On the field of definition of $$p$$ -torsion points on elliptic curves over the rationals

Lozano-Robledo, Álvaro

doi:10.1007/s00208-013-0906-5

Cited by 61 publications

(140 citation statements)

References 29 publications

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“…Rogers's method, meanwhile, sets a bound on the heights of the rationals x for which we compute the squarefree parts D f (x) of f (x); while this imposes an implicit bound on the size of the twists considered, it is a much looser one, and twists by squarefree integers of much greater magnitude occur. For instance, note that in our search, in which we considered rationals x of height up to 10 4 , one of the curves of rank 5 with a 21-isogeny was given by twisting by −531894503. This explicit restriction is, at least conjecturally, a real disadvantage: since the rank of an elliptic curve E with conductor N E is conjectured to be bounded by C log NE log log NE for some constant C, 2 and since the conductor of E (D) is essentially D 2 · N E , we want to allow twists by large integers in our search for twists of large rank.…”

Section: 3mentioning

confidence: 99%

On the ranks of elliptic curves with isogenies

Daniels

Goodwillie

2017

Int. J. Number Theory

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Abstract. In recent years, the question of whether the ranks of elliptic curves defined over Q are unbounded has garnered much attention. One can create refined versions of this question by restricting one's attention to elliptic curves over Q with a certain algebraic structure, e.g., with a rational point of a given order. In an attempt to gather data about such questions, we look for examples of elliptic curves over Q with an n-isogeny and rank as large as possible. To do this, we use existing techniques due to Rogers, Rubin, Silverberg, and Nagao and develop a new technique (based on an observation made by Mazur) that is more computationally feasible when the naive heights of the elliptic curves are large.

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Section: 3mentioning

confidence: 99%

On the ranks of elliptic curves with isogenies

Daniels

Goodwillie

2017

Int. J. Number Theory

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“…Throughout the paper, we exemplify our results with the elliptic curves E 27a4 /Q and E 121c2 /Q with Cremona labels "27a4" and "121c2", and the primes p = 3 and 11, respectively. In the last section of the article, Section 6, we discuss several other examples that correspond to non-cuspidal rational points on the modular curves X 0 (p n ), which appear in applications such as [5], and also we work out an example with an elliptic curve defined over a (quadratic) number field (see Example 6.2).…”

Section: Conjecture 14 ([6]mentioning

confidence: 99%

“…Conversely, following [1], each non-cuspidal Q-rational point on X 0 (p n ) comes from such a pair (E/Q, R ), with R ∈ E p n . The rational points on the modular curves X 0 (p n ) have been completely classified (see, for example, Section 9.1 and Tables 2, 3, and 4 of [5]). Here, in Table 1, we list every non-cuspidal Q-rational point on the modular curves X 0 (p n ) of genus ≥ 1, which correspond to elliptic curves with potential supersingular reduction at the prime p (and provide the Cremona labels for curves with the given jinvariant and least conductor).…”

Section: Examples From X 0 (P N )mentioning

confidence: 99%

Ramification in the division fields of elliptic curves with potential supersingular reduction

Lozano-Robledo

2016

Res. number theory

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Let d ≥ 1 be fixed. Let F be a number field of degree d, and let E/F be an elliptic curve. Let E(F) tors be the torsion subgroup of E(F). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E/F, such that the size of E(F) tors is bounded by B(d). Moreover, Merel gave a bound (exponential in d) for the largest prime that may be a divisor of the order of E(F) tors . In 1996, Parent proved a bound (also exponential in d) for the largest p-power order of a torsion point that may appear in E(F) tors . It has been conjectured, however, that there is a bound for the size of E(F) tors that is polynomial in d. In this article we show that if E/F has potential supersingular reduction at a prime ideal above p, then there is a linear bound for the largest p-power order of a torsion point defined over F, which in fact is linear in the ramification index of the prime of supersingular reduction.

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“…The growth of torsion subgroups of elliptic curves upon base change was further studied in [13,16]. Using Galois representations, Lozano-Robledo has explicitly characterized the set of primes for which an elliptic curve E/Q has a point of order over a number field of degree at most d, assuming a positive answer to Serre's uniformity problem [24]. The same was done by Najman for cyclic -isogenies of elliptic curves with rational j-invariant [27].…”

Section: Introductionmentioning

confidence: 99%

Cartan images and $$\ell $$ ℓ -torsion points of elliptic curves with rational j-invariant

Propp

2018

Res. number theory

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Let be an odd prime and d a positive integer. We determine when there exists a degree-d number field K and an elliptic curve E/K with j(E) ∈ Q \ {0, 1728} for which E(K ) tors contains a point of order , conditionally on a conjecture of Sutherland. We likewise determine when there exists such a pair (K, E) for which the image of the associated mod-Galois representation is contained in a Cartan subgroup or its normalizer. We do the same under the stronger assumption that E is defined over Q.

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On the field of definition of $$p$$ -torsion points on elliptic curves over the rationals

Cited by 61 publications

References 29 publications

On the ranks of elliptic curves with isogenies

On the ranks of elliptic curves with isogenies

Ramification in the division fields of elliptic curves with potential supersingular reduction

Cartan images and $$\ell $$ ℓ -torsion points of elliptic curves with rational j-invariant

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