Extra Seminar - Dr. P. Bruin (Leiden University)

TITLE:             Explicit methods in arithmetic geometry

ABSTRACT:

Arithmetic geometry connects number theory and (algebraic) geometry: it

is about varieties defined by polynomial equations over the rational

numbers and other rings of arithmetic importance. I will speak about

several algorithmic and quantitative aspects of this captivating area of

research. In the first part, I will introduce some of the main objects

of interest, such as elliptic curves and Galois representations, and a

number of natural questions that arise when working with these objects.

In the second part, I will present two of my recent projects in more

detail: a framework for efficiently computing with finite group schemes

and Galois representations, and joint work in progress with Filip Najman

on counting elliptic curves over number fields with prescribed level

structure.