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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 “class group” obstruction for the equation C y d = F ( x , z )

Denis Simon (2008)

Journal de Théorie des Nombres de Bordeaux

In this paper, we study equations of the form C y d = F ( x , z ) , where F [ x , z ] is a binary form, homogeneous of degree n , which is supposed to be primitive and irreducible, and d is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result...

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