On the Weak Leopoldt Conjecture and Coranks of Selmer Groups of Supersingular Abelian Varieties in p-adic Lie Extensions

摘要

Let A be an abelian variety defined over a number field F with supersingular reduction at all primes of F above p. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the classical Selmer group of A over a p-adic Lie extension (not necessarily containing the cyclotomic Z(p)-extension). As an application, we obtain the exactness of the defining sequence of the Selmer group. In the event that the p-adic Lie extension is one-dimensional, we show that the Pontryagin dual of the classical Selmer group has no nontrivial finite submodules. Finally, we show that the aforementioned conclusions carry over to the Selmer group of a non-ordinary cuspidal modular form.

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