UCLA

We examine the $p$-adically completed Mordell-Weil groups $\pJ(K)$ and $\calpJQ(K)$, where $K$ is a $p$-adic number field and $\pJ$ and $\calpJQ$ arise as Hecke components of certain $p$-adic completions of Jacobians of either a suitable elliptic modular curve or a suitable Shimura curve. The local ring $\mathbb{T}$ is an appropriate component of the Hecke algebra of a given level $N$ and $\mathbb{T}_Q$ is the corresponding local ring with level augmented by primes in $Q$. For elliptic modular curves, $Q$ refers to a finite set of primes $\{q_1,\dots,q_r\}$ at which the residual Galois representation determined by $\mathbb{T}$ is $q$-distinguished (and analogously for Shimura curves). The $\calpJQ$ carry a natural $\Delta_Q$-action, where $\Delta_Q$ is the $p$-Sylow subgroup of $(\mathbb{Z}/\prod_{i=1}^r q_i\mathbb{Z})^\times$. We study the action of $\Delta_Q$ on $\calpJQ$ and show that we can form a Taylor-Wiles system $(\mathbb{T}_Q,\calpJQ(K)^\ast)_{Q \in X}$, where $X$ ranges over an infinite set of finite sets $Q$ consisting of $q$-distinguished primes and $\ast$ denotes the $\mathcal{O}$-dual for $\mathcal{O}$ a discrete valuation ring, finite and flat, over $\mathbb{Z}_p$. As a consequence, we show that $\pJ(K)$ is free of finite rank $[K:\mathbb{Q}_p]$ over the local ring $\mathbb{T}$ of the Hecke algebra.