Eichler-Shimura congruence, and would eventually encompass the complements of sub-Shimura varieties in larger Shimura varieties, as well as Kuga-Sato varieties and other natural varieties fibered over Shimura varieties, the complements of Heegner cycles in such varieties, and so on. With this expanded notion of modularity, the program of characterising the non-semisimple Galois representations that are modular becomes richer and more subtle. See [BDP14] for a fragment of experimental mathematics that might be viewed as fitting into this program. The following question seems like it might repay further investigation, given the paucity of evidence, both theoretical and experimental, that has been gathered around it so far:

Question 1.2. Let V_{1} and V_{2} be Galois representations for which hom(V_{1}, V_{2}) is irreducible. Suppose that there is a non-trivial κ ∈ Ext_{fin}^{1}(V_{1}, V_{2}) arising as a subquotient of the cohomology of an open Shimura variety. Is Ext_{fin}^{1}(V_{1}, V_{2}) necessarily one-dimensional?

If the answer to this question were “yes”, it would imply that the open curve E −{P_{1}, P_{2}} discussed in Theorem 1.1 is never modular when rank(E(ℚ)) > 1. (But see the inspiring article [NS], as well as the striking ongoing work of Zhiwei Yun and Wei Zhang in the function field case, for some tantalizing ideas in the opposite, more optimistic direction.)

A second idea for enlarging the class of p-adic Galois representations deemed to be modular is to allow p-adic limits of Galois representations arising in the cohomology of (open) Shimura varieties. This idea is very natural in light of the classical work of Deligne-Serre on Artin representations attached to weight one forms, whereby such Artin representations are obtained by piecing together the Galois representations attached to modular forms of higher weights which are realised in the cohomology of Kuga-Sato varieties. It is via this broader notion of modularity that all odd, irreducible two-dimensional Artin representations of ℚ can be related to modular forms. The idea of realising automorphic Galois representations as p-adic limits has become pervasive in the subject, and led to important advances: for example, it plays a key role in the recent construction [HLTT] by Harris, Lan, Taylor, and Thorne of Galois representations attached to non-self-dual automorphic forms on GL_{n}. Even more germane to this article, p-adic limits of automorphic Galois representations appear to capture non-trivial extension classes going beyond settings of “multiplicity one”, as is illustrated by the following theorem of Skinner and Urban [SU06, Thm. B]:

Theorem 1.3. Let E be an elliptic curve over ℚ. If L(E, s) vanishes to even order ≥ 2 at s = 1, then the Selmer group Ext_{fin}^{1}(ℚ_{p}, H^{1}(E)) of E contains at least two linearly independent modular classes.

The modular classes in this theorem are constructed as p-adic limits of geometric Galois representations in the cohomology of Shimura varieties associated to the unitary group U(2, 2). Although these geometric Galois representations are believed to be semisimple, Theorem 1.3 rests on the fact that this feature need not persist in the limit.

The primary goal of this article is to discuss a different approach for constructing canonical extension classes of ϱ by H^{1}(E) for a large class of self-dual Artin representations ϱ of dimension 4 (and their lower-dimensional subrepresentations, in case ϱ is reducible) arising as the tensor product ϱ = ϱ_{1} ⊗ ϱ_{2} of a pair of odd, two-dimensional Artin representations. The construction of these classes is one of the main results of [DR2] (resp. [BDR2]) when both ϱ_{1} and ϱ_{2} are irreducible (resp. when exactly one of ϱ_{1} and ϱ_{2} is irreducible), and is based on p-adic limits of non-semisimple, but “geometrically modular” Galois representations. These limit classes are referred to as generalised Kato classes because their construction is inspired by the seminal work [Ka98] of Kato (cf. also [Sc], [BD14]) on BSD(E, χ) for χ a Dirichlet character. Like Heegner points in the setting of BSD(E, V_{ψ}), generalised Kato classes enjoy close relations to (p-adic) Hasse-Weil-Artin L-functions attached to E and ϱ, but unlike Heegner points, they are expected to generate a non-trivial subgroup of the Selmer group attached to E and ϱ precisely when ord_{s=1}L(E, ϱ, s) = 2. The formulae of [DR2] (cf. Corollary 3.6 below) relating the linear independence of two generalised Kato classes to the non-vanishing of certain p-adic L-series can thus be regarded as a p-adic Gross-Zagier formula “in analytic rank two”.

The main new contribution of this article is a conjecture expressing the same generalised Kato classes as canonical elements in (E(H) ⊗ V_{ϱ})^{G_{ℚ}} when this latter space is two-dimensional. This conjecture strengthens and refines the “elliptic Stark conjecture” of [DLR1], and provides a framework for understanding the results of [DLR1], [BDR2] and [DR2]. The settings in which ϱ is reducible often take on special arithmetic interest and are described in detail in the last chapter.

2. Hida families and periods for weight one forms

This section provides background on certain canonical structures associated to a weight one form g, arising from the Hida families specialising in weight one to (a p-stabilisation of) g. These are important for the conjectures of Section 3.4, but Section 2 can be skipped on a first reading by the reader wishing to get a quick feeling for the generalised Kato classes described in Sections 3.1 and 3.2. On the other hand, it is also worth noting that Section 2 is entirely self-contained. Conjecture 2.1, which can be viewed as a p-adic analogue of the Stark conjecture for the adjoint of the Galois representation attached to a weight one form, appears to be new and may be of independent interest.

Let g ∈ S_{1}(N, χ) be a newform of weight one and level N with Fourier coefficients in a field L, and let

ϱ : G_{ℚ} ⟶ Aut(V) ≃ GL_{2}(L)

be the Artin representation associated to it by the construction of Deligne and Serre. We view ϱ as acting on a two-dimensional L-vector space V, where L ⊂ ℂ can be chosen to be contained in a cyclotomic field.

Let H be the number field cut out by ϱ, so that ϱ factors through Gal (H/ℚ). Fix a rational prime p and choose a prime 𝔭 of H above p. The latter determines a canonical inclusion

H ⊂ H_{p} ⊂ ℚ̄_{p}

of H in its completion H_{p} at 𝔭. Assume that the pair (ϱ, p) satisfies the following conditions:

(I) The prime p splits completely in L/ℚ, so that L is equipped with an embedding into ℚ_{p} which will be fixed from now on. This assumption, which is made solely to lighten the notations and could easily be dispensed with, allows ϱ to be viewed as a ℚ_{p}-linear representation via the natural action of G_{ℚ} on the ℚ_{p}-vector space V ⊗_{L} ℚ_{p}.

(II) The representation V is unramified at p. There is then a well defined arithmetic frobenius

element

σ_{p} ∈ Gal (H/ℚ)

acting canonically on V, and the characteristic polynomial of ϱ(σ_{p}) is equal to the Hecke polynomial

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

attached to g. (III) The modular form g is regular at p, i.e., α_{g} ≠ β_{g}. After possibly enlarging L, it may also be

assumed that this coefficient field contains the roots of unity α_{g} and β_{g}. (IV) The representation ϱ_{g} is not induced from a character of a real quadratic field K in which the

prime p splits. The rationale for this condition, which seems to be essential for a number of the constructions and conjectures proposed in this paper, is explained in [DLR1, §1.1].

The p-stabilisations of g at p are the normalised eigenforms of weight one with Fourier coefficients in L defined by

g_{α} := g(z) − β_{g}g(pz), g_{β} := g(z) − α_{g}g(pz).

They are eigenvectors for the U_{p}-operator satisfying

U_{p}g_{α} = α_{g}g_{α}, U_{p}g_{β} = β_{g}g_{β}.

The Artin representation V decomposes naturally as a direct sum

V = V^{α} ⊕ V^{β}

into one-dimensional eigenspaces for σ_{p}, with eigenvalues α_{g} and β_{g} respectively.

By a theorem of Hida, there exists a finite flat extension Λ_{g} of the Iwasawa algebra Λ and a Hida family \underline{g} ∈ Λ\underline{g}[[q]] of tame level N and tame character χ passing through the p-stabilised weight one eigenform g_{α}. When g is cuspidal, the regularity hypothesis imposed on g implies that such a Hida family is unique, thanks to a recent result of Bellaïche and Dimitrov [BeDi].

The Hida family \underline{g} comes equipped with the following canonical structures:

(a) There is a locally free Λ_{g}-module 𝕍_{g} of rank two, affording Hida’s ordinary Λ-adic Galois

representation

ϱ\underline{g} : G_{ℚ} ⟶ Aut_{Λ_{g}}(𝕍_{g})

which is realised in the inverse limit of ordinary étale cohomology groups associated to the tower X_{1}(Np^{r}) of modular curves. This representation interpolates the Galois representations associated by Deligne to the classical specializations of \underline{g}.

(b) The restriction of 𝕍_{g} to G_{ℚ_{p}} admits a stable filtration

0 ⟶ 𝕌_{g} ⟶ 𝕍_{g} ⟶ 𝕎_{g} ⟶ 0,

where both 𝕌_{g} and 𝕎_{g} are flat Λ\underline{g}[G_{ℚ_{p}}]-modules that are locally free of rank one over Λ\underline{g}, and the quotient 𝕎_{g} is unramified, with σ_{p} acting on 𝕎_{g} as multiplication by the p-th Fourier coefficient a_{p}(\underline{g}).

(c) Let ℚ_{p}^{nr} denote the maximal unramified extension of ℚ_{p} and let ℚ̂_{p}^{nr} denote its p-adic completion. In [Oh95], Ohta constructs a canonical Λ_{g}-adic period

ω\underline{g} ∈ D(𝕎_{g}) := (ℚ̂_{p}^{nr} ˆ⊗𝕎_{g})^{G_{ℚp}} ,

corresponding to the normalised Λ-adic eigenform \underline{g} under the isomorphism in Theorem (A) of the introduction of [Oh95].

(d) There is a natural perfect Galois-equivariant duality, given in Theorem (B) of the introduction

of [Oh95],

𝕌_{g} × 𝕎_{g} ⟶ Λ_{g}(det(ϱ\underline{g})),

where G_{ℚ} acts on the module Λ_{g} of the right-hand side via the determinant of ϱ\underline{g}. Let

y_{g} : Λ_{g} ⟶ ℚ_{p}

be the specialization map attached to the p-stabilised weight one form g_{α}. By specialising the structures above attached to \underline{g} via the map y_{g}, we obtain

(a’) A non-canonical isomorphism of ℚ_{p}[G_{ℚ}]-modules

Φ_{g_{α}} : V_{g} := 𝕍_{g} ⊗_{y_{g}} ℚ_{p} ⟶^{∼} V ⊗_{L} ℚ_{p}.

(b’) A non-trivial G_{ℚ_{p}}-stable filtration

0 ⟶ U_{g} ⟶ V_{g} ⟶ W_{g} ⟶ 0

of V_{g} by one-dimensional subspaces, where U_{g} := 𝕌_{g} ⊗_{y_{g}} ℚ_{p} and W_{g} := 𝕎_{g} ⊗_{y_{g}} ℚ_{p}. The Frobenius element σ_{p} acts on W_{g} and U_{g} as multiplication by α_{g} and β_{g} respectively. Since these eigenvalues are assumed to be distinct, the exact sequence above splits canonically, leading to the identifications

U_{g} = V_{g}^{β},  W_{g} = V_{g}^{α},  V_{g} = U_{g} ⊕ W_{g} = V_{g}^{β} ⊕ V_{g}^{α}.

(c’) Specialising Ohta’s period leads to a canonical element

(6)  ω_{g_{α}} := y_{g}(ω\underline{g}) ∈ D(V_{g}^{α}) := (ℚ_{p}^{nr} ⊗ V_{g}^{α})^{G_{ℚp}} = (H_{p} ⊗ V_{g}^{α})^{G_{ℚp}} .

