Let E/ℚ be an elliptic curve having multiplicative reduction at a prime p. Let (g, h) be a pair of eigenforms of weight 1 arising as the theta series of an imaginary quadratic field K, and assume that the triple-product L-function L(f, g, h, s) is self-dual and does not vanish at the central critical point s=1 . The main result of this article is a formula expressing the value of the associated triple-product p-adic L-function at a point of weights (2, 1, 1) lying outside the region of classical interpolation to the Kolyvagin classes associated by Bertolini and Darmon to a system of Heegner points on E.

A classical point of the Coleman-Mazur eigencurve is said to be exceptional if the map to weight space is non & eacute;tale at that point. This paper revisits the p-adic elliptic Stark conjecture of Darmon et al. (Forum Math. Pi 3 (2015), art. id. e8) concerning a triple ( f, g, h) of classical modular forms of weights (2, 1, 1), and extends it to the setting where the p-stabilised eigenform g corresponds to such an exceptional point.

Let $f$ be a cuspidal eigenform of weight two and level $N$, let $p\nmid N$ be a prime at which $f$ is congruent to an Eisenstein series and let $V_f$ denote the $p$-adic Tate module of $f$. Beilinson constructed a class $\kappa_f\in H^1(\mathbb Q,V_f(1))$ arising from the cup-product of two Siegel units and proved a striking relationship with the first derivative $L'(f,0)$ at the near central point $s=0$ of the $L$-series of $f$, which led him to formulate his celebrated conjecture. In this note we prove two congruence formulae relating the motivic part of $L'(f,0) \,(\mathrm{mod} \, p)$ and $L''(f,0) \,(\mathrm{mod} \, p)$ with circular units. The proofs make use of delicate Galois properties satisfied by various integral lattices within $V_f$ and exploits Perrin-Riou's, Coleman's and Kato's work on the Euler systems of circular units and Beilinson--Kato elements and, most crucially, the work of Sharifi, Fukaya--Kato and Ohta.

We investigate Eisenstein congruences between the so-called Euler systems of Garrett--Rankin--Selberg type. This includes the cohomology classes of Beilinson--Kato, Beilinson--Flach and diagonal cycles. The proofs crucially rely on different known versions of the Bloch--Kato conjecture, and are based on the study of the Perrin-Riou formalism and the comparison between the different $p$-adic $L$-functions.

We study the action of the derived Hecke algebra on the space of weight one forms. By analogy with the topological case, we formulate a conjecture relating this to a certain Stark unit. We verify the truth of the conjecture numerically, for the weight one forms of level 23 and 31, and many derived Hecke operators at primes less than 200. Our computation depends in an essential way on Merel’s evaluation of the pairing between the Shimura and cuspidal subgroups of J0(q).

Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadratic field of the theory of complex multiplication is replaced by a real quadratic field $K$. They are constructed analytically as local points on elliptic curves with multiplicative reduction at a prime $p$ that remains inert in $K$, but are conjectured to be rational over ring class fields of $K$ and to satisfy a Shimura reciprocity law describing the action of $G_K$ on them. The main conjectures of \cite{darmon-hpxh} predict that any linear combination of Stark-Heegner points weighted by the values of a ring class character $\psi$ of $K$ should belong to the corresponding piece of the Mordell-Weil group over the associated ring class field, and should be non-trivial when $L'(E/K,\psi,1) \ne 0$. Building on the results on families of diagonal classes described in the remaining contributions to this volume, this note explains how such linear combinations arise from global classes in the idoneous pro-$p$ Selmer group, and are non-trivial when the first derivative of a weight-variable $p$-adic $L$-function associated to the Hida family passing through $f$ does not vanish at the point associated to $(E/K,\psi)$.

This note provides the construction of a three-variable family of cohomology classes arising from diagonal cycles on a triple product of towers of modular curves, and proves a reciprocity law relating it to the three variable triple-product p-adic L-function associated to a triple of Hida families by means of Perrin-Riou's Lambda-adic regulator.

Let $E/\mathbb{Q}$ be an elliptic curve having multiplicative reduction at a prime $p$. Let $(g,h)$ be a pair of eigenforms of weight $1$ arising as the theta series of an imaginary quadratic field $K$, and assume that the triple-product $L$-function $L(f,g,h,s)$ is self-dual and does not vanish at the central critical point $s=1$. The main result of this article is a formula expressing the $p$-adic iterated integrals introduced in [DLR] to the Kolyvagin classes associated by Bertolini and Darmon to a system of Heegner points on $E$.

Kings, Lei, Loeffler and Zerbes constructed a three-variable Euler system $\kappa({\bf g},{\bf h})$ of Beilinson-Flach elements associated to a pair of Hida families $({\bf g},{\bf h})$ and exploited it to obtain applications to the arithmetic of elliptic curves by specializing the Euler system to points of weights $(2,1,1)$. The aim of this article is showing that this Euler system also encodes arithmetic information at points of weights $(1,1,0)$, concerning the group of units of the associated number fields. The setting becomes specially novel and intriguing when ${\bf g}$ and ${\bf h}$ specialize in weight $1$ to $p$-stabilizations of eigenforms such that one is dual of another. We encounter an exceptional zero phenomenon which forces the specialization of $\kappa({\bf g}, {\bf h})$ at $(1,1,0)$ to vanish and we are led to study the derivative of this class. The main result we obtain is the proof of a conjecture of Darmon, Lauder and Rotger on iterated integrals and another conjecture of Darmon and Rotger for Beilinson-Flach elements in the adjoint setting. The main point of this paper is that the methods of previous works, where the above conjectures are proved when the weight $1$ eigenforms have CM, do not apply to our setting and new ideas are required. Here, a factorization of $p$-adic $L$-functions 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 are appearing in the literature.

