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Divisibility of orders of reductions of elliptic curves

Antigona Pajaziti, Mohammad Sadek

TL;DR

The paper addresses when the order of reductions $|\tilde{E}_p(\mathbb{F}_p)|$ is divisible by a fixed $m$ for a density-1 set of primes, linking this to torsion growth in $K$-isogenous families and using Serre–Katz framework to identify congruence obstructions. It develops local-to-global machinery over quadratic fields, deriving exact formulas for $|\tilde{E}_{\mathfrak{p}}(k_{\mathfrak{p}})|$ in terms of $|\tilde{E}_p(\mathbb{F}_p)|$ and its twists, and uses Chebotarev density to bound densities of primes with given divisibility properties. The authors provide explicit $E/\mathbb{Q}$ families with computable congruence classes modulo $|E(K)_{\mathrm{tors}}|$, and show how torsion growth over a quadratic field imposes modular constraints on Frobenius traces $a_p(E)$. Finally, they construct $E_t$ over $\mathbb{Q}(t)$ yielding divisibility patterns for reductions across primes of positive density, illustrating the breadth of attainable congruence behavior in non-CM settings and across rational-function families.

Abstract

Let $E$ be an elliptic curve defined over $\mathbb Q$ and $\widetilde{E}_p$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. The divisibility of $|\widetilde{E}_{p}(\mathbb{F}_p)|$ by an integer $m\ge 2$ for a set of primes $p$ of density $1$ is determined by the torsion subgroups of elliptic curves that are $\mathbb Q$-isogenous to $E$. In this work, we give explicit families of elliptic curves $E$ over $\mathbb Q$ together with integers $m_E$ such that the congruence class of $|\widetilde{E}_p(\mathbb{F}_p)|$ modulo $m_E$ can be computed explicitly. In addition, we can estimate the density of primes $p$ for which each congruence class occurs. These include elliptic curves over $\mathbb Q$ whose torsion grows over a quadratic field $K$ where $m_E$ is determined by the $K$-torsion subgroups in the $\mathbb Q$-isogeny class of $E$. We also exhibit elliptic curves over $\mathbb Q(t)$ for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than $1$ are divisible by given small integers.

Divisibility of orders of reductions of elliptic curves

TL;DR

The paper addresses when the order of reductions is divisible by a fixed for a density-1 set of primes, linking this to torsion growth in -isogenous families and using Serre–Katz framework to identify congruence obstructions. It develops local-to-global machinery over quadratic fields, deriving exact formulas for in terms of and its twists, and uses Chebotarev density to bound densities of primes with given divisibility properties. The authors provide explicit families with computable congruence classes modulo , and show how torsion growth over a quadratic field imposes modular constraints on Frobenius traces . Finally, they construct over yielding divisibility patterns for reductions across primes of positive density, illustrating the breadth of attainable congruence behavior in non-CM settings and across rational-function families.

Abstract

Let be an elliptic curve defined over and denote the reduction of modulo a prime of good reduction for . The divisibility of by an integer for a set of primes of density is determined by the torsion subgroups of elliptic curves that are -isogenous to . In this work, we give explicit families of elliptic curves over together with integers such that the congruence class of modulo can be computed explicitly. In addition, we can estimate the density of primes for which each congruence class occurs. These include elliptic curves over whose torsion grows over a quadratic field where is determined by the -torsion subgroups in the -isogeny class of . We also exhibit elliptic curves over for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than are divisible by given small integers.
Paper Structure (5 sections, 15 theorems, 24 equations, 1 table)

This paper contains 5 sections, 15 theorems, 24 equations, 1 table.

Table of Contents

  1. Introduction
  2. Density $1$ known results
  3. Local results over quadratic fields
  4. Congruence classes of orders of reduction
  5. Divisibility in families of elliptic curves

Key Result

Theorem 1.1

Let $K={\mathbb Q}(\sqrt{d})$, where $d$ is a square free integer. Let $E$ be an elliptic curve defined over $\mathbb Q$. Set $M_K(E)=\sup\{|E'(K)_{\operatorname{tors}}|: E'\textrm{ is ${\mathbb Q}$-isogenous to }E\}$. Assume that $M_K(E)>M_{\mathbb Q}(E)$. If $p\nmid 2d M_K(E)$ is a prime of good r where $E^d$ is the quadratic twist of $E$ by $d$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Proposition 4.1
  • Proposition 4.2
  • Theorem 4.3
  • proof
  • ...and 14 more