[Paper Review] The twisted second moment of modular half integral weight $L$--functions

Alexander R. Dunn, Alexandru Zaharescu|arXiv (Cornell University)|Mar 8, 2019

Analytic Number Theory Research48 references4 citations

TL;DR

This paper establishes an unconditional asymptotic formula with power-saving error term for the twisted second moment of half-integral weight $L$-functions attached to Kohnen newforms, using spectral analysis of unbalanced shifted convolutions and bounds for products of Salié sums. The result confirms a sharp Lindelöf-on-average estimate for $L$-functions without Euler products, resolving a conjecture of Hoffstein and extending non-archimedean moment theory to half-integral weight forms.

ABSTRACT

Given a half-integral weight holomorphic Kohnen newform $f$ on $Γ_0(4)$, we prove an asymptotic formula for large primes $p$ with power saving error term for \begin{equation*} \sideset{}{^*} \sum_{χ\hspace{-0.15cm} \pmod{p}} | L(1/2,f,χ) |^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 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 Polya--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 $αn^2 \pmod{1}$. Our treatment of this sequence is inspired by work of Rudnick--Sarnak and the second author on the local spacings of $αn^2$ modulo one.

Motivation & Objective

• To establish an asymptotic formula for the twisted second moment of $L$-functions associated with half-integral weight holomorphic cusp forms on $\Gamma_0(4)$.
• To prove this result unconditionally, without assuming the Ramanujan–Petersson conjecture for the form $f$.
• To extend the theory of $L$-function moments to non-Euler product $L$-series, particularly for half-integral weight forms.
• To achieve a power-saving error term in the asymptotic formula, matching the edge of current analytic number theory techniques.
• To provide a sharp Lindelöf-on-average estimate for $L$-functions without Euler products, confirming a conjecture of Hoffstein.

Proposed method

• Employ spectral methods to analyze a highly unbalanced shifted convolution sum involving Fourier coefficients of half-integral weight modular forms.
• Use a novel bound for sums of products of Salié sums in the Polya–Vinogradov range, exploiting the half-integrality of the weight to derive cancellation.
• Apply the closed-form evaluation of Salié sums to relate the problem to the equidistribution of $\alpha n^2 \pmod{1}$, leveraging ideas from Rudnick–Sarnak and Zaharescu on local spacing statistics.
• Perform a detailed dyadic decomposition and frequency localization to control the error terms in the shifted convolution sum.
• Use divisor bounds and character sum estimates to control the number of solutions to certain congruences, ensuring $O(p^\varepsilon)$-type bounds on solution counts.
• Combine spectral and exponential sum techniques to bound the main error terms, ultimately yielding a power-saving error term in the moment formula.

Experimental results

Research questions

• RQ1 Can an asymptotic formula with power-saving error term be established for the twisted second moment of half-integral weight $L$-functions without assuming the Ramanujan–Petersson conjecture?
• RQ2 How can spectral methods be applied to unbalanced shifted convolution problems arising from half-integral weight forms?
• RQ3 What bounds can be obtained for sums of products of Salié sums in the Polya–Vinogradov range when the form has half-integral weight?
• RQ4 To what extent can the equidistribution of $\alpha n^2 \pmod{1}$ be used to control exponential sums in the context of $L$-function moments?
• RQ5 Does the Lindelöf hypothesis on average hold for $L$-functions attached to half-integral weight Hecke eigenforms without Euler products?

Key findings

• An unconditional asymptotic formula is proven for $\sum_{\chi \pmod{p}}^{\ast} |L(1/2, f, \chi)|^2$ with power-saving error term, valid for large primes $p$.
• The error term is $O_{\varepsilon}(p^{1 - \delta})$ for some $\delta > 0$, explicitly quantified via the choice of parameters $\bm{\delta} = (11/288, 11/48, 25/36, 407/432, 11/96, 11/96)$.
• The result confirms a sharp Lindelöf-on-average estimate for $L$-functions without Euler products, supporting Hoffstein's conjecture.
• The method avoids reliance on the Ramanujan–Petersson conjecture, making the result unconditional.
• The analysis of the sequence $\alpha n^2 \pmod{1}$ is central, with its local spacing statistics used to control exponential sums.
• The bound for products of Salié sums is established via a novel application of the closed-form formula and diophantine approximation techniques.