Simplicity of twists of abelian varieties
We give some easy necessary and sufficient criteria for twists of abelian varieties by Artin representations to be simple.
We give some easy necessary and sufficient criteria for twists of abelian varieties by Artin representations to be simple.
We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves over a prime finite field of elements, such that the discriminant of the quadratic number field containing the endomorphism ring of over is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).
Let be an elliptic curve defined over , the finite field of elements. We show that for some constant depending only on , there are infinitely many positive integers such that the exponent of , the group of -rational points on , is at most . This is an analogue of a result of R. Schoof on the exponent of the group of -rational points, when a fixed elliptic curve is defined over and the prime tends to infinity.
Let be a finite extension of a global field. Such an extension can be generated over by a single element. The aim of this article is to prove the existence of a ”small” generator in the function field case. This answers the function field version of a question of Ruppert on small generators of number fields.
Let be an algebraic subvariety of a torus and denote by the complement in of the Zariski closure of the set of torsion points of . By a theorem of Zhang, is discrete for the metric induced by the normalized height . We describe some quantitative versions of this result, close to the conjectural bounds, and we discuss some applications to study of the class group of some number fields.
Let K be any quadratic field with its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over ℚ, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r+s+t = rst = 1 in . This Diophantine equation gives an elliptic curve defined over ℚ with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields...
Let k ∈ ℤ be such that is finite, where . We complete the determination of all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.
This article concerns the problem of solving diophantine equations in rational numbers. It traces the way in which the 19th century broke from the centuries-old tradition of the purely algebraic treatment of this problem. Special attention is paid to Sylvester’s work “On Certain Ternary Cubic-Form Equations” (1879–1880), in which the algebraico-geometrical approach was applied to the study of an indeterminate equation of third degree.
We perform descent calculations for the families of elliptic curves over with a rational point of order or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over may become arbitrarily large.