Descent and theta functions for metaplectic groups

Solomon Friedberg ; David Ginzburg

Journal of the European Mathematical Society, Tome 20 (2018) no. 8, pp. 1913-1957.

Résumé

There are few constructions of square-integrable automorphic functions on metaplectic groups. Such functions may be obtained by the residues of certain Eisenstein series on covers of groups, “theta functions,” but the Fourier coefficients of these residues are not well-understood, even for low degree covers of GL2​. Patterson and Chinta–Friedberg–Hoffstein proposed conjectured relations for the Fourier coefficients of the GL2​ quartic and sextic theta functions (resp.), each obtained from a conjectured equality of non-Eulerian Dirichlet series. In this article we propose a new framework for constructing specific L2 metaplectic functions and for understanding these conjectures: descent integrals. We study descent integrals which begin with theta functions on covers of larger rank classical groups and use them to construct certain L2 metaplectic functions on covers related to GL2​. We then establish information about the Fourier coefficients of these metaplectic automorphic functions, properties which are consistent with the conjectures of Patterson and Chinta–Friedberg–Hoffstein. In particular, we prove that Fourier coefficients of the descent functions are arithmetic for infinitely many primes p. We also show that they generate a representation with nonzero projection to the space of theta.We conjecture that the descents may be used to realize the quartic and sextic theta functions. Moreover, this framework suggests that each of the conjectures of Patterson and Chinta–Friedberg–Hoffstein is the first in a series of relations between certain Fourier coefficients of two automorphic forms on different covering groups.

Détail

DOI : 10.4171/jems/803

Classification : 11-XX
Mots-clés : Metaplectic group, theta representation, descent integral, unipotent orbit, Patterson conjecture, Chinta–Friedberg–Hoffstein conjecture, Gauss sum