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

2013 ◽  
Vol 357 (1) ◽  
pp. 279-305 ◽  
Author(s):  
Álvaro Lozano-Robledo
Keyword(s):  
Elliptic Curves ◽  
Torsion Points ◽  
Field Of Definition ◽  
Definition Of

scholarly journals On fields of definition of torsion points of elliptic curves with complex multiplication

2010 ◽  
Vol 139 (6) ◽  
pp. 1961-1969 ◽  
Author(s):  
Luis Dieulefait ◽  
Enrique González-Jiménez ◽  
Jorge Jiménez Urroz
Keyword(s):  
Elliptic Curves ◽  
Complex Multiplication ◽  
Torsion Points ◽  
Definition Of

SELF-POINTS ON ELLIPTIC CURVES OF PRIME CONDUCTOR

2009 ◽  
Vol 05 (05) ◽  
pp. 911-932
Author(s):  
CHRISTOPHE DELAUNAY ◽  
CHRISTIAN WUTHRICH
Keyword(s):  
Lower Bound ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Cyclic Subgroup ◽  
Infinite Order ◽  
Field Of Definition ◽  
Modular Parametrization ◽  
Weil Group ◽  
Definition Of

Let E be an elliptic curve of conductor p. Given a cyclic subgroup C of order p in E[p], we construct a modular point PC on E, called self-point, as the image of (E,C) on X0(p) under the modular parametrization X0(p) → E. We prove that the point is of infinite order in the Mordell–Weil group of E over the field of definition of C. One can deduce a lower bound on the growth of the rank of the Mordell–Weil group in its PGL 2(ℤp)-tower inside ℚ(E[p∞]).

scholarly journals Mazur’s rational torsion result for pointless genus one curves: examples

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Arjan Dwarshuis ◽  
Majken Roelfszema ◽  
Jaap Top
Keyword(s):  
Algebraic Geometry ◽  
Elliptic Curves ◽  
Rational Point ◽  
Torsion Points ◽  
Arbitrary Genus ◽  
Mazur’S Theorem ◽  
Genus One

AbstractThis note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

scholarly journals Torsion points of elliptic curves with good reduction

2008 ◽  
Vol 31 (3) ◽  
pp. 385-403
Author(s):  
Masaya Yasuda
Keyword(s):  
Elliptic Curves ◽  
Good Reduction ◽  
Torsion Points

scholarly journals The Walker Abel–Jacobi map descends

2021 ◽  
Author(s):  
Jeffrey D. Achter ◽  
Sebastian Casalaina-Martin ◽  
Charles Vial
Keyword(s):  
Abelian Variety ◽  
Projective Manifold ◽  
Field Of Definition ◽  
Definition Of ◽  
Complex Projective ◽  
Intermediate Jacobian ◽  
Cycle Classes ◽  
Coniveau Filtration

AbstractFor a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths’ Abel–Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths intermediate Jacobian. Recently Suzuki gave an alternate, Hodge-theoretic, construction of this Walker Abel–Jacobi map. We provide a third construction based on a general lifting property for surjective regular homomorphisms, and prove that the Walker Abel–Jacobi map descends canonically to any field of definition of the complex projective manifold. In addition, we determine the image of the l-adic Bloch map restricted to algebraically trivial cycle classes in terms of the coniveau filtration.

scholarly journals Fields of definition of elliptic curves with prescribed torsion

2017 ◽  
Vol 181 (1) ◽  
pp. 85-95 ◽  
Author(s):  
Peter Bruin ◽  
Filip Najman
Keyword(s):  
Elliptic Curves ◽  
Definition Of
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