scholarly journals Torsion points on elliptic curves over all quadratic fields. II

1986 ◽  
Vol 79 ◽  
pp. 119-122 ◽  
Author(s):  
Sheldon Kamienny
Keyword(s):  
Elliptic Curves ◽  
Quadratic Fields ◽  
Torsion Points

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

KUMMER GENERATORS AND TORSION POINTS OF ELLIPTIC CURVES WITH BAD REDUCTION AT SOME PRIMES

2013 ◽  
Vol 09 (07) ◽  
pp. 1743-1752 ◽  
Author(s):  
MASAYA YASUDA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Quadratic Fields ◽  
Torsion Subgroup ◽  
Torsion Point ◽  
Torsion Points ◽  
Root Of Unity ◽  
Fundamental Units

For a prime p, let ζp denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field K with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over K(ζp), and in this paper, we study the ramification of such Kummer extensions using the Kummer generators directly computed by Verdure in 2006. For quadratic fields K, we also give unramified Kummer extensions over K(ζp) generated from elliptic curves over K having a p-torsion point with bad reduction at certain primes. Many of these unramified Kummer extensions have not appeared in the previous work using fundamental units of quadratic fields.

scholarly journals Torsion points on elliptic curves defined over quadratic fields

1988 ◽  
Vol 109 ◽  
pp. 125-149 ◽  
Author(s):  
M. A. Kenku ◽  
F. Momose
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Prime Number ◽  
Prime Power ◽  
Quadratic Field ◽  
Rational Points ◽  
Negative Integer ◽  
Quadratic Fields ◽  
Prime Power Order ◽  
Torsion Points

Let k be a quadratic field and E an elliptic curve defined over k. The authors [8, 12, 13] [23] discussed the k-rational points on E of prime power order. For a prime number p, let n = n(k, p) be the least non negative integer such thatfor all elliptic curves E defined over a quadratic field k ([15]).

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.

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