Torsion growth of rational elliptic curves in sextic number fields
type: Item
instance of: scholarly article
title: Torsion growth of rational elliptic curves in sextic number fields
zbMATH Open document ID: 1469.14071
author: Tomislav Gužvić
published in: Journal of Number Theory
publication date: 2020-11-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1910.01561
review text: For some years now, an interest has arisen in the problem of studying the following problem: given an elliptic curve \(E\), defined over \({\mathbb Q}\), study how its torsion subgroup \(E_{tors} ({\mathbb Q})\) changes when the base field is expanded to a number field \(K\). This line of research has been pioneered by E. González-Jiménez and F. Najman, who have tackled themselves (also with a number of co-authors) many of the challenging problems arising in this context. The paper under review follows the footsteps of these previous works by giving a(n almost) complete result concerning the case where \(K\) is a sextic degree field. In particular, it narrows down the list of possible torsion groups appearing under these conditions to exactly \(34\) of them (\(18\) of them cyclic, the rest of them the direct sum of the form \(C_n \oplus C_m\)). All but one of the groups are known to occur, the sole exception being \(C_3 \oplus C_{18}\), which we still do not know if it appears. The paper is well-written, and the account on the previous landmarks is a nice introduction to the subject (even for non-specialists). The techniques are pretty much standard in this line of work, but it requires a high level of mastery to apply them in the way they are done here.
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11R21
zbMATH DE Number: 7276987
zbMATH Keywords: torsion group
zbMATH Keywords: elliptic curves
reviewed by: José María Tornero
describes a project that uses: Magma
describes a project that uses: ell-adic-galois-images
MaRDI profile type: Publication
OpenAlex ID: W3093901597
arXiv ID: 1910.01561
cites work: The Magma algebra system. I: The user language
cites work: Torsion Subgroups of CM Elliptic Curves over Odd Degree Number Fields:
cites work: Elliptic curves with 2-torsion contained in the 3-torsion field
cites work: Torsion of rational elliptic curves over quartic Galois number fields
cites work: Computation on elliptic curves with complex multiplication
cites work: Galois theory
cites work: On the torsion of rational elliptic curves over sextic fields
cites work: Torsion subgroups of rational elliptic curves over the compositum of all cubic fields
cites work: Torsion subgroups of elliptic curves over quintic and sextic number fields
cites work: Complete classification of the torsion structures of rational elliptic curves over quintic number fields
cites work: On the minimal degree of definition of \(p\)-primary torsion subgroups of elliptic curves
cites work: Growth of torsion groups of elliptic curves upon base change
cites work: Torsion of rational elliptic curves over cubic fields
cites work: Elliptic divisibility sequences over certain curves
cites work: ON THE TORSION OF ELLIPTIC CURVES OVER QUARTIC NUMBER FIELDS
cites work: On the torsion of elliptic curves over cubic number fields
cites work: Torsion points on elliptic curves and \(q\)-coefficients of modular forms
cites work: The modular curve X0(39) and rational isogeny
cites work: The Modular Curve X 0 (169) and Rational Isogeny
cites work: The modular curves X0(65) and X0(91) and rational isogeny
cites work: On the Modular Curves X 0 (125), X 1 (25), X 1 (49)
cites work: Torsion points on elliptic curves defined over quadratic fields
cites work: Modular curves and the Eisenstein ideal
cites work: Rational isogenies of prime degree. (With an appendix by D. Goldfeld)
cites work: Bounds for the torsion of elliptic curves over number fields
cites work: Composite images of Galois for elliptic curves over $\mathbf {Q}$ and entanglement fields
cites work: Complete classification of torsion of elliptic curves over quadratic cyclotomic fields
cites work: Torsion of rational elliptic curves over cubic fields and sporadic points on \(X_1(n)\)
cites work: Points of finite order on elliptic curves with complex multiplication
cites work: Elliptic curves over \(\mathbb {Q}\) and 2-adic images of Galois
cites work: COMPUTING IMAGES OF GALOIS REPRESENTATIONS ATTACHED TO ELLIPTIC CURVES
cites work: Modular curves of prime-power level with infinitely many rational points
cites work: Q5386130
DOI: 10.1016/J.JNT.2020.09.010
description: scientific article