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 vary in a Hida deformation by using the Kubota-Leopoldt p-adic L-function. Then in Corollary 1.7, we give a sufficient condition for the numbers of the isomorphic classes of stable lattices in Hida deformation to be infinite.is the canonical character and−1 the Teichmüller character. Let O ⊂ Q p be a commutative ring which is finite flat over Z p and let ψ be a Dirichlet character modulo M . We denote by S k (Γ 0 (M ) , ψ, O) the space of cusp forms of weight k, level M , Neben character ψ and Fourier coefficients in O. We also denote by the same symbol ψ the corresponding character of Gal (Q(µ M )/Q) ∼ = (Z/M ) × . For a group ∆ M which is isomorphic to Gal (Q(µ M )/Q), a Z p -module M which has a Z p -linear action of ∆ M and a character ε of ∆ M , we denote by M ε = M ⊗ Zp[∆M ]