Serre’s Constant of Elliptic Curves Over the Rationals

Abstract

Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre’s constant associated to E, that gives necessary conditions to conclude that ${\rho }_{E,m},$ the mod m Galois representation associated to E, is non-surjective. In particular, if there exists a prime factor p of m satisfying ${\text{val}}_p(m)\ge {\text{val}}_p(A(E))>0$ then ${\rho }_{E,m}$ is non-surjective. Conditionally under Serre’s Uniformity Conjecture, we determine all the Serre’s constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod p Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over $\mathbb{Q},$ and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of ${\text{GL}}_2({\mathbb{Z}}_p).$ Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre’s constants.

Keywords:

• Elliptic curves
• rationals
• Galois representation

2010 Mathematics Subject Classification:

• Primary: 11G05
• Secondary: 11F80
• Galois representation

Acknowledgements

The authors would like to thank Alina Cojocaru, Álvaro Lozano-Robledo, Filip Najman, Andrew Sutherland, Xavier Xarles and David Zywina for help in the preparation of this article. We would like to thank to John Cremona for providing access to computer facilities on the Number Theory Warwick Grid at University of Warwick, where the main part of the computations where done. The authors would also like to thank the anonymous referee for useful comments during the review process as well as the editors of this paper for a speedy review.

Declaration of interest

No potential conflict of interest was reported by the authors.

Notes

1 J. S. Balakrishnan, N. Dogra, J. S. Müller, J. Tuitman, and J. Vonk have recently announced a proof of this conjecture but have yet to make the results available publicly.

2 The definition of Serre’s constant that appears in this paper is a generalization of the one that Cojocaru defined at [13].

3 One might expect to see 3Ns and 5Ns on this list of groups, but due to the unique characteristics of 3 and 5 these groups are in fact not maximal. One can check that in these cases, $\mathtt{3}\mathtt{N}\mathtt{s}⊊\mathtt{3}\mathtt{N}\mathtt{n}$ and $\mathtt{5}\mathtt{N}\mathtt{s}⊊\mathtt{5}\mathtt{S}\mathtt{4}.$

4 A remarkable fact is that this genus 2 curve is new modular of level 90 and its jacobian is $\mathbb{Q}$-isogenous to the product of two elliptic curves (see [2]).

Additional information

Funding

Ministerio de Ciencia, Innovación y Universidades; Agencia Estatal de Investigación; and Fondo Europeo de Desarrollo Regional (FEDER).