Point générique et saut du rang du groupe de Mordell-Weil

2019/06/16 by Colliot-Thélène, Jean-Louis · 1 citation
#14D10 #14G05 #14G25 #14K15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1906.06670

Abstract

Let k be a number field and U a smooth integral k-variety. Let X → U be an abelian scheme. We consider the set R of rational points m ∈ U(k) such that the Mordell-Weil rank of the fibre U_m is strictly bigger than the Mordell-Weil rank of the generic fibre. We prove the following results. If the k-variety X is k-unirational, then R is dense for the Zariski topology on U. If X is k-rational, then R is not thin in U. This generalizes results of Billard and of Salgado. The main idea goes back to Néron's thesis: use the generic point of the generic fibre of the family.

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