# Freeness of $p$-adically Completed Modular Jacobians over a Hecke Algebra

## TL;DR

This paper constructs a Taylor-Wiles system using $p$-adically completed modular Jacobians over Hecke algebras and proves the freeness of certain Mordell-Weil groups over these algebras.

## Contribution

It introduces a new Taylor-Wiles system framework for $p$-adically completed modular Jacobians and establishes their Mordell-Weil groups are free modules over Hecke algebras.

## Key findings

- Freeness of $p$-adically completed Mordell-Weil groups over Hecke algebras.
- Construction of a Taylor-Wiles system in this context.
- Proof of finite rank of these Mordell-Weil groups.

## Abstract

We construct a Taylor-Wiles system using a family of $p$-adically completed modular Jacobians over suitable Hecke algebras and prove that certain $p$-adically completed Mordell-Weil groups of these Jacobians is free of finite rank over a Hecke algebra.