Elliptic curves of rank two and generalized Kato classes

Darmon and Rotger Res Math Sci (2016) 3:27 DOI 10.1186/s40687-016-0074-9 R E S E A R C H Open Access Elliptic curves of rank two and generalised Kato classes Henri Darmon1* and Victor Rotger2 In memory of Robert Coleman. *Correspondence: [email protected] 1Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montreal, Canada Full list of author information is available at the end of the article 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)andDarmon 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 κ(f, g, h) is associated to a triple (f, g, h) consisting of an eigenform fof weight two and classical p-stabilised eigenforms gand hof weight one, corresponding to odd two-dimensional Artin representations Vgand Vhof 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 Hfor the modular abelian variety Eover Qattached to f. One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016)isthat κ(f, g, h)liesinthepro-pSelmer group of Eover Hprecisely when L(E, Vgh,1) =0, where L(E, Vgh,s)istheL-function of Etwisted by Vgh :=Vg⊗Vh. In the setting of interest, parity considerations imply that L(E, Vgh,s)vanishestoevenorderats=1, and the Selmer class κ(f, g, h) is expected to be trivial when ords=1L(E, Vgh,s)>2. The main new contribution of this article is a conjecture expressing κ(f, g, h) as a canonical point in (E(H)⊗Vgh)GQwhen ords=1L(E, Vgh,s)=2. This conjecture strengthens and refines the main conjecture of Darmon et al. (Forum Math Pi 3:e8, 2015)andsuppliesaframework for understanding the results of Darmon et al. (2015), Bertolini et al. (2015)andDarmon and Rotger (J AMS 2016). Mathematics Subject Classification: 11G18, 14G35 Contents 1 Background and motivation ................................... 2 Hida families and periods for weight one forms ........................ 3 Generalised Kato classes ..................................... 3.1 Definition .......................................... 3.2 Basic properties ....................................... ©2016 The Author(s). This 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. 0123456789().,–: vol Darmon and Rotger Res Math Sci (2016) 3:27 Page 2 of 32 3.3 Enhanced regulators .................................... 3.4 The conjecture ....................................... 4 Special cases ........................................... 4.1 Beilinson–Kato classes ................................... 4.2 Beilinson–Flach classes .................................. 4.3 Complex multiplication classes and Heegner points .................. 4.4 Real multiplication classes and Stark–Heegner points ................. 4.5 Adjoint classes ....................................... References .............................................. 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 explainshowmuchoftheprogressachievedontheBirchand 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. One of the most celebrated modularity results is the statement that all elliptic curves over Qarise as quotients of suitable modular curves: more precisely, that an elliptic curve Eover Qof conductor Nis equipped with a surjective parameterisation πE:X0(N)−→ E, (1) where X0(N) is the modular curve attached to Hecke’s congruence subgroup 0(N). This was proved in [37,40], and [12] by showing that the p-adic representation H1(E):=H1 et(E¯ Q,Qp)(1) =(lim ←,n E[pn]) ⊗ZpQp of GQ:=Gal ( ¯ Q/Q) arises as a quotient of the étale cohomology group1 H1(X0(N)) :=H1 et(X0(N)¯ Q,Qp(1)). The existence of a Galois-equivariant projection πE:H1(X0(N)) −→ H1(E)(2) is the real content of the breakthrough in [40]and[37], 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 of Eover Q.Thep-adic Galois representation H1(E) sits in the middle of the short exact excision sequence 0−→ H1(E)−→ H1(E)−→ H0()0−→ 0 of étale cohomology groups, where the subscript of 0 denotes the degree 0 elements of H0(). By analogy with (2), the curve Eis(provisionally)saidtobemodular if H1(E) 1The systematic shorthand Hi(X):=Hi et(X¯ Q,Qp(i)) for any variety Xover Qis 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. Darmon and Rotger Res Math Sci (2016) 3:27 Page 3 of 32 arises as a subquotient of H1(Y), where Yis an open sub-Shimura variety of X0(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 X0(N) over Q, note that the latter is the coarse moduli space of elliptic curves Awith 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 O⊂X0(N) consists of CM points for Oand is the coarse moduli space of elliptic curves Awith level Nstructure 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 Ois isomorphic over K(at least, when the discriminant of Ois prime to N)toφK(N) copies of spec(HO), where φK(N) is the number of primitive ideals of Kof norm Nand HOis the ring class field of Kattached to O, whose Galois group over Kis canonically identified with the Picard group of Ovia global