(d’) The duality in (d) above specialises via y_{g} to a canonical pairing of ℚ_{p}-vector spaces

⟨ , ⟩ : V_{g}^{β} × V_{g}^{α} ⟶ ℚ_{p}(χ),

which induces a pairing by functoriality (denoted by the same symbol by a slight abuse of notation):

(7)  ⟨ , ⟩ : D(V_{g}^{β}) × D(V_{g}^{α}) ⟶ D(ℚ_{p}(χ)).

When this pairing is perfect, it can be used to define a period η_{g_{α}} ∈ D(V_{g}^{β}), as the unique element satisfying

⟨η_{g_{α}}, ω_{g_{α}}⟩ = 𝔤(χ) ⊗ 1,

where

𝔤(χ) := ∑_{j=1}^{𝔣_{χ})} χ(j)e^{2πij/𝔣_{χ}}

is the Gauss sum attached to the Dirichlet character χ, viewed as an element of H_{p} by assigning an 𝔣_{χ}-th root of unity in H_{p} to the complex number e^{2πi/𝔣_{χ}}.

Making use of the above arsenal we now turn to introduce certain p-adic periods associated to g and the choice of a L-structure on V_{g}. We assume for simplicity in the sequel that g is a cusp form, and thus V_{g} is irreducible.

Fix a G_{ℚ}-equivariant isomorphism j_{g} : V ⊗_{L} ℚ_{p} ⟶ V_{g} and let V_{g}^{L} := j_{g}(V) denote the associated L-rational structure on V_{g}, which by Schur’s lemma is well-defined up to scaling by ℚ_{p}^{×}. Since j_{g} induces isomorphisms V_{g}^{α} ≃ V^{α} ⊗_{L} ℚ_{p} and V_{g}^{β} ≃ V^{β} ⊗_{L} ℚ_{p}, we may choose L-bases v_{g}^{α} and v_{g}^{β} for V_{g}^{L} ∩ V_{g}^{α} and V_{g}^{L} ∩ V_{g}^{β} respectively, so that

V_{g}^{L} ∩ V_{g}^{α} = ⟨v_{g}^{α}⟩_{L} and V_{g}^{L} ∩ V_{g}^{β} = ⟨v_{g}^{β}⟩_{L}.

Define p-adic periods

(8)  Ω_{g_{α}} = Ω_{g_{α}}(V_{g}^{L}) ∈ H_{p}^{σ_{p}=α_{g}^{−1}},  Ξ_{g_{α}} = Ξ_{g_{α}}(V_{g}^{L}) ∈ H_{p}^{σ_{p}=β_{g}^{−1}}

by setting

Ω_{g_{α}} ⊗ v_{g}^{α} = ω_{g_{α}}, Ξ_{g_{α}} ⊗ v_{g}^{β} = η_{g_{α}}.

These periods depend on the choice of the basis (v_{g}^{α}, v_{g}^{β}) for V_{g}^{L} , but only up to multiplication by L^{×}. Furthermore, for all μ ∈ ℚ_{p}^{×},

Ω_{g_{α}}(μV_{g}^{L}) = μ^{−1} · Ω_{g_{α}}(V_{g}^{L}), Ξ_{g_{α}}(μ · V_{g}^{L}) = μ^{−1} · Ξ_{g_{α}}(V_{g}^{L}).

It follows that the ratio

(9)  L_{g_{α}} := \frac{Ω_{g_{α}}}{Ξ_{g_{α}}} ∈ (H_{p})^{σ_{p}= \frac{βg}{αg}}

is a number in H_{p}^{×} that is well-defined up to multiplication by elements in L^{×}.

This expression is a canonical p-adic period attached to the eigenform g_{α}, and can be viewed as a p-adic avatar of the Petersson norm of g.

Definition 1.8 of [DLR1, §1.2] associates in many cases a canonical p-adic Stark unit u_{g_{α}} attached to g_{α} as follows. Let V_{ad} := End_{0}(V) be the three dimensional adjoint representation attached to V consisting of trace zero endomorphisms of V. Since complex conjugation acts with eigenvalues −1, −1 and 1 on V_{ad}, it follows that hom_{G_{ℚ}}(V_{ad}, (O_{H}^{×})_{L}) = L · φ_{ad} for a suitable generator φ_{ad}. If one further assumes that α_{g} ≠ ±β_{g}, then the subspace of V_{ad} on which σ_{p} acts as multiplication by \frac{β_{g}}{α_{g}} is one-dimensional; after choosing an L-basis v_{g_{α}} for it, one lets

(10)  u_{g_{α}} := φ_{ad}(v_{g_{α}}).

This element is well defined up to multiplication by L^{×}, and hypothesis (IV) above guarantees that it is a non-zero vector of the V_{ad}-isotypic subspace of (O_{H}^{×})_{L}. See [DLR1, §1.2] for further details.

Conjecture 2.1. The period in (9) satisfies

L_{g_{α}} = log_{p}(u_{g_{α}}) (mod L^{×}).

Remark 2.2. It would be interesting to test this conjecture numerically. To the extent that L_{g_{α}} is a p-adic avatar of the Petersson norm of g, Conjecture 2.1 can be viewed as a p-adic analogue of the Stark conjecture for the L-function attached to the adjoint of g, in the form in which it is illustrated, for example, in the concluding paragraphs of [St75].

3. Generalized Kato classes

3.1. Definition. Let E be an elliptic curve over ℚ and let

ϱ_{1}, ϱ_{2} : Gal (H/ℚ) ⟶ GL_{2}(L)

be odd, irreducible two-dimensional Artin representations of G_{ℚ} satisfying

χ := det(ϱ_{1}) = det(ϱ_{2})^{−1},

where L and H are finite extensions of ℚ (and L is chosen, as before, to be contained in a cyclotomic field). Let V_{1} and V_{2} be L[Gal (H/ℚ]-modules which are two-dimensional over L and realise ϱ_{1} and ϱ_{2} respectively. Observe that

V_{12} := V_{1} ⊗_{L} V_{2}

is a four-dimensional L-linear representation of Gal (H/ℚ) with real traces, i.e., it is isomorphic to its contragredient representation.

Fix a rational prime p and continue to assume that hypotheses (I-IV) of the previous section hold for both the pairs (ϱ_{1}, p) and (ϱ_{2}, p).

The progress in modularity realised over the last two decades implies the existence of cusp forms f, g, and h attached to E, ϱ_{1} and ϱ_{2} respectively, whose associated p-adic representations, denoted V_{f}, V_{g} and V_{h}, satisfy

(11)  H^{1}(E) = V_{f},  V_{1} ⊗_{L} ℚ_{p} ≃ V_{g},  V_{2} ⊗_{L} ℚ_{p} ≃ V_{h}.

It is important to keep in mind that the last two isomorphisms, whose existence is proved by comparing traces on both sides, are only well-defined up to multiplication by a scalar in ℚ_{p}^{×} (by Schur’s lemma), and that the ℚ_{p}-vector spaces V_{g} and V_{h} therefore admit no natural L-rational structure. Let

V_{gh} := V_{g} ⊗ V_{h}, V_{fgh} := V_{f} ⊗ V_{gh}

denote the tensor products of ℚ_{p}-linear representations of G_{ℚ}, and write

(12)  j_{gh} : V_{12} ⊗_{L} ℚ_{p} ⟶ V_{gh}

for the isomorphism induced from (11). Let

(13)  V_{gh}^{L} := j_{gh}(V_{12})

denote the resulting G_{ℚ}-stable L-rational structure on V_{gh}, which is well-defined up to multiplication by a scalar in ℚ_{p}^{×}, even when V_{gh} is reducible, because V_{g} and V_{h} themselves are irreducible.

Because of (11), the Hasse-Weil and Artin L-functions attached to E, ϱ_{1} and ϱ_{2} are equal to the Hecke L-functions attached to f, g and h respectively:

L(E, s) = L(f, s),  L(ϱ_{1}, s) = L(g, s),  L(ϱ_{2}, s) = L(h, s),

and therefore admit functional equations and analytic continuations to the entire complex plane. By the theory of Rankin-Selberg and Garrett, the same is true of the degree 8 L-function L(V_{fgh}, s) attached to the convolution of f, g and h.

Let N = lcm(N_{f}, N_{g}, N_{h}) denote the least common multiple of the conductors of E, ϱ_{1} and ϱ_{2} and assume further that p does not divide N. As in the previous section, let

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

be the Hecke polynomials at p attached to g and h respectively, and assume that the coefficient field L contains the roots of unity α_{g}, β_{g}, α_{h} and β_{h}. Denote as before by

g_{α} := g(z) − β_{g}g(pz), g_{β} := g(z) − α_{g}g(pz), h_{α} := h(z) − β_{h}h(pz), h_{β} := h(z) − α_{h}h(pz)

the relevant p-stabilisations of g and h.

One of the running assumptions of [DR2] that is also enforced in this article is that the Artin conductor of V_{gh} is relatively prime to the conductor of E. Under this assumption, [Pr, Theorem 1.4] implies that the local root numbers that govern the sign in the functional equation for L(E, V_{gh}, s) are equal to 1 at all places of ℚ, and the Hasse-Weil-Artin L-function attached to E and V_{gh} therefore vanishes to even order at the symmetry point s = 1 for its functional equation.

The article [DR2] describes the construction of four canonical (a priori non-trivial, and distinct) generalised Kato classes

(14)  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_{fgh}).

These classes are essentially obtained as p-adic limits

(15)  κ(f, g_{α}, h_{α}) := lim_{k⟶1} κ(f, g_{k}, h_{k}),

as (g_{k}, h_{k}) range over the classical specialisations of weight k ≥ 2 of Hida families \underline{g} and \underline{h} specialising to g_{α} and h_{α} respectively in weight one, and κ(f, g_{k}, h_{k}) arises from a geometric construction whereby it is realised in the p-adic étale cohomology of some (open) variety over ℚ.

More precisely, let V_{f}(N) denote the f-isotypic component of H^{1}(X_{0}(N)), which is (non-canonically) isomorphic to a finite number of copies of V_{f}, indexed by the positive divisors of N/N_{f}. Let V_{g}(N) and V_{h}(N) denote the similar spaces occurring as the weight one specialisations of the \underline{g} and \underline{h}-isotypic parts of the inverse limits of the ordinary quotients of H^{1}(X_{1}(Np^{s})), which are abstractly isomorphic to a direct sum of finitely many copies of V_{g} and V_{h} respectively, endowed with all the structures described in (a’)-(d’) of Section 2. The classes in (14) of [DR2] take values in the Galois representation

V_{fgh}(N) = V_{f}(N) ⊗ V_{g}(N) ⊗ V_{h}(N)

and the classes of (14) are obtained by applying to them a suitable surjective G_{ℚ}-equivariant projection

(16)  π : V_{fgh}(N) ⟶ V_{fgh}

compatible with the L-structure, filtration, Ohta periods and dualities described in (a’)-(d’). The dependence of κ(f, g_{α}, h_{α}) on the choice of π is supressed from the notations but should be kept in mind.

The generalised Kato classes belong to the global cohomology group H^{1}(ℚ, V_{fgh}) = Ext_{G_{ℚ}}^{1}(ℚ_{p}, V_{fgh}), where ℚ_{p} stands for the one-dimensional p-adic representation of G_{ℚ} with trivial action and the Ext group is taken in the category of finite dimensional ℚ_{p}-vector spaces equipped with a continuous G_{ℚ}-action (whose restriction to G_{ℚ_{p}} need not be de Rham).

When \underline{g} and \underline{h} are cuspidal Hida families, the “weight two” classes κ(f, g_{2}, h_{2}) attached to weight two specialisations g_{2} and h_{2} of \underline{g} and \underline{h} are obtained from the p-adic étale Abel-Jacobi image of a Gross-Kudla-Schoen diagonal cycles in the Chow group of null-homologous codimension two cycles in the triple product of the modular curve X_{1}(Np^{s}). It is worth noting that when passing from k = 1 to k > 1, the local root number at ∞ attached to L(V_{fg_{k}h_{k}}, s) changes sign (while the other root numbers stay the same), so that this L-function vanishes to odd order at its center. The presence of Gross-Kudla-Schoen diagonal cycles in this range is consistent with the Beilinson-Bloch conjecture for L(V_{fg_{k}h_{k}}, s) and in fact provides evidence for it. (Cf. the preprint [YZZ] of Yuan-Zhang-Zhang, where the case k = 2 is studied.) The fact that the extension κ(f, g_{α}, h_{α}) does not arise directly in p-adic étale cohomology, but only as a p-adic limit of geometric Galois representations, explains why κ(f, g_{α}, h_{α}) need not be cristalline at p in general.

