p-Selmer ranks of CM abelian varieties

2022/08/30 by Bell, Jamie
#11G15 (Primary) 14K22 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2208.14563

Abstract

For an elliptic curve with complex multiplication over a number field, the p^∞--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.

Related

• Finding large Selmer rank via an arithmetic theory of local constants
• The p-parity conjecture for elliptic curves with a p-isogeny
• Selmer ranks in twists of CM abelian varieties
• A rank zero p-converse to a theorem of Gross--Zagier, Kolyvagin and Rubin
• Increasing the p-Selmer rank by twisting