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 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 Hyperelliptic modular curves and isogenies of elliptic curves over quadratic fields

2015 ◽  
Vol 18 (1) ◽  
pp. 578-602 ◽  
Author(s):  
Peter Bruin ◽  
Filip Najman
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Quadratic Field ◽  
Quadratic Extension ◽  
Quadratic Fields ◽  
Modular Curves ◽  
Modular Curve

We study elliptic curves over quadratic fields with isogenies of certain degrees. Let $n$ be a positive integer such that the modular curve $X_{0}(n)$ is hyperelliptic of genus ${\geqslant}2$ and such that its Jacobian has rank $0$ over $\mathbb{Q}$. We determine all points of $X_{0}(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to $\overline{\mathbb{Q}}$-isomorphism, every elliptic curve over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for some $d\mid n$, to the twist of its Galois conjugate by a quadratic extension $L$ of $K$. We determine $d$ and $L$ explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to $\overline{\mathbb{Q}}$-isomorphism, all elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb{Q}$-curves.

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 Galois groups of number fields generated by torsion points of elliptic curves

1986 ◽  
Vol 104 ◽  
pp. 43-53 ◽  
Author(s):  
Kay Wingberg
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Group ◽  
Number Field ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Number Fields ◽  
Galois Groups ◽  
Torsion Points ◽  
Special Case

Coates and Wiles [1] and B. Perrin-Riou (see [2]) study the arithmetic of an elliptic curve E defined over a number field F with complex multiplication by an imaginary quadratic field K by using p-adic techniques, which combine the classical descent of Mordell and Weil with ideas of Iwasawa’s theory of Zp-extensions of number fields. In a special case they consider a non-cyclotomic Zp-extension F∞ defined via torsion points of E and a certain Iwasawa module attached to E/F, which can be interpreted as an abelian Galois group of an extension of F∞. We are interested in the corresponding non-abelian Galois group and we want to show that the whole situation is quite analogous to the case of the cyclotomic Zp-extension (which is generated by torsion points of Gm).

scholarly journals A note on quadratic fields in which a fixed prime number splits completely

1985 ◽  
Vol 99 ◽  
pp. 63-71 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Elliptic Curve ◽  
Natural Number ◽  
Prime Number ◽  
Endomorphism Ring ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields

Throughout this note, p denotes a fixed prime number and f denotes a fixed natural number prime to p.It is easy to see and more or less known that for any natural number n, there exists an elliptic curve over p whose j-invariant is of degree n over Fp and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field. In this note, we consider a more precise problem: for any natural number n, decide whether or not there exists an elliptic curve over p whose j-invariant is of degree n over Fp and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field with conductor f.

scholarly journals Elliptic Curves over the Perfect Closure of a Function Field

2010 ◽  
Vol 53 (1) ◽  
pp. 87-94
Author(s):  
Dragos Ghioca
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Positive Characteristic ◽  
Rational Points ◽  
Finitely Generated

AbstractWe prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

scholarly journals p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

2015 ◽  
Vol 219 ◽  
pp. 269-302
Author(s):  
Kenichi Bannai ◽  
Hidekazu Furusho ◽  
Shinichi Kobayashi
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Theta Function ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Good Reduction ◽  
Ring Of Integers

AbstractConsider an elliptic curve defined over an imaginary quadratic fieldKwith good reduction at the primes abovep≥ 5 and with complex multiplication by the full ring of integersof K. In this paper, we constructp-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then provep-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

Number of rational points of elliptic curves

2021 ◽  
pp. 2250017
Author(s):  
Viliam Ďuriš ◽  
Timotej Šumný
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Current Knowledge ◽  
American Mathematical Society ◽  
Rational Points ◽  
Modern Theory ◽  
Cambridge University ◽  
Weil Theorem ◽  
A Finite Group

In the modern theory of elliptic curves, one of the important problems is the determination of the number of rational points on an elliptic curve. The Mordel–Weil theorem [T. Shioda, On the Mordell–Weil lattices, Comment. Math. University St. Paul. 39(2) (1990) 211–240] points out that the elliptic curve defined above the rational points is generated by a finite group. Despite the knowledge that an elliptic curve has a final number of rational points, it is still difficult to determine their number and the way how to determine them. The greatest progress was achieved by Birch and Swinnerton–Dyer conjecture, which was included in the Millennium Prize Problems [A. Wiles, The Birch and Swinnerton–Dyer conjecture, The Millennium Prize Problems (American Mathematical Society, 2006), pp. 31–44]. This conjecture uses methods of the analytical theory of numbers, while the current knowledge corresponds to the assumptions of the conjecture but has not been proven to date. In this paper, we focus on using a tangent line and the osculating circle for characterizing the rational points of the elliptical curve, which is the greatest benefit of the contribution. We use a different view of elliptic curves by using Minkowki’s theory of number geometry [H. F. Blichfeldt, A new principle in the geometry of numbers, with some applications, Trans. Amer. Math. Soc. 15(3) (1914) 227–235; V. S. Miller, Use of elliptic curves in cryptography, in Proc. Advances in Cryptology — CRYPTO ’85, Lecture Notes in Computer Science, Vol. 218 (Springer, Berlin, Heidelberg, 1985), pp. 417–426; E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Vol. 670, 1st edn. (Cambridge University Press, 2007)].

Signed-Selmer Groups over the ℤ2p-extension of an Imaginary Quadratic Field

2014 ◽  
Vol 66 (4) ◽  
pp. 826-843 ◽  
Author(s):  
Byoung Du (B. D.) Kim
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Selmer Groups

AbstractLet E be an elliptic curve over ℚ that has good supersingular reduction at p > 3. We construct what we call the ±/±-Selmer groups of E over the ℤ2p-extension of an imaginary quadratic field K when the prime p splits completely over K/ℚ, and prove that they enjoy a property analogous to Mazur's control theorem.Furthermore, we propose a conjectural connection between the±/±-Selmer groups and Loeffler's two-variable ±/±-p-adic L-functions of elliptic curves.

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