The analogy with the work of Kato [Ka98], [Sc] arises when the cuspidal Hida families \underline{g} and \underline{h} are replaced by Hida families of Eisenstein series. A global class κ_{BK} (f, g_{α}, h_{α}), designated as the Beilinson-Kato class attached to (f, g_{α}, h_{α}), is then defined as in (15), but replacing the étale Abel-Jacobi images κ(f, g_{2}, h_{2}) by p-adic étale regulators of Beilinson elements in the higher Chow group K_{2}(X_{1}(Np^{s})) = CH^{2}(X_{1}(Np^{s}), 2) attached to a pair of modular units whose logarithmic derivatives give rise to g_{2} and h_{2}. We refer the reader to [BD14] for more details in this setting.

In the intermediate setting where exactly one of \underline{g} and \underline{h} (say, \underline{g}) is cuspidal (and thus \underline{h} is Eisenstein), global classes κ(f, g_{2}, h_{2}) can be constructed geometrically as p-adic étale regulators of suitable Beilinson-Flach elements in the higher Chow group K_{1}(X_{1}(Np^{s})^{2}) = CH^{2}(X_{1}(Np^{s})^{2}, 1) attached to a modular unit whose logarithmic derivative is h_{2}. The limit cohomology class arising in (15) is then denoted κ_{BF} (f, g_{α}, h_{α}), and called the Beilinson-Flach class attached to the triple (f, g_{α}, h_{α}). The Beilinson-Flach classes in p-adic families were introduced and studied in [BCDDPR], [BDR1] and [BDR2]. See also [LLZ] and [KLZ] for more recent work leading to substantial extensions and refinements of the results of loc.cit. in the setting of Beilinson-Flach elements.

The p-adic Selmer group

(17)  H_{fin}^{1}(ℚ, V_{fgh}) := Ext_{cris}^{1}(ℚ_{p}, V_{fgh})

attached to E and V_{gh} are the group of extensions of ℚ_{p} by V_{fgh} in the category of ℚ_{p}-linear representations of G_{ℚ} that are cristalline at p. Let

(E(H)_{L} ⊗ V_{12})^{Gal (H/ℚ)}, where E(H)_{L} := E(H) ⊗_{ℤ} L

denote the ϱ_{12}-isotypic part of the Mordell-Weil group of E. It is a finite-dimensional L-vector space by the Mordell-Weil theorem, and is equipped with a natural inclusion

(18)  (E(H)_{L} ⊗ V_{12})^{Gal (H/ℚ)} ⊂ H_{fin}^{1}(H, V_{p}(E) ⊗_{ℚ_{p}} V_{gh})^{Gal (H/ℚ)} = H_{fin}^{1}(ℚ, V_{fgh})

induced from the connecting homomorphism δ of Kummer theory for E(H) and the map j_{gh} of (13).

When L(E, ϱ_{gh}, s) has a double zero at s = 1, Conjecture BSD(E, ϱ_{gh}) described in the previous section predicts that the associated Mordell-Weil and Selmer group in (18) are 2-dimensional over L and ℚ_{p} respectively. The finiteness of the relevant Shafarevich-Tate group furnishes the Selmer group with a natural L-rational structure

Hom_{G_{ℚ}}(V_{12}, E(H)_{L}) ⊂ H_{Sel}^{1}(ℚ, V_{fgh}).

As mentioned above, the vanishing of the central critical value L(E, ϱ_{gh}, 1) implies that the generalised Kato classes in (14) are cristalline at p and thus belong to H_{fin}^{1}(ℚ, V_{fgh}). The main goal of this article is to give a precise conjectural description of the position of the generalised Kato classes in H_{fin}^{1}(ℚ, V_{fgh}) relative to the L-structure given by (18), in a way that recovers older conjectures of Kato and Perrin-Riou in the setting of Beilinson-Kato classes when g and h are Eisenstein series, and is consistent with the theorems and conjectures of [DR2] and [DLR1].

3.2. Basic properties. In this section we recall some of the main properties of the generalised Kato classes already established in [DR2] and [DLR1].

Restricting (17) to G_{ℚ_{p}}, let

H_{fin}^{1}(ℚ_{p}, V_{fgh}) := Ext_{fin,ℚ_{p}}^{1}(ℚ_{p}, V_{fgh})

denote the group of cristalline extensions of ℚ_{p} by V_{fgh} in the category of ℚ_{p}-linear representations of G_{ℚ_{p}}, and let H_{sing}^{1}(ℚ_{p}, V_{fgh}) := H^{1}(ℚ_{p}, V_{fgh})/H_{fin}^{1}(ℚ_{p}, V_{fgh}) denote the “singular quotient” of the local cohomology at p. Recall that L has been chosen to be large enough to contain the frobenius eigenvalues α_{g}, β_{g}, α_{h} and β_{h}, which therefore belong to ℚ_{p}. Let

(19)  V_{g}^{α}, V_{g}^{β} ⊂ V_{g},  V_{h}^{α}, V_{h}^{β} ⊂ V_{h}

be the eigenspaces in V_{g} and V_{h} respectively associated to these eigenvalues, and set

(20) V_{gh}^{αα} := V_{g}^{α} ⊗ V_{h}^{α},  V_{gh}^{αβ} := V_{g}^{α} ⊗ V_{h}^{β},  V_{gh}^{βα} := V_{g}^{β} ⊗ V_{h}^{α},  V_{gh}^{ββ} := V_{g}^{β} ⊗ V_{h}^{β}

of V_{gh}. Even though V_{g} and V_{h} are both assumed to be regular at p, the same need not be true for V_{gh}, and in this case

(21)  V_{gh} = V_{gh}^{αα} ⊕ V_{gh}^{αβ} ⊕ V_{gh}^{βα} ⊕ V_{gh}^{ββ}.

gives a strict refinement of the decomposition of V_{gh} into frobenius eigenspaces. In fact, some of the most interesting arithmetic applications of the generalised Kato classes (notably those spelled out in sections 4.3, 4.4, and 4.5) arise when V_{gh} is not regular at p.

The first basic result extends Kato’s explicit reciprocity law (corresponding to the case where g and h are both Eisenstein series) to the setting where both g and h are cuspidal (Theorem C of [DR2]) as well as to the intermediate Beilinson-Flach setting (Theorem 3.10 of [BDR2]).

Theorem 3.1. The natural image of κ(f, g_{α}, h_{α}) in H_{sing}^{1}(ℚ_{p}, V_{fgh}) belongs to H_{sing}^{1}(ℚ_{p}, V_{f} ⊗ V_{gh}^{ββ}), and analogously for the remaining classes of (14). Moreover, the following are equivalent:

(1) For all choices of π in (16), the generalised Kato classes of (14) belong to the Bloch-Kato Selmer group of V_{fgh}, i.e., their images in H_{sing}^{1}(ℚ_{p}, V_{fgh}) are trivial;

(2) the central critical value L(E, V_{gh}, 1) vanishes.

Assume from now on that L(E, V_{gh}, 1) = 0, so that (1) the L-series L(E, V_{gh}, s) has a zero of even order ≥ 2 at s = 1;

(2) the generalised Kato classes of (14) belong to the Selmer group attached to E and V_{gh}. One is then naturally interested in a formulating non-vanishing criterion for these Selmer classes:

Conjecture 3.2. The generalised Kato classes in (14) generate a non-trivial subgroup of the Selmer group of V_{fgh} for a suitable choice of π in (16), if and only if the following equivalent conditions are satisfied:

(a) The L-series L(E, V_{gh}, s) has a double zero at s = 1;

(b) the Mordell-Weil group (E(H)_{L} ⊗ V_{12})^{G_{Q}} is two-dimensional over L;

(c) the Selmer group H_{fin}^{1}(ℚ, V_{fgh}) is two-dimensional over ℚ_{p}.

Remark 3.3. Although the equivalence of conditions (a), (b) and (c) certainly lies very deep, it is part of a well-established conjecture, namely BSD(E, V_{gh}). The main novelty of Conjecture 3.2 is in providing a criterion for the non-triviality of the space generated by the generalised Kato classes. Note that Conjecture 3.2 does not predict that all four of the classes in (14) are non-trivial, nor even that these four classes generate the Selmer group, when (a), (b), and/or (c) are satisfied. These stronger conclusions are expected to be false in general, as illustrated by some of the examples in Chapter 4.

Let

κ_{p}(f, g_{α}, h_{α}) = res_{p}(κ(f, g_{α}, h_{α}))

denote the image of the global class κ(f, g_{α}, h_{α}) in the local cohomology group

H_{fin}^{1}(ℚ_{p}, V_{fgh}) = (H_{fin}^{1}(H_{p}, V_{f}) ⊗ V_{gh})^{Gal (H_{p}/ℚ_{p})} = (E(H_{p}) ⊗ V_{gh})^{Gal (H_{p}/ℚ_{p})}.

As we describe more explicitly below, Theorem D of [DR2] asserts that this image is controlled by suitable p-adic avatars of the second derivative of the classical L-series L(f, V_{gh}, s) at the central critical point s = 1.

These p-adic values were defined and explored in [DR2] and [DLR1] and are denoted

(22)  ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆),  ℒ_{p} ^{g_{β}}(f̆, ğ^{∗}, h̆),  ℒ_{p} ^{h_{α}}(f̆, ğ, h̆^{∗}),  ℒ_{p} ^{h_{β}}(f̆, ğ, h̆^{∗}).

They depend on the choice of certain test vectors

(f̆, ğ, h̆) ∈ S_{2}(N; L) × M_{1}(N, χ; L) × M_{1}(N, χ^{−1}; L)

with the same system of Hecke eigenvalues as f, g and h respectively, and with fourier coefficients in L, and on the choice of dual test vectors

(ğ^{∗}, h̆^{∗}) ∈ Hom(M_{1}(N, χ^{−1}; L), L) × Hom(M_{1}(N, χ; L), L)

with the same system of Hecke eigenvalues as g and h. We refer to the introduction of [DR2] for more details on their definition, contenting ourselves with remark that the p-adic L-value ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆) is defined essentially as the p-adic limit of central critical values

ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆) := lim_{ℓ→1} E(f, g_{ℓ}, h) × C(f̆, ğ^{∗}, h̆) × \frac{L(V_{f} ⊗ V_{g_{ℓ}} ⊗ V_{h}, (ℓ + 1)/2)}{⟨g_{ℓ}, g_{ℓ}⟩},

as g_{ℓ} ranges over the specialisations of (odd) weight ℓ ≥ 3 of the Hida family \underline{g} specialising to g_{α} in weight one. Here E(f, g_{ℓ}, h) is a p-adic multiplier arising from a recipe of Panciskin, whose presence allows the p-adic interpolation of the special values above, and C(f̆, ğ^{∗}, h̆) is a product over the primes dividing N · ∞ of local terms which depend in a simple way on the choice of test vectors.

Choose a basis of V_{gh} (over ℚ_{p}, for now) which is compatible with the decomposition (21), i.e., choose non-zero vectors

(23)  v_{gh}^{αα} ∈ V_{gh}^{αα}, v_{gh}^{αβ} ∈ V_{gh}^{αβ}, v_{gh}^{βα} ∈ V_{gh}^{βα}, v_{gh}^{ββ} ∈ V_{gh}^{ββ}.

