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2021 On the Weak Leopoldt Conjecture and Coranks of Selmer Groups of Supersingular Abelian Varieties in $p$-adic Lie Extensions
Meng Fai LIM
Tokyo J. Math. Advance Publication 1-18 (2021). DOI: 10.3836/tjm/1502179341

Abstract

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 $\mathbb{Z}p$-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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Meng Fai LIM. "On the Weak Leopoldt Conjecture and Coranks of Selmer Groups of Supersingular Abelian Varieties in $p$-adic Lie Extensions." Tokyo J. Math. Advance Publication 1 - 18, 2021. https://doi.org/10.3836/tjm/1502179341

Information

Published: 2021
First available in Project Euclid: 23 March 2021

Digital Object Identifier: 10.3836/tjm/1502179341

Subjects:
Primary: 11G05
Secondary: 11R23, 11S25

Rights: Copyright © 2021 Publication Committee for the Tokyo Journal of Mathematics

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