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$2^k$-Selmer groups and Goldfeld's conjecture, II

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Alexander Smith
Harvard University
Tue, 2019-06-04 15:00 - 16:00
MPIM Lecture Hall

Choose some positive $k$ and a rational elliptic curve $E$, and choose $k$ pairs of primes $(p_i, p'_i)$. Take $d_0$ to be $p_1 p_2 \dots p_k$, and consider the family of $d$ given by replacing $p_i$ with $p'_i$ for some set of $i$. Under special circumstances, we show that $2^k$-Selmer elements of a twist $E^{d_0}$ can be constructed from $2^k$-Selmer elements of the remaining twists $E^d$. By elaborating on this strategy we show that, in a grid of twists of $E$, some information about the distribution of $2^k$-Selmer groups over this grid can be found from symbols whose values are subject to analytic control.

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