Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

One-level density of zeros of $Γ_1(q)$ $L$-functions

Arijit Paul

TL;DR

The paper studies the one-level density of zeros for the large unitary family of $\Gamma_1(q)$ $L$-functions associated to holomorphic cusp forms. Using the explicit formula and a delicate off-diagonal analysis, it extends the admissible Fourier support to $(-\frac{8}{3},\frac{8}{3})$ under GRH and confirms the Katz–Sarnak unitary prediction for this family, i.e., $D_{\mathcal{F}}(\varphi) = \int_{-\infty}^{\infty} \varphi(x)\,dx + o(1)$. As an application, the work yields a lower bound of $\frac{5}{8}$ (62.5%) for the proportion of non-vanishing central values, the best known for any unitary family. The results indicate that the underlying GL2 structure and off-diagonal arithmetic play a decisive role in extending spectral-density support, more so than the symmetry group alone.

Abstract

We study the one-level density of zeros for a family of $Γ_1(q)$ $L$-functions. Assuming GRH, we are able to extend the support of the Fourier transform of the test function to $\left(-\frac{8}{3},\frac{8}{3}\right)$ and verify the Katz-Sarnak prediction for our unitary family. As an application, we obtain that the proportion of forms in the family with non-vanishing at the central point is at least $62.5\%$, assuming GRH. This is the highest non-vanishing proportion for any family associated with a unitary group. Moreover, this result indicates that the structural properties of $L$-functions play a more important role in extending the support than the associated symmetry group.

One-level density of zeros of $Γ_1(q)$ $L$-functions

TL;DR

The paper studies the one-level density of zeros for the large unitary family of -functions associated to holomorphic cusp forms. Using the explicit formula and a delicate off-diagonal analysis, it extends the admissible Fourier support to under GRH and confirms the Katz–Sarnak unitary prediction for this family, i.e., . As an application, the work yields a lower bound of (62.5%) for the proportion of non-vanishing central values, the best known for any unitary family. The results indicate that the underlying GL2 structure and off-diagonal arithmetic play a decisive role in extending spectral-density support, more so than the symmetry group alone.

Abstract

We study the one-level density of zeros for a family of -functions. Assuming GRH, we are able to extend the support of the Fourier transform of the test function to and verify the Katz-Sarnak prediction for our unitary family. As an application, we obtain that the proportion of forms in the family with non-vanishing at the central point is at least , assuming GRH. This is the highest non-vanishing proportion for any family associated with a unitary group. Moreover, this result indicates that the structural properties of -functions play a more important role in extending the support than the associated symmetry group.
Paper Structure (17 sections, 14 theorems, 140 equations)

This paper contains 17 sections, 14 theorems, 140 equations.

Table of Contents

  1. INTRODUCTION
  2. Background and main result
  3. Application
  4. Outline of the Proof
  5. PRELIMINARY LEMMAS
  6. PROOF OF THEOREM \ref{['Main Thm']}
  7. Initial setup
  8. Proof of Theorem \ref{['Main Thm']}
  9. PROOF OF LEMMA (\ref{['lemma 4.2 a']}) -- CONTRIBUTION FROM THE TRIVIAL CHARACTER
  10. Proof of Equation (\ref{['Equation 6.2']}) — Bound for S_1,case_1
  11. Proof of Equation (\ref{['Equation 6.3']}) — Bound for S_1,case_2
  12. Proof of Equation (\ref{['Equation 6.4']}) — Bound for S_1,case_3
  13. PROOF OF LEMMA \ref{['Lemma 4.3']} -- CONTRIBUTION FROM THE NON-TRIVIAL CHARACTERS
  14. Proof of Lemma \ref{['Lemma 4.3']}
  15. Proof of Equation (\ref{['equation 7.6']}) — Bound for D(P,S,T)_case_1
  16. ...and 2 more sections

Key Result

Theorem 1.1

Assume GRH. Let $\varphi$ be an even Schwartz function with $\operatorname{supp}\widehat{\varphi} \subset (-\frac{8}{3},\frac{8}{3})$. Then, with the notation above, for $k\geq 3$ odd,

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 12 more