Elliptic curves of rank two and generalised Kato classes
- Henri Darmon1Email author and
- Victor Rotger2
https://doi.org/10.1186/s40687-016-0074-9
© The Author(s) 2016
Received: 3 July 2015
Accepted: 4 February 2016
Published: 24 August 2016
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 Bertolini et al. (J Algebr Geom 24:569–604, 2015) and Darmon and Rotger (J AMS 2016), 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 centre. The generalised Kato class denoted \(\kappa (f,g,h)\) 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 \(\mathrm {Gal\,}(H/\mathbb {Q})\) 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 \(\mathbb {Q}\) attached to f. One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that \(\kappa (f,g,h)\) 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 \(V_{gh}:= V_g\otimes V_h\). 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 \(\kappa (f,g,h)\) is expected to be trivial when \({\mathrm {ord}}_{s=1}L(E,V_{gh},s) >2\). The main new contribution of this article is a conjecture expressing \(\kappa (f,g,h)\) as a canonical point in \((E(H)\otimes V_{gh})^{G_\mathbb {Q}}\) when \({\mathrm {ord}}_{s=1} L(E,V_{gh},s)=2\). This conjecture strengthens and refines the main conjecture of Darmon et al. (Forum Math Pi 3:e8, 2015) and supplies a framework for understanding the results of Darmon et al. (2015), Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016).
Mathematics Subject Classification
1 Background and motivation
The theme of modularity of p-adic Galois representations has occupied centre 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 Coleman–Mazur eigencurve parameterising 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 [11, 15] and [19], can be viewed as part of the larger programme of understanding the modularity of (non-semisimple) p-adic Galois representations.
To completely describe the open sub-Shimura varieties of the modular curve \(X_0(N)\) over \(\mathbb {Q}\), 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 \({\mathcal O}\subset K\), the associated closed sub-Shimura variety \(\Sigma _{{\mathcal O}}\subset X_0(N)\) consists of CM points for \({\mathcal O}\) and is the coarse moduli space of elliptic curves A with level N structure equipped with an optimal embedding \(\iota :{\mathcal O}\longrightarrow \mathrm {End}(A)\) (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 \(\Sigma _{{\mathcal O}}\) is isomorphic over K (at least, when the discriminant of \({\mathcal O}\) is prime to N) to \(\phi _K(N)\) copies of \({{\mathrm{spec}}}(H_{\mathcal O})\), where \(\phi _K(N)\) is the number of primitive ideals of K of norm N and \(H_{{\mathcal O}}\) is the ring class field of K attached to \({\mathcal O}\), whose Galois group over K is canonically identified with the Picard group of \({\mathcal O}\) via global class field theory.
Theorem 1.1
- (a)
The curve \(E' = E{\setminus } \{ P_1,P_2\}\) 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 \(E(\mathbb {Q})\otimes \mathbb {Q}\) and \(L\!\!L\!\!I(E/\mathbb {Q})\) is finite;
- (d)
for all primes p, the group \({{\mathrm{Ext}}}^1_\mathrm{fin}(\mathbb {Q}_p,H^1(E))\) of extensions of p-adic representations of the Galois group of \(\mathbb {Q}\) that are cristalline at p is one-dimensional over \(\mathbb {Q}_p\).
Sketch of proof
The implication (b) \(\Rightarrow \) (c) was subsequently proved by Kolyvagin [25], who parlayed the non-triviality of \(P_{E,K}\) into a bound on the Mordell–Weil rank and the Selmer group of E over K.
The implication (c) \(\Rightarrow \) (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) \(\Rightarrow \) (a) follows from Skinner’s “converse of the Gross–Zagier–Kolyvagin Theorem” [33]. 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 \(\mathbb {Q}\) [35], an important variant explored by Xin Wan in his Ph.D. thesis [39], and the p-adic analogue of [21] formulated and proved in [8].
