The twisted second moment of modular half integral weight $L$--functions (1903.03416v3)

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 \begin{equation*} \sideset{}{^*} \sum_{\chi \hspace{-0.15cm} \pmod{p}} | L(1/2,f,\chi) |^2. \end{equation*} Our result is unconditional, it does not rely on the Ramanujan--Petersson conjecture for the form $f$. This gives a very sharp Lindel\"{o}f on average result for Dirichlet series attached to Hecke eigenforms without an Euler product. The Lindel\"{o}f 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 Sali\'{e} sums in the Polya--Vinogradov range. Half--integrality is fully exploited to establish such an estimate. We use the closed form evaluation of the Sali\'{e} sum to relate our problem to the sequence $\alpha n^2 \pmod{1}$. 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 one.