Stable lattices in modular Galois representations and Hida deformation

TL;DR

This paper investigates how the count of stable lattice classes in modular Galois representations varies within Hida deformations, linking it to $p$-adic $L$-functions and establishing conditions for their infinitude.

Contribution

It introduces a new analysis of stable lattice variation in Hida deformations using $p$-adic $L$-functions and provides criteria for infinite classes.

Findings

Variation of stable lattice classes is linked to Hida deformation parameters.

A sufficient condition for infinite stable lattice classes is established.

Uses Kubota-Leopoldt $p$-adic $L$-function in the analysis.

Abstract

In this paper, we discuss the variation of the numbers of the isomorphic classes of stable lattices when the weight and the level varies in a Hida deformation by using the Kubota-Leopoldt $p$-adic $L$-function. As a corollary, we give a sufficient condition for the numbers of the isomorphic classes of stable lattices in Hida deformation to be infinite.