##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Alex Smith

#### Stanford

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

##### Abstract:

Take $E$ to be an elliptic curve over a number field whose four torsion obeys certain technical conditions. In this talk, we will outline a proof that 100% of the quadratic twists of $E$ have rank at most one. To do this, we will find the distribution of $2^k$-Selmer ranks in this family for every positive $k$. We will also show how are techniques may be applied to find the distribution of $2^k$-class groups of quadratic fields.

The pre-talk will focus on the definition of Selmer groups. We will also give some context for the study of the arithmetic statistics of these groups.

### February 3, 2022

### 2:00 PM

Pre-talk at 1:20 PM

APM 6402 and Zoom;

See https://www.math.ucsd.edu/~nts

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