##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Kristin Lauter

#### Microsoft Research

## Class invariants for genus two

##### Abstract:

Evaluating certain Siegel modular functions at CM points on the moduli space of principally polarized abelian surfaces give algebraic numbers which we call class invariants. The construction of class invariants is motivated by explicit class field theory, specifically, the construction of units with possible applications to Stark conjectures. Class invariants can also be viewed as invariants of the binary sextic defining a genus 2 curve whose Jacobian corresponds to the CM point on the moduli space. The explicit construction of genus two curves with CM is motivated by cryptographic applications. When evaluating certain Siegel modular functions at CM points, the coefficients of the minimal polynomials have striking factorizations. In joint work with Eyal Goren, we studied primes that appear in the factorization of the denominators, and proved a bound on such primes closely related to the discriminant of the CM field. In more recent work, we study the primes appearing simultaneously in the numerators of CM values of certain Siegel modular functions in dimension 2. This work generalizes the work of Gross and Zagier for the modular j-function and is related to a conjecture of Bruinier and Yang on intersection numbers.

Host: Cristian Popescu

### March 1, 2007

### 1:00 PM

### AP&M 7321

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