Symbolic Computation Group

Polynomials are a convenient way of dealing with mathematical objects, algebraic number fields, finite fields, etc. They also encode many properties of algebraic curves. The aim of this talk is to describe two families of polynomials that exist in any genus, namely division polynomials and modular polynomials. The genus 0 case will be used as a starting point and we will rediscover cyclotomic polynomials. In higher genus, modular polynomials are the key to fast point counting algorithms. We will explain how to compute these polynomials, and how they are used, insisting on the genus 1 and 2 cases.