Tetrahedral elliptic curves and the local-global principle for isogenies

S., Banwait, Barinder; Cremona, John

doi:10.2140/ant.2014.8.1201

Cited by 28 publications

(52 citation statements)

References 18 publications

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“…As far as we are aware, such methods have only been applied to curves of genus 2 and 3, and our computations for the three curves in (2) pushes these methods to their limit. In Sections 12, 13, 14 we determine the quadratic points on the three modular curves in (2), and deduce modularity of the non-cuspidal real quadratic points.…”

Section: Summary Of Resultsmentioning

confidence: 99%

Elliptic curves over real quadratic fields are modular

2014

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Section: Summary Of Resultsmentioning

confidence: 99%

Elliptic curves over real quadratic fields are modular

2014

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“…We would like to show that J(Q) = C, and for this it is enough to show that J(Q) [2] = C[2] = (Z 2Z) 3 . However for all primes p ∤ N we tried J(F p ) [2] = (Z 2Z) 6 or (Z 2Z) 4 , and so it does not seem to be possible to prove the desired conclusion using reduction modulo primes. Instead we will compute the entire mod 2 representation of J ρ J,2 ∶ Gal(Q Q) → Sp 6 (F 2 ) and use this to deduce that J(Q) [2] = (Z 2Z) 3 .…”

Section: 5mentioning

confidence: 99%

Quadratic points on modular curves

Ozman

Siksek

2018

Math. Comp.

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In this paper we determine the quadratic points on the modular curves X 0 (N ), where the curve is non-hyperelliptic, the genus is 3, 4 or 5, and the Mordell-Weil group of J 0 (N ) is finite. The values of N are 34, 38, 42, 44, 45, 51, 52, 54, 55, 56, 63, 64, 72, 75, 81. As well as determining the non-cuspidal quadratic points, we give the j-invariants of the elliptic curves parametrized by those points, and determine if they have complex multiplication or are quadratic Q-curves.In this paper we focus on non-hyperelliptic X 0 (N) of genera 3, 4, 5 where the Mordell-Weil group J 0 (N)(Q) is finite. We determine the quadratic points on these modular curves, and supply the modular interpretation of the points. In the forthcoming second part of this work [30] we deal with those values of N for which the genus is 3, 4, 5 but J 0 (N)(Q) is infinite, using a version of Chabauty for symmetric powers of curves as in [35].Lemma 1.1. The values of N for which X 0 (N) is non-hyperelliptic, of genus g where 3 ≤ g ≤ 5 and for which J 0 (N)(Q) is finite are genus 3:: 34, 45, 64; genus 4:: 38, 44, 54, 81; genus 5:: 42, 51, 52, 55, 56, 63, 72, 75.Let X be a curve defined over Q. A point P ∈ X is called quadratic if the field Q(P ) is a quadratic extension of Q.Main Theorem. For the values of N listed in Lemma 1.1 the quadratic points on X 0 (N) are as given in the tables of Section 8.For the non-cuspidal quadratic points, we compute j-invariants of the elliptic curves parametrized by them. In addition, we check whether those points are related via any Atkin-Lehner involutions and decide whether or not they have complex multiplication or are Q-curves.One motivation for this work is the current interest in the Fermat equation over quadratic fields and similar Diophantine problems. The approach via modularity and level-lowering requires the irreducibility of the mod p representation of a Frey elliptic curve defined over the given quadratic field, K say. This Frey elliptic curve often has extra level structure in the form of a K-rational 2 or 3-isogeny. If the mod p representation is reducible, then the Frey curve gives rise to a K-rational point on X 0 (2p) or X 0 (3p), and thus having a parametrization of quadratic points is useful in establishing irreducibility for small values of p. The results of the current paper have already proved useful in that context [15].A Theoretical Approach to the Problem. Let X Q be a non-hyperelliptic curve of genus ≥ 3 with J(Q) finite where J is the Jacobian of X, and suppose for convenience that X has at least one rational point P 0 . For example X could be any of the curves X 0 (N) for the values of N in Lemma 1.1. There is a straightforward theoretical method (see for instance [14]) of computing all effective degree 2 rational divisors on X, and hence all points defined over quadratic extensions, provided we are able to enumerate all the elements of J(Q). Let X (2) denote the second symmetric product of X. A Q-rational point on X (2) can be represented by unordered pair {P 1 , P 2 } where P 1...

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“…Banwait and Cremona [BC14] have shown that X G 7,13 (13) is isomorphic to the genus 3 curve C ′ defined in P 2 Q with equation 4x 3 y − 3x 2 y 2 + 3xy 3 − x 3 z + 16x 2 yz − 11xy 2 z + 5y 3 z + 3x 2 z 2 + 9xyz 2 + y 2 z 2 + xz 3 + 2yz 3 = 0.…”

Section: Analysis Of Rational Points -Higher Genusmentioning

confidence: 99%

Composite images of Galois for elliptic curves over $\mathbf {Q}$ and entanglement fields

Morrow

2019

Math. Comp.

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Let E be an elliptic curve defined over Q without complex multiplication. For each prime ℓ, there is a representation ρ E,ℓ : Gal(Q/Q) → GL 2 (Z/ℓZ) that describes the Galois action on the ℓ-torsion points of E. Building on recent work of Rouse-Zureick-Brown and Zywina, we find models for composite level modular curves whose rational points classify elliptic curves over Q with simultaneously non-surjective, composite image of Galois. We also provably determine the rational points on almost all of these curves. Finally, we give an application of our results to the study of entanglement fields. 7

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Tetrahedral elliptic curves and the local-global principle for isogenies

Cited by 28 publications

References 18 publications

Elliptic curves over real quadratic fields are modular

Elliptic curves over real quadratic fields are modular

Quadratic points on modular curves

Composite images of Galois for elliptic curves over $\mathbf {Q}$ and entanglement fields

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