##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Dr. Kristin Lauter

#### Microsoft

## Computing Igusa class polynomials

##### Abstract:

This talk will present a new method for computing the Igusa class polynomials of a primitive quartic $CM$ field. For a primitive quartic $CM$ field, $K$, we compute the Igusa class polynomials modulo $p$ for certain small primes $p$ and then use the Chinese remainder theorem and a conjectural bound on the denominators to construct the class polynomials. We also provide an extension to genus $2$ of the algorithm for determining endomorphism rings of elliptic curves. Our algorithm can be used to generate genus $2$ curves over a finite field with a given zeta function. \vskip .1in \noindent Joint work with Kirsten Eisentraeger, University of Michigan.

Host: Audrey Terras

### February 24, 2005

### 2:00 PM

### AP&M 7321

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