Conference "Arithmetic, Geometry and Coding Theory"

March 14 - 18, 2011

CIRM, Marseille, France

Organisers: Yves Aubry (Université de la Méditerranée, IML), Christophe Ritzenthaler (Université de la Méditerranée, IML), Alexey Zykin (Mathematical Department of HSE, Laboratoire Poncelet, IITP)

Bounding the denominators of Igusa class polynomials using arithmetic intersection theory

Bianca Viray

Abstract

The CM methods for constructing genus 2 curves require as input a bound on the size of the denominators of the Igusa class polynomials. Moreover, a sharper bound results in a faster algorithm. We build on work of Bruinier-Yang and Yang to give a very sharp bound on the denominators, under some assumptions on the CM field and away from the 2-primary part. We will discuss how bounding the denominators is related to the problem of counting embeddings of the ring of integers of a quartic CM eld into maximal orders into a division algebra, and how we use this to obtain a bound. If time permits, we will also discuss work in progress to remove the assumptions mentioned above. This is joint work with Kristin Lauter.

Web page of the conference at CIRM