On the refined class number formula for global function fields

We investigate a conjecture of Gross regarding a congruence relation of the Stickelberger element. We consider the case when $k$ is a global function field of characteristic $p$ and $\mathrm{Gal}(K/k)$ is an abelian $l$-group where $l$ is a prime number different from $p$. Under the additional assumption that $k$ does not contain a primitive $l$-th root of unity and that the divisor class number of $k$ is prime to $l$, we prove that the conjecture of Gross holds. This result generalizes the author’s previous result on the elementary abelian case (cf. \cite{lee}).

Full Text (PDF format)

Published 1 January 2004