Entanglement for elliptic curve group orders
Abstract
For an elliptic curve E defined over a number field K and p a prime of good reduction of E/K, we let np=#E(kp) be the order of the group of points over the residue class field kp. Divisibility of np by a prime ℓ depends on the splitting behavior of p in the ℓ-division field K(E[ℓ](K̅)) of E and, generically, divisibility of np by different primes ℓ1 and ℓ2 are `independent events' on the set of primes p of good reduction. However, there do exist elliptic curves with `entangled division fields' for which infinitely many np are coprime to ℓ1 and infinitely many np are coprime to ℓ2, but no np is coprime to to ℓ1 ℓ2. For ℓ1=2, we classify in group theoretical terms the entanglement giving rise to this behavior, and count up to isomorphism the number of `special 2ℓ-division groups' exhibiting the phenomenon. A final section shows how such groups may be realized as the Galois group of 2ℓ division fields for elliptic curves E/Q.
Related
- Divisibility Estimates for Gaussian Norms and Almost-Prime Orders of a CM Elliptic Curve
- Algebra of the Infrared with Elliptic Curve-Valued Potential
- Counterexamples to Dujella's conjecture on integral points on the elliptic curve attached to a Diophantine triple
- A Criterion for classifying Elliptic curve that are unsolvable in the set integers
- Elliptic curve rank growth in multi-quadratic extensions
BibTeX
@misc{campagna2026entanglement,
title = {{Entanglement for elliptic curve group orders}},
author = {Francesco Campagna and Nathan Jones and Francesco Pappalardi and Peter Stevenhagen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01204},
eprint = {2611.01204},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01204},
}
RIS
TY - UNPB
TI - Entanglement for elliptic curve group orders
AU - Francesco Campagna
AU - Nathan Jones
AU - Francesco Pappalardi
AU - Peter Stevenhagen
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01204
UR - https://arxiv.org/abs/2611.01204
AB - For an elliptic curve E defined over a number field K and p a prime of good reduction of E/K, we let n_p=#E(k_p) be the order of the group of points over the residue class field k_p. Divisibility of n_p by a prime ℓ depends on the splitting behavior of p in the ℓ-division field K(E[ℓ](K̅)) of E and, generically, divisibility of n_p by different primes ℓ_1 and ℓ_2 are `independent events' on the set of primes p of good reduction. However, there do exist elliptic curves with `entangled division fields' for which infinitely many n_p are coprime to ℓ_1 and infinitely many n_p are coprime to ℓ_2, but no n_p is coprime to to ℓ_1 ℓ_2. For ℓ_1=2, we classify in group theoretical terms the entanglement giving rise to this behavior, and count up to isomorphism the number of `special 2ℓ-division groups' exhibiting the phenomenon. A final section shows how such groups may be realized as the Galois group of 2ℓ division fields for elliptic curves E/Q.
ER -