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Mathematische Annalen

, Volume 357, Issue 1, pp 279–305 | Cite as

On the field of definition of \(p\)-torsion points on elliptic curves over the rationals

  • Álvaro Lozano-Robledo
Article

Abstract

Let \(S_\mathbb Q (d)\) be the set of primes \(p\) for which there exists a number field \(K\) of degree \(\le d\) and an elliptic curve \(E/\mathbb Q \), such that the order of the torsion subgroup of \(E(K)\) is divisible by \(p\). In this article we give bounds for the primes in the set \(S_\mathbb Q (d)\). In particular, we show that, if \(p\ge 11\), \(p\ne 13,37\), and \(p\in S_\mathbb Q (d)\), then \(p\le 2d+1\). Moreover, we determine \(S_\mathbb Q (d)\) for all \(d\le 42\), and give a conjectural formula for all \(d\ge 1\). If Serre’s uniformity problem is answered positively, then our conjectural formula is valid for all sufficiently large \(d\). Under further assumptions on the non-cuspidal points on modular curves that parametrize those \(j\)-invariants associated to Cartan subgroups, the formula is valid for all \(d\ge 1\).

Keywords

Elliptic Curve Rational Point Elliptic Curf Isomorphism Class Galois Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

This work was motivated by an earlier collaboration with Benjamin Lundell, where we described bounds on fields of definition in terms of ramification indices [30]. The author would like to thank Benjamin Lundell, Robert Pollack, Jeremy Teitelbaum, Ravi Ramakrishna, and the anonymous referee for their helpful suggestions and comments.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA

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