- (b’)
The Hasse–Weil–Artin L-series \(L(E, V_\psi ,s)\) has a simple zero at \(s=1\);
- (c’)
the representation \(V_\psi \) occurs with multiplicity one in \(E(H)\otimes \bar{\mathbb {Q}}_p\), and the \(V_\psi \)-isotypic component of the \(L\!\!L\!\!I(E/H)\) is finite;
- (d’)
the group \({{\mathrm{Ext}}}^1_\mathrm{fin}(V_\psi ,H^1(E)\otimes \bar{\mathbb {Q}}_p)\) is one-dimensional over \(\bar{\mathbb {Q}}_p\), and generated by \(\kappa \).
- (1)
Very few Artin representations arise in the cohomology of the 0-dimensional Shimura varieties \(\Sigma _{\mathcal O}\), which are not even rich enough to capture all of the irreducible two-dimensional Artin representations of \(\mathbb {Q}\). The open Shimura varieties \(Y_{{\mathcal O}}(N)\) thus appear to give no purchase on \({{\mathrm{BSD}}}(E,\varrho )\) when \(\varrho \) is not induced from a ring class character of an imaginary quadratic field.
- (2)
Theorem 1.1 suggests that the modularity of elements of \({{\mathrm{Ext}}}^1_{{{\mathrm{fin}}}}(V_\psi ,H^1(E))\) 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 \(H^1(Y_{{\mathcal O}}(N))\).
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 \(\kappa \in {{\mathrm{Ext}}}^1_{{{\mathrm{fin}}}}(V_1,V_2)\) arising as a subquotient of the cohomology of an open Shimura variety. Is \({{\mathrm{Ext}}}^1_{{{\mathrm{fin}}}}(V_1,V_2)\) 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\((E(\mathbb {Q}))>1\). (But see the inspiring article [29], as well as the striking ongoing work of Zhiwei Yun and Wei Zhang in the function field case, for some tantalising 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 the 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 \(\mathbb {Q}\) 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 [22] by Harris, Lan, Taylor, and Thorne of Galois representations attached to non-self-dual automorphic forms on \(\mathrm {GL}_n\). 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 [34, Thm. B]:
Theorem 1.3
Let E be an elliptic curve over \(\mathbb {Q}\). If L(E, s) vanishes to even order \(\ge 2\) at \(s=1\), then the Selmer group \({{\mathrm{Ext}}}^1_{{{\mathrm{fin}}}}(\mathbb {Q}_p, H^1(E))\) 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 \(\varrho \) by \(H^1(E)\) for a large class of self-dual Artin representations \(\varrho \) of dimension 4 (and their lower-dimensional subrepresentations, in case \(\varrho \) is reducible) arising as the tensor product \(\varrho =\varrho _1\otimes \varrho _2\) of a pair of odd, two-dimensional Artin representations. The construction of these classes is one of the main results of [19] (resp. [11]) when both \(\varrho _1\) and \(\varrho _2\) are irreducible (resp. when exactly one of \(\varrho _1\) and \(\varrho _2\) 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 [23] of Kato (cf. also [6, 32]) on \({{\mathrm{BSD}}}(E,\chi )\) for \(\chi \) a Dirichlet character. Like Heegner points in the setting of \({{\mathrm{BSD}}}(E,V_\psi )\), generalised Kato classes enjoy close relations to (p-adic) Hasse–Weil–Artin L-functions attached to E and \(\varrho \), but unlike Heegner points, they are expected to generate a non-trivial subgroup of the Selmer group attached to E and \(\varrho \) precisely when \({\mathrm {ord}}_{s=1}L(E,\varrho ,s) = 2\). The formulae of [19] (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 \((E(H)\otimes V_{\varrho })^{G_\mathbb {Q}}\) when this latter space is two-dimensional. This conjecture strengthens and refines the “elliptic Stark conjecture” of [15], and provides a framework for understanding the results of [11, 15] and [19]. The settings in which \(\varrho \) 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 Sect. 3.4, but Sect. 2 can be skipped on a first reading by the reader wishing to get a quick feeling for the generalised Kato classes described in Sects. 3.1 and 3.2. On the other hand, it is also worth noting that Sect. 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.
