The eighth Takagi Lectures
 November 23, 2010 (Tue) 11:30--12:30, 15:20--16:20 Lecture Hall (Room No. 420) Research Institute for Mathematical Sciences Kyoto University, Kyoto, Japan

The BC-system and L-functions
Alain Connes
(Collège de France, IHÉS)

Abstract

In these lectures we survey some relations between L-functions and the BC-system, including new results obtained in collaboration with C. Consani. For each prime p and embedding σ of the multiplicative group of an algebraic closure of $\mathbb{F}_p$ as complex roots of unity, we construct a p-adic irreducible representation $\pi_\sigma$ of the integral BC-system. This construction is done using the identification of the big Witt ring of $\bar{\mathbb{F}}_p$ and by implementing the Artin--Hasse exponentials. The obtained representations are the p-adic analogues of the complex, extremal KMS$_\infty$ states of the BC-system. We use the theory of p-adic L-functions to determine the partition function. Together with the analogue of the Witt construction in characteristic one, these results provide further evidence towards the construction of an analogue, for the global field of rational numbers, of the curve which provides the geometric support for the arithmetic of function fields.