class field theory. The complements YO(N):=X0(N)−O thus provide an exhaustive list of the open sub-Shimura varieties of X0(N). Given the modularity of E, the modularity of Eamounts to the existence of a Galois-equivariant inclusion i:H0()0−→ H0(O)0 for suitable O,realisingH1(E) as a subquotient of H1(YO(N)) via the pushforward under πEand the pullback under ιof the first row in the following diagram with exact rows: 0H1(X0(N)) πE H1(YO(N)) ? H0(O)00 0H1(E)H1(E)H0()0 i 0. (3) Consider the simplest non-trivial setting where ={P1,P 2}⊂E(Q) consists of two points defined over Q, so that H0()0=Qpwith trivial Galois action. The resulting extension 0H1(E)H1(E)Qp0(4) encodes the image of the point P2−P1∈E(Q) under the connecting homomorphism δ:E(Q)−→ H1(Q,H1(E)) :=Ext1 GQ(Qp,H1(E)) of Kummer theory, where the Ext group is taken in the category of continuous p-adic representations of GQ. The following statement, which gives a “modularity criterion” for Eand encapsulates many of the deepest theorems on the Birch and SwinnertonDyer 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 Ebe a semistable elliptic curve having at least one odd prime of non-split multiplicative reduction or at least two odd primes of split multiplicative reduction. Darmon and Rotger Res Math Sci (2016) 3:27 Page 4 of 32 Theorem 1.1 Assume that the point P2−P1is of infinite order in E(Q). Then the following are equivalent: (a) The curve E=E\{P1,P 2}is modular; (b) the Hasse-Weil L-series L(E, s)has a simple zero at s =1; (c) the point P2−P1generates E(Q)⊗Qand LLI(E/Q)is finite; (d) for all primes p, the group Ext1 fin(Qp,H1(E)) of extensions of p-adic representations of the Galois group of Qthat are cristalline at p is one-dimensional over Qp. Sketch of proof The modularity of Eamounts to the statement that there exists an order Oin an imaginary quadratic field Ksuch that the extension (4) can be obtained as the pullback of (3) via an inclusion i:Qp−→ H0(O)GQ, whose image contains a degree 0 divisor DK∈Div0(O)GQ⊂Div0(X0(N))(Q). This means that the point P1−P2∈E(Q) is a nonzero multiple of the Heegner point PE,K :=πE(DK). The implication (a)⇒(b) therefore follows from the Gross–Zagier formula [21] expressing the height of PE,K as a nonzero multiple of L(E/K, 1) =L(E, 1) ·L(EK,1), where EKis the quadratic twist of Eby K. The existence of a suitable Kfor which L(EK,1) = 0 follows from a non-vanishing result of Waldspurger or can be deduced from analytic number theory techniques (cf. [28]). The implication (b) ⇒(c) was subsequently proved by Kolyvagin [25], who parlayed the non-triviality of PE,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” [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 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]. More precisely, choose a prime p≥5 of good ordinary reduction for Esuch that E[p] is an irreducible GQ-representation and the image of the restriction map Selp(E)−→ E(Qp)/pE(Qp) does not lie in the image of E(Qp)[p]. A result of Waldspurger ensures the existence of an odd quadratic character χsuch that L(E, χ,1) = 0, which can be chosen so that χ(2) =χ(p)=1. Let Kdenote the imaginary quadratic field associated to χ. The p-adic Selmer group Ext1 K,fin(Qp,H1(E)) of Eover K(defined as an Ext group in the category of cristalline representations of GK) decomposes as a direct sum of eigenspaces Ext1 K,fin(Qp,H1(E)) ≃Ext1 fin(Qp,H1(E)) ⊕Ext1 K,fin(Qp,H1(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 Ext1 K,fin(Qp,H1(E))−. Assumption (d) Darmon and Rotger Res Math Sci (2016) 3:27 Page 5 of 32 therefore implies that Ext1 K,fin(Qp,H1(E)) is one-dimensional over Qp. One can then argue as in [33]. 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 [39] and [8] (which interpolates critical values of the L-series of the Rankin convolution of the modular form fassociated to Ewith suitable Hecke characters of Kof higher infinity-type) does not vanish at the trivial point, which lies outside its region of classical interpolation. This in turn implies, in the light of [33, Corollary 2.6.2] resting on the variant of the Gross–Zagier formula of [8], that the Heegner point PE,K has non-trivial p-adic formal group logarithm and is therefore non-torsion. As already explained, the non-triviality of PE,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 :Gal(H/Q)→Aut(V)≃GLn(¯ Qp) be an n-dimensional representation of the Galois group of a finite extension H/Q,asocalled Artin representation, viewed as having coefficients in ¯ Qp. The pair (E, ) gives rise to the Hasse–Weil–Artin L-series L(E, ,s):=  det(1 −−s(Fr−1 )(H1(E)⊗V)I)−1, where the product is taken over the rational primes , the arithmetic frobenius element at is denoted by Fr,andIdenotes the inertia group at . The equivariant Birch and Swinnerton-Dyer conjecture for Eand , denoted BSD(E, ), asserts that ords=1L(E, ,s)=dim ¯ Qp(E(H)⊗V)GQ.