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Elliptic curves and spin

Peter Koymans, Peter Vang Uttenthal

Abstract

In the early 2000s, Ramakrishna asked the question: For the elliptic curve $$ E: y^2 = x^3 - x, $$ what is the density of primes $p$ for which the Fourier coefficient $a_p(E)$ is a cube modulo $p$? As a generalization of this question, Weston--Zaurova formulated conjectures concerning the distribution of power residues of degree $m$ of the Fourier coefficients of elliptic curves $E/\mathbb{Q}$ with complex multiplication. In this paper, we prove their conjecture for cubic residues using the analytic theory of spin. Our proof works for all elliptic curves $E$ with complex multiplication.

Elliptic curves and spin

Abstract

In the early 2000s, Ramakrishna asked the question: For the elliptic curve what is the density of primes for which the Fourier coefficient is a cube modulo ? As a generalization of this question, Weston--Zaurova formulated conjectures concerning the distribution of power residues of degree of the Fourier coefficients of elliptic curves with complex multiplication. In this paper, we prove their conjecture for cubic residues using the analytic theory of spin. Our proof works for all elliptic curves with complex multiplication.
Paper Structure (18 sections, 18 theorems, 67 equations, 1 table)

This paper contains 18 sections, 18 theorems, 67 equations, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. General spin symbols
  4. Power residues of degree m and spin
  5. Spin and inertial degrees of ramified primes
  6. Analytic prerequisites
  7. Cubic residue symbols
  8. Definition of the spin symbol
  9. Field lowering
  10. A fundamental domain
  11. Vinogradov's sieve
  12. Sums of type I
  13. Sums of type II
  14. Proof of main theorems
  15. Proof of Theorem \ref{['sieve']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $K$ be an imaginary quadratic field satisfying $\gcd(3, w_K) = 1$, or equivalently $K \neq \mathbb{Q}(\zeta_3)$. For any prime $p$ that splits completely in $K$ as $(\pi \overline{\pi})$, where $\overline{\pi}$ is the conjugate of $\pi$, and where $\pi$ lies below a prime ideal $\mathfrak{p}$ in Furthermore, there is a constant $C > 0$ such that for all $X \geqslant 100$

Theorems & Definitions (32)

  • Conjecture 1: Weston--Zaurova Weston-Zaurova
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3: Splitting in number fields that depend on the prime
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • ...and 22 more