Arithmetic Geometry and Number Theory RTG Seminar: $2^k$-Selmer groups, $2^k$-class groups, and Goldfeld's conjecture

Seminar | March 12 | 3:10-5 p.m. | 748 Evans Hall

 Alexander Smith, Harvard University

 Department of Mathematics

Take $E/\mathbb Q$ to be an elliptic curve with full rational 2-torsion (satisfying some extra technical assumptions). In this talk, we will show that $100\%$ of the quadratic twists of $E$ have rank less than two, thus proving that the BSD conjecture implies Goldfeld's conjecture in these families. To do this, we will extend Kane's distributional results on the 2-Selmer groups in these families to $2^k$-Selmer groups for any $k >1$. In addition, using the close analogy between $2^k$-Selmer groups and $2^{k+1}$-class groups, we will prove that the $2^k$-class groups of the quadratic imaginary fields are distributed as predicted by the Cohen-Lenstra heuristics for all $k > 1$.

Seminar Format: The seminar consists of two 50-minute talks, a pre-talk (3:10-4:00) and an advanced talk (4:10-5:00), with a 10-minute break (4:00-4:10) between them. The advanced talk is a regular formal presentation about recent research results to general audiences in arithmetic geometry and number theory; the pre-talk (3:10-4:00) is to introduce some prerequisites or background for the advanced talk to audiences consisting of graduate students.