##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Ronen Mukamel

#### Stanford University

## Billiards, Hilbert modular forms and algebraic models for Teichmuller curves

##### Abstract:

For each real quadratic order O, there is a Weierstrass curve W in the Hilbert modular surface parametrizing Jacobians with real multiplication by O. The curve W emerges from the study of billiards in polygons and is important in Teichmuller theory because its natural immersion into the moduli space of curves is isometric. Such an immersion is called a Teichmuller curve. We will present explicit algebraic models for Weierstrass curves obtained by studying Hilbert modular forms. We will also present evidence from our examples that suggest a rich arithmetic associated to Teichmuller curves. This work is joint with A. Kumar.

Host: Kiran Kedlaya

### February 13, 2014

### 12:00 PM

### AP&M 7321

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