p-adic measures and square roots of special values of triple product L-functions (original) (raw)
Related papers
2-ADIC Properties of Certain Modular Forms and Their Applications to Arithmetic Functions
It is a classical observation of Serre that the Hecke algebra acts locally nilpo- tently on the graded ring of modular forms modulo 2 for the full modular group. Here we consider the problem of classifying spaces of modular forms for which this phenomenon continues to hold. We give a number of consequences of this investigation as they relate to quadratic forms, partition functions, and central values of twisted modular L-functions.
p-Adic Aspects of Modular Forms, 2016
These are the expanded notes of a mini-course of four lectures by the same title given in the workshop "p-adic aspects of modular forms" held at IISER Pune, in June, 2014. We give a brief introduction of p-adic L-functions attached to certain types of automorphic forms on GLn with the specific aim to understand the p-adic symmetric cube L-function attached to cusp forms on GL 2 over rational numbers. Contents 1. What is a p-adic L-function? 2 2. The symmetric power L-functions 11 3. p-adic L-functions for GL 4 16 4. p-adic L-functions for GL 3 × GL 2 22 References 27 The aim of this survey article is to bring together some known constructions of the p-adic L-functions associated to cohomological, cuspidal automorphic representations on GL n /Q. In particular, we wish to briefly recall the various approaches to construct p-adic L-functions with a focus on the construction of the p-adic L-functions for the Sym 3 transfer of a cuspidal automorphic representation π of GL 2 /Q. We note that p-adic L-functions for modular forms or automorphic representations are defined using p-adic measures. In almost all cases, these p-adic measures are constructed using the fact that the L-functions have integral representations, for example as suitable Mellin transforms. Candidates for distributions corresponding to automorphic forms can be written down using such integral representations of the L-functions at the critical points. The well-known Prop. 2 is often used to prove that they are indeed distributions, which is usually a consequence of the defining relations of the Hecke operators. Boundedness of these distributions are shown by proving certain finiteness or integrality properties, giving the sought after p-adic measures. In Sect. 1, we discuss general notions concerning p-adic L-functions, including our working definition of what we mean by a p-adic L-function. As a concrete example, we discuss the construction of the p-adic L-functions that interpolate critical values of L-functions attached to modular forms. Manin [47]
Gross–Stark units and p-adic iterated integrals attached to modular forms of weight one
Annales mathématiques du Québec, 2016
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. Keywords Eisenstein series • p-adic modular forms • p-adic iterated integrals • Gross-Stark units Résumé Cet article peut se lire comme un supplément à l'article des mêmes auteurs sur les "points de Stark" et les intégrales itérées p-adiques. Dans cet article antérieur, il est conjecturé que les intégrales itérées p-adiques associées à un triplet (f, g, h) de formes modulaires de poids (2, 1, 1) s'expriment au moyen de points de Stark sur la variété abélienne associée à f , définis sur le corps de nombres découpé par le produit tensoriel des représentations d'Artin B Henri Darmon
$p$-adic interpolation of special values of Hecke L-functions
1992
© Foundation Compositio Mathematica, 1992, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Stark points and p-adic iterated integrals attached to modular forms of weight one, submitted
2016
To our families Abstract. Let E be an elliptic curve over Q and let % [ and %] be odd two-dimensional Artin representations for which %[ %] is self-dual. The progress on modularity achieved in the last decades ensures the existence of normalised eigenforms f, g and h of respective weights 2, 1 and 1, giving rise to E, %[, and %] via the constructions of Eichler-Shimura and Deligne-Serre. This article examines certain p-adic iterated integrals attached to the triple (f; g; h), which are p-adic avatars of the leading term of the Hasse-Weil-Artin L-series L(E; %[ %]; s) when it has a double zero at the center. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on E|referred to as Stark points|which are de ned over the number eld 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 g and h are binary theta series attached to a ...
Stark points and p-adic iterated integrals attached to modular forms of weight one
2016
To our families Let E be an elliptic curve over Q, 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 f, g, and h of respective weights two, one, and one, giving rise to E, %[, and %] via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain p-adic iterated integrals attached to the triple ( f, g, h), which are p-adic avatars of the leading term of the Hasse–Weil–Artin L-series L(E, %[⊗%], s) 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 E—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 g and h are binary theta series ...
p-Adic L-Functions for Unitary Shimura Varieties I: Construction of the Eisenstein Measure
We construct the Eisenstein measure in several variables on a quasi-split unitary group, as a first step towards the construction of p-adic L-functions of families of ordinary holomorphic modular forms on unitary groups. The construction is a direct generalization of Katz' construction of p-adic L-functions for CM fields, and is based on the theory of p-adic modular forms on unitary Shimura varieties developed by Hida, and on the explicit calculation of non-degenerate Fourier coefficients of Eisenstein series.
The p-Adic Eisenstein Measure and Fourier coefficients for SL(2)
2010
We introduce an analog of part of the Langlands-Shahidi method to the p-adic setting, constructing reciprocals of certain p-adic L-functions using the nonconstant terms of the Fourier expansions of Eisenstein series. We carry out the method for the group SL 2 , and give explicit p-adic measures whose Mellin transforms are reciprocals of Dirichlet L-functions. The formulas for these measures involve Fourier coefficients of Eisenstein series, plus a delicately chosen multiple of Haar measure necessary for boundedness.