- (I)
The prime p splits completely in \(L/\mathbb {Q}\), so that L is equipped with an embedding into \(\mathbb {Q}_p\) which will be fixed from now on. This assumption, which is made solely to lighten the notations and could easily be dispensed with, allows \(\varrho \) to be viewed as a \(\mathbb {Q}_p\)-linear representation via the natural action of \(G_\mathbb {Q}\) on the \(\mathbb {Q}_p\)-vector space \(V\otimes _L\mathbb {Q}_p\).
- (II)The representation V is unramified at p. There is then a well-defined arithmetic frobenius elementacting canonically on V, and the characteristic polynomial of \(\varrho (\mathrm {Fr}_p)\) is equal to the Hecke polynomial$$\begin{aligned} \mathrm {Fr}_p\in \mathrm {Gal\,}(H/\mathbb {Q}) \end{aligned}$$attached to g.$$\begin{aligned} x^2-a_p(g) x + \chi (p) =: (x-\alpha _g)(x-\beta _g) \end{aligned}$$
- (III)
The modular form g is regular at p, i.e. \(\alpha _g\ne \beta _g\). After possibly enlarging L, it may also be assumed that this coefficient field contains the roots of unity \(\alpha _g\) and \(\beta _g\).
- (IV)
The representation \(\varrho _g\) 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 [15, §1.1].
By a theorem of Hida, there exists a finite flat extension \(\Lambda _g\) of the Iwasawa algebra \(\Lambda \) and a Hida family \({\underline{g}}\in \Lambda _{\underline{g}}[[q]]\) of tame level N and tame character \(\chi \) passing through the p-stabilised weight one eigenform \(g_\alpha \). 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 [1].
- (a)There is a locally free \(\Lambda _g\)-module \({\mathbb {V}}_{g}\) of rank two, affording Hida’s ordinary \(\Lambda \)-adic Galois representationwhich 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 specialisations of \({\underline{g}}\).$$\begin{aligned} \varrho _{{\underline{g}}}: G_\mathbb {Q}\longrightarrow \mathrm {Aut}_{\Lambda _g}({\mathbb {V}}_{g}) \end{aligned}$$
- (b)The restriction of \({\mathbb {V}}_{g}\) to \(G_{\mathbb {Q}_p}\) admits a stable filtrationwhere both \({\mathbb {U}}_{g} \) and \({\mathbb {W}}_{g}\) are flat \(\Lambda _{\underline{g}}[G_{\mathbb {Q}_p}]\)-modules that are locally free of rank one over \(\Lambda _{\underline{g}}\), and the quotient \({\mathbb {W}}_{g}\) is unramified, with \(\mathrm {Fr}_p\) acting on \({\mathbb {W}}_{g}\) as multiplication by the p-th Fourier coefficient \(a_p({\underline{g}})\).$$\begin{aligned} 0 \longrightarrow {\mathbb {U}}_{g} \longrightarrow {\mathbb {V}}_{g} \longrightarrow {\mathbb {W}}_{g} \longrightarrow 0, \end{aligned}$$
- (c)Let \({\mathbb {Q}_p^{{{\mathrm{nr}}}}}\) denote the maximal unramified extension of \(\mathbb {Q}_p\) and let \({\widehat{\mathbb {Q}}_p^{{{\mathrm{nr}}}}}\) denote its p-adic completion. In [30], Ohta constructs a canonical \(\Lambda _g\)-adic periodcorresponding to the normalised \(\Lambda \)-adic eigenform \({\underline{g}}\) under the isomorphism in Theorem (A) of the introduction of [30].$$\begin{aligned} \omega _{{\underline{g}}} \in D({\mathbb {W}}_{g}) := ( {\widehat{\mathbb {Q}}_p^{{{\mathrm{nr}}}}} {\hat{\otimes }} {\mathbb {W}}_{g})^{G_{\mathbb {Q}_p}}, \end{aligned}$$
