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arxiv: 2603.24915 · v2 · submitted 2026-03-26 · 🧮 math.NT

On the Density of Coprime Reductions of Elliptic Curves

Asimina S. Hamakiotes , Sung Min Lee , Jacob Mayle , Tian Wang This is my paper

Pith reviewed 2026-05-15 01:11 UTC · model grok-4.3

classification 🧮 math.NT
keywords elliptic curvescoprime ordersnatural densitySerre pairsKoblitz conjecturepoint countsEuler productmoments
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The pith

Non-CM elliptic curves over Q have a conjectured positive density of primes p where their reductions have coprime point counts.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the natural density of primes p for which the orders of E1(F_p) and E2(F_p) are coprime, for non-CM elliptic curves E1, E2 over Q. This is an elliptic analogue of the density of coprime pairs of integers. The authors formulate a conjecture for this density as a specific constant. They prove the defining series converges and has an almost Euler product expansion. For Serre pairs they derive a closed formula and prove a moments result on the distribution of the constant as the pair varies.

Core claim

The natural density of primes p of good reduction such that the orders of E1(F_p) and E2(F_p) are coprime is conjectured to exist and equal a constant C(E1, E2) defined by a series. This series is shown to converge and admit an almost Euler product expansion. When (E1, E2) is a Serre pair, a closed formula for C(E1, E2) is given and used to describe the distribution via moments as (E1, E2) varies.

What carries the argument

The constant C(E1, E2) defined by the convergent series with almost Euler product expansion, which measures the local probabilities that a prime divides neither or one but not both point counts.

If this is right

  • The density is positive for any such pair of curves.
  • For Serre pairs the constant admits an explicit closed-form expression.
  • The moments of the distribution of these constants over varying Serre pairs can be computed explicitly.
  • The series converges absolutely, permitting numerical approximation of the density for any given pair.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The almost Euler product structure would allow efficient numerical computation of the density for arbitrary pairs.
  • The moments result implies the constants vary in a controlled way that could be compared to random models of point counts.
  • Verification for concrete curve pairs would require counting primes up to large bounds and comparing the observed proportion to the series value.

Load-bearing premise

The elliptic curves are assumed to be non-CM over the rationals, with pairs being Serre pairs for the closed formula and moments.

What would settle it

For a fixed pair of elliptic curves, compute the proportion of primes p less than 10^6 where gcd(#E1(F_p), #E2(F_p)) = 1 and check if it approaches the conjectured constant.

read the original abstract

Given non-CM elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$, we study the natural density of primes $p$ of good reduction for which the orders of the groups $E_1(\mathbb{F}_p)$ and $E_2(\mathbb{F}_p)$ are coprime. This problem may be viewed as an elliptic curve analogue of the classical question concerning the density of coprime integer pairs. Motivated by Zywina's refinement of the Koblitz conjecture, we formulate a conjecture for the density of such primes. We prove that the series defining this constant converges and admits an almost Euler product expansion. In the case of Serre pairs, we give a closed formula for the constant and use it to prove a moments result describing the distribution of these constants as $(E_1, E_2)$ varies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. Given non-CM elliptic curves E1 and E2 over Q, the paper studies the natural density of primes p of good reduction such that the orders of E1(F_p) and E2(F_p) are coprime. Motivated by Zywina's refinement of the Koblitz conjecture, they formulate a conjecture for this density as a certain series. They prove that this series converges and admits an almost Euler product expansion. For Serre pairs, they provide a closed formula for the constant and prove a moments result on the distribution of these constants as the pair varies.

Significance. If the conjecture holds, the work supplies an explicit density for coprime reductions of elliptic curves, providing an elliptic analogue of the classical density of coprime integer pairs. The rigorously established convergence of the defining series, its almost-Euler-product form, the closed formula for Serre pairs, and the resulting moments statement constitute concrete analytic-number-theoretic contributions that can be used independently of the conjecture itself. These results strengthen the toolkit for arithmetic statistics of elliptic curves and may inform further refinements of the Koblitz conjecture.

minor comments (2)
  1. The abstract introduces 'Serre pairs' without a one-sentence definition or reference; adding a brief gloss would improve accessibility for readers outside the immediate subfield.
  2. In the statement of the moments result, the precise measure on the space of pairs (E1, E2) should be recalled explicitly (e.g., height ordering or natural density in the moduli space) to make the limiting statement fully self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We are grateful to the referee for their careful reading of the manuscript and for recommending minor revision. We appreciate their accurate summary of our results on the density of coprime reductions for non-CM elliptic curves and the analytic contributions regarding the convergence of the associated series and the moments result for Serre pairs.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines the density constant via an explicit series, proves its convergence and almost-Euler-product expansion independently of the conjecture, derives a closed formula for Serre pairs, and obtains a moments result from that formula. These proofs rely on the stated hypotheses (non-CM curves over Q) rather than reducing to fitted parameters, self-definitions, or load-bearing self-citations. The conjecture is motivated by Zywina but the proven statements remain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions about non-CM elliptic curves and the existence of natural densities; the density constant itself is defined by a convergent series rather than fitted parameters.

axioms (2)
  • domain assumption Non-CM elliptic curves over Q have good reductions whose point counts satisfy the expected distribution properties from the Lang-Trotter and Koblitz frameworks.
    Invoked to formulate the conjecture and to restrict to the non-CM case.
  • standard math Natural densities exist for the sets of primes defined by the coprimality condition on reduction orders.
    Standard assumption in analytic number theory when studying such prime sets.

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Reference graph

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