On the Ramanujan-Petersson conjecture for modular forms of half-integral weight

Abstract

It is shown that a “character twist” of a growth estimate for certain weighted infinite sums of Kloosterman sums which is equivalent to the Ramanujan-Petersson conjecture for modular forms of half-integral weight, can easily be proved using Deligne’s theorem (previously the Ramanujan-Petersson conjecture for modular forms of integral weight).