(5) As a first step to understanding BSD(E, ), it is natural to ask which κ∈Ext1 fin(V,H1(E)) can be realised as a subquotient of a suitable H1(YO(N)). The Artin representation H0(O)0which appears in the upper rightmost term of the diagram (3) is readily analysed using the theory of complex multiplication. Namely, the slightly larger Artin representation H0(O) decomposes as a direct sum H0(O)⊗¯ Qp=⊕ φK(N) j=1Wj,where Wj=⊕ ψVj(ψ), with Vj(ψ)⊂Vψ:=IndQ Kψ. In this equation, the second direct sum is taken over the non-trivial, ¯ Qp-valued, finite order characters ψof Gal (HO/K) modulo the involution ψ→ ψ−1,andVj(ψ) is a non-trivial irreducible constituent of the two-dimensional representation Vψobtained by inducing the Galois character ψfrom GKto GQ. The representation Vψis irreducible precisely when ψ= ψ−1, and in this case a non-trivial class κ∈Ext1 fin(Vψ,H1(E)⊗¯ Qp) 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, Vψ,s) has a simple zero at s=1; (c’) the representation Vψoccurs with multiplicity one in E(H)⊗¯ Qp, and the Vψ-isotypic component of the LLI(E/H) is finite; (d’) the group Ext1 fin(Vψ,H1(E)⊗¯ Qp) is one-dimensional over ¯ Qp, and generated by κ. Darmon and Rotger Res Math Sci (2016) 3:27 Page 6 of 32 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 O, which are not even rich enough to capture all of the irreducible two-dimensional Artin representations of Q. The open Shimura varieties YO(N) thus appear to give no purchase on BSD(E, ) when is not induced from a ring class character of an imaginary quadratic field. (2) Theorem 1.1 suggests that the modularity of elements of Ext1 fin(Vψ,H1(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 H1(YO(N)). Inordertorelatealargerclassofnon-semisimple Galoisrepresentationsto 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 YO(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 fibred over Shimura varieties, the complements of Heegner cycles in such varieties, and so on. With this expanded notion of modularity, the programme of characterising the nonsemisimple Galois representations that are modular becomes richer and more subtle. See [9] for a fragment of experimental mathematics that might be viewed as fitting into this programme. 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 V1and V2be Galois representations for which hom(V1,V 2) is irreducible. Suppose that there is a non-trivial κ∈Ext1 fin(V1,V 2) arising as a subquotient of the cohomology of an open Shimura variety. Is Ext1 fin(V1,V 2) necessarily one-dimensional? If the answer to this question were “yes”, it would imply that the open curve E−{P1,P 2} discussed in Theorem 1.1 is never modular when rank(E(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 twodimensional Artin representations of Qcan 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 GLn. Even more germane to this article, p-adic limits Darmon and Rotger Res Math Sci (2016) 3:27 Page 7 of 32 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 Q.IfL(E, s)vanishes to even order ≥2at s =1, then the Selmer group Ext1 fin(Qp,H1(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 by H1(E) for a large class of self-dual Artin representations  of dimension 4 (and their lower-dimensional subrepresentations, in case is reducible) arising as the tensor product =1⊗2of 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 1and 2are irreducible (resp. when exactly one of 1and 2is 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]ofKato(cf.also[6,32]) on BSD(E, χ) for χa Dirichlet character. Like Heegner points in the setting of BSD(E, Vψ), generalised Kato classes enjoy close relations to (p-adic) Hasse–Weil–Artin L-functions attached to E and , but unlike Heegner points, they are expected to generate a non-trivial subgroup of the Selmer group attached to Eand precisely when ords=1L(E, ,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)⊗V)GQwhen 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 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. 2can 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. 2is 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∈S1(N, χ) be a newform of weight one and level Nwith Fourier coefficients in a field L,andlet Darmon and Rotger Res Math Sci (2016) 3:27 Page 8 of 32 :GQ−→ Aut(V)≃GL2(L) be the Artin representation associated to it by the construction of Deligne and Serre. We view as acting on a two-dimensional L-vector space V, where L⊂Ccan be chosen to be contained in a cyclotomic field. Let Hbe the number field cut out by , so that factors through Gal (H/Q). Fix a rational prime pand choose a prime pof Habove p. The latter determines a canonical inclusion H⊂Hp⊂¯ Qp of Hin its completion Hpat p. Assume that the pair (,p) satisfies the following conditions: (I) The prime psplits completely in L/Q, so that Lis equipped with an embedding into Qpwhich will be fixed from now on. This assumption, which is made solely to lighten the notations and could easily be dispensed with, allows to be viewed as a Qp-linear representation via the natural action of GQon the Qp-vector space V⊗LQp. (II) The representation Vis unramified at p. There is then a well-defined arithmetic frobenius element Frp∈Gal (H/Q) acting canonically on V, and the characteristic polynomial of (Frp) is equal to the Hecke polynomial x2−ap(g)x+χ(p)=:(x−αg)(x−βg) attached to g. (III) The modular form gis regular at p, i.e. αg= βg. After possibly enlarging L,itmay also be assumed that this coefficient field contains the roots of unity αgand βg. (IV) The representation gis not induced from a character of a real quadratic field K in which the prime psplits. 