Murmurations of Elliptic Curves over Function Fields

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.