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On the torsion growth in quadratic number fields for elliptic curves defined over the rationals

Arias-de-Reyna, Sara, Pineda, Miguel, Tornero, José M

TL;DR

The paper investigates torsion growth of elliptic curves over $\mathbb{Q}$ under quadratic base change. It analyzes the problem through mod-$\ell$ Galois representations for $\ell\in\{2,3,5,7\}$, combined with inertia considerations and Tate normal forms, to derive constraints relating the curve conductor $N_E$ to the extension conductor via ramification data. The main result shows that if a quadratic field $K=\mathbb{Q}(\sqrt{d})$ yields new torsion, then any prime dividing $d$ must lie in $\{2,3\}\cup\{p: p|N_E\}$, with detailed, case-by-case obstruction results for $\ell=2,3,5,7$ and explicit examples illustrating when growth occurs. This work provides a structured path toward a complete sieve of quadratic fields that can cause torsion growth and highlights the roles of reduction type and Galois-image constraints in eliminating most ramified cases.

Abstract

Given an elliptic curve defined over the field of rational numbers, it is known how its torsion subgroup may grow when we make a base change to a quadratic number field. In this paper we consider the inverse question: if we have the elliptic curve defined over the rationals and we know how the torsion subgroup grows, what can we say about the field? Our main result gives an explicit relationship between the primes dividing the conductor of the curve and the conductor of the extension as a first approach to a better understanding of this problem.

On the torsion growth in quadratic number fields for elliptic curves defined over the rationals

TL;DR

The paper investigates torsion growth of elliptic curves over under quadratic base change. It analyzes the problem through mod- Galois representations for , combined with inertia considerations and Tate normal forms, to derive constraints relating the curve conductor to the extension conductor via ramification data. The main result shows that if a quadratic field yields new torsion, then any prime dividing must lie in , with detailed, case-by-case obstruction results for and explicit examples illustrating when growth occurs. This work provides a structured path toward a complete sieve of quadratic fields that can cause torsion growth and highlights the roles of reduction type and Galois-image constraints in eliminating most ramified cases.

Abstract

Given an elliptic curve defined over the field of rational numbers, it is known how its torsion subgroup may grow when we make a base change to a quadratic number field. In this paper we consider the inverse question: if we have the elliptic curve defined over the rationals and we know how the torsion subgroup grows, what can we say about the field? Our main result gives an explicit relationship between the primes dividing the conductor of the curve and the conductor of the extension as a first approach to a better understanding of this problem.

Paper Structure

This paper contains 10 sections, 15 theorems, 190 equations, 2 tables.

Table of Contents

  1. Introduction: The problem
  2. The prime $\ell=2$: The strict case
  3. The prime $\ell=2$: The mixed case
  4. The prime $\ell=3$
  5. The primes $\ell=5$ and $\ell=7$
  6. Ordinary reduction with nontrivial wild inertia action
  7. Ordinary Reduction with Trivial Wild Inertia Action
  8. Supersingular Reduction
  9. Conclusion and final remarks
  10. Appendix: Matrix groups appearing in Section 6

Key Result

Proposition 1

Let $E/\mathbb{Q}$ be an elliptic curve. If $K$ is a quadratic number field such that $E_\mathrm{tors}(\mathbb{Q}) \not= E_\mathrm{tors}(K)$, then the primes ramifying in $K$ belong to the set $\{2, 3, 5, 7\}\cup \{p: p\vert N_E\}$.

Theorems & Definitions (24)

  • Conjecture 1
  • Proposition 1
  • Proposition 2
  • Example 1
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Lemma 1
  • Lemma 2
  • ...and 14 more