Selmer ranks in twists of CM abelian varieties

Jie Shu

Selmer ranks in twists of CM abelian varieties

Number Theory 2025-08-01 v2

Authors: Jie Shu

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.

Keywords

Selmer ranks CM abelian varieties twists symplectic distributions unitary distributions Fermat curves rank growth cyclic extensions

Cite

@article{arxiv.2504.21274,
  title  = {Selmer ranks in twists of CM abelian varieties},
  author = {Jie Shu},
  journal= {arXiv preprint arXiv:2504.21274},
  year   = {2025}
}

Comments

Revised and extended

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