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Murmurations and ratios conjectures

Alex Cowan

Abstract

We introduce a new method for studying murmurations, based on random matrix theory. With this method, we exhibit murmurations or similar phenomena: assuming ratios conjectures, for elliptic curves ordered by height, quadratic twists of a fixed elliptic curve, and the inverse Mellin transform of the shifted second moment of $ζ'/ζ$ on vertical lines; assuming GRH, for primitive quadratic Dirichlet characters, and holomorphic modular forms of prime level tending to infinity with sign and weight fixed; and unconditionally the inverse Mellin transform of the shifted second moment of $ζ$ on vertical lines. We also present a generalization of our approach which relies only on the approximate functional equation in place of ratios conjectures.

Murmurations and ratios conjectures

Abstract

We introduce a new method for studying murmurations, based on random matrix theory. With this method, we exhibit murmurations or similar phenomena: assuming ratios conjectures, for elliptic curves ordered by height, quadratic twists of a fixed elliptic curve, and the inverse Mellin transform of the shifted second moment of on vertical lines; assuming GRH, for primitive quadratic Dirichlet characters, and holomorphic modular forms of prime level tending to infinity with sign and weight fixed; and unconditionally the inverse Mellin transform of the shifted second moment of on vertical lines. We also present a generalization of our approach which relies only on the approximate functional equation in place of ratios conjectures.
Paper Structure (14 sections, 25 theorems, 136 equations, 1 figure, 1 table)

This paper contains 14 sections, 25 theorems, 136 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Murmuration densities
  3. Quadratic characters
  4. Elliptic curves ordered by height
  5. Ratios conjecture
  6. Proof of \ref{['elliptic_curve_murmuration_rho']}
  7. Other examples
  8. Quadratic characters --- Ratios conjecture, smoothed
  9. Quadratic characters --- GRH
  10. Holomorphic modular forms in the level aspect --- GRH
  11. Quadratic twists of an elliptic curve --- Ratios conjecture
  12. $\zeta'/\zeta$ --- Ratios conjecture
  13. $\zeta$ --- Unconditional
  14. Approximate functional equations

Key Result

Theorem 1.1

Fix $\omega \in \{\pm 1\}$ and define $\mathcal{F}(H)^\omega \coloneqq \{E \in \mathcal{F}(H) \,:\, \omega_E = \omega\}$. Assume that conductor_distribution, small_conductors, and the "ratios conjecture" DHP hold with $\mathcal{F}(H)$ replaced with $\mathcal{F}(H)^\omega$. For any $H, y, T, \varepsi where $A(\alpha,\gamma)$ is the product from A_def involving traces of Hecke operators, and $F_N'$

Figures (1)

  • Figure :

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8: Special case of quadratictwists
  • Theorem 2.1
  • proof
  • Theorem 3.1: Conrey--Snaith conrey_snaith
  • ...and 44 more