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]. The p-stabilisations of gat pare the normalised eigenforms of weight one with Fourier coefficients in Ldefined by gα:=g(z)−βgg(pz),g β:=g(z)−αgg(pz). They are eigenvectors for the Up-operator satisfying Upgα=αggα,U pgβ=βggβ. The Artin representation Vdecomposes naturally as a direct sum V=Vα⊕Vβ into one-dimensional eigenspaces for Frp, with eigenvalues αgand βg, respectively. By a theorem of Hida, there exists a finite flat extension gof the Iwasawa algebra and aHidafamilyg∈g[[q]] of tame level Nand tame character χpassing through the pstabilised weight one eigenform gα. When gis cuspidal, the regularity hypothesis imposed on gimplies that such a Hida family is unique, thanks to a recent result of Bellaïche and Dimitrov [1]. Darmon and Rotger Res Math Sci (2016) 3:27 Page 9 of 32 The Hida family gcomes equipped with the following canonical structures: (a) There is a locally free g-module Vgof rank two, affording Hida’s ordinary -adic Galois representation g:GQ−→ Autg(Vg) which is realised in the inverse limit of ordinary étale cohomology groups associated to the tower X1(Npr) of modular curves. This representation interpolates the Galois representations associated by Deligne to the classical specialisations of g. (b) The restriction of Vgto GQpadmits a stable filtration 0−→ Ug−→ Vg−→ Wg−→ 0, where both Ugand Wgare flat g[GQp]-modules that are locally free of rank one over g, and the quotient Wgis unramified, with Frpacting on Wgas multiplication by the p-th Fourier coefficient ap(g). (c) Let Qnr pdenote the maximal unramified extension of Qpand let  Qnr pdenote its p-adic completion. In [30], Ohta constructs a canonical g-adic period ωg∈D(Wg):=( Qnr pˆ ⊗Wg)GQp, corresponding to the normalised -adic eigenform gunder the isomorphism in Theorem (A) of the introduction of [30]. (d) There is a natural perfect Galois-equivariant duality, given in Theorem (B) of the introduction of [30], Ug×Wg−→ g(det(g)), where GQacts on the module gof the right-hand side via the determinant of g. Let yg:g−→ Qp be the specialisation map attached to the p-stabilised weight one form gα. By specialising the structures above attached to gvia the map yg,weobtain (a’) A non-canonical isomorphism of Qp[GQ]-modules gα:Vg:=Vg⊗ygQp ∼ −→ V⊗LQp. (b’) A non-trivial GQp-stable filtration 0−→ Ug−→ Vg−→ Wg−→ 0 of Vgby one-dimensional subspaces, where Ug:=Ug⊗ygQpand Wg:=Wg⊗ygQp. The Frobenius element Frpacts on Wgand Ugas multiplication by αgand βg, respectively. Since these eigenvalues are assumed to be distinct, the exact sequence above splits canonically, leading to the identifications Ug=Vβ g,W g=Vα g,V g=Ug⊕Wg=Vβ g⊕Vα g. (c’) Specialising Ohta’s period leads to a canonical element ωgα:=yg(ωg)∈D(Vα g):=(Qnr p⊗Vα g)GQp=(Hp⊗Vα g)GQp.(6) Darmon and Rotger Res Math Sci (2016) 3:27 Page 16 of 32 Let κp(f, gα,h α)=resp(κ(f, gα,h α)) denote the image of the global class κ(f, gα,h α) in the local cohomology group H1 fin(Qp,V fgh)=(H1 fin(Hp,V f)⊗Vgh)Gal (Hp/Qp)=(E(Hp)⊗Vgh)Gal (Hp/Qp). As we describe more explicitly below, Theorem D of [19] asserts that this image is controlled by suitable p-adic avatars of the second derivative of the classical L-series L(f, Vgh,s) at the central critical point s=1. These p-adic values were defined and explored in [19]and[15] and are denoted Lpgα(˘ f,˘ g∗,˘ h),Lpgβ(˘ f,˘ g∗,˘ h),Lphα(˘ f,˘ g, ˘ h∗),Lphβ(˘ f,˘ g, ˘ h∗).(22) They depend on the choice of certain test vectors (˘ f,˘ g, ˘ h)∈S2(N;L)×M1(N, χ;L)×M1(N, χ−1;L) with the same system of Hecke eigenvalues as f,g,andh, respectively, and with fourier coefficients in L, and on the choice of dual test vectors (˘ g∗,˘ h∗)∈Hom(M1(N, χ−1;L),L)×Hom(M1(N, χ;L),L) with the same system of Hecke eigenvalues as gand h. We refer to the introduction of [19] for more details on their definition, contenting ourselves with remark that the p-adic L-value Lpgα(˘ f,˘ g∗,˘ h) is defined essentially as the p-adic limit of central critical values Lpgα(˘ f,˘ g∗,˘ h):=lim →1 E(f, g,h)×C(˘ f,˘ g∗,˘ h)×L(Vf⊗Vg⊗Vh,(+1)/2) g,g , as granges over the specialisations of (odd) weight ≥3oftheHidafamilygspecialising to gαin weight one. Here E(f, g,h)isap-adic multiplier arising from a recipe of Panciskin, whose presence allows the p-adic interpolation of the special values above, and C(˘ f,˘ g∗,˘ h) 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 Vgh (over Qp, for now) which is compatible with the decomposition (21), i.e. choose nonzero vectors vαα gh ∈Vαα gh ,v αβ gh ∈Vαβ gh ,v βα gh ∈Vβα gh ,v ββ gh ∈Vββ gh .