The number of solutions of the Mordell equation
Dimitrios Poulakis (1999)
Acta Arithmetica
J. Cheon, S. Hahn (1999)
Acta Arithmetica
Remke Kloosterman (2005)
Journal de Théorie des Nombres de Bordeaux
In this paper we show that for every prime the dimension of the -torsion in the Tate-Shafarevich group of can be arbitrarily large, where is an elliptic curve defined over a number field , with bounded by a constant depending only on . From this we deduce that the dimension of the -torsion in the Tate-Shafarevich group of can be arbitrarily large, where is an abelian variety, with bounded by a constant depending only on .
Pete L. Clark (2010)
Journal de Théorie des Nombres de Bordeaux
Let be a complete discretely valued field with perfect residue field . Assuming upper bounds on the relation between period and index for WC-groups over , we deduce corresponding upper bounds on the relation between period and index for WC-groups over . Up to a constant depending only on the dimension of the torsor, we recover theorems of Lichtenbaum and Milne in a “duality free” context. Our techniques include the use of LLR models of torsors under abelian varieties with good reduction and...
Ayhan Günaydın, Philipp Hieronymi (2011)
Fundamenta Mathematicae
We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets definable in that structure are semialgebraic.
D.R. Heath-Brown (1993)
Inventiones mathematicae
D.R. Heath-Brown (1994)
Inventiones mathematicae
Duke, W., Tóth, Á. (2002)
Experimental Mathematics
Graham Everest, Patrick Ingram, Valéry Mahé, Shaun Stevens (2008)
Acta Arithmetica
Gerhard Frey (2009)
Annales de la faculté des sciences de Toulouse Mathématiques
John Coates (1984/1985)
Séminaire Bourbaki
Bernadette Perrin-Riou (1990)
Inventiones mathematicae
Jordi Guàrdia (2007)
Publicacions Matemàtiques
We will show the utility of the classical Jacobi Thetanullwerte for the description of certain period lattices of elliptic curves, providing equations with good arithmetical properties. These equations will be the starting point for the construction of families of elliptic curves with everywhere good reduction.[Proceedings of the Primeras Jornadas de Teoría de Números (Vilanova i la Geltrú (Barcelona), 30 June - 2 July 2005)].
Dino Lorenzini (2011)
Annales de l’institut Fourier
Let be a number field, and let be an abelian variety. Let denote the product of the Tamagawa numbers of , and let denote the finite torsion subgroup of . The quotient is a factor appearing in the leading term of the -function of in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over or quadratic extensions , and for abelian surfaces . The smallest possible ratio...
Pierre Parent (2000)
Annales de l'institut Fourier
On donne la liste (à un élément près) des nombres premiers qui sont l’ordre d’un point de torsion d’une courbe elliptique sur un corps de nombres de degré trois.
Sheldon Kamienny, Filip Najman (2012)
Acta Arithmetica
A. Agboola (1996)
Inventiones mathematicae
S. Kamienny (1992)
Inventiones mathematicae
Yasutsugu Fujita (2004)
Acta Arithmetica
Bernadette Perrin-Riou (1989/1990)
Séminaire Bourbaki