- (d)There is a natural perfect Galois-equivariant duality, given in Theorem (B) of the introduction of [30],where \(G_\mathbb {Q}\) acts on the module \(\Lambda _g\) of the right-hand side via the determinant of \(\varrho _{{\underline{g}}}\).$$\begin{aligned} {\mathbb {U}}_{g} \times {\mathbb {W}}_{g} \longrightarrow \Lambda _g(\det (\varrho _{{\underline{g}}})), \end{aligned}$$
- (a’)A non-canonical isomorphism of \(\mathbb {Q}_p[G_\mathbb {Q}]\)-modules$$\begin{aligned} \Phi _{g_\alpha }: V_g:= \mathbb {{\mathbb {V}}}_{g} \otimes _{y_g} \mathbb {Q}_p \mathop {\longrightarrow }\limits ^{\sim } V\otimes _L \mathbb {Q}_p. \end{aligned}$$
- (b’)A non-trivial \(G_{\mathbb {Q}_p}\)-stable filtrationof \(V_g\) by one-dimensional subspaces, where \(U_{g} := {\mathbb {U}}_{g}\otimes _{y_g} \mathbb {Q}_p\) and \(W_{g} := {\mathbb {W}}_{g}\otimes _{y_g} \mathbb {Q}_p\). The Frobenius element \(\mathrm {Fr}_p\) acts on \(W_{g}\) and \(U_g\) as multiplication by \(\alpha _g\) and \(\beta _g\), respectively. Since these eigenvalues are assumed to be distinct, the exact sequence above splits canonically, leading to the identifications$$\begin{aligned} 0 \longrightarrow U_{g} \longrightarrow V_{g} \longrightarrow W_{g} \longrightarrow 0 \end{aligned}$$$$\begin{aligned} U_g = V_g^\beta , \qquad W_g = V_g^\alpha , \qquad V_g = U_g \oplus W_g = V_g^\beta \oplus V_g^\alpha . \end{aligned}$$
- (c’)Specialising Ohta’s period leads to a canonical element$$\begin{aligned} \omega _{g_\alpha } := y_g(\omega _{{\underline{g}}}) \in D(V_g^{\alpha }) := (\mathbb {Q}_p^{{{\mathrm{nr}}}} \otimes V_{g}^\alpha )^{G_{\mathbb {Q}_p}} = (H_p \otimes V_g^\alpha )^{G_{\mathbb {Q}_p}}. \end{aligned}$$(6)
- (d’)The duality in (d) above specialises via \(y_g\) to a canonical pairing of \(\mathbb {Q}_p\)-vector spaceswhich induces a pairing by functoriality (denoted by the same symbol by a slight abuse of notation):$$\begin{aligned} \langle \ , \rangle : V_{g}^\beta \times V_{g}^\alpha \longrightarrow \mathbb {Q}_p(\chi ), \end{aligned}$$When this pairing is perfect, it can be used to define a period \(\eta _{g_\alpha } \in D(V_g^\beta )\), as the unique element satisfying$$\begin{aligned} \langle \ , \ \rangle : D(V_g^\beta ) \times D(V_g^\alpha ) \longrightarrow D(\mathbb {Q}_p(\chi )). \end{aligned}$$(7)where$$\begin{aligned} \langle \eta _{g_\alpha }, \omega _{g_\alpha } \rangle = {\mathfrak {g}}(\chi ) \otimes 1, \end{aligned}$$is the Gauss sum attached to the Dirichlet character \(\chi \), viewed as an element of \(H_p\) by assigning an \({\mathfrak {f}}_\chi \)-th root of unity in \(H_p\) to the complex number \(e^{2\pi i/{\mathfrak {f}}_\chi }\).$$\begin{aligned} {\mathfrak {g}}(\chi ) := \sum _{j=1}^{{\mathfrak {f}}_\chi )} \chi (j) e^{2\pi i j/{\mathfrak {f}}_\chi } \end{aligned}$$
This expression is a canonical p-adic period attached to the eigenform \(g_\alpha \) and can be viewed as a p-adic avatar of the Petersson norm of g.
