Derived Beilinson-Flach elements and the arithmetic of the adjoint of a modular form

Kings, Lei, Loeffler and Zerbes constructed in [LLZ], [KLZ1] a three-variable Euler system ¿(g,h) of Beilinson–Flach elements associated to a pair of Hida families (g,h) and exploited it to obtain applications to the arithmetic of elliptic curves, extending the earlier work [BDR]. The aim of this article is to show that this Euler system also encodes arithmetic information concerning the group of units of the associated number fields. The setting becomes specially novel and intriguing when g and h specialize in weight 1 to p-stabilizations of eigenforms such that one is dual to the other. We encounter an exceptional zero phenomenon which forces the specialization of ¿(g,h) to vanish and we are led to study the derivative of this class. The main result we obtain is the proof of the main conjecture of [DLR4] on iterated integrals and the main conjecture of [DR1] for Beilinson–Flach elements in the adjoint setting. The main point of this paper is that the methods of [DLR1], [DLR4] and [CH], where the above conjectures are proved when the weight 1 eigenforms have CM, do not apply to our setting and new ideas are required. In the previous works, a crucial ingredient is a factorization of p-adic L-functions, which in our scenario is not available due to the lack of critical points. Instead we resort to the principle of improved Euler systems and p-adic L-functions to reduce our problems to questions which can be resolved using Galois deformation theory. We expect this approach may be adapted to prove other cases of the Elliptic Stark Conjecture and of its generalizations that appear in the literature.Both authors were supported by Grant MTM2015-63829-P. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 682152). The first author has also received financial support through “la Caixa” Fellowship Grant for Doctoral Studies (grant LCF/BQ/ES17/ 11600010). The second author gratefully acknowledges Icrea for financial support through an Icrea Academia award.Peer ReviewedPostprint (published version

Elliptic curves of rank two and generalized Kato classes

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 ¿(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 Vg and Vh 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 Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that ¿(f,g,h) lies in the pro-p Selmer group of E over H precisely when L(E,Vgh,1)=0, where L(E,Vgh,s) is the L-function of E twisted by Vgh:=Vg¿Vh. In the setting of interest, parity considerations imply that L(E,Vgh,s) vanishes to even order at s=1, and the Selmer class ¿(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 ¿(f,g,h) as a canonical point in (E(H)¿Vgh)GQ when ords=1L(E,Vgh,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).Peer ReviewedPostprint (published version

Beilinson-Flach elements, Stark units and p-adic iterated integrals

We study weight one specializations of the Euler system of Beilinson–Flach elements introduced by Kings, Loeffler and Zerbes, with a view towards a conjecture of Darmon, Lauder and Rotger relating logarithms of units in suitable number fields to special values of the Hida–Rankin p-adic L-function. We show that the latter conjecture follows from expected properties of Beilinson–Flach elements and prove the analogue of the main theorem of Castella and Hsieh about generalized Kato classes.Peer ReviewedPostprint (author's final draft

Beilinson-Flach elements and Euler systems I: syntomic regulators and p-adic Rankin L-series

This article is the first in a series devoted to the Euler system arising from p-adic families of Beilinson-Flach elements in the first K-group of the product of two modular curves. It relates the image of these elements under the p-adic syntomic regulator (as described by Besser (2012)) to the special values at the near-central point of Hida's p-adic Rankin L-function attached to two Hida families of cusp forms.Peer ReviewedPostprint (author’s final draft

Pour une prospective Baudelairienne

This article has been published in a revised form in Forum of Mathematics Pi, http://dx.doi.org/10.1017/fmp.2015.7. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © 2015.Let be an elliptic curve over , and let and be odd two-dimensional Artin representations for which is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms , , and of respective weights two, one, and one, giving rise to , , and via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain -adic iterated integrals attached to the triple , which are -adic avatars of the leading term of the Hasse–Weil–Artin -series when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on —referred to as Stark points—which are defined over the number field cut out by . This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weight-one forms. It is proved when and are binary theta series attached to a common imaginary quadratic field in which splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing -adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell–Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark–Heegner points attached to Shintani-type cycles on ), and extensions of with Galois group a central extension of the dihedral group or of one of the exceptional subgroups , , and ofPeer ReviewedPostprint (published version

Heegner points on Hijikata–Pizer–Shemanske curves and the Birch and Swinnerton-Dyer conjecture

We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rathergeneral type of quaternionic orders. We address several questions arising from the Birch and Swinnerton-Dyer (BSD) conjecture in this general context. In particular, under mild technical conditions, we show the existence of non-torsion Heegner points on elliptic curves in all situations in which the BSD conjecture predicts their existence.We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rathergeneral type of quaternionic orders. We address several questions arising from the Birch and Swinnerton-Dyer (BSD) conjecture in this general context. In particular, under mild technical conditions, we show the existence of non-torsion Heegner points on elliptic curves in all situations in which the BSD conjecture predicts their existence

Special values of triple-product -adic L-functions and non-crystalline diagonal classes

The main purpose of this note is to understand the arithmetic encoded in the special value of the $p$-adic $L$-function $E_p^g$ (f, $\left.\mathbf{g}, \mathbf{h}\right)$ associated to a triple of modular forms $(f, g, h)$ of weights $(2,1,1)$, in the case where the classical $L$-function $L(f \otimes g \otimes h, s)$ (which typically has sign +1$)$ does not vanish at its central critical point $s=1$. When $f$ corresponds to an elliptic curve $E / \mathbb{Q}$ and the classical $L$-function vanishes, the Elliptic Stark Conjecture of Darmon-Lauder-Rotger predicts that $E_p^g$ (f, $\left.\mathbf{g}, \mathbf{h}\right)(2,1,1)$ is either 0 (when the order of vanishing of the complex $L$-function is $>2$ ) or related to logarithms of global points on $E$ and a certain Gross-Stark unit associated to $g$ (when the order of vanishing is exactly 2). We complete the picture proposed by the Elliptic Stark Conjecture by providing a formula for the value $E_p^g(\mathbf{f}, \mathbf{g}, \mathbf{h})(2,1,1)$ in the case where $L(f \otimes g \otimes h, 1) \neq 0$

A survey on recent p-adic integration theories and arithmetic applications

"Vegeu el resum a l'inici del document del fitxer adjunt"

On rigid analytic uniformizations of Jacobians of Shimura curves

The main goal of this article is to give an explicit rigid analytic uniformization of the maximal toric quotient of the Jacobian of a Shimura curve over Q at a prime dividing exactly the level. This result can be viewed as complementary to the classical theorem of Cerednik and Drinfeld which provides rigid analytic uniformizations at primes dividing the discriminant. As a corollary, we offer a proof of a conjecture formulated by M. Greenberg in hispaper on Stark-Heegner points and quaternionic Shimura curves, thus making Greenberg's construction of local points on elliptic curves over Q unconditional

Heegner points on Hijikata-Pizer-Shemanske curves and the Birch and Swinnerton-Dyer conjecture

We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rathergeneral type of quaternionic orders. We address several questions arising from the Birch and Swinnerton-Dyer (BSD) conjecture in this general context. In particular, under mild technical conditions, we show the existence of non-torsion Heegner points on elliptic curves in all situations in which the BSD conjecture predicts their existence