Write

(24)  κ_{p}(f, g_{α}, h_{α}) = R_{αα} ⊗ v_{gh}^{ββ} + R_{αβ} ⊗ v_{gh}^{βα} + R_{βα} ⊗ v_{gh}^{αβ} + R_{ββ} ⊗ v_{gh}^{αα}.

The coordinate R_{ξ} belongs to E(H_{p})_{ℚ_{p}}^{σ_{p}=ξ}, where ξ ranges over the index set

{αα = α_{g}α_{h}, αβ = α_{g}β_{h}, βα = β_{g}α_{h}, ββ = β_{g}β_{h}}.

Note that R_{ξ} is even the image of a global point in E(H)_{ℚ_{p}}, assuming the finiteness of the Shafarevich-Tate group of E over H. Let

(25)  log_{p} : E(H_{p})_{ℚ_{p}} ⟶ H_{p}

denote the formal group logarithm attached to an invariant differential on E/ℚ. The following theorem is stated in Section 6.4 of [DR2]:

Theorem 3.4. When L(E, V_{gh}, 1) = 0, there exists a choice of π in (16) and of test vectors for f, g and h such that the coordinates in (24) satisfy

(26)  log_{p}(R_{αβ}) ∼ ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆),  log_{p}(R_{βα}) ∼ ℒ_{p} ^{h_{α}}(f̆, ğ, h̆^{∗}),  log_{p}(R_{ββ}) = 0,

where ∼ denotes equality up to a non-zero p-adic period in H_{p}^{×}.

Remark 3.5. This theorem says nothing about the quantity log_{p}(R_{αα}), which does not bear any direct relationship with p-adic L-values introduced above. We expect that log_{p}(R_{αα}) may rather be connected with the first derivative of a putative refinement of ℒ_{p} ^{f}(f, g_{α}, h_{α}) in which all three modular forms would be made to vary in a Hida family.

As explained in the introduction and in Section 6.3. of [DR2], Theorem 3.4 has the following corollary which can be viewed as a p-adic Gross-Zagier formula in “analytic rank two”:

Corollary 3.6. If L(E, V_{gh}, 1) = 0 and ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆) ≠ 0 for a suitable choice (f̆, ğ^{∗}, h̆) of test vectors, then the two global classes

κ(f, g_{α}, h_{α}), κ(f, g_{α}, h_{β})

are linearly independent in the Selmer group H_{fin}^{1}(ℚ, V_{fgh}) attached to E and V_{gh}, for a suitable choice of π in (16).

Theorem 3.4 and its corollary motivated the experimental study undertaken in [DLR1] of the special values of p-adic L-functions appearing in (26). This led to a precise conjecture for these values up to a factor of L^{×} rather than ℚ_{p}^{×}.

To formulate this conjecture, recall that the class κ(f, g_{α}, h_{α}) is expected to be trivial when ord_{s=1}L(E, V_{gh}, s) > 2. Assume that this L-function has a double zero at the center, which implies, by Conjecture BSD(E, V_{gh}), that (E(H)_{L} ⊗ V_{12})^{G_{ℚ}} is a two-dimensional L-vector space.

Fix vectors v_{gh}^{αα}, …, v_{gh}^{ββ} chosen as in (23), with the difference that they belong to L-vector space V_{12} rather than the ℚ_{p}-vector space V_{gh}. Choose a basis (P, Q) for this L-vector space, and write

P = P_{αα} ⊗ v_{gh}^{ββ} + P_{αβ} ⊗ v_{gh}^{βα} + P_{βα} ⊗ v_{gh}^{αβ} + P_{ββ} ⊗ v_{gh}^{αα},
Q = Q_{αα} ⊗ v_{gh}^{ββ} + Q_{αβ} ⊗ v_{gh}^{βα} + Q_{βα} ⊗ v_{gh}^{αβ} + Q_{ββ} ⊗ v_{gh}^{αα},

where P_{ξ}, Q_{ξ} are points in E(H)_{L}^{σ_{p}=ξ} for every ξ ∈ {αα = α_{g}α_{h}, αβ = α_{g}β_{h}, βα = β_{g}α_{h}, ββ = β_{g}β_{h}}.

These points can be used to define a regulator attached to g_{α}, whose entries are the p-adic formal group logarithms of the coordinates attached to the vectors v_{gh}^{αα} and v_{gh}^{αβ} (and similarly for h_{α}):

Definition 3.7. The regulators attached to E and V_{12} are

Reg_{g_{α}}(E, V_{12}) = det ( \begin{matrix}log_{p} P_{ββ} & log_{p} P_{βα} \\ log_{p} Q_{ββ} & log_{p} Q_{βα} \end{matrix} ) = log_{p} P_{ββ} · log_{p} Q_{βα} − log_{p} Q_{ββ} · log_{p} P_{βα},

Reg_{h_{α}}(E, V_{12}) = det ( \begin{matrix}log_{p} P_{ββ} & log_{p} P_{αβ} \\ log_{p} Q_{ββ} & log_{p} Q_{αβ} \end{matrix} ) = log_{p} P_{ββ} · log_{p} Q_{αβ} − log_{p} Q_{ββ} · log_{p} P_{αβ}.

The main conjecture of [DLR1] is the following, assuming \frac{α_{g}}{β_{g}} ≠ ±1 (resp. \frac{α_{h}}{β_{h}} ≠ ±1) so that the Stark unit u_{g_{α}} (resp. u_{h_{α}}) is well-defined:

Conjecture 3.8. Assume that L(E, V_{gh}, s) vanishes to order 2 at s = 1. Then there exists a choice of test vectors (f̆, ğ^{∗}, h̆) and (f̆, ğ, h̆^{∗}) such that

ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆) = \frac{Reg_{g_{α}}(E, V_{12})}{log_{p} u_{g_{α}}}, ℒ_{p} ^{h_{α}}(f̆, ğ, h̆^{∗}) = \frac{Reg_{h_{α}}(E, V_{12})}{log_{p} u_{h_{α}}} (mod L^{×}).

Remark 3.9. Conjecture 3.8 lends itself to numerical verification and has been extensively tested in [DLR1]. This is because the p-adic L-values ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆) and ℒ_{p} ^{h_{α}}(f̆, ğ, h̆^{∗}) can be expressed in terms of the rather concrete p-adic iterated integrals of loc.cit., which can be computed efficiently using Alan Lauder’s [La] fast ordinary projection algorithms on the space of overconvergent modular forms. In contrast, the generalised Kato classes themselves (like many objects constructed in étale cohomology) seem difficult to compute in practice, even though their theoretical usefulness is amply illustrated in [BDR2] and [DR2].

3.3. Enhanced regulators. The goal of this article is to combine the insights arising from Theorem 3.4 and Conjecture 3.8 to formulate a conjecture on the position of the generalised Kato classes themselves in (E(H)⊗V_{gh})^{G_{ℚ}}, specifying this position up to an ambiguity of L^{×} rather than the less precise ℚ_{p}^{×} ambiguity of Theorem 3.4.

The most important ingredients in the formulation of this conjecture are the so-called enhanced regulators

R̃eg(E, V_{12}) ∈ (E(H)_{L} ⊗ V_{12})^{G_{ℚ}} ⊗ (E(H)_{L} ⊗ V_{12})^{G_{ℚ}},
R̃eg_{αα}(E, V_{12}) ∈ (H_{p})^{σ_{p}=β_{g}β_{h}} ⊗ (E(H)_{L} ⊗ V_{12})^{G_{ℚ}},
R̃eg(E, V_{gh}) ∈ (E(H)_{L} ⊗ V_{gh})^{G_{ℚ}} ⊗ (E(H)_{L} ⊗ V_{gh})^{G_{ℚ}},
R̃eg_{αα}(E, V_{gh}) ∈ D(V_{gh}^{αα}) ⊗ (E(H)_{L} ⊗ V_{gh})^{G_{ℚ}},

whose definition is somewhat in the spirit of the regulator R_{S} defined in equation (2) of [Da92], and which we now proceed to describe. As in (6), here D(V_{gh}^{αα}) := (ℚ_{p}^{nr} ⊗ V_{gh}^{αα})^{G_{ℚp}} = (H_{p} ⊗ V_{gh}^{αα})^{G_{ℚp}}.

Definition 3.10. Choose an L-basis (P, Q) of the two-dimensional vector space (E(H) ⊗ V_{12})^{G_{ℚ}}, and set

(27)  R̃eg(E, V_{12}) := det ( \begin{matrix}P & P \\ Q & Q \end{matrix} ) := P ⊗ Q − Q ⊗ P.

It does not depend on the choice of basis that was made to define it, up to multiplication by L^{×}. The function log_{αα} : (E(H)_{L} ⊗ V_{12})^{G_{ℚ}} ⟶ (H_{p})^{σ_{p}=β_{g}β_{h}} defined by

log_{αα}(P) := log_{p}(P_{ββ})

induces a linear map

log_{αα} ⊗1 : (E(H)_{L} ⊗ V_{12})^{G_{ℚ}} ⊗ (E(H)_{L} ⊗ V_{12})^{G_{ℚ}} ⟶ (H_{p})^{σ_{p}=β_{g}β_{h}} ⊗ (E(H)_{L} ⊗ V_{12})^{G_{ℚ}},

and we set

(28)  R̃eg_{αα}(E, V_{12}) := (log_{αα} ⊗1)(R̃eg(E, V_{12})) = log_{p}(P_{ββ}) ⊗ Q − log_{p}(Q_{ββ}) ⊗ P.

Recall the embedding j_{gh} : V_{12} ⟶ V_{gh}^{L} ⊂ V_{gh} of (12). Although this embedding is completely non-canonical and only defined up to scaling by ℚ_{p}^{×}, there is a canonical way of embedding V_{12}^{⊗2} into V_{gh}^{⊗2}. This is done by exploiting the canonical dualities on V_{g} and V_{h} described in Section 2, which gives rise to perfect pairings

V_{g} × V_{g} ⟶ ℚ_{p}(χ),  V_{h} × V_{h} ⟶ ℚ_{p}(χ^{−1}),  V_{gh} × V_{gh} ⟶ ℚ_{p}.

These pairings allow us to define L-rational structures V_{g}^{L∗}, V_{h}^{L∗} and V_{gh}^{L∗} which are dual to V_{g}^{L}, V_{h}^{L} and V_{gh}^{L} respectively, by letting V_{g}^{L∗} be the L-dual of V_{g}^{L} in V_{g}, and likewise for V_{h}^{L∗} and V_{gh}^{L∗}. We may then choose G_{ℚ}-equivariant embeddings

j_{g}^{∗} : V_{1} ⟶ V_{g}^{L∗},  j_{h}^{∗} : V_{2} ⟶ V_{h}^{L∗},  j_{gh}^{∗} := j_{g}^{∗} ⊗ j_{h}^{∗} : V_{12} ⟶ V_{gh}^{L∗},

which are well-defined up to scaling by L^{×}. Replacing j_{gh} by μ · j_{gh}, for any μ ∈ ℚ_{p}^{×}, has the effect of replacing j_{gh}^{∗} by μ^{−1} · j_{gh}^{∗}. Hence the map

j_{gh} ⊗ j_{gh}^{∗} : V_{12} ⊗ V_{12} ⟶ V_{gh} ⊗ V_{gh}

is well defined up to scaling by L^{×}.

Definition 3.11. The enhanced regulator R̃eg(E, V_{gh}) associated to E and V_{gh} is

(29)  R̃eg(E, V_{gh}) := (j_{gh} ⊗ j_{gh}^{∗})(R̃eg(E, V_{12})) ∈ (E(H) ⊗ V_{gh})^{G_{ℚ}} ⊗ (E(H) ⊗ V_{gh})^{G_{ℚ}}.