Conjecture 2.1
Remark 2.2
It would be interesting to test this conjecture numerically. To the extent that \(\mathcal {L}_{g_\alpha }\) 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 [36].
3 Generalised Kato classes
3.1 Definition
Fix a rational prime p and continue to assume that hypotheses (I–IV) of the previous section hold for both the pairs \((\varrho _1,p)\) and \((\varrho _2,p)\).
One of the running assumptions of [19] 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, Prasad [31, 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 \(\mathbb {Q}\), 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 generalised Kato classes belong to the global cohomology group \(H^1(\mathbb {Q},V_{fgh}) = {{\mathrm{Ext}}}^1_{G_\mathbb {Q}}(\mathbb {Q}_p, V_{fgh})\), where \(\mathbb {Q}_p\) stands for the one-dimensional p-adic representation of \(G_\mathbb {Q}\) with trivial action and the Ext group is taken in the category of finite-dimensional \(\mathbb {Q}_p\)-vector spaces equipped with a continuous \(G_\mathbb {Q}\)-action (whose restriction to \(G_{\mathbb {Q}_p}\) need not be de Rham).
When \({\underline{g}}\) and \({\underline{h}}\) are cuspidal Hida families, the “weight two” classes \(\kappa (f,g_2,h_2)\) 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 \(\infty \) attached to \(L(V_{fg_kh_k},s)\) changes sign (while the other root numbers stay the same), so that this L-function vanishes to odd order at its centre. The presence of Gross–Kudla–Schoen diagonal cycles in this range is consistent with the Beilinson–Bloch conjecture for \(L(V_{fg_kh_k},s)\) and in fact provides evidence for it. (Cf. the preprint [38] of Yuan–Zhang–Zhang, where the case \(k=2\) is studied.) The fact that the extension \(\kappa (f,g_\alpha ,h_\alpha )\) does not arise directly in p-adic étale cohomology, but only as a p-adic limit of geometric Galois representations, explains why \(\kappa (f,g_\alpha ,h_\alpha )\) need not be cristalline at p in general.
The analogy with the work of Kato [23, 32] arises when the cuspidal Hida families \({\underline{g}}\) and \({\underline{h}}\) are replaced by Hida families of Eisenstein series. A global class \(\kappa _{_{BK}}(f,g_\alpha ,h_\alpha )\), designated as the Beilinson–Kato class attached to \((f,g_\alpha ,h_\alpha )\), is then defined as in (15), but replacing the étale Abel–Jacobi images \(\kappa (f,g_2,h_2)\) by p-adic étale regulators of Beilinson elements in the higher Chow group \(K_2(X_1(Np^s))={\mathrm {CH}}^2(X_1(Np^s),2)\) attached to a pair of modular units whose logarithmic derivatives give rise to \(g_2\) and \(h_2\). We refer the reader to [6] 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 \(\kappa (f, g_2,h_2)\) can be constructed geometrically as p-adic étale regulators of suitable Beilinson–Flach elements in the higher Chow group \(K_1(X_1(Np^s)^2)={\mathrm {CH}}^2(X_1(Np^s)^2,1)\) attached to a modular unit whose logarithmic derivative is \(h_2\). The limit cohomology class arising in (15) is then denoted \(\kappa _{_{BF}}(f,g_\alpha ,h_\alpha )\) and called the Beilinson–Flach class attached to the triple \((f,g_\alpha , h_\alpha )\). The Beilinson–Flach classes in p-adic families were introduced and studied in [2, 10] and [11]. See also [27] and [24] for more recent work leading to substantial extensions and refinements of the results of loc.cit. in the setting of Beilinson–Flach elements.
