Glasgow Mathematical Journal

Research Article

NOTE ON A PAPER OF J. HOFFSTEIN

S. J. PATTERSONa1

a1 Mathematisches Institut, Bunsenstr. 3–5, 37073 Göttingen, Germany e-mail: sjp@uni-math.gwdg.de

Abstract

The concept of a metaplectic form was introduced about 40 years ago by T. Kubota. He showed how Jacobi-Legendre symbols of arbitrary order give rise to characters of arithmetic groups. Metaplectic forms are the automorphic forms with these characters. Kubota also showed how higher analogues of the classical theta functions could be constructed using Selberg's theory of Eisenstein series. Unfortunately many aspects of these generalized theta series are still unknown, for example, their Fourier coefficients. The analogues in the case of function fields over finite fields can in principle be calculated explicitly and this was done first by J. Hoffstein in the case of a rational function field. Here we shall return to his calculations and clarify a number of aspects of them, some of which are important for recent developments.

(Received May 12 2006)

(Accepted November 30 2006)

Key Words:

  • 11F27;
  • 11L05;
  • 11T24

Footnotes

DEDICATED TO THE MEMORY OF BOB ODONI