The Metaplectic Case of the Weil-Siegel Formula

Abstract

The Weil-Siegel formula, in the form developed by Weil, asserts the equality of a special value of an Eisenstein series with the integral of a related theta series. Recently, Kudla and Rallis have extended the formula into the range in which the Eisenstein series fails to converge at the required special value, so that Langlands' meromorphic analytic continuation must be used. In the case addressed by Kudla and Rallis, both the Eisenstein series and the integral of the theta series are automorphic forms on the adelic symplectic group. This thesis concentrates on extending the Weil-Siegel formula in the case in which both functions are automorphic forms on the two-fold metaplectic cover of the adelic symplectic group. First of all, a concrete model of the global metaplectic cover mentioned above is constructed by modifying the local formulas of Rao. Next, the meromorphic analytic continuation of the Eisenstein series is shown to be holomorphic at the special value in question. In the course of this work, we develop the functional equation and find all poles of an interesting family of local zeta-integrals similar to those studied in a paper of Igusa. Finally, the Weil-Siegel formula is proven in many cases by the methods of Kudla and Rallis.