The modular curve <i>X</i><sub>0</sub>(39) and rational isogeny

Kenku, M. A.

doi:10.1017/s0305004100055444

Cited by 30 publications

(21 citation statements)

References 5 publications

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“…Theorem 4 (Mazur [31] and Kenku [23,24,25,26]). Let E/Q be an elliptic curve with a rational n-isogeny.…”

Section: Background Informationmentioning

confidence: 99%

On the torsion of rational elliptic curves over sextic fields

Daniels¹,

González–Jiménez²

2019

Math. Comp.

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Given an elliptic curve E/Q with torsion subgroup G = E(Q)tors we study what groups (up to isomorphism) can occur as the torsion subgroup of E base-extended to K, a degree 6 extension of Q. We also determine which groups H = E(K)tors can occur infinitely often and which ones occur for only finitely many curves. This article is a first step towards a complete classification of torsion growth over sextic fields.

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“…Theorem 4 (Mazur [31] and Kenku [23,24,25,26]). Let E/Q be an elliptic curve with a rational n-isogeny.…”

Section: Background Informationmentioning

confidence: 99%

On the torsion of rational elliptic curves over sextic fields

Daniels¹,

González–Jiménez²

2019

Math. Comp.

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“…The next sieve comes from the classification of rational n-cyclic isogenies given by Mazur and Kenku (cf. [34,25,26,27,28]) and torsion structure given by Mazur (cf. [33]).…”

Section: Exceptional Pairsmentioning

confidence: 99%

Serre’s Constant of Elliptic Curves Over the Rationals

Daniels

González–Jiménez

2019

Experimental Mathematics

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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 ρE,m, the mod m Galois representation associated to E, is non-surjective. In particular, if there exists a prime factor p of m satisfying valp(m) ≥ valp(A(E)) > 0 then ρE,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 Q, and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of GL2(Zp). Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre's constants.Serre's Uniformity Question. If E/Q is a non-CM elliptic curve, then must it be that ρ E,p is surjective for any prime p ≥ 41?Nowadays, an affirmative answer to the above question has received the name of Serre's Uniformity Conjecture (or sometimes just Uniformity Conjecture) despite the fact that Serre himself never conjectured it to be true.

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“…For example, there is a complete determination of the elliptic curves A and integers m such that the image of a conjugate of G Q in GL 2 (Z/mZ) is contained in a subgroup of the group of upper-triangular matrices 36 (cf. [Maz,Ken,Me 3]). Given any homomorphism ƒ: A -> B of abelian varieties defined over K, one can restrict ƒ to m-division points to get a G^-equivariant homomorphism (between finite abelian groups) …”

Section: G Timesmentioning

confidence: 99%

Arithmetic on curves

Mazur¹

1986

Bull. Amer. Math. Soc.

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The modular curve X₀(39) and rational isogeny

Abstract: Recently (3) Mazur proved that if N is a prime number such that some elliptic curve E over Q admits a Q-rational isogeny then N is one of 2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67 or 163.

Cited by 30 publications

References 5 publications

On the torsion of rational elliptic curves over sextic fields

On the torsion of rational elliptic curves over sextic fields

Serre’s Constant of Elliptic Curves Over the Rationals

Arithmetic on curves

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The modular curve X0(39) and rational isogeny

Abstract: Recently (3) Mazur proved that if N is a prime number such that some elliptic curve E over Q admits a Q-rational isogeny then N is one of 2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67 or 163.

Cited by 30 publications

References 5 publications

On the torsion of rational elliptic curves over sextic fields

On the torsion of rational elliptic curves over sextic fields

Serre’s Constant of Elliptic Curves Over the Rationals

Arithmetic on curves

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Product

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About

The modular curve X₀(39) and rational isogeny