Explicit computations of Hida families via overconvergent modular symbols

TL;DR

This paper extends algorithms for overconvergent modular symbols to compute entire Hida families, enabling the calculation of p-adic families of eigenvalues, L-functions, and algebraic structures.

Contribution

It generalizes existing algorithms to families of overconvergent modular symbols, allowing comprehensive computations of p-adic families and algebraic structures.

Findings

Computed p-adic families of Hecke-eigenvalues

Calculated two-variable p-adic L-functions and L-invariants

Analyzed the structure of ordinary Hida-Hecke algebras

Abstract

In [Pollack-Stevens 2011], efficient algorithms are given to compute with overconvergent modular symbols. These algorithms then allow for the fast computation of $p$-adic $L$-functions and have further been applied to compute rational points on elliptic curves (e.g. [Darmon-Pollack 2006, Trifkovi\'c 2006]). In this paper, we generalize these algorithms to the case of families of overconvergent modular symbols. As a consequence, we can compute $p$-adic families of Hecke-eigenvalues, two-variable $p$-adic $L$-functions, $L$-invariants, as well as the shape and structure of ordinary Hida-Hecke algebras.