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arXiv · 2504.21274

Selmer ranks in twists of CM abelian varieties

Jie Shu
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Abstract

We prove the Selmer ranks in certain families of $p$-th twists of CM abelian varieties obey the symplectic or unitary distributions. As an application, for a prime $p\geq 3$, we obtain that the twisted Fermat curves $X^p+Y^p=\delta$ over a number field containing a primitive $p$-th root of unity are ``largely" unsolvable as $\delta$ varies. We also discuss the rank growth in cyclic extensions of prime degree for CM abelian varieties.

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Jie Shu. 2025-04-30. Selmer ranks in twists of CM abelian varieties. https://arxiv.org/abs/2504.21274

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