Finally, let

Log_{p} : (E(H) ⊗ V_{gh})^{G_{ℚ}} ⟶ (H_{p} ⊗ V_{gh})^{G_{ℚp}} = D(V_{gh})

be the canonical p-adic logarithm map induced from the p-adic logarithm of (25) via the fixed embedding H ⊂ H_{p}, and let

Log_{αα} : (E(H) ⊗ V_{gh})^{G_{ℚ}} ⟶ D(V_{gh}^{αα})

be its composition with the functorial projection D(V_{gh}) ⟶ D(V_{gh}^{αα}). This logarithm map is just the more canonical counterpart of the map log_{αα}: the latter depends on the choice of a basis vector v_{gh}^{αα} for V^{αα} and is related to Log_{αα} by the rule

Log_{αα} := log_{αα} ⊗v_{gh}^{αα}.

We set

(30)  R̃eg_{αα}(E, V_{gh}) := (Log_{αα} ⊗1)(R̃eg(E, V_{gh})) = Log_{αα}(P) ⊗ Q − Log_{αα}(Q) ⊗ P.

It is worth noting that the enhanced regulator R̃eg_{αα}(E, V_{gh}) is a canonical invariant associated to E and V_{gh}, i.e., it is well-defined up to multiplication by L^{×}, while the less canonical R̃eg_{αα}(E, V_{12}) depends on the choice of a basis v_{gh}^{αα} for V_{gh}^{αα}. The two regulators are related by

(31)  R̃eg_{αα}(E, V_{gh}) = R̃eg_{αα}(E, V_{12}) ⊗ v_{gh}^{αα}.

3.4. The conjecture. Recall the periods

ω_{g_{α}} ∈ D(V_{g}^{α}), ω_{h_{α}} ∈ D(V_{h}^{α})

constructed in (6). The main conjecture of this note is:

Conjecture 3.12. Assume that r(E, V_{gh}) = 2. The generalised Kato class κ(f, g_{α}, h_{α}) belongs to (E(H) ⊗ V_{gh})^{G_{ℚ}} and satisfies the relation

ω_{g_{α}}ω_{h_{α}} ⊗ κ(f, g_{α}, h_{α}) ∼_{L} R̃eg_{αα}(E, V_{gh})

in D(V_{gh}^{αα}) ⊗ (E(H) ⊗ V_{gh})^{G_{ℚ}}, where ∼_{L} denotes an equality up to scaling by a factor in L which is non-zero for a suitable choice of π in (16).

The following proposition shows that, under Conjecture 2.1 (relating the canonical period attached to g to the Stark unit u_{g_{α}}) and Conjecture 3.2 (a mild strengthening of BSD(E, ϱ_{gh})), Conjecture 3.12 implies the main conjecture of [DLR1]. Before dismissing this proposition as mere conjectural relations between conjectures, the reader is reminded that Conjecture 3.8 lends itself to experiment and has been extensively tested numerically in [DLR1], while the strengthening described in Conjecture 3.12 lies for the moment beyond the range of explicit calculations (cf. Remark 3.9).

Proposition 3.13. Assume Conjectures 2.1 and 3.2. Then Conjecture 3.12 implies Conjecture 3.8.

Proof. Consider the product of periods

η_{g_{α}}ω_{h_{α}} = (Ξ_{g_{α}} ⊗ v_{g}^{β}) · (Ω_{h_{α}} ⊗ v_{h}^{α}) = Ξ_{g_{α}} · Ω_{h_{α}} ⊗ v_{gh}^{βα} ∈ D(V_{gh}^{βα})

defined in Section 2.

The pairing introduced in (7) gives rise to a pairing

⟨ , ⟩ : D(V_{gh}^{αβ}) × D(V_{gh}^{βα}) ⟶ D(ℚ_{p}) = ℚ_{p}.

As shown in the proof of [DR2, Theorem 6.10 (ii)],

(32)  ⟨Log_{αβ} κ(f, g_{α}, h_{α}), η_{g_{α}}ω_{h_{α}}⟩ = ℒ_{p} ^{g_{α}}(f, g, h) (mod L^{×}).

On the other hand, by the definition of the enhanced regulator,

Log_{αβ} R̃eg_{αα}(E, V_{gh}) = (log_{p} P_{ββ} log_{p} Q_{βα} − log_{p} Q_{ββ} log_{p} P_{βα}) ⊗ v_{gh}^{αα} ⊗ v_{gh}^{∗αβ}
= Reg_{g_{α}}(E, V_{12}) ⊗ v_{gh}^{αα} ⊗ v_{gh}^{∗αβ} (mod L^{×}).

Hence the following equality holds in D(V_{gh}^{αα}):

(33)  ⟨Log_{αβ} R̃eg_{αα}(E, V_{gh}), η_{g_{α}}ω_{h_{α}}⟩ = Ξ_{g_{α}} · Ω_{h_{α}} · Reg_{g_{α}}(E, V_{gh}) ⊗ v_{gh}^{αα} (mod L^{×}).

By pairing the value of Log_{αβ} at both sides of the displayed identity in Conjecture 3.12 with the class η_{g_{α}}ω_{h_{α}} and invoking (32) and (33), we obtain

ω_{g_{α}}ω_{h_{α}} ⊗ ℒ_{p} ^{g_{α}}(f, g, h) = Ξ_{g_{α}} · Ω_{h_{α}} · Reg_{g_{α}}(E, V_{12}) ⊗ v_{gh}^{αα} ∈ D(V_{gh}^{αα}) (mod L^{×}).

Since

ω_{g_{α}}ω_{h_{α}} = Ω_{g_{α}} · Ω_{h_{α}} · v_{gh}^{αα} (mod L^{×}),

it follows that

Ω_{g_{α}}ℒ_{p} ^{g_{α}}(f, g, h) = Ξ_{g_{α}} Reg_{g_{α}}(E, V_{12}) (mod L^{×}),

and therefore that

ℒ_{p} ^{g_{α}}(f, g, h) = \frac{Reg_{g_{α}}(E, V_{12})}{L_{g_{α}}} (mod L^{×}).

Conjecture 3.8 now follows directly from this equality after invoking Conjecture 2.1. □

Remark 3.14. As explained in a number of the examples covered in Chapter 4 below, it may happen that all four of the p-adic iterated integrals in (22) are equal to zero even when some of the generalised Kato classes are non-trivial. This suggests that Conjecture 3.12 is a genuine strengthening of Conjecture 3.8.

4. Special cases

This section examines Conjecture 3.12, and the special forms taken by the enhanced regulators

R̃eg_{αα}(E; V_{12}),  R̃eg_{αβ}(E; V_{12}),  R̃eg_{βα}(E; V_{12}),  R̃eg_{ββ}(E; V_{12}),

in the arithmetically interesting cases where V_{gh} is reducible. According to [DLR2, §2], the following is a complete list of scenarios where this occurs:

(1) The original Beilinson-Kato setting where V_{g} and V_{h} are both reducible, i.e., where g and h are both Eisenstein series of weight one;

(2) the Beilinson-Flach setting where exactly one of V_{g} or V_{h} is reducible, i.e., where exactly one of g or h is cuspidal;

(3) the complex multiplication case where V_{g} and V_{h} are both induced from characters of a common imaginary quadratic field;

(4) the real multiplication case where V_{g} and V_{h} are induced from characters of mixed signature of a common real quadratic field;

(5) the adjoint case where h is (a twist of) the dual of g, so that V_{gh} is the direct sum of a one-dimensional representation and a twist of the adjoint of V_{g}.

The reader will notice that some of the above settings arise when ϱ_{g} and/or ϱ_{h} are reducible, while in §2 and §3 these representations were assumed to be irreducible. This assumption was imposed to a large extent for the sake of simplicity of the exposition, and the statement (and presumed validity) of Conjecture 3.12 does not rely on it. For completeness, we have therefore described the enhanced regulators that appear in Conjecture 3.12 in all of the above cases.

4.1. Beilinson-Kato classes. Assume that g and h are both Eisenstein series. After possibly twisting g or h, there is no real loss of generality in assuming that there exist Dirichlet characters χ_{1}, χ_{2} such that g and h are given by

g = E_{1}(χ_{1}, χ_{2}),  h = E_{1}(1, χ_{12}^{−1}),  where χ_{12} = χ_{1}χ_{2}.

We refer to e.g. [BDR1, §2.1.2] for the definition of these weight one Eisenstein series in terms of their q-expansions. The Galois representations attached to g and h are reducible, namely,

(34) V_{1} = L(χ_{1}) ⊕ L(χ_{2}),  V_{2} = L ⊕ L(χ_{12}^{−1}),  V_{12} = L(χ_{1}) ⊕ L(χ_{1}^{−1}) ⊕ L(χ_{2}) ⊕ L(χ_{2}^{−1}),

where the coefficient field L is the cyclotomic field generated by the images of χ_{1} and χ_{2}. These representations factor through the Galois group Gal (H/ℚ) of an abelian extension H of ℚ. We may set

α_{g} = χ_{1}(p), β_{g} = χ_{2}(p), α_{h} = 1, β_{h} = χ_{12}^{−1}(p).

The regularity asumption implies that V_{1} and V_{2} decompose uniquely as a direct sum of two G_{ℚ_{p}}-stable lines, which are also stable under G_{ℚ}. More precisely,

V_{12}^{αα} = L · v_{χ_{1}},  V_{12}^{ββ} = L · vχ̄_{1},  V_{12}^{αβ} = L · vχ̄_{2},  V_{12}^{βα} = L · v_{χ_{2}},

where (v_{χ_{1}}, vχ̄_{1}, vχ̄_{2}, v_{χ_{2}}) is a basis for V_{12} on which G_{ℚ} acts via the characters χ_{1}, χ̄_{1}, χ̄_{2}, and χ_{2} respectively.

The class κ(f, g_{α}, h_{α}) = κ_{BK}(f, g_{α}, h_{α}) was constructed by Kato as a p-adic limit of Beilinson elements attached to pairs of modular units whose logarithmic derivatives are weight two Eisenstein series. Theorem 3.1 in this case boils down to Kato’s reciprocity law, which asserts that κ(f, g_{α}, h_{α}) belongs to the Selmer group of E over H if and only if the L-function

L(E, V_{gh}, s) = L(E, χ_{1}, s)L(E, χ̄_{1}, s)L(E, χ_{2}, s)L(E, χ̄_{2}, s)

vanishes at s = 1. In this case, it clearly vanishes to even order, and vanishes to order two if and only if (after eventually interchanging the characters χ_{1} and χ_{2})

ord_{s=1}L(E, χ_{1}, s) = ord_{s=1}L(E, χ̄_{1}, s) = 1, L(E, χ_{2}, 1), L(E, χ̄_{2}, 1) ≠ 0.

Assuming that this is the case, Conjectures BSD(E, χ_{1}) and BSD(E, χ_{2}) predict that (E(H)_{L} ⊗V_{12})^{G_{ℚ}} is two-dimensional over L and that a basis for it can be chosen to be

P := Pχ̄_{1} ⊗ v_{χ_{1}}, Q := Q_{χ_{1}} ⊗ vχ̄_{1},

where Pχ̄_{1} and Q_{χ_{1}} are global points in E(H)_{L} generating the χ̄_{1} and χ_{1} eigenspaces respectively for the natural action of G_{ℚ}. With these notations, we have

P_{αα} = P_{αβ} = P_{βα} = 0, P_{ββ} = Pχ̄_{1},
Q_{αβ} = Q_{βα} = Q_{ββ} = 0, Q_{αα} = Q_{χ_{1}}.

This immediately implies that

R̃eg_{αα}(E, V_{12}) = log_{p}(Pχ̄_{1}) · Q,  R̃eg_{αβ}(E, V_{12}) = 0,
R̃eg_{βα}(E, V_{12}) = 0,  R̃eg_{ββ}(E; V_{12}) = log_{p}(Q_{χ_{1}}) · P.

It follows that

Reg_{g_{α}}(E; V_{12}) = Reg_{g_{β}}(E; V_{12}) = Reg_{h_{α}}(E; V_{12}) = Reg_{h_{β}}(E; V_{12}) = 0.

