# Stable $\mathbb{I}$-free lattices and the two-variable algebraic $p$-adic $L$-functions in residually reducible Hida deformation

## TL;DR

This paper investigates the structure of stable lattices in residually reducible Hida deformations and explores the variation of associated two-variable algebraic p-adic L-functions, extending previous work and answering a question by Ochiai.

## Contribution

It introduces the concept of stable $ ext{I}$-free lattices in Hida deformations and analyzes their impact on two-variable algebraic p-adic L-functions, providing new insights into their variation.

## Key findings

- Characterization of $ ext{I}$-free lattices in Hida deformations.
- Analysis of the variation of two-variable algebraic p-adic L-functions.
- Answer to Ochiai's question on lattice classes in residually reducible cases.

## Abstract

This paper is a continuation of the author's previous work, where we studied the variation of the number of isomorphic classes of $G_{\mathbb{Q}}$-stable lattices in p-adic families of residually reducible ordinary Galois representations. In this paper, we study the number of isomorphic classes of $G_{\mathbb{Q}}$-stable free lattices in a residually reducible Hida deformation and the variation of their related two-variable algebraic $p$-adic $L$-functions. This result gives an answer of a question asked by Ochiai.