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.

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In T-cell Acute Lymphocytic Leukemia (T-ALL), the inhibitors of cyclin-dependent kinases (CDK) 4 and 6, p16 and p15, are inactivated almost universally at the DNA, RNA and protein levels. This suggests that CDK-targeting may be an effective therapeutic approach for T-ALL and other cancers. In this study, we tested 3 inhibitors of CDK4, 3-aminothioacridone (3-ATA), thioacridone (TA), and oxindole, for their effects on DNA synthesis and viability in primary T-ALL. Each compound was an effective inhibitor, with overall IC50s in similar ranges. In colony formation assay, leukemic cells were approximately 10-fold more sensitive to 3-ATA than normal bone marrow cells. When sorted by G1 protein status of T-ALL, p16(+), p15(+) or pRb(-) samples were significantly less sensitive to 3-ATA and TA, but not to oxindole, than p16(-), p15(-) or pRb(+) samples. There was no relationship of sensitivity with ARF expression. Despite their in vitro function as inhibitors of CDK4, 3-ATA did not inhibit pRb phosphorylation or cause G1 arrest, but did cause DNA damage and result in the induction and phosphorylation of p53. We conclude that 3-ATA efficacy can be predicted by p16 status in T-ALL, but the mechanism of action may be distinct from their in vitro ability to regulate CDK4 kinase activity.