Lenny Taelman

Special L-values of Drinfeld modules

We state and prove a formula for a certain value of the Goss L-function of a Drinfeld module. This gives characteristic-p-valued function eld analogues of the class number formula and of the Birch and Swinnerton-Dyer conjecture. The formula and its proof are presented in an entirely selfcontained fashion.

A Dirichlet unit theorem for Drinfeld modules

We show that the module of integral points on a Drinfeld module satisfies an analogue of Dirichlet’s unit theorem, despite its failure to be finitely generated. As a consequence, we obtain a construction of a canonical finitely generated sub-module of the module of integral points. We use the results to give a precise formulation of a conjectural analogue of the class number formula.

Arithmetic of characteristic

Recently, the second author has associated a finite Fq[T]-module H to the Carlitz module over a finite extension of Fq(T). This module is an analogue of the ideal class group of a number field. In this paper, we study the Galois module structure of this module H for ‘cyclotomic’ extensions of Fq(T). We obtain function field analogues of some classical results on cyclotomic number fields, such as the p-adic class number formula, and a theorem of Mazur and Wiles about the Fitting ideal of ideal class groups. We also relate the Galois module H to Andersons module of circular units, and give a negative answer to Andersons Kummer-Vandiver-type conjecture. These results are based on a kind of equivariant class number formula which refines the second authors class number formula for the Carlitz module.

p

We prove a function field analogue of the Herbrand-Ribet theorem on cyclotomic number fields. The Herbrand-Ribet theorem can be interpreted as a result about cohomology with μp-coefficients over the splitting field of μp, and in our analogue both occurrences of μp are replaced with the

special

\mathfrak{p}

L

-torsion scheme of the Carlitz module for a prime

-values

\mathfrak{p}

A Herbrand-Ribet theorem for function fields

in Fq[t].

Artin t-motifs

Abstract We show that analytically trivial t -motifs satisfy a Tannakian duality, without restrictions on the base field, save for that it be of generic characteristic. We show that the group of components of the t -motivic Galois group coincides with the absolute Galois group of the base field.

K3 surfaces over finite fields with given L-function

The zeta function of a K3 surface over a finite field satisfies a number of obvious (archimedean and l-adic) and a number of less obvious (p-adic) constraints. We consider the converse question, in the style of Honda-Tate: given a function Z satisfying all these constraints, does there exist a K3 surface whose zeta-function equals Z? Assuming semi-stable reduction, we show that the answer is yes if we allow a finite extension of the finite field. An important ingredient in the proof is the construction of complex projective K3 surfaces with complex multiplication by a given CM field.