### 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(*x*^{2}+*y*^{2}+*u*^{2}+*v*^{2}) + 2*xu* +*xv*+*yu*-2*yv* =*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

*Y*^{2}+*Y* =*X*^{3} -*X*^{2}
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.