$p$进李扩展中超奇异阿贝尔变的Selmer群的弱Leopoldt猜想和Coranks

Pub Date : 2020-03-19 DOI:10.3836/tjm/1502179341
M. Lim
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引用次数: 0

摘要

设$A$是定义在数字域$F$上的一个阿贝尔变量,它在$F$大于$p$的所有素数上具有超奇异约简。我们建立了弱Leopoldt猜想与经典Selmer群在$p$- li扩展(不一定包含环切$ Zp$-扩展)上的corank期望值之间的等价性。作为一个应用,我们得到了Selmer群定义序列的准确性。在$p$进Lie扩展是一维的情况下,我们证明了对偶Selmer群没有非平凡的有限子模。最后,我们证明了上述结论可以推广到非常倒模形式的Selmer群。
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On the Weak Leopoldt Conjecture and Coranks of Selmer Groups of Supersingular Abelian Varieties in $p$-adic Lie Extensions
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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