Class groups and L-series of function fields
The author considers Galois coverings Y/X of projective non-singular algebraic curves defined over a perfect field K of prime characteristic p.. Throughout the paper, the group G of Y/X is assumed to be prime to p. First, in considering the decomposition (with respect to characters \(\chi\) of G) of the group \(C^ 0(Y)(p)\) of divisor classes of Y annihilated by p, the author refines the notion of Hasse-Witt invariant, and describes the relation with the Cartier operator C on Y. For being able to formulate his principal results, he assumes now Y/X to be defined over a finite field \({\mathbb{F}}_ q\), and he gives a short review of the relevant L-series. The main results are contained in section 4, namely the relation between ''reductions'' of L-series modulo p and the characteristic polynomial of C (theorem 4.1) and the p-class number formula (theorem 4.2). These results refine and extend analogous results of Manin and others in more special situations (e.g. G \(= trivial\), or abelian with certain ramification conditions).
Galois coverings of projective non-singular algebraic curves
scientific article; zbMATH DE number 3929168
scientific article