Mathematics Colloquium

Jens Funke

(University of Durham)

"Hecke and Langlands: A little bit of number theory"

Date: Mon, February 21, 2011
Time: 17:15
Place: Research II Lecture Hall

Abstract: A very classical problem in number theory is `In how many ways can one write a given integer as the sum of 4 squares?'. A variant of this question is the following. How many integral solutions has the equation

2(x2+y2+u2+v2) + 2xu +xv+yu-2yv =p

for a prime number p? Moreover, if one takes two such quadratic equations in 4 variables, how is their number of solutions related?

On the other hand, consider the cubic equation

Y2+Y =X3 -X2

in the XY-plane. This is an example for an integral elliptic curve. What are the integral or rational solutions for this equation? Or, how many solutions are there modulo a prime p?

Questions over questions. But most astonishingly, these two seemingly completely different equations are in fact closely related!

In this talk we outline how the explanation of such a relationship leads naturally to questions in modern number theory. In particular, we will outline the role of the classical theory of modular forms as developed by Hecke. This offers a glimpse of the imminent Langlands program, which asserts far reaching generalizations of the examples presented in this talk.

The talk will be very elementary, suitable for the entire mathematical family, in particular students.