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Repulsion of zeros close to $s=1/2$ for L-functions

Nicolás Coloma, Maria Espericueta Sandoval, Erika Lopez, Francisco Ponce, Gustavo Rama, Nathan C. Ryan, Alejandro Vargas-Altamirano

Abstract

In this paper we present results of several experiments in which we model the repulsion of low-lying zeros of L-functions using random matrix theory. Previous work has typically focused on the twists of L-functions associated to elliptic curves and on families that can be modeled by unitary and orthogonal matrices. We consider families of L-function of modular forms of weight greater than 2 and we consider families that can be modeled by symplectic matrices. Additionally, we explore a model for low-lying zeros of twists that incorporates a discretization arising from the Kohnen--Zagier theorem. Overall, our numeric evidence supports the expectation that the repulsion of zeros decreases as the conductor of the twist increases. Surprisingly, though, it appears that using the discretization that arises from the Kohnen--Zagier theorem does not model the data better than if the discretization is not used for forms of weight 4 or above.

Repulsion of zeros close to $s=1/2$ for L-functions

Abstract

In this paper we present results of several experiments in which we model the repulsion of low-lying zeros of L-functions using random matrix theory. Previous work has typically focused on the twists of L-functions associated to elliptic curves and on families that can be modeled by unitary and orthogonal matrices. We consider families of L-function of modular forms of weight greater than 2 and we consider families that can be modeled by symplectic matrices. Additionally, we explore a model for low-lying zeros of twists that incorporates a discretization arising from the Kohnen--Zagier theorem. Overall, our numeric evidence supports the expectation that the repulsion of zeros decreases as the conductor of the twist increases. Surprisingly, though, it appears that using the discretization that arises from the Kohnen--Zagier theorem does not model the data better than if the discretization is not used for forms of weight 4 or above.
Paper Structure (21 sections, 2 theorems, 29 equations, 7 figures, 1 table)

This paper contains 21 sections, 2 theorems, 29 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Modular forms
  4. Admissible discriminants and families of L-functions
  5. Justification of the congruence conditions
  6. Theorem of Kohnen--Zagier
  7. Repulsion of the lowest zero
  8. Random matrices
  9. Matrix sizes and cutoffs
  10. Computing the first few zeros of L-functions
  11. The standard model
  12. Distributions
  13. Repulsion
  14. The excised model
  15. Computing central values
  16. ...and 6 more sections

Key Result

Theorem 3

Let $f\in S_k(\Gamma_0(M),\chi)$ with $\chi$ nontrivial and let $\diamondsuit$ be an integer so that $1\leq \diamondsuit < M$. Then all the central values $L(f,1/2,\psi)$ for $L(f,s,\psi)\in \mathcal{F}_U(f,\diamondsuit)$, lie on a line through the origin.

Figures (7)

  • Figure 1: Density plots of 100,000 $50\times 50$ matrices in (from the left) the special orthogonal group, the unitary and the unitary symplectic group.
  • Figure 2: Distributions of lowest zeros of admissible twists of 3.8.a.a and lowest mean eigenvalues from $SO(2N)$ (left), distributions of lowest zeros of admissible twists of 13.2.e.a and lowest mean eigenvalues from $USp(2N)$ (center), and distributions of lowest zeros of admissible twists of 7.3.b.a and lowest mean eigenvalues from $U(N)$ (right). The data have been normalized to have a mean of one.
  • Figure 3: Distributions of lowest zeros of admissible twists of forms in Table \ref{['tbl:mfs']}. From left to right the top plots in the first row are distributions of lowest zeros for admissible quadratic twists of the forms 11.2.a.a, 7.4.a.a, 3.6.a.a and the top plots in the second row are distributions of lowest zeros for admissible quadratic twists of the form 3.8.a.a. The bottom plots are the lowest eigenvalues from $SO(2N)$. Both plots have been normalized to have means of 1.
  • Figure 4: Distributions of lowest zeros of admissible twists of 3.8.a.a separated into those of small and large conductor (left), distributions of lowest zeros of admissible twists of 13.2.e.a separated into those of small and large conductor (center), and distributions of lowest zeros of admissible twists of 7.3.b.a separated into those of small and large conductor (right). The dashed vertical lines in each graph are the means of the data; and, again, before splitting into small and large conductors, the data were normalized to have a mean of one.
  • Figure 5: Distributions of lowest zeros of admissible twists of 11.2.a.a separated into those of small and large conductor (top row, left), distributions of lowest zeros of admissible twists of 7.4.a.a separated into those of small and large conductor (top row, center), distributions of lowest zeros of admissible twists of 3.6.a.a separated into those of small and large conductor (top row, right), distributions of lowest zeros of admissible twists of 3.6.8.a separated into those of small and large conductor (bottom row, left) The dashed vertical lines in each graph are the means of the data; and, again, before splitting into small and large conductors, the data were normalized to have a mean of one.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Theorem 3
  • proof
  • Corollary 4