The Mordell-Weil Theorem

1990 ◽  
pp. 319-338
Author(s):  
Kenneth Ireland ◽  
Michael Rosen
Keyword(s):  
Weil Theorem

scholarly journals There are genus one curves of every index over every infinite, finitely generated field

2019 ◽  
Vol 2019 (749) ◽  
pp. 65-86
Author(s):  
Pete L. Clark ◽  
Allan Lacy
Keyword(s):  
Elliptic Curve ◽  
Abelian Variety ◽  
Finitely Generated ◽  
Weil Theorem ◽  
Genus One

Abstract We show that a nontrivial abelian variety over a Hilbertian field in which the weak Mordell–Weil theorem holds admits infinitely many torsors with period any given n>1 that is not divisible by the characteristic. The corresponding statement with “period” replaced by “index” is plausible but open, and it seems much more challenging. We show that for every infinite, finitely generated field K, there is an elliptic curve E_{/K} which admits infinitely many torsors with index any given n>1 .

A Freudenthal–Weil theorem for pro-Lie groups

2015 ◽  
Vol 22 (4) ◽  
pp. 546-549 ◽  
Author(s):  
A. I. Shtern
Keyword(s):  
Lie Groups ◽  
Weil Theorem

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)].

scholarly journals A Mordell–Weil theorem for cubic hypersurfaces of high dimension

2017 ◽  
Vol 11 (8) ◽  
pp. 1953-1965 ◽  
Author(s):  
Stefanos Papanikolopoulos ◽  
Samir Siksek
Keyword(s):  
High Dimension ◽  
Weil Theorem ◽  
Cubic Hypersurfaces

scholarly journals Beyond Lebesgue and Baire IV: Density topologies and a converse Steinhaus–Weil theorem

2018 ◽  
Vol 239 ◽  
pp. 274-292 ◽  
Author(s):  
N.H. Bingham ◽  
A.J. Ostaszewski
Keyword(s):  
Weil Theorem

scholarly journals Chern-Weil theorem, Lovelock Lagrangians in critical dimensions, and boundary terms in gravity actions

2018 ◽  
Vol 98 (4) ◽  
Author(s):  
Nathalie Deruelle ◽  
Nelson Merino ◽  
Rodrigo Olea
Keyword(s):  
Critical Dimensions ◽  
Boundary Terms ◽  
Weil Theorem

Freudenthal–Weil theorem for pro-Lie groups

2016 ◽  
Vol 23 (1) ◽  
pp. 115-117 ◽  
Author(s):  
A. I. Shtern
Keyword(s):  
Lie Groups ◽  
Weil Theorem
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