(23) Write κp(f, gα,h α)=Rαα ⊗vββ gh +Rαβ ⊗vβα gh +Rβα ⊗vαβ gh +Rββ ⊗vαα gh .(24) The coordinate Rξbelongs to E(Hp)Frp=ξ Qp, where ξranges over the index set {αα =αgαh,αβ =αgβh,βα =βgαh,ββ =βgβh}. Note that Rξis even the image of a global point in E(H)Qp, assuming the finiteness of the Shafarevich–Tate group of Eover H.Let logp:E(Hp)Qp−→ Hp(25) denote the formal group logarithm attached to an invariant differential on E/Q.The following theorem is stated in Section 6.4 of [19]: Darmon and Rotger Res Math Sci (2016) 3:27 Page 17 of 32 Theorem 3.4 When L(E, Vgh,1) =0, there exists a choice of πin (16)and of test vectors forf,g,andhsuchthatthecoordinatesin(24)satisfy logp(Rαβ)∼Lpgα(˘ f,˘ g∗,˘ h),logp(Rβα)∼Lphα(˘ f,˘ g, ˘ h∗),logp(Rββ)=0,(26) where ∼denotes equality up to a nonzero p-adic period in H× p. Remark 3.5 This theorem says nothing about the quantity logp(Rαα), which does not bear any direct relationship with p-adic L-values introduced above. We expect that logp(Rαα) may rather be connected with the first derivative of a putative refinement of Lpf(f, gα,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 [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 If L(E, Vgh,1) =0and Lpgα(˘ f,˘ g∗,˘ h)= 0for a suitable choice (˘ f,˘ g∗,˘ h)of test vectors, then the two global classes κ(f, gα,h α),κ(f, gα,h β) are linearly independent in the Selmer group H1 fin(Q,V fgh)attached to E and Vgh,fora suitable choice of πin (16). Theorem 3.4 and its corollary motivated the experimental study undertaken in [15]ofthe 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, gα,h α) is expected to be trivial when ords=1L(E, Vgh,s)>2. Assume that this L-function has a double zero at the centre, which implies, by Conjecture BSD(E, Vgh), that (E(H)L⊗V12)GQis a two-dimensional L-vector space. Fix vectors vαα gh ,...,vββ gh chosen as in (23), with the difference that they belong to Lvector space V12 rather than the Qp-vector space Vgh. Choose a basis (P, Q) for this L-vector space, and write P=Pαα ⊗vββ gh +Pαβ ⊗vβα gh +Pβα ⊗vαβ gh +Pββ ⊗vαα gh , Q=Qαα ⊗vββ gh +Qαβ ⊗vβα gh +Qβα ⊗vαβ gh +Qββ ⊗vαα gh , where Pξ,Q ξare points in E(H)Frp=ξ Lfor every ξ∈{αα =αgαh,αβ =αgβh,βα = βgαh,ββ =βgβh}. These points can be used to define a regulator attached to gα, whose entries are the p-adic formal group logarithms of the coordinates attached to the vectors vαα gh and vαβ gh (and similarly for hα): Definition 3.7 The regulators attached to Eand V12 are Reggα(E, V12)=det logpPββ logpPβα logpQββ logpQβα  =logpPββ ·logpQβα −logpQββ ·logpPβα, Reghα(E, V12)=det logpPββ logpPαβ logpQββ logpQαβ  =logpPββ ·logpQαβ −logpQββ ·logpPαβ. Darmon and Rotger Res Math Sci (2016) 3:27 Page 18 of 32 The main conjecture of [15] is the following,3assuming αg βg=±1 (resp. αh βh=±1) so that the Stark unit ugα(resp. uhα) is well defined: Conjecture 3.8 Assume that L(E, Vgh,s)vanishes to order 2at s =1. Then there exists a choice of test vectors (˘ f,˘ g∗,˘ h)and (˘ f,˘ g, ˘ h∗)such that Lpgα(˘ f,˘ g∗,˘ h)=Reggα(E, V12) logpugα ,Lphα(˘ f,˘ g, ˘ h∗)=Reghα(E, V12) logpuhα (mod L×). 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 Lpgα(˘ f,˘ g∗,˘ h)andLphα(˘ f,˘ g, ˘ 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)⊗Vgh)GQ, specifying this position up to an ambiguity of L×rather than the less precise Q× pambiguity of Theorem 3.4. The most important ingredients in the formulation of this conjecture are the so-called enhanced regulators  Reg(E, V12)∈(E(H)L⊗V12)GQ⊗(E(H)L⊗V12)GQ,  Regαα(E, V12)∈(Hp)Frp=βgβh⊗(E(H)L⊗V12)GQ,  Reg(E, Vgh)∈(E(H)L⊗Vgh)GQ⊗(E(H)L⊗Vgh)GQ,  Regαα(E, Vgh)∈D(Vαα gh )⊗(E(H)L⊗Vgh)GQ, whose definition is somewhat in the spirit of the regulator RSdefined in equation (2) of [13], and which we now proceed to describe. As in (6), here D(Vαα gh ):=(Qnr p⊗Vαα gh )GQp= (Hp⊗Vαα gh )GQp. Definition 3.10 Choose an L-basis (P, Q) of the two-dimensional vector space (E(H)⊗ V12)GQ,andset  Reg(E, V12):=det PP QQ :=P⊗Q−Q⊗P. (27) It does not depend on the choice of basis that was made to define it, up to multiplication by L×. The function logαα :(E(H)L⊗V12)GQ−→ (Hp)Frp=βgβhdefined by logαα(P):=logp(Pββ) 3We warn the reader that here in this note we have chosen to state the main conjecture of [15] in terms of the arithmetic frobenius Frpat p,whilein[15] we rather employ the geometric frobenius σp=Fr−1 p. It is for this reason that the roles of αand βare swapped in both formulations. Darmon and Rotger Res Math Sci (2016) 3:27 Page 19 of 32 induces a linear map logαα ⊗1:(E(H)L⊗V12)GQ⊗(E(H)L⊗V12)GQ −→ (Hp)Frp=βgβh⊗(E(H)L⊗V12)GQ, and we