This accounts for the fact that the p-adic iterated integrals

ℒ_{p} ^{g_{α}}(f, g, h), ℒ_{p} ^{g_{β}}(f, g, h), ℒ_{p} ^{h_{α}}(f, g, h), ℒ_{p} ^{h_{β}}(f, g, h)

systematically vanish^{3} when g and h are Eisenstein series that are regular at p. Conjecture 3.12 makes the stronger prediction that the generalised Kato classes themselves are non-trivial, and is consistent with a Conjecture of Perrin-Riou, since it predicts that

log_{ββ}(κ(f, g_{α}, h_{α})) = log_{αα}(κ(f, g_{β}, h_{β})) = log_{p}(Pχ̄_{1}) log_{p}(Q_{χ_{1}}) (mod L^{×}).

4.2. Belinson-Flach classes. In the Beilinson-Flach setting, it can be assumed without loss of generality that g is a weight one cusp form with nebentypus character χ and Galois representation V_{g} = V_{1} ⊗_{L} ℚ_{p}, and that h := E_{1}(1, χ^{−1}) is the weight one Eisenstein series attached to the pair (1, χ^{−1}) of Dirichlet characters. The relevant four-dimensional representations are then equal to

V_{gh} = V_{g} ⊕ Vḡ; V_{12} = V_{1} ⊕ V̄_{1},

and the Hasse-Weil-Artin L-series

L(E, V_{gh}, s) = L(E, V_{g}, s)L(E, V̄_{g}, s)

has a double zero at s = 1 precisely when each of the primitive L-series L(E, V_{g}, s) and L(E, Vḡ, s) have a simple zero at s = 1. Conjecture BSD(E, V_{g}) then implies that each of the L-vector spaces on the right-hand side of

(E(H)_{L} ⊗ V_{12})^{G_{ℚ}}) = (E(H)_{L} ⊗ V_{1})^{G_{ℚ}} ⊕ (E(H)_{L} ⊗ V̄_{1})^{G_{ℚ}},

3But see the experiments described in [DLR1, §7] in the case where g is irregular at p, which suggest that the irregular setting of Conjecture 3.12 would merit further investigation.

is one-dimensional. Let P be an L-basis for (E(H)_{L} ⊗ V_{1})^{G_{ℚ}} and let P̄ be the associated L-basis for (E(H)_{L} ⊗ V̄_{1})^{G_{ℚ}}, obtained by applying complex conjugation to the coefficients in L.

After fixing an ordering α_{g}, β_{g} ∈ L for the eigenvalues of σ_{p} on V_{1}, and setting

α_{h} = 1, β_{h} = χ^{−1}(p) = (α_{g}β_{g})^{−1},

we have

V_{12}^{αα} = V_{1}^{α_{g}}, V_{12}^{αβ} = V̄_{1}^{β_{g}^{−1}}, V_{12}^{βα} = V_{1}^{β_{g}}, V_{12}^{ββ} = V̄_{1}^{α_{g}^{−1}},

and hence

\begin{matrix}P_{αα} = P_{α_{g}}, & P_{βα} = P_{β_{g}}, & P_{αβ} = 0 & P_{ββ} = 0, \\ P̄_{αα} = 0 & P̄_{βα} = 0, & P̄_{αβ} = P̄_{β_{g}^{−1}}, & P̄_{ββ} = P̄_{α_{g}^{−1}}. \end{matrix}

A direct calculation reveals that, up to multiplication by L^{×},

R̃eg_{αα}(E, V_{12}) = log_{p}(P̄_{α_{g}^{−1}}) · P,  R̃eg_{αβ}(E, V_{12}) = log_{p}(P_{β_{g}}) · P̄,
R̃eg_{βα}(E, V_{12}) = log_{p}(P̄_{β_{g}^{−1}}) · P,  R̃eg_{ββ}(E; V_{12}) = log_{p}(P_{α_{g}}) · P̄.

It follows that

Reg_{g_{α}}(E, V_{12}) = log_{p}(P̄_{α_{g}^{−1}}) · log_{p}(P_{β_{g}}), Reg_{g_{β}}(E, V_{12}) = log_{p}(P̄_{β_{g}^{−1}}) · log_{p}(P_{α_{g}}),
Reg_{h_{α}}(E, V_{12}) = 0,  Reg_{h_{β}}(E, V_{12}) = 0.

as described in [DLR1, §6].

4.3. Complex multiplication classes and Heegner points. In this chapter we consider the setting where g and h are theta series attached to characters ψ_{g} and ψ_{h} of the same imaginary quadratic field K, and with inverse nebentypus character. Given any character ψ of G_{K}, let ψ′ denote the character obtained by conjugating it with the involution in Gal (K/ℚ). Then

V_{g} = Ind_{K}^{ℚ} ψ_{g} = Ind_{K}^{ℚ} ψ′_{g}, V_{h} = Ind_{K}^{ℚ} ψ_{h} = Ind_{K}^{ℚ} ψ′_{h},

and therefore

V_{gh} = Ind_{K}^{Q} ψ_{•} ⊕ Ind_{K}^{ℚ} ψ_{∘},  where ψ_{•} = ψ_{g}ψ_{h},  ψ_{∘} = ψ_{g}ψ′_{h}.

The self-duality assumption implies that ψ_{•} and ψ_{∘} are are ring class characters, i.e., they satisfy

ψ′_{•} = ψ−1_{•}, ψ′_{∘} = ψ−1_{∘}.

Assume that the induced representations

V_{•} := Ind_{K}^{ℚ} ψ_{•}, V_{∘} := Ind_{K}^{ℚ} ψ_{∘}

appearing in the decomposition

(35)  V_{12} = V_{•} ⊕ V_{∘}

(viewed as representations with coefficients in the number field L) are irreducible, which is always the case unless ψ_{•} or ψ_{∘} is a quadratic, i.e., a genus character. (The more degenerate case where this arises can be subsumed under the “adjoint setting” considered in Section 4.5.)

The Hasse-Weil-Artin L-series

L(E, V_{gh}, s) = L(E, V_{•}, s)L(E, V_{∘}, s) = L(E/K, ψ_{•}, s)L(E/K, ψ_{∘}, s)

has a double zero at s = 1 in one of the following two cases:

(1) the primitive L-series L(E, V_{•}, s) and L(E, V_{∘}, s) each have a simple zero at s = 1. This setting, which resembles more closely the phenomena described in the previous two sections on Beilinson-Kato and Beilinson-Flach elements, will be referred to as the rank (1,1) setting of Conjecture 3.12.

(2) Exactly one of the primitive L-series L(E, V_{•}, s) or L(E, V_{∘}, s) has a double zero at s = 1, and the other is non-vanishing at the center. This case shall be referred to as the rank (2,0) setting of Conjecture 3.12. The possible non-triviality of the generalised Kato classes in the presence of a “genuine” double zero of a primitive Hasse-Weil-Artin L-function represents a novel feature that did not arise in the setting of Beilinson-Kato or Beilinson-Flach elements.

4.3.1. The rank (1, 1) setting. In this case, Conjectures BSD(E, V_{•}) and BSD(E, V_{∘}) predict that the Mordell-Weil groups (E(H)_{L} ⊗ V_{•})^{G_{ℚ}} and (E(H)_{L} ⊗ V_{∘})^{G_{ℚ}} are both one-dimensional L-vector spaces, with generators P_{•} and P_{∘} respectively. It is natural to write

(36)  P_{•} = P_{ψ_{•}} ⊗ v_{ψ′_{•}} + P_{ψ′_{•}} ⊗ v_{ψ_{•}} ,  P_{∘} = P_{ψ_{∘}} ⊗ v_{ψ′_{∘}} + P_{ψ′_{∘}} ⊗ v_{ψ_{∘}} ,

where P_{ψ_{•}}, P_{ψ′_{•}}, P_{ψ_{∘}}, and P_{ψ′_{∘}} are generators for the one-dimensional subspaces of E(H)_{L} on which G_{K} acts via the characters ψ_{•}, ψ′_{•}, ψ_{∘}, and ψ′_{∘} respectively.

The description of the enhanced regulators attached to V_{12} and to (P_{•}, P_{∘}) can be further subdivided into two cases, with markedly different features: the case where the prime p is split in K, and the case where it is inert in K.

a) The case where p is split in K. In this case, let p = 𝔭 𝔭′ be the factorisation of p into distinct primes of K. We can then set

α_{g} = ψ_{g}(𝔭), β_{g} = ψ_{g}(𝔭′), α_{h} = ψ_{h}(𝔭), β_{h} = ψ_{h}(𝔭′),

so that

α_{g}α_{h} = ψ_{•}(𝔭), α_{g}β_{h} = ψ_{∘}(𝔭), β_{g}α_{h} = ψ_{∘}(𝔭′), β_{g}β_{h} = ψ_{•}(𝔭′).

The decomposition of the G_{K_{𝔭}} = G_{ℚ_{p}} representations attached to (35) into σ_{p}-eigenspaces is also stable under the action of the global Galois group G_{K}, and is described by:

V_{12}^{αα} = V_{•}^{ψ_{•}} , V_{12}^{αβ} = V_{∘}^{ψ_{∘}} , V_{12}^{βα} = V_{∘}^{ψ′_{∘}} , V_{12}^{ββ} = V_{•}^{ψ′_{•}} .

It follows that, up to multiplication by L^{×},

R̃eg_{αα}(E, V_{12}) = log_{𝔭}(P_{ψ′_{•}}) · P_{∘},  R̃eg_{αβ}(E, V_{12}) = log_{𝔭}(P_{ψ′_{∘}}) · P_{•},
R̃eg_{βα}(E, V_{12}) = log_{𝔭}(P_{ψ_{∘}}) · P_{•},  R̃eg_{ββ}(E, V_{12}) = log_{𝔭}(P_{ψ_{•}}) · P_{∘},

and therefore that

Reg_{g_{α}}(E, V_{12}) = log_{𝔭}(P_{ψ′_{•}}) · log_{𝔭}(P_{ψ′_{∘}}), Reg_{g_{β}}(E, V_{12}) = log_{𝔭}(P_{ψ_{•}}) · log_{𝔭}(P_{ψ_{∘}}),
Reg_{h_{α}}(E, V_{12}) = log_{𝔭}(P_{ψ′_{•}}) · log_{𝔭}(P_{ψ_{∘}}), Reg_{h_{β}}(E, V_{12}) = log_{𝔭}(P_{ψ_{•}}) · log_{𝔭}(P_{ψ′_{∘}}).

The corresponding formulae for the p-adic iterated integrals ℒ_{p} ^{g_{α}}(f, g, h), ℒ_{p} ^{g_{β}}(f, g, h), ℒ_{p} ^{h_{α}}(f, g, h), and ℒ_{p} ^{h_{β}}(f, g, h) were proved in [DLR1, §3], by using the p-adic Gross-Zagier formula of [BDP13] to express these L-values in terms of products of p-adic logarithms of Heegner points. Theorem 3.3 of loc.cit. is one of the few pieces of theoretical evidence in support of Conjecture 3.12.

b) The case where p is inert in K. In this case, the eigenvalues of the Frobenius automorphism σ_{p} acting on V_{g} and V_{h} are of the form

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

Let (v_{ψ_{g}}, v_{ψ′_{g}}) be a eigenbasis of V_{g} for the action of G_{K} relative to the distinct characters ψ_{g} and ψ′_{g}, and let (v_{ψ_{h}}, v_{ψ′_{h}}) be a similar basis for V_{h}. These vectors can be scaled so that σ_{p} acts on them as

σ_{p}(v_{ψ_{g}}) = α_{g} · v_{ψ′_{g}},  σ_{p}(v_{ψ′_{g}}) = α_{g} · v_{ψ_{g}},  σ_{p}(v_{ψ_{h}}) = α_{g}^{−1} · v_{ψ′_{h}},  σ_{p}(v_{ψ′_{h}}) = α_{g}^{−1} · v_{ψ_{h}},

and therefore we may set

V_{g}^{α} = L · (v_{ψ_{g}} + v_{ψ′_{g}}),  V_{g}^{β} = L · (v_{ψ_{g}} − v_{ψ′_{g}}),  V_{h}^{α} = L · (v_{ψ_{h}} + v_{ψ′_{h}}),  V_{h}^{β} = L · (v_{ψ_{h}} − v_{ψ′_{h}}).

