$p$-Selmer growth in extensions of degree $p$

Abstract

There is a known analogy between growth questions for class groups and for Selmer groups. If $p$ is a prime, then the $p$-torsion of the ideal class group grows unboundedly in $\mathbb{Z}/p\mathbb{Z}$-extensions of a fixed number field $K$, so one expects the same for the $p$-Selmer group of a nonzero abelian variety over $K$. This Selmer group analogue is known in special cases and we prove it in general, along with a version for arbitrary global fields.

19 pages; final version, to appear in Journal of the London Mathematical Society

References in corpus (3)

• Selmer groups as flat cohomology groups
• Selmer groups and class groups
• Potential Sha for abelian varieties

Cited by in corpus (3)

• Artin-Mazur-Milne duality for fppf cohomology
• Ranks of abelian varieties in cyclotomic twist families
• A positive density of elliptic curves are diophantine stable in certain Galois extensions