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

Given a half-integral weight holomorphic Kohnen newform $f$ on ${\mathrm{\Gamma }}_0(4)$, 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 $\alpha n^2(\mathrm{m}\mathrm{o}\mathrm{d}1)$. 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.