Class-groups of function fields
This paper explores the relationship between crystalline cohomology and the first author's theory of zeta-functions for function fields over finite fields. This relationship is based on the fact that, 1) the unit- root piece of the classical L-series of the Teichmüller character is the inverse characteristic polynomial of the Frobenius on the p-adic Tate-module; and 2) the reduction modulo p of this unit root piece is the reduction modulo a prime of the function field of the ''polynomial part'' of the above zeta-function. With this tool, the authors are able to describe the variation of the p- adic Tate-modules as the prime of the function field varies. Moreover, a converse to the ''Herbrand'' criterion for function fields is established. Finally, a criterion for cyclicity of the Galois components of p-class groups of ''cyclotomic function fields'' is given. This criterion amounts to having the Frobenius have at most one non 1-unit root on the appropriate Tate-module. In another paper, the first author has given a second criterion for cyclicity based on the values at the positive integers of his zeta- function. In particular, this second criterion implies the one given in the paper being reviewed. Moreover, because of the very simple form of the special values involved, this second criterion can be shown for all primes and some components. It is of interest that the components that arise in this manner are precisely those components that arise in the experimental evidence for a functional equation associated to these zeta- functions.
zeta-functions for function fields over finite fields
characteristic polynomial of the Frobenius
scientific article; zbMATH DE number 3910508
scientific article