set  Regαα(E, V12):=(logαα ⊗1)( Reg(E, V12)) =logp(Pββ)⊗Q−logp(Qββ)⊗P. (28) Recall the embedding jgh :V12 −→ VL gh ⊂Vgh 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⊗2 12 into V⊗2 gh . This is done by exploiting the canonical dualities on Vgand Vhdescribed in Sect. 2, which gives rise to perfect pairings Vg×Vg−→ Qp(χ),V h×Vh−→ Qp(χ−1),V gh ×Vgh −→ Qp. These pairings allow us to define L-rational structures VL∗ g,VL∗ hand VL∗ gh which are dual to VL g,VL hand VL gh, respectively, by letting VL∗ gbe the L-dual of VL gin Vg, and likewise for VL∗ hand VL∗ gh . We may then choose GQ-equivariant embeddings j∗ g:V1−→ VL∗ g,j ∗ h:V2−→ VL∗ h,j ∗ gh :=j∗ g⊗j∗ h:V12 −→ VL∗ gh , which are well defined up to scaling by L×. Replacing jgh by μ·jgh, for any μ∈Q× p,has the effect of replacing j∗ gh by μ−1·j∗ gh. Hence, the map jgh ⊗j∗ gh :V12 ⊗V12 −→ Vgh ⊗Vgh is well defined up to scaling by L×. Definition 3.11 The enhanced regulator  Reg(E, Vgh) associated to Eand Vgh is  Reg(E, Vgh):=(jgh ⊗j∗ gh)( Reg(E, V12)) ∈(E(H)⊗Vgh)GQ⊗(E(H)⊗Vgh)GQ.(29) Finally, let Logp:(E(H)⊗Vgh)GQ−→ (Hp⊗Vgh)GQp=D(Vgh) be the canonical p-adic logarithm map induced from the p-adic logarithm of (25)viathe fixed embedding H⊂Hp,andlet Logαα :(E(H)⊗Vgh)GQ−→ D(Vαα gh ) be its composition with the functorial projection D(Vgh)−→ D(Vαα gh ). This logarithm map is just the more canonical counterpart of the map logαα: the latter depends on the choice of a basis vector vαα gh for Vαα and is related to Logαα by the rule Logαα :=logαα ⊗vαα gh . We set  Regαα(E, Vgh):=(Logαα ⊗1)( Reg(E, Vgh)) =Logαα(P)⊗Q−Logαα(Q)⊗P. (30) It is worth noting that the enhanced regulator  Regαα(E, Vgh) is a canonical invariant associated to Eand Vgh, i.e. it is well defined up to multiplication by L×, while the less canonical  Regαα(E, V12) depends on the choice of a basis vαα gh for Vαα gh . The two regulators are related by  Regαα(E, Vgh)= Regαα(E, V12)⊗vαα gh .(31) Darmon and Rotger Res Math Sci (2016) 3:27 Page 20 of 32 3.4 The conjecture Recall the periods ωgα∈D(Vα g),ωhα∈D(Vα h) constructed in (6). The main conjecture of this note is: Conjecture 3.12 Assume that r(E, Vgh)=2. The generalised Kato class κ(f, gα,h α)belongs to (E(H)⊗Vgh)GQand satisfies the relation ωgαωhα⊗κ(f, gα,h α)∼L Regαα(E, Vgh) in D(Vαα gh )⊗(E(H)⊗Vgh)GQ,where∼Ldenotes an equality up to scaling by a factor in L which is nonzero for a suitable choice of πin (16). The following proposition shows that, under Conjecture 2.1 (relating the canonical period attached to gto the Stark unit ugα) and Conjecture 3.2 (a mild strengthening of BSD(E, 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). Proposition 3.13 Assume Conjectures 2.1 and 3.2. Then Conjecture 3.12 implies Conjecture 3.8. Proof Consider the product of periods ηgαωhα=(gα⊗vβ g)·(hα⊗vα h)=gα·hα⊗vβα gh ∈D(Vβα gh ) defined in Sect. 2. The pairing introduced in (7) gives rise to a pairing ,:D(Vαβ gh )×D(Vβα gh )−→ D(Qp)=Qp. As shown in the proof of [19, Theorem 6.10 (ii)], Logαβ κ(f, gα,h α),ηgαωhα=Lpgα(f, g, h) (mod L×).(32) On the other hand, by the definition of the enhanced regulator, Logαβ  Regαα(E, Vgh)=(logpPββ logpQβα −logpQββ logpPβα)⊗vαα gh ⊗v∗αβ gh =Reggα(E, V12)⊗vαα gh ⊗v∗αβ gh (mod L×). Hence, the following equality holds in D(Vαα gh ): Logαβ  Regαα(E, Vgh),ηgαωhα=gα·hα·Reggα(E, Vgh)⊗vαα gh (mod L×).(33) By pairing the value of Logαβ at both sides of the displayed identity in Conjecture 3.12 with the class ηgαωhαand invoking (32)and(33), we obtain ωgαωhα⊗Lpgα(f, g, h)=gα·hα·Reggα(E, V12)⊗vαα gh ∈D(Vαα gh ) (mod L×). Since ωgαωhα=gα·hα·vαα gh (mod L×), Darmon and Rotger Res Math Sci (2016) 3:27 Page 21 of 32 it follows that gαLpgα(f, g, h)=gαReggα(E, V12) (mod L×), and therefore that Lpgα(f, g, h)=Reggα(E, V12) Lgα (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 Sect. 4below, 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  Regαα(E;V12), Regαβ(E;V12), Regβα(E;V12), Regββ(E;V12), in the arithmetically interesting cases where Vgh is reducible. According to Darmon et al. [16, §2], the following is a complete list of scenarios where this occurs: (1) The original Beilinson–Kato setting where Vgand Vhare both reducible, i.e. where gand hare both Eisenstein series of weight one; (2) the Beilinson–Flach setting where exactly one of Vgor Vhis reducible, i.e. where exactly one of gor his cuspidal; (3) the complex multiplication case where Vgand Vhare both induced from characters of a common imaginary quadratic field; (4) the real multiplication case where Vgand Vhare induced from characters of mixed signature of a common real quadratic field; (5) the adjoint case where his (a twist of) the dual of g, so that Vgh is the direct sum of a one-dimensional representation and a twist of the adjoint of Vg. The reader will notice that some of the above settings arise when gand/or hare reducible, while in Sects. 