A mean value theorem for class numbers of quadratic extensions of function fields.

摘要

In this thesis we study a zeta function associated with the space of binary quadratic forms with coefficients in a function field of characteristic different from two. We establish the convergence, analytic continuation, and the functional equation for this zeta function. The method we use is that of T. Shintani as illustrated in the work of B. Datskovsky and D. J. Wright using adelic analysis.; As an application of studying this adelic zeta function, we obtain a mean value theorem for class numbers of quadratic extensions of a function field. This will be achieved by first conducting some local analysis. This local analysis amounts to studying certain integrals, which we call orbital zeta functions, that appear in a natural way as local factors of the adelic: zeta function we started with. Next we put together the global and local information we obtained to construct a sequence of Dirichlet series. Studying some analytic properties of this sequence of Dirichlet series will yield the mean value theorem.