The twisted second moment of modular half-integral weight $L$-functions

Abstract

Given a half-integral weight holomorphic Kohnen newform $f$ on ${\Gamma }_0(4)$, we prove an asymptotic formula for large primes $p$ with power saving error term for

Our result is unconditional, it does not rely on the Ramanujan–Petersson conjecture for the form $f$. This gives a very sharp Lindelöf-on-average result for Dirichlet series attached to Hecke eigenforms without an Euler product. The Lindelöf hypothesis for such series was originally conjectured by Hoffstein. There are two main inputs. The first is a careful spectral analysis of a highly unbalanced shifted convolution problem involving the Fourier coefficients of half-integral weight forms. The second input is a bound for sums of products of Salié sums in the Pólya–Vinogradov range. Half-integrality is fully exploited to establish such an estimate. We use the closed form evaluation of the Salié sum to relate our problem to the sequence $\alpha n^2(\mathrm{mod}1)$. Our treatment of this sequence is inspired by work of Rudnick–Sarnak and the second author on the local spacings of $\alpha n^2$ modulo 1.