2and 3these 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 gand hare both Eisenstein series. After possibly twisting gor h, there is no real loss of generality in assuming that there exist Dirichlet characters χ1,χ2such that g and hare given by g=E1(χ1,χ2),h=E1(1,χ−1 12 ),where χ12 =χ1χ2. We refer to e.g. [10, §2.1.2] for the definition of these weight one Eisenstein series in terms of their q-expansions. The Galois representations attached to gand hare reducible, Darmon and Rotger Res Math Sci (2016) 3:27 Page 22 of 32 namely V1=L(χ1)⊕L(χ2),V 2=L⊕L(χ−1 12 ), V12 =L(χ1)⊕L(χ−1 1)⊕L(χ2)⊕L(χ−1 2),(34) where the coefficient field Lis the cyclotomic field generated by the images of χ1and χ2. These representations factor through the Galois group Gal (H/Q) of an abelian extension Hof Q.Wemayset αg=χ1(p),βg=χ2(p),αh=1,βh=χ−1 12 (p). The regularity assumption implies that V1and V2decompose uniquely as a direct sum of two GQp-stable lines, which are also stable under GQ. More precisely, Vαα 12 =L·vχ1,V ββ 12 =L·v¯χ1,V αβ 12 =L·v¯χ2,V βα 12 =L·vχ2, where (vχ1,v¯χ1,v¯χ2,v χ2)isabasisforV12 on which GQacts via the characters χ1,¯χ1,¯χ2, and χ2, respectively. The class κ(f, gα,h α)=κBK(f, gα,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, gα,h α) belongs to the Selmer group of Eover Hif and only if the L-function L(E, Vgh,s)=L(E, χ1,s)L(E, ¯χ1,s)L(E, χ2,s)L(E, ¯χ2,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 χ1and χ2) ords=1L(E, χ1,s)=ords=1L(E, ¯χ1,s)=1,L(E, χ2,1),L(E, ¯χ2,1) = 0. Assuming that this is the case, Conjectures BSD(E, χ1) and BSD(E, χ2)predictthat (E(H)L⊗V12)GQis two-dimensional over Land that a basis for it can be chosen to be P:=P¯χ1⊗vχ1,Q:=Qχ1⊗v¯χ1, where P¯χ1and Qχ1are global points in E(H)Lgenerating the ¯χ1and χ1eigenspaces, respectively, for the natural action of GQ. With these notations, we have Pαα =Pαβ =Pβα =0,P ββ =P¯χ1, Qαβ =Qβα =Qββ =0,Q αα =Qχ1. This immediately implies that  Regαα(E, V12)=logp(P¯χ1)·Q,  Regαβ(E, V12)=0,  Regβα(E, V12)=0, Regββ(E;V12)=logp(Qχ1)·P. It follows that Reggα(E;V12)=Reggβ(E;V12)=Reghα(E;V12)=Reghβ(E;V12)=0. This accounts for the fact that the p-adic iterated integrals Lpgα(f, g, h),Lpgβ(f, g, h),Lphα(f, g, h),Lphβ(f, g, h) Darmon and Rotger Res Math Sci (2016) 3:27 Page 23 of 32 systematically vanish4when gand hare Eisenstein series that are regular at p. Conjecture 3.12 makes the stronger prediction that the generalised Kato classes themselves are nontrivial, and is consistent with a Conjecture of Perrin-Riou, since it predicts that logββ(κ(f, gα,h α)) =logαα(κ(f, gβ,h β)) =logp(P¯χ1)logp(Qχ1) (mod L×). 4.2 Beilinson–Flach classes IntheBeilinson–Flachsetting,itcanbeassumedwithoutlossofgeneralitythatgisaweight one cusp form with nebentypus character χand Galois representation Vg=V1⊗LQp, and that h:=E1(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 Vgh =Vg⊕V¯ g;V12 =V1⊕¯ V1, and the Hasse–Weil–Artin L-series L(E, Vgh,s)=L(E, Vg,s)L(E, ¯ Vg,s) has a double zero at s=1 precisely when each of the primitive L-series L(E, Vg,s)and L(E, V¯ g,s) have a simple zero at s=1. Conjecture BSD(E, Vg) then implies that each of the L-vector spaces on the right-hand side of (E(H)L⊗V12)GQ)=(E(H)L⊗V1)GQ⊕(E(H)L⊗¯ V1)GQ, is one-dimensional. Let Pbe an L-basis for (E(H)L⊗V1)GQand let ¯ Pbe the associated L-basis for (E(H)L⊗¯ V1)GQ, obtained by applying complex conjugation to the coefficients in L. After fixing an ordering αg,βg∈Lfor the eigenvalues of Frpon V1, and setting αh=1,βh=χ−1(p)=(αgβg)−1, we have Vαα 12 =Vαg 1,V αβ 12 =¯ Vβ−1 g 1,V βα 12 =Vβg 1,V ββ 12 =¯ Vα−1 g 1, and hence Pαα =Pαg,P βα =Pβg,P αβ =0Pββ =0, ¯ Pαα =0¯ Pβα =0,¯ Pαβ =¯ Pβ−1 g,¯ Pββ =¯ Pα−1 g. A direct calculation reveals that, up to multiplication by L×,  Regαα(E, V12)=logp(¯ Pα−1 g)·P,  Regαβ(E, V12)=logp(Pβg)·¯ P,  Regβα(E, V12)=logp(¯ Pβ−1 g)·P,  Regββ(E;V12)=logp(Pαg)·¯ P. It follows that Reggα(E, V12)=logp(¯ Pα−1 g)·logp(Pβg),Reggβ(E, V12)=logp(¯ Pβ−1 g)·logp(Pαg), Reghα(E, V12)=0,Reghβ(E, V12)=0. as described in [15,§6]. 4But see the experiments described in [15, §7] in the case where gis irregular at p, which suggest that the irregular setting of Conjecture 3.12 would merit further investigation. Darmon and Rotger Res Math Sci (2016) 3:27 Page 24 of 32 4.3 Complex multiplication classes and Heegner points In this chapter we consider the setting where gand hare theta series attached to characters ψgand ψhof the same imaginary quadratic field K, and with inverse nebentypus character. Given any character ψof GK,letψdenote the character obtained by conjugating it with the involution in Gal (K/Q). Then Vg=IndQ Kψg=IndQ Kψ g,V h=IndQ Kψh=IndQ Kψ h, and therefore Vgh =IndQ Kψ•⊕IndQ Kψ◦,where ψ•=ψgψh,ψ◦=ψgψ 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•:=IndQ Kψ•,V ◦:=IndQ Kψ◦ appearing in the decomposition V12 =V•⊕V◦(35) (viewed as representations with coefficients in the number field L)areirreducible, 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 