After setting

v_{ψ_{•}} := v_{ψ_{g}} ⊗ v_{ψ_{h}},  v_{ψ′_{•}} := v_{ψ′_{g}} ⊗ v_{ψ′_{h}},  v_{ψ_{∘}} := v_{ψ_{g}} ⊗ v_{ψ′_{h}},  v_{ψ′_{∘}} := v_{ψ′_{g}} ⊗ v_{ψ_{h}},

and letting

v_{•}^{+} := v_{ψ_{•}} + v_{ψ′_{•}} ,  v_{•}^{−} := v_{ψ_{•}} − v_{ψ′_{•}} ,  v_{∘}^{+} := v_{ψ_{∘}} + v_{ψ′_{∘}} ,  v_{•}^{−} := v_{ψ_{∘}} − v_{ψ′_{∘}} ,

it is easy to see that

(37)  \begin{matrix}V_{12}^{αα} = L · (v_{•}^{+} + v_{∘}^{+}), & V_{12}^{αβ} = L · (v_{•}^{−} + v_{∘}^{−}), \\ V_{12}^{βα} = L · (v_{•}^{−} − v_{∘}^{−}), & V_{12}^{ββ} = L · (v_{•}^{+} − v_{∘}^{+}). \end{matrix}

Note that σ_{p} acts on both V_{12}^{αα} and V_{12}^{ββ} with the eigenvalue 1, and on V_{12}^{αβ} and V_{12}^{βα} with the eigenvalue −1. In particular, while V_{g} and V_{h} are always regular at p, the tensor product V_{gh} ≃ V_{12} never enjoys this property, even though the vector spaces described in (37) are one dimensional.

Keeping the same notations as in (36), let

P_{•}^{+} := P_{ψ_{•}} + P_{ψ′_{•}} ,  P_{•}^{−} := P_{ψ_{•}} − P_{ψ′_{•}} ,  P_{∘}^{+} := P_{ψ_{∘}} + P_{ψ′_{∘}} ,  P_{∘}^{−} := P_{ψ_{∘}} − P_{ψ′_{∘}} .

With these notations, the enhanced regulators describing the associated generalised Kato classes are given by

R̃eg_{αα}(E, V_{12}) = log_{p}(P_{•}^{+}) · P_{∘} − log_{p}(P_{∘}^{+}) · P_{•},  R̃eg_{αβ}(E, V_{12}) = log_{p}(P_{•}^{−}) · P_{∘} − log_{p}(P_{∘}^{−}) · P_{•},
R̃eg_{βα}(E, V_{12}) = log_{p}(P_{•}^{−}) · P_{∘} + log_{p}(P_{∘}^{−}) · P_{•},  R̃eg_{ββ}(E, V_{12}) = log_{p}(P_{•}^{+}) · P_{∘} + log_{p}(P_{∘}^{+}) · P_{•}.

The four regulators Reg_{g_{α}}(E, V_{12}), Reg_{g_{β}}(E, V_{12}), Reg_{h_{α}}(E, V_{12}), and Reg_{h_{β}}(E, V_{12}) are all seen to be simple L^{×}-multiples of the expression

(38)  log_{p}(P_{•}^{+}) · log_{p}(P_{∘}^{−}) − log_{p}(P_{∘}^{+}) · log_{p}(P_{•}^{−}).

The resulting formula for ℒ_{p} ^{g_{α}}(f, g, h) predicted by Conjecture 3.8 has been extensively tested numerically in [DLR1, §3.3].

Remark 4.1. Even though the points P_{•}^{+}, P_{•}^{−}, P_{∘}^{+}, and P_{∘}^{−} that figure in the generalised Kato classes are in principle expressed as linear combinations of Heegner points, the methods used to prove Conjecture 3.8 when p is split in K, which are based on the p-adic Gross-Zagier formula of [BDP13] and on properties of the Katz p-adic L-function attached to K, seem to break down completely when p is inert in K. A theoretical understanding of the p-adic iterated integrals of [DLR1] in this setting would seem to require a new idea.

Remark 4.2. It is worth contrasting the expressions arising in (38) with the simpler formulae

log_{αα} R̃eg_{ββ}(E, V_{12}) = log_{ββ} R̃eg_{αα}(E, V_{12}) = log_{p}(P_{•}^{+}) · log_{p}(P_{∘}^{+}),
log_{αβ} R̃eg_{βα}(E, V_{12}) = log_{βα} R̃eg_{αβ}(E, V_{12}) = log_{p}(P_{•}^{−}) · log_{p}(P_{∘}^{−}).

In certain very special settings–notably, when the elliptic curve E has multiplicative reduction at p–these expressions arise as the leading terms of the p-adic L-series

ℒ_{p} ^{f}(f, g_{α}, h_{α}), …, ℒ_{p} ^{f}(f, g_{β}, h_{β}).

Methods based on the Cerednik-Drinfeld theory of p-adic uniformisation of Shimura curves make it possible to relate these leading terms to the p-adic logarithms of Heegner points, leading to some indirect theoretical evidence for Conjecture 3.12 in the setting where p is inert in K. See [DR3] for a description of this approach.

4.3.2. The rank (2, 0) setting. Assume, after possibly interchanging V_{•} and V_{∘}, that L(E, V_{•}, s) has a double zero at s = 1 and that L(E, V_{∘}, 1) ≠ 0. In this case, Conjectures BSD(E, V_{•}) and BSD(E, V_{∘}) predict that

(E(H)_{L} ⊗ V_{12})^{G_{ℚ}} = (E(H)_{L} ⊗ V_{•})^{G_{ℚ}}

is two-dimensional over L. Choose a basis (P, Q) for this vector space. It is natural to write

P = P_{ψ_{•}} ⊗ v_{ψ′_{•}} + P_{ψ′_{•}} ⊗ v_{ψ_{•}} , Q = Q_{ψ_{•}} ⊗ v_{ψ′_{•}} + Q_{ψ′_{•}} ⊗ v_{ψ_{•}} ,

where (P_{ψ_{•}} , Q_{ψ_{•}}) and (P_{ψ′_{•}} , Q_{ψ′_{•}}) are bases for the two-dimensional subspaces of E(H)_{L} on which G_{K} acts via the characters ψ_{•} and ψ′_{•} respectively.

As in the rank (1, 1) setting, the shape of the enhanced regulators attached to V_{12} and to the basis (P, Q) depend very much on whether the prime p is split or inert in K.

a) The case where p is split in K. In this case, letting p = 𝔭 𝔭′ be the factorisation of p into distinct primes of K, we can set

α_{g} = ψ_{g}(𝔭), β_{g} = ψ_{g}(𝔭′), α_{h} = ψ_{h}(𝔭), β_{h} = ψ_{h}(𝔭′),

so that

α_{g}α_{h} = ψ_{•}(𝔭), α_{g}β_{h} = ψ_{∘}(𝔭), β_{g}α_{h} = ψ_{∘}(𝔭′), β_{g}β_{h} = ψ_{•}(𝔭′).

The decomposition of the G_{K_{𝔭}} = G_{ℚ_{p}} representations attached to (35) into σ_{p}-eigenspaces is also stable under the action of the global Galois group G_{K}, and is described by:

V_{12}^{αα} = V_{•}^{ψ_{•}} , V_{12}^{αβ} = V_{∘}^{ψ_{∘}} , V_{12}^{βα} = V_{∘}^{ψ′_{∘}} , V_{12}^{ββ} = V_{•}^{ψ′_{•}} .

It follows that, up to multiplication by L^{×}, (39)

R̃eg_{αα}(E, V_{12}) = log_{𝔭}(P_{ψ′_{•}}) · Q − log_{𝔭}(Q_{ψ′_{•}}) · P,  R̃eg_{αβ}(E, V_{12}) = 0,
R̃eg_{βα}(E, V_{12}) = 0,  R̃eg_{ββ}(E, V_{12}) = log_{𝔭}(P_{ψ_{•}}) · Q − log_{𝔭}(Q_{ψ_{•}}) · P.

This suggests that the generalised Kato classes κ(f, g_{α}, h_{α}) and κ(f, g_{β}, h_{β}) give non-trivial elements of the two-dimensional vector space (E(H)_{L} ⊗ V_{•})^{G_{ℚ}}, while the generalised Kato classes κ(f, g_{α}, h_{β}) and κ(f, g_{β}, h_{α}) should vanish. Furthermore, a direct calculation shows that

Reg_{g_{α}}(E, V_{12}) = Reg_{g_{β}}(E, V_{12}) = Reg_{h_{α}}(E, V_{12}) = Reg_{h_{β}}(E, V_{12}) = 0,

which is consistent with the fact, proved in [DLR1, §3.2], that all the p-adic iterated integrals attached to (f, g, h) vanish in the rank (2, 0) setting when p is split in K. In this case the generalised Kato classes carry more arithmetic information that the p-adic iterated integrals which describe (certain of) their p-adic logarithms. This represents yet another setting where Conjecture 3.12 is a genuine strengthening of Conjecture 3.8 of [DLR1].

b) The case where p is inert in K. After scaling the points P_{ψ_{•}}, and P_{ψ′_{•}}, Q_{ψ_{•}} and Q_{ψ′_{•}} in such a way that

σ_{p}(P_{ψ_{•}}) = P_{ψ′_{•}} , σ_{p}(P_{ψ′_{•}}) = P_{ψ_{•}} , σ_{p}(Q_{ψ_{•}}) = Q_{ψ′_{•}} , σ_{p}(Q_{ψ′_{•}}) = Q_{ψ_{•}}

and letting

P^{±} := P_{ψ_{•}} ± P_{ψ′_{•}} , Q^{±} := Q_{ψ_{•}} ± Q_{ψ′_{•}} ,

the enhanced regulators are given by (40)

R̃eg_{αα}(E, V_{12}) = log_{p}(P^{+}) · Q − log_{p}(Q^{+}) · P,  R̃eg_{αβ}(E, V_{12}) = log_{p}(P^{−}) · Q − log_{p}(Q^{−}) · P,
R̃eg_{βα}(E, V_{12}) = log_{p}(P^{−}) · Q − log_{p}(Q^{−}) · P,  R̃eg_{ββ}(E, V_{12}) = log_{p}(P^{+}) · Q − log_{p}(Q^{+}) · P.

In this case, the four regulators Reg_{g_{α}}(E, V_{12}), …, Reg_{h_{β}}(E, V_{12}) attached to (f, g, h) are explicit multiples of the expression

(41)  log_{p}(P^{+}) · log_{p}(Q^{−}) − log_{p}(Q^{+}) · log_{p}(P^{−}).

See [DLR1, Ex. 3.14] for some numerical verifications of the agreement between this value and the p-adic iterated integrals attached to (f, g, h).

Equations (39) and (40) combined with Conjecture 3.12 suggest that the generalised Kato classes always generate the Mordell-Weil group (E(H)_{L} ⊗ V_{•})^{G_{ℚ}} (tensored over L with ℚ_{p}) in the rank (2, 0) setting. Since the irreducible representation V_{•} occurs with multiplicity two in E(H)_{L}, none of the V_{•}- isotypic part of the Mordell-Weil group is expected to be accounted for by Heegner points, as discussed in the introduction.

4.4. Real multiplication classes and Stark-Heegner points. In this chapter we consider the setting where g and h are theta series attached to characters ψ_{g} and ψ_{h} of mixed signature of the same real quadratic field K. In that case, we have, exactly as in Section 4.3,

V_{g} = Ind_{K}^{ℚ} ψ_{g}, V_{h} = Ind_{K}^{ℚ} ψ_{h},

and

V_{gh} = V_{•} ⊕ V_{∘} := Ind_{K}^{ℚ} ψ_{•} ⊕ Ind_{K}^{ℚ} ψ_{∘}, where ψ_{•} = ψ_{g}ψ_{h}, ψ_{∘} = ψ_{g}ψ′_{h}.

