Lectures on Abelian Varieties with Complex Multiplication
04 Mayıs 2015 - 07 Mayıs 2015
IMBM Seminar Room, Bogazici University South Campus
May 4,Monday, 13:00, IMBM Seminar Room, Bogazici University South Campus
Speaker: Ekin Ozman, Boğaziçi University
Title: Abelian Varieties and Complex Multiplication
Abstract: This talk will be about the basics of Abelian Varieties and complex multiplication(CM). After defining these terms, we will continue with the discussion of good and bad reduction of CM abelian varieties. The talk should be accessible to students and will introduce the notions needed in the following talks of E. Garcia and R. Newton.
May 6, Wednesday, 13:00, IMBM Seminar Room, Bogazici University South Campus
Speaker: Elisa Lorenzo Garcia, Leiden University
Title: A Gross-Zagier formula and some generalizations.
Abstract: The aim of this talk is to introduce the original results of Gross and Zagier about the factorization of products of differences of singular moduli. We will also describe some generalizations to the problem for CM curves of genus 2 due to the work of Goren, Lauter and Viray. These results motivate the partial generalizations for genus 3 that R. Newton will explain in the following talk.
May 7, Thursday, 13:00, IMBM Seminar Room, Bogazici University South Campus
Speaker: Rachel Newton, Max Planck Institute for Mathematics, Bonn
Title: Bad reduction of genus 3 curves with complex multiplication.
Abstract: If we take a polynomial defining a curve and reduce its coefficients modulo a prime, what happens to the curve? There are two kinds of behaviour: either the curve remains irreducible, or it breaks down into multiple components. We call the latter case 'bad reduction'. Cryptographers need a supply of curves with good reduction in order to build secure encryption algorithms. Complex multiplication (or CM) is an additional structure which allowed Goren and Lauter to give a bound on the primes of bad reduction for curves of genus 2 with CM. I will describe a partial generalisation of this work to curves of genus 3 with CM. This is joint work with I. Bouw, J. Cooley, K. Lauter, E. Lorenzo Garcia, M. Manes and E. Ozman.