##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Daniel Kane

#### Stanford University

## Ranks of 2-Selmer Groups of Twists of an Elliptic Curve

##### Abstract:

Let $E/\mathbb{Q}$ be an elliptic curve with full 2-torsion over $\mathbb{Q}$. We wish to study the distribution of the ranks of the 2-Selmer groups of twists of $E$ as we vary the twist parameter. A recent result of Swinnerton-Dyer shows that if $E$ has no cyclic 4-isogeny defined over $\mathbb{Q}$, then the density of twists with given rank approaches a particular distribution. Unfortunately Swinnerton-Dyer used an unusual notion of density essentially given as the number of primes dividing the twist parameter goes to infinity. We extend this result to cover density in the natural sense.

Host: Kiran Kedlaya

### February 6, 2014

### 12:00 PM

### AP&M 7402

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