Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Murmurations of Elliptic Curves over Function Fields

Dane Wachs

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.

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.
Paper Structure (30 sections, 6 theorems, 15 equations, 3 figures, 3 tables)

This paper contains 30 sections, 6 theorems, 15 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The main result
  3. Murmurations as motivation
  4. Outline
  5. Background and notation
  6. Elliptic curves over Fq(t)
  7. The L-polynomial
  8. BSD over function fields
  9. The Brauer group and the Artin--Tate conjecture
  10. Computational methods
  11. Enumeration and point counting
  12. Functional equation prediction
  13. Data summary
  14. Murmurations over function fields
  15. Existence of murmurations
  16. ...and 15 more sections

Key Result

Proposition 1.1

If $L(E_D, T) = L(E_{D'}, T)$, then $|\mathcal{X}_D(\mathbb{F}_{q^n})| = |\mathcal{X}_{D'}(\mathbb{F}_{q^n})|$ for all $n \geq 1$, and consequently $|\text{{Sh}}(E_D)| = |\text{{Sh}}(E_{D'})|$.

Figures (3)

  • Figure 1: Murmurations over $\mathbb{F}_q(t)$ at $q = 11$, $\deg D = 5$. Mean Frobenius traces by rank at place degrees $d = 1, \ldots, 4$. Shaded: $\pm 2\,\mathrm{SE}$.
  • Figure 2: Discriminant factor degrees: $q = 11$ (left, all even) vs. $q = 13$ (right, $67\%$ odd). Rank $0$, $L$-degree $\geq 6$.
  • Figure 3: Frobenius eigenvalues on the unit circle for four representative $L$-polynomial classes at $q = 11$. Numbers on dots indicate multiplicity. Blue: square factorization (all even multiplicities); red: non-square. $|\text{{Sh}}| = \prod \Phi_n(1)^{e_n}$: the root positions determine $|\text{{Sh}}|$ via cyclotomic evaluation at $z = 1$.

Theorems & Definitions (13)

  • Proposition 1.1: Zeta function determination
  • proof
  • Proposition 1.2: Kronecker universality
  • Theorem 2.1: Grothendieck Groth
  • Theorem 2.2: Artin--Tate; Milne Milne
  • Example 3.1: A worked computation
  • proof : Proof of \ref{['prop:kronecker']}
  • Conjecture 5.1: Convergence to Haar measure
  • Remark 5.2
  • Theorem 5.3: Finite-sum murmuration identity
  • ...and 3 more