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arXiv · 2003.08527

On the weak Leopoldt conjecture and coranks of Selmer groups of supersingular abelian varieties in $p$-adic Lie extensions

Meng Fai Lim
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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 neccesasily containing the cyclotomic $\Zp$-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 dual 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. 2020-03-19. On the weak Leopoldt conjecture and coranks of Selmer groups of supersingular abelian varieties in $p$-adic Lie extensions. https://doi.org/10.3836/tjm%2F1502179341

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