Arithmetic properties of the modular polylogarithm
(Un exposé de Guido Kings au séminaire de
théorie des nombres de Chevaleret le 13 déembre 2005)
Résumé :
The polylogarithm is a very successful tool in the study of
special values of L-functions. So far, only the polylogarithm
on the multiplicative group and on elliptic curves were subject
of deeper investigation. In this talk we will consider the polylogarithm
on modular curves and establish some of its properties.
We show in particular, that its periods are given by Goncharov's
polylogarithmic functions, which are described in terms of Green
functions, and show
that the dilogarithm is related to the L-value at 2 of a cusp
form of weight 2 (this depends on calculations by Brunault
in his thesis). Moreover, we show that one gets in a natural way
an Euler system for the Galois representation associated to
such a cusp form. In the last part of the lecture we discuss work
in progress about relations of the modular polylog to Kato's Euler
system and the elliptic Zagier conjecture.