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A Bogomolov property for curves modulo algebraic subgroups

Philipp Habegger (2009)

Bulletin de la Société Mathématique de France

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 2 . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.

A dynamical Shafarevich theorem for twists of rational morphisms

Brian Justin Stout (2014)

Acta Arithmetica

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.

Algebraic leaves of algebraic foliations over number fields

Jean-Benoît Bost (2001)

Publications Mathématiques de l'IHÉS

We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field K embedded in C , a smooth algebraic variety X over K , equipped with a K - rational point P , and F an algebraic subbundle of the its tangent bundle T X , defined over K . Assume moreover that the vector bundle F is involutive, i.e., closed under Lie bracket. Then it defines an holomorphic foliation of the analytic manifold X ( C ) , and one may consider its leaf F through P . We prove...

An annihilator for the p -Selmer group by means of Heegner points

Massimo Bertolini (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let E / Q be a modular elliptic curve, and let K be an imaginary quadratic field. We show that the p -Selmer group of E over certain finite anticyclotomic extensions of K , 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 Z p -extension of K . This refines in the current contest a result of [1].

An elliptic surface of Mordell-Weil rank 8 over the rational numbers

Charles F. Schwartz (1994)

Journal de théorie des nombres de Bordeaux

Néron showed that an elliptic surface with rank 8 , and with base B = P 1 , and geometric genus = 0 , may be obtained by blowing up 9 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 9 points. Manin also describes bases of the Mordell-Weil groups of these elliptic surfaces, in terms of the 9 points ; we observe that, relative to the Weierstrass form of the equation, Y 2 = X 3 + A X 2 + B X + C (with deg ( A ) 2 , deg ( B ) 4 , and deg ( C ) 6 ) a basis ( X 1 , Y 1 ) , , ( X 8 , Y 8 ) can be found...

An explicit algebraic family of genus-one curves violating the Hasse principle

Bjorn Poonen (2001)

Journal de théorie des nombres de Bordeaux

We prove that for any t 𝐐 , the curve 5 x 3 + 9 y 3 + 10 z 3 + 12 t 2 + 82 t 2 + 22 3 ( x + y + z ) 3 = 0 in 𝐏 2 is a genus 1 curve violating the Hasse principle. An explicit Weierstrass model for its jacobian E t is given. The Shafarevich-Tate group of each E t contains a subgroup isomorphic to 𝐙 / 3 × 𝐙 / 3 .

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