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)
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.