##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Sorina Ionica

#### ENS Paris

## Isogeny graphs with maximal real multiplication

##### Abstract:

An isogeny graph is a graph whose vertices are principally polarized abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel described the structure of isogeny graphs for elliptic curves and showed that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth first search algorithm in the graph. In dimension 2, the structure of isogeny graphs is less understood and existing algorithms for computing endomorphism rings are very expensive. We fully describe the isogeny graphs between genus 2 jacobians with complex multiplication, with the assumptions that the real multiplication subring is maximal and has class number one. We derive a depth first search algorithm for computing endomorphism rings locally at prime numbers, if the real multiplication is maximal. To the best of our knowledge, this is the first DFS-based algorithm in genus 2. (Joint work with Emmanuel ThomÃ©).

Host: Kiran Kedlaya

### January 23, 2014

### 12:00 PM

### AP&M 7321

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