Murmurations of Elliptic Curves over Function Fields

TL;DR

This paper investigates oscillatory patterns in Frobenius traces of elliptic curves over function fields, revealing how Murmurations differentiate ranks and relate to Shafarevich-Tate groups, with extensive computational data and theoretical insights.

Contribution

It introduces the first murmurations for elliptic curves over function fields, linking oscillatory trace patterns to arithmetic invariants and providing explicit formulas for Murmurations stratified by |Sha|.

Findings

Murmurations distinguish rank-0 from rank-1 curves.

All L-polynomials factor into cyclotomic polynomials.

|Sha| modulation affects murmuration patterns through a reweighting identity.

Abstract

We compute the first murmurations for elliptic curves over function fields F_q(t): oscillatory patterns in average Frobenius traces that separate rank-0 from rank-1 curves, with z-scores up to 256. For the family E_D: y^2 = x^3 + x + D(t) with D monic squarefree of degree 5, we enumerate 534,745 curves across q = 7, 11, 13 with exact BSD invariants. All L-polynomials factor into cyclotomic polynomials -- a weight-2 consequence of the Weil conjectures and Kronecker's theorem, independent of CM. Since |Sha| = L(1/q) in this family (a consequence of BSD with trivial torsion and Tamagawa numbers), the |Sha| modulation of murmurations is entirely a composition effect: different |Sha| strata have different mixtures of L-polynomial types, and hence different mean traces. This yields an exact reweighting identity for the |Sha|-stratified murmuration density: M_s(d,q) = -sum_lambda f_{lambda,s} p_d(lambda), where lambda ranges over cyclotomic types, f_{lambda,s} is the type composition of the |Sha| = s stratum, and p_d(lambda) is the degree-d power sum of the unitarized roots. Within each |Sha| stratum, joint cells -- distinct L-polynomial types sharing the same |Sha| -- show that the murmuration profile carries arithmetic information strictly finer than |Sha| alone.