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 |

Alain Connes

(Collège de France, IHÉS)

(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.