Stable lattices in modular Galois representations and Hida deformation

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.