The characters ψ_{•} and ψ_{∘} are also ring class characters of K, with one totally even, and the other totally odd. Once again, it is convenient to assume that V_{•} and V_{∘} are both irreducible, i.e., that neither ψ_{•} nor ψ_{∘} is a genus character of K.

As in the case where K is imaginary, the study of the generalised Kato classes divides naturally into the rank (1, 1) and rank (2, 0) settings, depending on the orders of vanishing of L(E/K, ψ_{•}, s) and L(E/K, ψ_{∘}, s) (or, alternately, on the dimensions of (E(H)_{L}⊗V_{•})^{G_{ℚ}} and (E(H)_{L}⊗V_{∘})^{G_{ℚ}}), and continue to depend in a crucial way on whether p is split or inert in K. In all four cases, the formulae for the enhanced regulators are identical to those obtained in Section 4.3, so it is unecessary to reproduce them here, contenting ourselves with the following comments in connection with the rank (1, 1) setting.

a) The case where p is split in K. This setting, where the greatest amount of theoretical evidence was available when K is imaginary quadratic, thanks to the theory of Heegner points, is a lot more mysterious when K is real quadratic. With notations being the same as in Section 4.3.1, we have

(42) R̃eg_{αα}(E, V_{12}) = log_{𝔭}(P_{ψ′_{•}}) · P_{∘},  R̃eg_{αβ}(E, V_{12}) = log_{𝔭}(P_{ψ′_{∘}}) · P_{•}, R̃eg_{βα}(E, V_{12}) = log_{𝔭}(P_{ψ_{∘}}) · P_{•},  R̃eg_{ββ}(E, V_{12}) = log_{𝔭}(P_{ψ_{•}}) · P_{∘},

and

(43) Reg_{g_{α}}(E, V_{12}) = log_{𝔭}(P_{ψ′_{•}}) · log_{𝔭}(P_{ψ′_{∘}}), Reg_{g_{β}}(E, V_{12}) = log_{𝔭}(P_{ψ_{•}}) · log_{𝔭}(P_{ψ_{∘}}), Reg_{h_{α}}(E, V_{12}) = log_{𝔭}(P_{ψ′_{•}}) · log_{𝔭}(P_{ψ_{∘}}), Reg_{h_{β}}(E, V_{12}) = log_{𝔭}(P_{ψ_{•}}) · log_{𝔭}(P_{ψ′_{∘}}).

This setting has special appeal in connection with an eventual (for now, highly conjectural, and not even clearly fomulated) theory of Stark-Heegner points over ring class fields of real quadratic fields. It would be of great interest to relate (conjecturally, at least) the regulators in (42) and in (43) to generalised Kato classes and p-adic iterated integrals, respectively.

The obstruction to doing this is that the modular forms g and h (more precisely, their stabilisations) fail to obey Hypothesis IV in Section 2. When g is a modular form of RM type which is regular at a prime p which splits in K, the Stark unit u_{g_{α}} is also unavailable, and an analogue of Conjecture 3.8 has yet to be formulated precisely in this setting. Because of the tantalising connection with Stark-Heegner points defined over ring class fields of K, it would be of great interest to extend the Conjectures of [DLR1], as well as Conjecture 3.12, to the real quadratic context. A first step has been made in [DLR3] towards understanding the periods of §2 in this setting.

b) The case where p is inert in K. The formulae for the enhanced regulators are identical to those in Part b) of Section 4.3.1, namely:

R̃eg_{αα}(E, V_{12}) = log_{p}(P_{•}^{+}) · P_{∘} − log_{p}(P_{∘}^{+}) · P_{•},  R̃eg_{αβ}(E, V_{12}) = log_{p}(P_{•}^{−}) · P_{∘} − log_{p}(P_{∘}^{−}) · P_{•},
R̃eg_{βα}(E, V_{12}) = log_{p}(P_{•}^{−}) · P_{∘} + log_{p}(P_{∘}^{−}) · P_{•},  R̃eg_{ββ}(E, V_{12}) = log_{p}(P_{•}^{+}) · P_{∘} + log_{p}(P_{∘}^{+}) · P_{•}.

The p-adic logarithms of these enhanced regulators ought to involve linear combinations of products of logarithms of so-called Stark-Heegner points. This prediction has been extensively tested numerically in [DLR1, §4.2].

Remark 4.3. The logarithms of the generalised Kato classes that are not amenable to expressions in terms of p-adic iterated integrals are expected to admit particularly simple expressions, as suggested by the formulae

log_{αα} R̃eg_{ββ}(E, V_{12}) = log_{ββ} R̃eg_{αα}(E, V_{12}) = log_{p}(P_{•}^{+}) · log_{p}(P_{∘}^{+}),
log_{αβ} R̃eg_{βα}(E, V_{12}) = log_{βα} R̃eg_{αβ}(E, V_{12}) = log_{p}(P_{•}^{−}) · log_{p}(P_{∘}^{−}).

In the special case where E has multiplicative reduction at p, the article [DR3] in progress proves a formula of the shape

log_{αα} κ(f, g_{α}, h_{α}) = log_{p}(P_{•}^{?±}) · log_{p}(P_{∘}^{?±}),

where P_{•}^{?±} and P_{∘}^{?±} are the Stark-Heegner points of [Da01], and the sign that arises depends on whether E has split or non-split multiplicative reduction at p.

4.5. Adjoint classes. The case where h is dual to g is of considerable arithmetic interest, since in that case the representation

V_{gh} = ℚ_{p} ⊕ M_{g}, (where M_{g} := Ad^{0}(V_{g})),

admits the trivial representation as a constituent. The generalised Kato classes attached to g and h may then, in appropriate circumstances, contribute to the Mordell-Weil group E(ℚ), and it is interesting to understand when this occurs.

The Hasse-Weil-Artin L-series

L(E, V_{gh}, s) = L(E, s)L(E, M_{g}, s)

has a double zero at s = 1 in one of the following three cases:

(1) the rank (0, 2) setting where L(E, 1) ≠ 0 and L(E, M_{g}, s) has a double zero at s = 1;

(2) the rank (1, 1) case where L(E, s) and L(E, M_{g}, s) each vanish to order 1 at s = 1;

(3) the rank (2, 0) setting where L(E, s) has a double zero at the center and L(E, M_{g}, 1) ≠ 0. This case is particularly intriguing for its direct connection with the arithmetic of elliptic curves of rank two over ℚ.

In all the examples that will be treated below, we always set

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

so that

α_{g}α_{h} = 1, α_{g}β_{h} = α_{g}/β_{g}, β_{g}α_{h} = \frac{β_{g}}{α_{g}} , β_{g}β_{h} = 1.

4.5.1. The rank (0, 2) setting. Let (P, Q) be an L-basis for (E(H)_{L} ⊗ V_{12})^{G_{ℚ}} = (E(H)_{L} ⊗ M_{g})^{G_{ℚ}}, and write

P = P\frac{α}{β} ⊗ v_{β/α} + P_{1} ⊗ v_{1} + P_{\frac{β}{α}} ⊗ v_{α/β},
Q = Q\frac{α}{β} ⊗ v_{β/α} + Q_{1} ⊗ v_{1} + Q_{\frac{β}{α}} ⊗ v_{α/β},

where v_{β/α}, v_{1}, and v_{α/β} are bases for the σ_{p}-eigenspaces of W_{g} attached to the eigenvalues \frac{β_{g}}{α_{g}}, 1, and α_{g}/β_{g} respectively. Then the four enhanced regulators are given, up to L^{×}, as follows:

R̃eg_{αα}(E, V_{12}) = log_{p}(P_{1}) · Q − log_{p}(Q_{1}) · P,  R̃eg_{αβ}(E, V_{12}) = log_{p}(P_{\frac{β}{α}}) · Q − log_{p}(Q_{\frac{β}{α}}) · P,
R̃eg_{βα}(E, V_{12}) = log_{p}(P\frac{α}{β}) · Q − log_{p}(Q\frac{α}{β}) · P,  R̃eg_{ββ}(E, V_{12}) = log_{p}(P_{1}) · Q − log_{p}(Q_{1}) · P.

Conjecture 3.12 suggests in this case that the generalised Kato class κ(f, g_{α}, h_{β}) generates the kernel of the map log_{αβ} in the two dimensional ℚ_{p}-vector space (E(H)_{L} ⊗ M_{g})^{G_{ℚ}} ⊗_{L} ℚ_{p}.

We refer the reader to [DLR1, Example 5.4] for the numerical verification of Conjecture 3.8 for two different instances in this setting.

4.5.2. The rank (1, 1) setting. Let P be a generator of E(ℚ)_{L} and let Q be a generator of (E(H)_{L} ⊗ M_{g})^{G_{ℚ}}. With the same notational conventions as before, we find: (44)

R̃eg_{αα}(E, V_{12}) = log_{p}(P) · Q − log_{p}(Q_{1}) · P,  R̃eg_{αβ}(E, V_{12}) = log_{p}(Q_{\frac{β}{α}}) · P,
R̃eg_{βα}(E, V_{12}) = log_{p}(Q\frac{α}{β}) · P,  R̃eg_{ββ}(E, V_{12}) = log_{p}(P) · Q − log_{p}(Q_{1}) · P.

In contrast, we have

(45)

Reg_{g_{α}}(E, V_{12}) = log_{p}(P) · log_{p}(Q_{\frac{β}{α}})  Reg_{g_{β}}(E, V_{12}) = log_{p}(P) · log_{p}(Q\frac{α}{β}),
Reg_{h_{α}}(E, V_{12}) = log_{p}(P) log_{p}(Q\frac{α}{β}),  Reg_{h_{β}}(E, V_{12}) = log_{p}(P) · log_{p}(Q_{\frac{β}{α}}).

Many numerical examples where the p-adic iterated integrals atttached to (f, g, h) are seen to agree with these regulators are described in in [DLR1, §5]. It is worth noting that the expression log_{p}(Q_{1}) that appears in the enhanced regulators of (44) disappears from the regulators (45) that arose in [DLR1].

4.5.3. The rank (2, 0) setting. Let (P, Q) be an L-basis of the two-dimensional L-vector space E(ℚ)_{L}. With the same notational conventions as before, we find:

R̃eg_{αα}(E, V_{12}) = log_{p}(P) · Q − log_{p}(Q) · P,  R̃eg_{αβ}(E, V_{12}) = 0,
R̃eg_{βα}(E, V_{12}) = 0,  R̃eg_{ββ}(E, V_{12}) = log_{p}(P) · Q − log_{p}(Q) · P.

In other words, the generalised Kato classes κ(f, g_{α}, h_{α}) and κ(f, g_{β}, h_{β}) give (essentially, the same, up to L^{×}-multiples) canonical element of E(ℚ)_{ℚ_{p}}, which is expected to be non-trivial precisely when

L′′(E, 1) ≠ 0, L(E, M_{g}, 1) ≠ 0.

Note that on the other hand

Reg_{g_{α}}(E, V_{12}) = Reg_{g_{β}}(E, V_{12}) = Reg_{h_{α}}(E, V_{12}) = Reg_{h_{β}}(E, V_{12}) = 0.

This last example gives yet another instance where Conjecture 3.12 represents a genuine strengthening of the elliptic Stark conjectures of [DLR1]. It predicts that that generalised Kato classes of the form κ(f, g_{α}, ḡ_{1/α}) ought to give non-trivial elements in the pro-p-Selmer groups of elliptic curves of rank two over ℚ, when the auxiliary L-value L(E, M_{g}, 1) is non-zero. Testing this prediction experimentally seems to present an interesting challenge.

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

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H. D.: Montreal, Canada

   E-mail address: darmon@math.mcgill.ca

V. R.: Departament de Matemàtiques, Universitat Politècnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain

   E-mail address: victor.rotger@upc.edu