Distribution of Selmer ranks in prime cyclic extensions

Abstract

Using modifications to work of Klagsbrun, Mazur, and Rubin, we study (assuming the Extended Riemann Hypothesis) the distribution of Selmer ranks of twist families of some given even-dimensional Galois modules satisfying some mild technical conditions. As a corollary, we study the probability with which a fixed elliptic curve gains (or does not gain) rank in $p$-cyclic extensions, obtaining bounds for this distribution. Likewise, for some superelliptic curves $C$, we bound the average size of $C(L)$ as $L$ ranges over $p$-cyclic extensions over a number field $K$ containing primitive $p$-th roots of unity. Lastly, we study the probability with which a fixed hyperelliptic curve gains (or does not gain) rank in quadratic extensions, also obtaining bounds for this distribution. In all three cases, the extensions under consideration are ordered by the product of ramified primes.