Lower bounds for the rank of families of abelian varieties under base\n change

2017/12/15 by Marc Hindry, Cecília Salgado, Hindry, Marc +1
Arts and Humanities · Mathematics · Social Sciences · #11G 30 #11G05 #11G10 #14D10 #14H40 #14K15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #North African History and Literature #Number Theory (math.NT) #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.1712.05858

openalex publication_date 2017/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the following question : given a family over abelian varieties\n\A over a curve B defined over a number field k, how does the\nrank of the Mordell-Weil group of the fibres \A_t(k) vary? A\nspecialisation theorem of Silverman guarantees that, for almost all t in\nC(k), the rank of the fibre is at least the generic rank, that is the rank of\n\A(k(B)). When the base curve B is rational, we show, at least in\nmany cases and under some geometric conditions, that there are infinitely many\nfibres for which the rank is larger than the generic rank. This paper is a\nsequel to a paper of the second author where the case of elliptic surfaces is\ntreated.\n

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