Average values of \(L\)-functions in characteristic two
Let \(k={\mathbb F}_q(t)\) be the rational function field over the finite field of \(q\) elements. For a non-square polynomial \(M\) in the ring \({\mathbf A} = {\mathbb F}_q[t]\), consider the \({\mathbf A}\)-order \({\mathcal O}_M = {\mathbf A} + {\mathbf A}\sqrt M\) in the ring of integers of the field \(k(\sqrt M)\). Let \(h_M\) denote the class number of \({\mathcal O}_M\). \textit{J.\ Hoffstein} and \textit{M.\ Rosen} [J. Reine Angew. Math. 426, 117--150 (1992; Zbl 0754.11036)] proved a formula for the mean value of \(h_M\), assuming that \(q\) is odd. Here \(M\) varies over monic polynomials of fixed degree. This formula is an analogue of the corresponding formula in the number field case, conjectured by Gauss and proven by Lipschitz and Siegel. The present author extends the formula to the case of even \(q\). To this end, he first proves an analogue of Dirichlet's class number formula for the \(L\)-functions associated to \({\mathcal O}_M\) for even \(q\). A basic difficulty now is that there is no bijection between quadratic fields and their discriminants. Therefore the problem of averaging \(h_M\) over all fundamental discriminants must be left open. However, the author has found a suitable parametrization of the fields which enables him to obtain nice averaging formulas.
Arithmetic theory of algebraic function fields
scientific article; zbMATH DE number 5379240
scientific article