Extremal p-Adic L-Functions

Santiago Molina

doi:10.3390/math9030234

Abstract

In this note, we propose a new construction of cyclotomic p-adic L-functions that are attached to classical modular cuspidal eigenforms. This allows for us to cover most known cases to date and provides a method which is amenable to generalizations to automorphic forms on arbitrary groups. In the classical setting of GL2 over Q, this allows for us to construct the p-adic L-function in the so far uncovered extremal case, which arises under the unlikely hypothesis that p-th Hecke polynomial has a double root. Although Tate’s conjecture implies that this case should never take place for GL2/Q, the obvious generalization does exist in nature for Hilbert cusp forms over totally real number fields of even degree, and this article proposes a method that should adapt to this setting. We further study the admissibility and the interpolation properties of these extremal p-adic L-functionsLpext(f,s), and relate Lpext(f,s) to the two-variable p-adic L-function interpolating cyclotomic p-adic L-functions along a Coleman family.

Highlights

IntroductionLet f ∈ Sk+2 (Γ1 ( N ), e) be a modular cuspidal eigeform for Γ1 ( N ) with nebentypus e and weight k + 2
Our approach exploits the theory of automorphic representations and, in that sense, it is similar to the construction that was provided by Spiess in [4] for weights strictly greater than 2
This opens the door to possible generalizations of p-adic measures attached to automorphic representations of GL2 (AF ) of any weight, for any number field F

Summary

Introduction

Let f ∈ Sk+2 (Γ1 ( N ), e) be a modular cuspidal eigeform for Γ1 ( N ) with nebentypus e and weight k + 2. The study of the complex L-function L(s, π ) attached to the automorphic representation π of GL2 (A) generated by f is a very important topic in modern Number Theory. Understanding this complex valued analytic function is the key point for some of the most important problems in mathematics, such as the Birch and Swinnerton–Dyer conjecture. Back in the middle of the seventies, Vishik [1] and Amice-Vélu [2] defined a p-adic measure μ f ,p of Z×

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Conclusion