3.2 Basic properties
In this section we recall some of the main properties of the generalised Kato classes already established in [19] and [15].
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 [19]) as well as to the intermediate Beilinson–Flach setting (Theorem 3.10 of [11]).
Theorem 3.1
- (1)
the L-series \(L(E,V_{gh},s)\) has a zero of even order \(\ge 2\) at \(s=1\);
- (2)
the generalised Kato classes of (14) belong to the Selmer group attached to E and \(V_{gh}\).
Conjecture 3.2
- (a)
The L-series \(L(E,V_{gh},s)\) has a double zero at \(s=1\);
- (b)
the Mordell–Weil group \((E(H)_L\otimes V_{12})^{G_Q}\) is two-dimensional over L;
- (c)
the Selmer group \(H^1_{{{\mathrm{fin}}}}(\mathbb {Q},V_{fgh})\) is two-dimensional over \(\mathbb {Q}_p\).
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 \({{\mathrm{BSD}}}(E,V_{gh})\). 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 Sect. 4.
Theorem 3.4
Remark 3.5
This theorem says nothing about the quantity \(\log _p(R_{\alpha \alpha })\), which does not bear any direct relationship with p-adic L-values introduced above. We expect that \(\log _p(R_{\alpha \alpha })\) may rather be connected with the first derivative of a putative refinement of \({\mathscr {L}_p}^{f}(f,g_\alpha ,h_\alpha )\) 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 [19], 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
Theorem 3.4 and its corollary motivated the experimental study undertaken in [15] 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 \(L^\times \) rather than \(\mathbb {Q}_p^\times \).
To formulate this conjecture, recall that the class \(\kappa (f,g_\alpha ,h_\alpha )\) is expected to be trivial when \({\mathrm {ord}}_{s=1} L(E,V_{gh},s)>2\). Assume that this L-function has a double zero at the centre, which implies, by Conjecture \({{\mathrm{BSD}}}(E,V_{gh})\), that \((E(H)_L\otimes V_{12})^{G_\mathbb {Q}}\) is a two-dimensional L-vector space.
These points can be used to define a regulator attached to \(g_\alpha \), whose entries are the p-adic formal group logarithms of the coordinates attached to the vectors \(v_{gh}^{\alpha \alpha }\) and \(v_{gh}^{\alpha \beta }\) (and similarly for \(h_\alpha \)):
Definition 3.7
The main conjecture of [15] is the following,3 assuming \(\frac{\alpha _g}{\beta _g} \ne \pm 1\) (resp. \(\frac{\alpha _h}{\beta _h} \ne \pm 1\)) so that the Stark unit \(u_{g_\alpha }\) (resp. \(u_{h_\alpha }\)) is well defined:
Conjecture 3.8
Remark 3.9
Conjecture 3.8 lends itself to numerical verification and has been extensively tested in [15]. This is because the p-adic L-values \({\mathscr {L}_p}^{g_\alpha }(\breve{f},\breve{g}^*,\breve{h})\) and \({\mathscr {L}_p}^{h_\alpha }(\breve{f},\breve{g},\breve{h}^*)\) 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 [26] 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 [11] and [19].
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 \((E(H)\otimes V_{gh})^{G_\mathbb {Q}}\), specifying this position up to an ambiguity of \(L^\times \) rather than the less precise \(\mathbb {Q}_p^\times \) ambiguity of Theorem 3.4.
Definition 3.10
Definition 3.11
3.4 The conjecture
Conjecture 3.12
The following proposition shows that, under Conjecture 2.1 (relating the canonical period attached to g to the Stark unit \(u_{g_\alpha }\)) and Conjecture 3.2 (a mild strengthening of \({{\mathrm{BSD}}}(E,\varrho _{gh})\)), Conjecture 3.12 implies the main conjecture of [15]. 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 [15], while the strengthening described in Conjecture 3.12 lies for the moment beyond the range of explicit calculations (cf. Remark 3.9).
