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