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

Alexander Dunn, Alexandru Zaharescu

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.

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

Abstract

Given a half-integral weight holomorphic Kohnen newform on , we prove an asymptotic formula for large primes 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 . 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 . Our treatment of this sequence is inspired by work of Rudnick--Sarnak and the second author on the local spacings of modulo one.

Paper Structure

This paper contains 32 sections, 23 theorems, 461 equations.

Table of Contents

  1. Introduction and statement of results
  2. High level sketch
  3. Conventions
  4. Acknowledgement
  5. Automorphic preliminaries I (half integral weight)
  6. Holomorphic cusp forms and $L$-functions
  7. Voronoi summation
  8. Half integral weight Kloosterman sums and Salié sums
  9. Eisenstein series and Rankin--Selberg $L$-functions
  10. Functional equation for the second moment
  11. The core argument
  12. Main terms
  13. Error terms from $\mathcal{D}_1$, $\mathcal{D}_2$ and $\mathcal{E}$
  14. Coefficients in residue classes
  15. Proof of Theorem \ref{['secondmoment']}
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $\varepsilon>0$, $j \in \mathbb{N}$, and $f$ be a holomorphic cuspidal newform of weight $k:=\frac{1}{2}+2 j$ on $\Gamma_0(4)$ such that As $p \rightarrow \infty$ through primes $p \equiv 1 \pmod{4}$, we have where $\psi(p):=p-2$ is the number of primitive Dirichlet characters modulo $p$, and $c_1(f), c_2(f) \in \mathbb{R}$ are constants depending only on $f$. Furthermore, we have that $c_1(

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 6.1
  • Lemma 6.2
  • proof
  • Lemma 6.3
  • proof
  • Lemma 6.4
  • ...and 31 more