Sect. 4.5.) The Hasse–Weil–Artin L-series L(E, Vgh,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)andL(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)orL(E, V◦,s)hasadoublezeroats=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 Lfunction 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•)GQand (E(H)L⊗V◦)GQare both one-dimensional L-vector spaces, with generators P•and P◦, respectively. It is natural to write P•=Pψ•⊗vψ •+Pψ •⊗vψ•,P ◦=Pψ◦⊗vψ ◦+Pψ ◦⊗vψ◦,(36) where Pψ•,Pψ •,Pψ◦,andPψ ◦are generators for the one-dimensional subspaces of E(H)L on which GKacts via the characters ψ•,ψ •,ψ◦,andψ ◦, respectively. Darmon and Rotger Res Math Sci (2016) 3:27 Page 25 of 32 The description of the enhanced regulators attached to V12 and to (P•,P ◦) can be further subdivided into two cases, with markedly different features: the case where the prime pis 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=pp be the factorisation of pinto distinct primes of K.Wecanthenset αg=ψg(p),βg=ψg(p),αh=ψh(p),βh=ψh(p), so that αgαh=ψ•(p),αgβh=ψ◦(p),βgαh=ψ◦(p),βgβh=ψ•(p). The decomposition of the GKp=GQprepresentations attached to (35) into Frpeigenspaces is also stable under the action of the global Galois group GK, and is described by: Vαα 12 =Vψ• •,V αβ 12 =Vψ◦ ◦,V βα 12 =Vψ ◦ ◦,V ββ 12 =Vψ • •. It follows that, up to multiplication by L×,  Regαα(E, V12)=logp(Pψ •)·P◦, Regαβ(E, V12)=logp(Pψ ◦)·P•,  Regβα(E, V12)=logp(Pψ◦)·P•, Regββ(E, V12)=logp(Pψ•)·P◦, and therefore that Reggα(E, V12)=logp(Pψ •)·logp(Pψ ◦),Reggβ(E, V12)=logp(Pψ•)·logp(Pψ◦), Reghα(E, V12)=logp(Pψ •)·logp(Pψ◦),Reghβ(E, V12)=logp(Pψ•)·logp(Pψ ◦). The corresponding formulae for the p-adic iterated integrals Lpgα(f, g, h), Lpgβ(f, g, h), Lphα(f, g, h), and Lphβ(f, g, h) were proved in [15,§3],byusingthep-adic Gross–Zagier formula of [8] 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 Frpacting on Vgand Vhare of the form αg,βg=−αg,αh=α−1 g,βh=−α−1 g. Let (vψg,v ψ g) be a eigenbasis of Vgfor the action of GKrelative to the distinct characters ψgand ψ g,andlet(vψh,v ψ h) be a similar basis for Vh. These vectors can be scaled so that Frpacts on them as Frp(vψg)=αg·vψ g,Frp(vψ g)=αg·vψg,Frp(vψh)=α−1 g·vψ h, Frp(vψ h)=α−1 g·vψh, and therefore we may set Vα g=L·(vψg+vψ g),V β g=L·(vψg−vψ g),V α h=L·(vψh+vψ h), Vβ h=L·(vψh−vψ h). After setting vψ•:=vψg⊗vψh,v ψ •:=vψ g⊗vψ h,v ψ◦:=vψg⊗vψ h,v ψ ◦:=vψ g⊗vψh, and letting v+ •:=vψ•+vψ •,v − •:=vψ•−vψ •,v + ◦:=vψ◦+vψ ◦,v − •:=vψ◦−vψ ◦, Darmon and Rotger Res Math Sci (2016) 3:27 Page 32 of 32 24. Kings, G., Loeffler, D., Zerbes, S.: Rankin–Selberg Euler systems and p-adic interpolation (submitted) 25. Kolyvagin, V.: Finiteness of E(Q)andX(E, Q) for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52(3) 670–671 (1988); translation in Math. USSR-Izv. 32(3) 523–541 (1989) 26. Lauder,A.:EfficientcomputationofRankinp-adicL-functions.In:Böckle,G.Wiese,G.(eds.)ComputationswithModular Forms:Proceedings of a SummerSchool andConference,Heidelberg, August/September 2011, pp.181–200. Springer (2014) 27. Lei, A., Loeffler, D., Zerbes, S.L.: Euler systems for Rankin–Selberg convolutions of modular forms. Ann. Math. 180(2), 653–771 (2014) 28. Murty, M.R., Murty, V.K.: Non-vanishing of L-values and applications. In: Progress Math, vol. 157. Birkhäuser Verlag, Basel (1997) 29. Neková˘r, J., Scholl, A.J.: Introduction to plectic cohomology. In: Jiang, D., Shahidi, F., Soudry, D. (eds.) Advances in the Theory of Automorphic Forms and Their L-functions. Contemporary Mathematics 664, pp. 321–337. American Mathematical Society, Providence, RI (2016) 30. Ohta, M.: On the p-adic Eichler–Shimura isomorphism for -adic cusp forms. J. Reine Angew. Math. 463, 49–98 (1995) 31. Prasad, D.: Trilinear forms for representations of GL2and local epsilon factors. Comput. Math. 75, 1–46 (1990) 32. Scholl, A.J.: An introduction to Kato’s Euler systems. In: Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Mathematical Society Lecture Note Series 254, pp. 379–460. Cambridge University Press (1998) 33. Skinner, C.: A converse to a theorem of Gross, Zagier, and Kolyvagin, preprint, in arXiv:1405.7294 34. Skinner, C., Urban, E.: Vanishing of L-functions and ranks of Selmer groups. Int. Congr. Math. 2, 473–500 (2006) 35. Skinner, C., Urban, E.: The Iwasawa main conjectures for GL2. Invent. Math. 195(1), 1–277 (2014) 36. Stark, H.M.: L-functions at s=1. II. Artin L-functions with rational characters. Adv. Math. 17, 60–92 (1975) 37. Taylor, R., Wiles, A.: Ring-theoretic properties of certain Hecke algebras. Ann. Math. 141, 553–572 (1995) 38. Yuan, X., Zhang, S., Zhang, W.: Triple product L-series and Gross–Schoen cycles, preprint 39. Wan, X.: Iwasawa main conjecture for Rankin–Selberg p-adic L-functions, submitted, 2014. http://www.math. columbia.edu/~xw2295/paper2 40. Wiles, A.: Modular elliptic curves and Fermat’s last theorem. Ann. Math. (2) 141(3), 443–551 (1995)