Onp-adic analogues of the conjectures of Birch and Swinnerton-Dyer

2012

Abstract. This article is the first in a series devoted to Kato’s Euler system arising from p-adic families of Beilinson elements in the K-theory of modular curves. It proves a p-adic Beilinson formula relating the syntomic regulator (in the sense of Coleman-de Shalit and Besser) of certain distinguished elements in the K-theory of modular curves to the special values at integer points ≥ 2 of the Mazur-Swinnerton-Dyer p-adic L-function attached to cusp forms of weight 2. When combined with the explicit relation between syntomic regulators and p-adic étale cohomology, this leads to an alternate proof of the main results of [Br] and

Pacific Journal of Mathematics, 2003

We derive the existence of a specific two-variable p-adic Lfunction by means of a method provided by Washington. This two-variable function is a generalization of the one-variable p-adic L-function of Kubota and Leopoldt, yielding the onevariable function when the second variable vanishes.

<i>p</i>-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]

2014

In [12], McCarthy defined a function n G n [• • •] using the Teichmüller character of finite fields and quotients of the p-adic gamma function, and expressed the trace of Frobenius of elliptic curves in terms of special values of 2 G 2 [• • •]. We establish two different expressions for the traces of Frobenius of elliptic curves in terms of the function 2 G 2 [• • •]. As a result, we obtain two transformation formulas of the function 2 G 2 [• • •] with different parameters.

2006

The object of this paper is to give several properties and applications of the multiple p-adic q-L-function of two variables L (r) p,q (s, z, χ). The explicit formulas relating higher order q- Bernoulli polynomials, which involve sums of products of higher order q-zeta function and higher order Dirichlet q-L-function are given. The value of higher order Dirichlet p-adic q-L-function for positive integers is also calculated. Furthermore, the Kummer-type congruences for multiple generalized q-Bernoulli polynomials are derived by making use of the difference theorem of higher order Dirichlet p-adic q-L-function.