Torsion points on elliptic curves andq-coefficients of modular forms

1992 ◽  
Vol 109 (1) ◽  
pp. 221-229 ◽  
Author(s):  
S. Kamienny
Keyword(s):  
Elliptic Curves ◽  
Modular Forms ◽  
Torsion Points

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 Quasi-modular forms attached to elliptic curves, I

2012 ◽  
Vol 19 (2) ◽  
pp. 307-377 ◽  
Author(s):  
Hossein Movasati
Keyword(s):  
Elliptic Curves ◽  
Modular Forms

scholarly journals p-Torsion points on elliptic curves defined over quadratic fields

1984 ◽  
Vol 96 ◽  
pp. 139-165 ◽  
Author(s):  
Fumiyuki Momose
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Quadratic Fields ◽  
Finite Degree ◽  
Torsion Points

Let p be a prime number and k an algebraic number field of finite degree d. Manin [14] showed that there exists an integer n = n(k,p) (≧0) which satisfies the condition

scholarly journals Higher reciprocity law, modular forms of weight 1 and elliptic curves

1985 ◽  
Vol 98 ◽  
pp. 109-115 ◽  
Author(s):  
Masao Koike
Keyword(s):  
Elliptic Curves ◽  
Modular Forms ◽  
Cusp Forms ◽  
Close Connection ◽  
Irreducible Polynomials ◽  
Reciprocity Law

In this paper, we study higher reciprocity law of irreducible polynomials f(x) over Q of degree 3, especially, its close connection with elliptic curves rational over Q and cusp forms of weight 1. These topics were already studied separately in a special example by Chowla-Cowles [1] and Hiramatsu [2]. Here we bring these objects into unity.

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