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Square-free orders for CM elliptic curves modulo $p$ in short intervals

Peng-Jie Wong

TL;DR

This work studies primes $p$ for CM elliptic curves $E/\mathbb{Q}$ for which the reduction $\bar E(\mathbb{F}_p)$ has square-free order, improving Cojocaru's unconditionally strength and providing short-interval analogues. The author converts the square-free condition into ray-class/Artin reciprocity congruences and harnesses a Bombieri–Vinogradov theorem for number fields, together with a short-interval variant, to control error terms. A refined number-field Brun–Titchmarsh inequality overcomes the precarious middle-range estimates, yielding unconditional asymptotics $h_E(x,\mathbb{Q})=\delta_E\mathrm{Li}(x)+O_A( N_E/(\log x)^A )$ and short-interval analogues for $0\le\delta<1/5$; the results also extend to a CM cyclicity problem with an explicit constant $\mathfrak{c}_E$. A key practical outcome is an unconditional bound on the smallest ordinary prime with square-free order, $p_E=O_{\varepsilon}(\exp(N_E^{\varepsilon}))$, and corresponding short-interval cyclicity results, significantly strengthening the elliptic-curve analogue of prime-distribution questions without assuming GRH.

Abstract

Let $E$ be a CM elliptic curve over $\Bbb{Q}$. We refine the work of Cojocaru on the asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of $E$ is of square-free order. Also, we derive an unconditional short interval variant for the asymptotics. Compared to the estimate derived from the generalised Riemann hypothesis, the presented result is valid for even shorter intervals. Furthermore, we improve the short interval variant of the cyclicity problem for CM elliptic curves previously obtained by the author.

Square-free orders for CM elliptic curves modulo $p$ in short intervals

TL;DR

This work studies primes for CM elliptic curves for which the reduction has square-free order, improving Cojocaru's unconditionally strength and providing short-interval analogues. The author converts the square-free condition into ray-class/Artin reciprocity congruences and harnesses a Bombieri–Vinogradov theorem for number fields, together with a short-interval variant, to control error terms. A refined number-field Brun–Titchmarsh inequality overcomes the precarious middle-range estimates, yielding unconditional asymptotics and short-interval analogues for ; the results also extend to a CM cyclicity problem with an explicit constant . A key practical outcome is an unconditional bound on the smallest ordinary prime with square-free order, , and corresponding short-interval cyclicity results, significantly strengthening the elliptic-curve analogue of prime-distribution questions without assuming GRH.

Abstract

Let be a CM elliptic curve over . We refine the work of Cojocaru on the asymptotic formulae for the number of primes for which the reduction modulo of is of square-free order. Also, we derive an unconditional short interval variant for the asymptotics. Compared to the estimate derived from the generalised Riemann hypothesis, the presented result is valid for even shorter intervals. Furthermore, we improve the short interval variant of the cyclicity problem for CM elliptic curves previously obtained by the author.
Paper Structure (7 sections, 7 theorems, 97 equations)

This paper contains 7 sections, 7 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Overview of the proofs of Theorems \ref{['main-thm1']} and \ref{['main-thm2']}
  3. Proof of Theorem \ref{['main-thm1']}
  4. Proof of the first part of Theorem \ref{['main-thm1']}
  5. Proof of the second part of Theorem \ref{['main-thm1']}
  6. Proof of Theorem \ref{['main-thm2']}
  7. Proof of Theorem \ref{['main-thm-cyclicity']}

Key Result

Theorem 1

Let $E/\Bbb{Q}$ be an elliptic curve of conductor $N_E$ and with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$. Then for any $A>0$, we have where $\delta_E$ is defined as in delta. Moreover, assuming GRH, for any $\eta\in(0, \frac{1}{3})$, we have uniformly in $N_E\le x^\eta$.

Theorems & Definitions (11)

  • Theorem 1
  • Remark
  • Corollary 1.1
  • proof
  • Theorem 2
  • Remark
  • Theorem 3
  • Remark
  • Theorem 4: Huxley
  • Theorem 5
  • ...and 1 more