Isogeny graphs with maximal real multiplication

TL;DR

This paper characterizes isogeny graphs of genus 2 Jacobians with maximal real multiplication and introduces a novel depth first search algorithm for computing endomorphism rings in this setting.

Contribution

It provides a complete description of isogeny graphs for genus 2 Jacobians with maximal real multiplication and presents the first DFS-based algorithm for endomorphism ring computation in genus 2.

Findings

Full description of isogeny graphs in the genus 2 case with maximal real multiplication.

First DFS-based algorithm for local endomorphism ring computation in genus 2.

Algorithm applicable over finite fields for prime-specific endomorphism rings.

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. Our setting considers genus 2 jacobians with complex multiplication, with the assumptions that the real multiplication subring is maximal and has class number one. We fully describe the isogeny graphs in that case. Over finite fields, 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.