Monday, May 11

Speaker: Cornelius Greither, Univ. der Bundeswehr Munchen

Title: Fitting ideals of Tate modules of Jacobians, and of class groups of curves over finite fields

Using the theory of 1-motives and their duality, we sharpen existing results on the equivariant Fitting ideals attached to degree zero class groups of function fields. Let L/K be an abelian extension of function fields in one variable over the finite field F_q. This corresponds to a G-covering D --> C of curves, with G=Gal(L/K). We show: The G-equivariant Fitting ideal of the ell-adic realisation of Deligne's 1-motive attached to the curve D (and some technical data) is principal and generated by a standard Stickelberger element at infinite level. From this one may obtain the Fitting ideal of the ell-Tate module of J_C(\bar F_q) for any prime ell not dividing q, and then by descent the Fitting ideal of the dual of the degree zero class group of C, outside the q-part. One underlying idea is to find out, roughly speaking, how far these Tate modules are removed from being projective over Z[G]. These results are sharper than the annihilation results of Deligne, which can be found for example in Tate's classic book on the Stark conjectures. They also should have some influence on the number of F_q-rational points on C.