Coleman maps and the p-adic regulator (original) (raw)
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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]
Wach modules and Iwasawa theory for modular forms
2009
We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules. Let f = anq n be a normalized new eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f , we define Coleman maps Col i for i = 1, 2 with values in Q p ⊗ Zp Λ, where Λ is the Iwasawa algebra of Z × p . Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case ap = 0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a Λ-cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture.
An introduction to the theory of p-adic representations
Geometric Aspects of Dwork Theory, 2004
The purpose of this informal article is to introduce the reader to some of the objects and methods of the theory of p-adic representations. My hope is that students and mathematicians who are new to the subject will find it useful as a starting point. It consists mostly of an expanded version of the notes for my two lectures at the "Dwork trimester" in June 2001.
Refined Iwasawa theory for p-adic representations and the structure of Selmer groups
2013
In this paper, we develop the idea in [16] to obtain finer results on the structure of Selmer modules for p-adic representations than the usual main conjecture in Iwasawa theory. We determine the higher Fitting ideals of the Selmer modules under several assumptions. Especially, we describe the structure of the classical Selmer group of an elliptic curve over Q, using the ideals defined from modular symbols. We also develop the theory of Euler systems and Kolyvagin systems of Gauss sum type. 0
Katz p-adic L-functions, congruence modules and deformation of Galois representations
L-Functions and Arithmetic, 1991
Although the two-variable main conjecture for imaginary quadratic fields has been successfully proven by Rubin [R] using brilliant ideas found by Thaine and Kolyvagin, we still have some interest in studying the new proof of a special case of the conjecture, i.e., the anticyclotomic case given by Mazur and the second named author of the present article ([M-T], [Tl]). Its interest lies firstly in surprizing amenability of the method to the case of CM fields in place of imaginary quadratic fields and secondly in its possible relevance for non-abelian cases. In this short note, we begin with a short summary of the result in [M-T] and [Tl] concerning the Iwasawa theory for imaginary quadratic fields, and after that, we shall give a very brief sketch of how one can generalize every step of the proof to the general CM-case. At the end, coming back to the original imaginary quadratic case, we remove some restriction of one of the main result in [M-T]. The idea for this slight amelioration to [M-T] is to consider deformations of Galois representations not only over finite fields but over any finite extension of Q p. Throughout the paper, we assume that p > 2.
Kato's Euler system and rational points on ELLIPTIC CURVES I: A p-ADIC BEILINSON FORMULA
2014
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 [Br2] and [Ge] which is independent of Kato’s explicit reciprocity law.
31 p-adic L-functions for modular forms
2019
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