Modular curves of prime-power level with infinitely many rational points

Sutherland, Andrew V.; Zywina, David

doi:10.2140/ant.2017.11.1199

Cited by 43 publications

(116 citation statements)

References 15 publications

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“…An enumeration in Magma of the subgroups of GL 2 (Z/27Z) finds that every such G is conjugate to a subgroup of the full inverse image of of two subgroups of GL 2 (Z/9Z) whose intersection with SL 2 (Z/9Z) gives the congruence subgroups 9I 0 and 9J 0 . Explicit rational parameterizations for the genus zero modular curves X H 1 and X H 2 appear in [43]; these curves both admit rational maps to X 0 (3), allowing us to explicitly construct a rational model for X H as the fiber product of these maps over X 0 (3). This model is isomorphic to the elliptic curve 27a3, which has just 3 rational points, which is equal to the number of rational cusps on X H , so there are no non-cuspidal rational points.…”

Section: 2mentioning

confidence: 99%

Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

et al. 2017

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Abstract. Let E/Q be an elliptic curve and let Q(3 ∞ ) be the compositum of all cubic extensions of Q. In this article we show that the torsion subgroup of E(Q(3 ∞ )) is finite and determine 20 possibilities for its structure, along with a complete description of the Q-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many Q-isomorphism classes of elliptic curves, and a complete list of j-invariants for each of the 4 that do not.

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Section: 2mentioning

confidence: 99%

Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

et al. 2017

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“…In particular, we take advantage of the impressive results of Rouse and Zureick-Brown in [30], where all the possible 2-adic images of Galois representations attached to elliptic curves are classified. We also use the equally impressive classifications of mod p representations attached to rational elliptic curves given in [35] and the classification of modular curves of primes-power level with infinitely many points given in [34].…”

Section: Introductionmentioning

confidence: 99%

Torsion subgroups of rational elliptic curves over the compositum of all D4 extensions of the rational numbers

Daniels

2018

Journal of Algebra

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Let E/Q be an elliptic curve and let Q(D ∞ 4 ) be the compositum of all extensions of Q whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of D4. In this article we first show that Q(D ∞ 4 ) is infact the compostium of all D4 extensions of Q and then we prove that the torsion subgroup of E(Q(D ∞ 4 )) is finite and determine the 24 possibilities for its structure. We also give a complete classification of the elliptic curves that have each possible torsion structure in terms of their j-invariants.

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“…(The Galois representation computations in LMFDB were carried out using the algorithm from [Sut16]. ) Sutherland and Zywina's classification of modular curves of prime-power level with infinitely many points [SZ17] shows that there are only finitely many rational j-invariants corresponding to elliptic curves as in (3), and suggests that in fact they do not exist. Table 7.1 gives, for each prime ℓ, the maximal prime-power level for which there exists a modular curve of that level with infinitely many rational points.…”

Section: Sporadic Points With Rational J-invariantmentioning

confidence: 99%

On the level of modular curves that give rise to isolated j-invariants

Bourdon

Ejder

Liu

et al. 2019

Advances in Mathematics

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We say a closed point x on a curve C is sporadic if C has only finitely many closed points of degree at most deg (x). Motivated by well-known classification problems concerning rational torsion of elliptic curves, we study sporadic points on the modular curves X 1 (N ). In particular, we show that any non-cuspidal non-CM sporadic point x ∈ X 1 (N ) maps down to a sporadic point on a modular curve X 1 (d), where d is bounded by a constant depending only on j(x). Conditionally, we show that d is bounded by a constant depending only on the degree of Q(j(x)), so in particular there are only finitely many j-invariants of bounded degree that give rise to sporadic points. 1In this paper, we study sporadic points of arbitrary degree, focusing particularly on sporadic points corresponding to non-CM elliptic curves. We prove that non-CM non-cuspidal sporadic points on X 1 (n) map to sporadic points on X 1 (d), for d some bounded divisor of n.Theorem 1.1. Fix a non-CM elliptic curve E over k := Q(j(E)), and let m be an integer divisible by 2, 3 and all primes ℓ where the ℓ-adic Galois representation is not surjective. Let M = M(E, m) be the level of the m-adic Galois representation of E. If x ∈ X 1 (n) is a sporadic point with j(x) = j(E), then π(x) ∈ X 1 (gcd(n, M)) is a sporadic point, where π denotes the natural map X 1 (n) → X 1 (gcd(n, M)).For many elliptic curves, we may take both m and M to be quite small. For instance, let E be the set of elliptic curves over Q where the ℓ-adic Galois representation is surjective for all ℓ > 3 and where the 6-adic Galois representation has level dividing 24. Note that E contains all Serre curves [Ser72, Proof of Prop. 22] (that is, elliptic curves over Q whose adelic Galois representation is as large as possible) and hence contains almost all elliptic curves over Q [Jon10]. For E ∈ E, we may apply Theorem 1.1 with m = 6 and M|24.The curve X 1 (24) has infinitely many quartic points, but no rational or quadratic points, nor cubic points corresponding to elliptic curves over Q [Maz77, KM88, Mor]. Therefore X 1 (24) has no sporadic points with Q-rational j-invariant. For M a proper divisor of 24, the curves X 1 (M) have genus 0, and so also have no sporadic points. Hence Theorem 1.1 yields the following corollary.Corollary 1.2. For all n, there are no sporadic points on X 1 (n) corresponding to elliptic curves in E. In particular, there are no sporadic points corresponding to Serre curves.

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Modular curves of prime-power level with infinitely many rational points

Cited by 43 publications

References 15 publications

Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

Torsion subgroups of rational elliptic curves over the compositum of all D4 extensions of the rational numbers

On the level of modular curves that give rise to isolated j-invariants

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