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We are studying the infinite family of elliptic curves associated with simplest cubic fields. If the rank of such curves is 1, we determine the whole structure of the Mordell-Weil group and find all integral points on the original model of the curve. Note however, that we are not able to find them on the Weierstrass model if the parameter is even. We have also obtained similar results for an infinite subfamily of curves of rank 2. To our knowledge, this is the first time that so much information...
We construct isotrivial and non-isotrivial elliptic curves over with an arbitrarily large set of separable integral points. As an application of this construction, we prove that there are isotrivial log-general type varieties over with a Zariski dense set of separable integral points. This provides a counterexample to a natural translation of the Lang-Vojta conjecture to the function field setting. We also show that our main result provides examples of elliptic curves with an explicit and arbitrarily...
We give a family of elliptic curves, depending on two nonzero rational parameters and , such that the following statement holds: let be an elliptic curve and let be its 3-torsion subgroup. This group verifies if and only if belongs to .Furthermore, we consider the problem of the local-global divisibility by 9 for points of elliptic curves. The number 9 is one of the few exceptional powers of primes, for which an answer to the local-global divisibility is unknown in the case of such...
Let p be a prime number, p ≠ 2,3 and Fp the finite field with p elements. An elliptic curve E over Fp is a projective nonsingular curve of genus 1 defined over Fp. Each one of these curves has an isomorphic model given by an (Weierstrass) equation E: y2 = x3 + Ax + B, A,B ∈ Fp with D = 4A3 + 27B2 ≠ 0. The j-invariant of E is defined by j(E) = 1728·4A3/D.The aim of this note is to establish some results concerning the cardinality of the group of points on elliptic curves over Fp with j-invariants...
The question of embedding fields into central simple algebras over a number field was the realm of class field theory. The subject of embedding orders contained in the ring of integers of maximal subfields of such an algebra into orders in that algebra is more nuanced. The first such result along those lines is an elegant result of Chevalley [6] which says that with the ratio of the number of isomorphism classes of maximal orders in into which the ring of integers of can be embedded...
We give necessary and sufficient conditions for an orthogonal group defined over a global field of characteristic to contain a maximal torus of a given type.
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