Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents

We study the probability with which an elliptic curve $E\mathrm{/}k$, subject to some technical conditions, gains rank upon base extension to an $S_3$-cubic extension $K\mathrm{/}k$ with quadratic resolvent field $F\mathrm{/}k$, all three fields of which are subject to some mild technical conditions. To do so, we determine the distribution (under a non-standard ordering) of Selmer ranks of an auxiliary abelian variety associated to $E$ and $S_3$-cubic extensions $K\mathrm{/}k$ following ideas of Klagsbrun, Mazur, and Rubin. One corollary of this distribution is that $E$ gains rank by at most one upon base extension to $K$ with probability at least $31.95\mathrm{\%}$.