Extremal p-adic L-functions

TL;DR

This paper introduces a new method for constructing extremal p-adic L-functions associated with modular forms, extending to cases previously thought impossible and suggesting potential for broader automorphic generalizations.

Contribution

It proposes a novel construction of extremal p-adic L-functions for modular forms, including cases with double roots in Hecke polynomials, and discusses their properties and generalizations.

Findings

Constructed extremal p-adic L-functions in previously inaccessible cases.

Analyzed admissibility and interpolation properties of these functions.

Connected extremal p-adic L-functions to two-variable p-adic L-functions in Coleman families.

Abstract

In this note we propose a new construction of cyclotomic p-adic L-functions attached to classical modular cuspidal eigenforms. This allows 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 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 which should adapt to this setting. We further study the admissibility and the interpolation properties of these extremal p-adic L-functions, and relate them to the two-variable p-adic L-function interpolating cyclotomic p-adic L-functions along a Coleman family.