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

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Speaker:
Alexander Smith
Affiliation:
Harvard University
Date:
Tue, 2019-06-04 15:00 - 16:00
Location:
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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