Proof
Remark 3.14
As explained in a number of the examples covered in Sect. 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
- (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 \(V_g\).
4.1 Beilinson–Kato classes
4.2 Beilinson–Flach classes
4.3 Complex multiplication classes and Heegner points
- (1)
The primitive L-series \(L(E,V_{\!\bullet },s)\) and \(L(E,V_{\!\circ },s)\) 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 \(L(E,V_{\!\bullet },s)\) or \(L(E,V_{\!\circ },s)\) has a double zero at \(s=1\), and the other is non-vanishing at the centre. 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
The description of the enhanced regulators attached to \(V_{12}\) and to \((P_{\!\bullet },P_{\!\circ })\) 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.
Remark 4.1
Even though the points \(P_{\!\bullet }^+\), \(P_{\!\bullet }^-\), \(P_{\!\circ }^+\), and \(P_{\!\circ }^-\) 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 [8] 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 [15] in this setting would seem to require a new idea.
Remark 4.2
4.3.2 The rank (2, 0) setting
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.
Equations (39) and (40) combined with Conjecture 3.12 suggest that the generalised Kato classes always generate the Mordell–Weil group \((E(H)_L\otimes V_{\!\bullet })^{G_\mathbb {Q}}\) (tensored over L with \(\mathbb {Q}_p\)) in the rank (2, 0) setting. Since the irreducible representation \(V_{\!\bullet }\) occurs with multiplicity two in \(E(H)_L\), none of the \(V_{\!\bullet }\)-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
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 \(L(E/K,\psi _{_{\!\bullet }},s)\) and \(L(E/K,\psi _{_{\!\circ }},s)\) (or, alternately, on the dimensions of \((E(H)_L\otimes V_{\!\bullet })^{G_\mathbb {Q}}\) and \((E(H)_L\otimes V_{\!\circ })^{G_\mathbb {Q}}\)), 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 Sect. 4.3, so it is unecessary to reproduce them here, contenting ourselves with the following comments in connection with the rank (1, 1) setting.
The obstruction to doing this is that the modular forms g and h (more precisely, their stabilisations) fail to obey Hypothesis IV in Sect. 2. When g is a modular form of RM type which is regular at a prime p which splits in K, the Stark unit \(u_{g_\alpha }\) 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 [15], as well as Conjecture 3.12, to the real quadratic context. A first step has been made in [17] towards understanding the periods of §2 in this setting.
Remark 4.3
4.5 Adjoint classes
- (1)
the rank (0, 2) setting where \(L(E,1)\ne 0\) 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 centre and \(L(E,M_g,1)\ne 0\). This case is particularly intriguing for its direct connection with the arithmetic of elliptic curves of rank two over \(\mathbb {Q}\).
4.5.1 The rank (0, 2) setting
We refer the reader to [15, 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
4.5.3 The rank (2, 0) setting
Acknowledgements
The first author was supported by an NSERC Discovery Grant, and the second author was supported by Grant MTM2012-34611.
The systematic shorthand \(H^i(X) := H^i_{{{\mathrm{et}}}}(X_{\bar{\mathbb {Q}}},\mathbb {Q}_p(i))\) for any variety X over \(\mathbb {Q}\) is adopted henceforth to lighten the notations.
Subsequently, (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.
We warn the reader that here in this note we have chosen to state the main conjecture of [15] in terms of the arithmetic frobenius \(\mathrm {Fr}_p\) at p, while in [15] we rather employ the geometric frobenius \(\sigma _p=\mathrm {Fr}_p^{-1}\). It is for this reason that the roles of \(\alpha \) and \(\beta \) are swapped in both formulations.
Notes
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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