The main purpose of this note is to understand the arithmetic encoded in the special value of the p-adic L-function L-p(g)(f, , g, h) 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 circle times g circle times 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/Q and the classical L-function vanishes, the Elliptic Stark Conjecture of Darmon-Lauder-Rotger predicts that L-p(g)(f, g, h)(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 El-liptic Stark Conjecture by providing a formula for the value L-p(g)(f, g, h)(2, 1, 1) in the case where L(f circle times g circle times h, 1) not equal 0.

The purpose of this article is proving the equality of two natural L-invariants attached to the adjoint representation of a weight one cusp form, each defined by purely analytic, respectively, algebraic means. The proof departs from Greenberg's definition of the algebraic L-invariant as a universal norm of a canonical Zp-extension of Qp associated to the representation. We relate it to a certain 2x2 regulator of p-adic logarithms of global units by means of class field theory, which we then show to be equal to the analytic L-invariant computed in Rivero and Rotger [J. Eur. Math. Soc., to appear].

Abstract 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.

1.1. The BSD Conjecture. The goal of these lectures is to describe some recent progress towards the Birch–Swinnerton-Dyer conjecture in cases when the ground field is not Q and in the rank 2 case. These are cases where Heegner points are not available, so they are not covered by the methods of Gross–Zagier [GZ86] and Kolyvagin [Kol90]. We start with a review of the conjecture and these classical methods. Let E/Q be an elliptic curve of conductor NE. For ` prime, the Tate module is

In [DLR.], Darmon, Lauder, and Rotger formulated a padic elliptic Stark conjecture for the twist of an elliptic curve E/Q by the self-dual tensor product rho(1) circle times rho(2) of two odd and two-dimensional Artin representations. These authors provided abundant numerical evidence and proved the conjecture in the special setting where p is a prime of good reduction for E and rho(1) and rho(2) are induced from finite-order characters psi(g), psi(h) of the same imaginary quadratic field. The key step in their proof is a factorization of one-variable p-adic L-functions, where psi(g) varies in a p-adic family of Hecke characters. The main goal of this article is to prove a new case of the conjecture, placing ourselves in the setting where p is a prime of multiplicative reduction for E. In order to achieve our theorem, we need to work with two-variable p-adic L-functions, where the weight 2 cusp form associated with E also moves independently along a Hida family. Our main result then follows from a factorization of p-adic L-series extending to two variables the one obtained in [DLR]. On the way we also generalize to our setting the results obtained in [CR].

We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rather general 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.

This article is devoted to the elliptic Stark conjecture formulated by Darmon (Forum Math Pi 3:e8, 2015), which proposes a formula for the transcendental part of a p-adic avatar of the leading term at \(s=1\) of the Hasse–Weil–Artin L-series \(L(E,\varrho _1\otimes \varrho _2,s)\) of an elliptic curve \(E/\mathbb {Q}\) twisted by the tensor product \(\varrho _1\otimes \varrho _2\) of two odd 2-dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a \(2\times 2\) p-adic regulator involving the p-adic formal group logarithm of suitable Stark points on E. This conjecture was proved by Darmon (Forum Math Pi 3:e8, 2015) in the setting where \(\varrho _1\) and \(\varrho _2\) are induced from characters of the same imaginary quadratic field K. In this note, we prove a refinement of this result that was discovered experimentally by Darmon (Forum Math Pi 3:e8, 2015, [Remark 3.4]) in a few examples. Namely, we are able to determine the algebraic constant up to which the main theorem of Darmon (Forum Math Pi 3:e8, 2015) holds in a particular setting where the Hida–Rankin p-adic L-function associated to a pair of Hida families can be exploited to provide an alternative proof of the same result. This constant encodes local and global invariants of both E and K.

This article studies the first-order p-adic deformations of classical weight one newforms, relating their fourier coefficients to the p-adic logarithms of algebraic numbers in the field cut out by the associated projective Galois representation.

This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank 0, for elliptic curves over $ \mathbb{Q}$ viewed over the fields cut out by certain self-dual Artin representations of dimension at most $ 4$. When the associated $ L$-function vanishes (to even order $ \ge 2$) at its central point, two canonical classes in the corresponding Selmer group are constructed and shown to be linearly independent assuming the non-vanishing of a Garrett-Hida $ p$-adic $ L$-function at a point lying outside its range of classical interpolation. The key tool for both results is the study of certain $ p$-adic families of global Galois cohomology classes arising from Gross-Kudla-Schoen diagonal cycles in a tower of triple products of modular curves.

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

We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rather general type of quaternionic or- ders closely related to those introduced by HijikataPizerShemanske in the 80's. We address several questions arising from the Birch and Swinnerton-Dyer (BSD) conjecture in this general context. In par- ticular, 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.