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arXiv · 2208.14563

p-Selmer ranks of CM abelian varieties

Jamie Bell
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Abstract

For an elliptic curve with complex multiplication over a number field, the $p^{\infty}$--Selmer rank is even for all $p$. \v{C}esnavi\v{c}ius proved this using the fact that $E$ admits a $p$-isogeny whenever $p$ splits in the complex multiplication field, and invoking known cases of the $p$-parity conjecture. We give a direct proof, and generalise the result to abelian varieties.

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Jamie Bell. 2022-08-30. p-Selmer ranks of CM abelian varieties. https://arxiv.org/abs/2208.14563

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