The number of solutions of the Mordell equation
The orders of the reductions of a point in the Mordell-Weil group of an elliptic curve
The -part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large
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 .
The period-index problem in WC-groups IV: a local transition theorem
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...
The real field with the rational points of an elliptic curve
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.
The size of Selmer groups for the congruent number problem.
The size of Selmer groups for the congruent number problem, II.
The splitting of primes in division fields of elliptic curves.
The uniform primality conjecture for elliptic curves
The Way to the Proof of Fermat’s Last Theorem
The work of Gross and Zagier on Heegner points and the derivatives of -series
Théorie d'Iwasawa p-adique locale et globale.
Thetanullwerte: from periods to good equations.
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)].
Torsion and Tamagawa numbers
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...
Torsion des courbes elliptiques sur les corps cubiques
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.
Torsion groups of elliptic curves over quadratic fields
Torsion points on elliptic curves and galois module structure.
Torsion points on elliptic curves and q-coeffincients of modular forms.
Torsion subgroups of elliptic curves with non-cyclic torsion over ℚ in elementary abelian 2-extensions of ℚ
Travaux de Kolyvagin et Rubin