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p-Selmer ranks of CM abelian varieties

Jamie Bell

Abstract

For an elliptic curve with complex multiplication over a number field, the $p^{\infty}$--Selmer rank is even for all $p$. Česnavičius 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.

p-Selmer ranks of CM abelian varieties

Abstract

For an elliptic curve with complex multiplication over a number field, the --Selmer rank is even for all . Česnavičius proved this using the fact that admits a -isogeny whenever splits in the complex multiplication field, and invoking known cases of the -parity conjecture. We give a direct proof, and generalise the result to abelian varieties.
Paper Structure (2 sections, 9 theorems, 22 equations)

This paper contains 2 sections, 9 theorems, 22 equations.

Table of Contents

  1. Self-isogenies
  2. Complex Multiplication

Key Result

Theorem 1

Suppose $A/K$ is an abelian variety with complex multiplication by $M$, and $p$ a prime. Then $\mathrm{rk}_p(A)$ is even.

Theorems & Definitions (20)

  • Theorem 1
  • Corollary 2
  • Definition 3: Rosati involution
  • Definition 4: Complex multiplication
  • Theorem 5: Milne, Proof of I.7.3, I.7.3.1
  • Corollary 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 10 more