Murmurations and explicit formulas

TL;DR

This paper investigates unexpected oscillations in elliptic curve data, proposing a heuristic explanation based on the explicit formula and zero distributions of associated L-functions, supported by empirical analysis of various families.

Contribution

It introduces a heuristic model linking oscillations in elliptic curve data to the quasi-periodic structure of L-function zeros, supported by empirical evidence.

Findings

Oscillations observed in elliptic curve data and Dirichlet characters.

Quasi-periodic zero distribution explains the oscillations.

Empirical results support the heuristic model.

Abstract

Unexpected oscillations in $a_p$ values in a family of elliptic curves were observed experimentally by He, Lee, Oliver, and Pozdnyakov. We propose a heuristic explanation for these oscillations based on the "explicit formula" from analytic number theory. A crucial ingredient in this heuristic is that the distribution of the zeros of the associated $L$-functions has a quasi-periodic structure. We present empirical results for a family of elliptic curves, a family of quadratic Dirichlet characters whose values exhibit similar oscillations, and a family of Dirichlet characters whose values do not.