Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves

Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius elements, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.

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On the cohomological coprimality of Galois representations associated with elliptic curves

Proceedings of the Japan Academy Series A Mathematical Sciences ◽

10.3792/pjaa.91.141 ◽

2015 ◽

Vol 91 (10) ◽

pp. 141-146

Author(s):

Jerome Tomagan Dimabayao

Keyword(s):

Elliptic Curves ◽

Galois Representations

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On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves

Canadian Mathematical Bulletin ◽

10.4153/cmb-2005-002-x ◽

2005 ◽

Vol 48 (1) ◽

pp. 16-31 ◽

Cited By ~ 13

Author(s):

Alina Carmen Cojocaru ◽

Ernst Kani

Keyword(s):

Positive Integer ◽

Elliptic Curve ◽

Elliptic Curves ◽

Galois Representations ◽

Complex Multiplication ◽

Galois Representation ◽

Upper Bounds

AbstractLet E be an elliptic curve defined over ℚ, of conductor N and without complex multiplication. For any positive integer l, let ϕl be the Galois representation associated to the l-division points of E. From a celebrated 1972 result of Serre we know that ϕl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which ϕl is not surjective.

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Multiplicative isogeny estimates

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700001683 ◽

1998 ◽

Vol 64 (2) ◽

pp. 178-194 ◽

Cited By ~ 4

Author(s):

David Masser

Keyword(s):

Elliptic Curves ◽

Galois Representations ◽

Abelian Varieties

AbstractThe theory of isogeny estimates for Abelian varieties provides ‘additive bounds’ of the form ‘d is at most B’ for the degrees d of certain isogenies. We investigate whether these can be improved to ‘multiplicative bounds’ of the form ‘d divides B’. We find that in general the answer is no (Theorem 1), but that sometimes the answer is yes (Theorem 2). Further we apply the affirmative result to the study of exceptional primes ℒ in connexion with modular Galois representations coming from elliptic curves: we prove that the additive bounds for ℒ of Masser and Wüstholz (1993) can be improved to multiplicative bounds (Theorem 3).

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