##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Ronald van Luijk

#### Mathematical Sciences Research Institute

## Nontrivial Sha for curves of genus 2 arising from K3 surfaces

##### Abstract:

When doing a $2$-descent on the Jacobian $J$ of a curve of genus $2$, one wishes to determine whether or not certain twists of $J$ have rational points. As $J$ and its twists are unwieldy, we consider the quotient $K$ of a twist by the involution induced by multiplication by $-1$ on $J$. We construct an explicit curve $C$ and corresponding twist for which there is a Brauer-Manin obstruction to the existence of rational points on $K$. This yields infinitely many twists of $C$ with nontrivial Tate-Shafarevich group. This is joint work with Adam Logan at Waterloo.

Host: Cristian Popescu

### May 4, 2006

### 3:00 PM

### AP&M 7321

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