### 11 december 2006,
Francis Brown (Institut Mittag-Leffler),
Arithmetic of zeta values and the irrationality of $\zeta(2)$
and $\zeta(3)$.

**Abstract.**
A folklore conjecture states that the values of the Riemann zeta
function at odd positive integers are algebraically independent over
$\Q$. The main results in this direction are that $\zeta(3)$ is
irrational (Ap\'ery), and that there are infinitely many irrational
numbers amongst $\zeta(2n+1)$ (Rivoal). It is not known, however,
whether $\zeta(5)$ is irrational or not.
In this talk, I will begin by surveying the known and conjectured
diophantine properties of multiple zeta values. I will then explain
Beukers' elementary proof that $\zeta(3)$ is irrational, and
describe the group structures corresponding to $\zeta(2)$ and
$\zeta(3)$, which give the best bounds for their irrationality
measures known to date. In the last part of the talk I will explain
how the moduli spaces of curves of genus 0 give a unifying geometric
interpretation of these proofs.