$${{\mathbb {Q}}}$$-curves over odd degree number fields

Cremona, John; Najman, Filip

doi:10.1007/s40993-021-00270-0

Cited by 10 publications

(6 citation statements)

References 33 publications

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“…The non-CM cubic points on X 0 (N ) for N ∈ {53, 57, 61, 67, 73} do not give rise to Q-curves: this follows Theorem 1.1 in [CN] and that given a non-CM point P ∈ X 0 (N )(K) one may always find an elliptic curve defined over K with a K-rational N -isogeny. For N = 65, the same may be deduced from combining [CN,Theorem 2.7] and [CN,Corollary 3.4].…”

Section: Resultsmentioning

confidence: 84%

Cubic and quartic points on modular curves using generalised symmetric Chabauty

Box¹,

Gajović²,

Goodman³

2021

Preprint

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Answering a question of Zureick-Brown, we determine the cubic points on the modular curves X 0 (N ) for N ∈ {53, 57, 61, 65, 67, 73} as well as the quartic points on X 0 (65). To do so, we develop a "partially relative" symmetric Chabauty method. Our results generalise current symmetric Chabauty theorems, and improve upon them by lowering the involved prime bound. For our curves a number of novelties occur. We prove a "higher order" Chabauty theorem to deal with these cases. Finally, to study the isolated quartic points on X 0 (65), we rigorously compute the full rational Mordell-Weil group of its Jacobian.

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Section: Resultsmentioning

confidence: 84%

Cubic and quartic points on modular curves using generalised symmetric Chabauty

Box¹,

Gajović²,

Goodman³

2021

Preprint

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“…For most values of j we will have F = Q( j), meaning that j is not isogenous to any of its conjugates (except itself, trivially). If F = Q then (by definition), j is a Q-number, in the sense that elliptic curves with j-invariant j are Q-curves; in general j is a K-number (with an analogous definition): see [15] and [13]. By the theory of K-curves, if there are any isogenies between j and its conjugates of degree divisible by ℓ, they all factor through a unique ℓ-isogeny, so that m ℓ ( j) = 1; otherwise m ℓ ( j) = 0.…”

Section: Proposition 5 Let H Be a Hilbert Class Polynomial With H + R...mentioning

confidence: 99%

Computing the endomorphism ring of an elliptic curve over a number field

Cremona¹,

Sutherland²

2023

Preprint

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We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H ∈ Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given algebraic integer is the j-invariant of an elliptic curve with complex multiplication (CM), and if so, the associated CM discriminant. More generally, given an elliptic curve E over a number field, one can use them to compute the endomorphism ring of E. Our algorithms admit simple implementations that are asymptotically and practically faster than existing approaches.

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“…or when K is Galois. In all cases, Algorithm QCurveTest of [17] is required (as well as a lot of interesting properties of isogenous curves and extensions of [26]).…”

Section: Algorithms To Simplify the Descriptionmentioning

confidence: 99%