Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.
Let K denote a number field, S a finite set of places of K, and ϕ: ℙⁿ → ℙⁿ a rational morphism defined over K. The main result of this paper states that there are only finitely many twists of ϕ defined over K which have good reduction at all places outside S. This answers a question of Silverman in the affirmative.
We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field embedded in , a smooth algebraic variety over , equipped with a rational point , and an algebraic subbundle of the its tangent bundle , defined over . Assume moreover that the vector bundle is involutive, i.e., closed under Lie bracket. Then it defines an holomorphic foliation of the analytic manifold , and one may consider its leaf through . We prove...
Let be a modular elliptic curve, and let be an imaginary quadratic field. We show that the -Selmer group of over certain finite anticyclotomic extensions of , modulo the universal norms, is annihilated by the «characteristic ideal» of the universal norms modulo the Heegner points. We also extend this result to the anticyclotomic -extension of . This refines in the current contest a result of [1].
Néron showed that an elliptic surface with rank , and with base , and geometric genus , may be obtained by blowing up points in the plane. In this paper, we obtain parameterizations of the coefficients of the Weierstrass equations of such elliptic surfaces, in terms of the points. Manin also describes bases of the Mordell-Weil groups of these elliptic surfaces, in terms of the points ; we observe that, relative to the Weierstrass form of the equation,(with , and a basis can be found...
We prove that for any , the curvein is a genus curve violating the Hasse principle. An explicit Weierstrass model for its jacobian is given. The Shafarevich-Tate group of each contains a subgroup isomorphic to .