Metaplectic Ramanujan conjecture over function fields with applications to quadratic forms

TL;DR

This paper proves an analogue of the Ramanujan Conjecture for half-integral weight modular forms over function fields, applying it to quadratic forms and developing related theories like theta functions and L-functions.

Contribution

It formulates and proves the Ramanujan Conjecture analogue in the function field setting, extending the theory of half-integral weight forms and their applications.

Findings

Effective solution to representation problems for quadratic forms

Establishment of equidistribution of roots of polynomial L-functions

Key estimates for Fourier coefficients of cusp forms

Abstract

We formulate and prove the analogue of the Ramanujan Conjectures for modular forms of half-integral weight subject to some ramification restriction in the setting of a polynomial ring over a finite field. This is applied to give an effective solution to the problem of representations of elements of the ring by ternary quadratic forms. Our proof develops the theory of half-integral weight forms and Siegel's theta functions in this context as well as the analogue of an explicit Waldspurger formula. As in the case over the rationals, the half-integral weight Ramanujan Conjecture is in this way converted into a question of estimating special values of members of a special family of L-functions. These polynomial functions have a growing number of roots (all on the unit circle thanks to Drinfeld and Deligne's work) which are shown to become equidistributed. This eventually leads to the key estimate for Fourier coefficients of half-integral weight cusp forms.