scholarly journals Corrigendum to: “Torsion subgroups of rational elliptic curves over the compositum of all D4 extensions of the rational numbers” [J. Algebra 509 (2018) 535–565]

2021 ◽  
Author(s):  
Harris B. Daniels
Keyword(s):  
Elliptic Curves ◽  
Rational Numbers ◽  
Torsion Subgroups

scholarly journals Elliptic curves with a torsion point

1977 ◽  
Vol 66 ◽  
pp. 99-108 ◽  
Author(s):  
Toshihiro Hadano
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Rational Numbers ◽  
Torsion Subgroup ◽  
Torsion Point ◽  
Weil Group ◽  
Torsion Subgroups

Let E be an elliptic curve defined over the field Q of rational numbers, then the torsion subgroup of the Mordell-Weil group E(Q) is finite and it is known that there exist the elliptic curves whose torsion subgroups E(Q)t are of the following types: (1), (2), (3), (2, 2), (4), (5), (2, 3), (7), (2, 4), (8), (9), (2, 5), (2, 2, 3), (3, 4) and (2, 8). It has been conjectured from various reasons that E(Q)t is exhausted by the above types only. If E has a torsion point of order precisely n, then it is known that E has an n-isogeny, that is to say, an isogeny of degree n.

The Joint Universality for L-Functions of Elliptic Curves

2004 ◽  
Vol 9 (4) ◽  
pp. 331-348
Author(s):  
V. Garbaliauskienė
Keyword(s):  
Elliptic Curves ◽  
Rational Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
Joint Universality Theorem

A joint universality theorem in the Voronin sense for L-functions of elliptic curves over the field of rational numbers is proved.

p-ADIC PERIODS OF MODULAR ELLIPTIC CURVES AND THE LEVEL-LOWERING THEOREM

2008 ◽  
Vol 04 (01) ◽  
pp. 15-23 ◽  
Author(s):  
S. TAKAHASHI
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Cusp Form ◽  
Rational Numbers ◽  
Fermat's Last Theorem ◽  
Complex Torus ◽  
Essential Tool ◽  
Fermat’S Last Theorem ◽  
Level Lowering

An elliptic curve defined over the field of rational numbers can be considered as a complex torus. We can describe its complex periods in terms of integration of the weight-2 cusp form corresponding to the elliptic curve. In this paper, we will study an analogous description of the p-adic periods of the elliptic curve, considering the elliptic curve as a p-adic torus. An essential tool for the proof of such a description is the level-lowering theorem of Ribet, which is one of the main ingredients used in the proof of Fermat's Last Theorem.

scholarly journals On class numbers, torsion subgroups, and quadratic twists of elliptic curves

2021 ◽  
pp. 1
Author(s):  
Talia Blum ◽  
Caroline Choi ◽  
Alexandra Hoey ◽  
Jonas Iskander ◽  
Kaya Lakein ◽  
...  
Keyword(s):  
Elliptic Curves ◽  
Class Numbers ◽  
Quadratic Twists ◽  
Torsion Subgroups

scholarly journals Elliptic Curves with Long Arithmetic Progressions Have Large Rank

2020 ◽  
Author(s):  
Natalia Garcia-Fritz ◽  
Hector Pasten
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Progression ◽  
Nevanlinna Theory ◽  
Uniform Boundedness ◽  
Rational Numbers ◽  
Rational Points ◽  
Arithmetic Progressions ◽  
Large Rank ◽  
Arithmetic Statistics

Abstract For any family of elliptic curves over the rational numbers with fixed $j$-invariant, we prove that the existence of a long sequence of rational points whose $x$-coordinates form a nontrivial arithmetic progression implies that the Mordell–Weil rank is large, and similarly for $y$-coordinates. We give applications related to uniform boundedness of ranks, conjectures by Bremner and Mohanty, and arithmetic statistics on elliptic curves. Our approach involves Nevanlinna theory as well as Rémond’s quantitative extension of results of Faltings.

scholarly journals Families of elliptic curves with prescribed torsion subgroups over dihedral quartic fields

2015 ◽  
Vol 147 ◽  
pp. 342-363 ◽  
Author(s):  
Daeyeol Jeon ◽  
Chang Heon Kim ◽  
Yoonjin Lee
Keyword(s):  
Elliptic Curves ◽  
Torsion Subgroups
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