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

Marc Hindry, Cecília Salgado

Research output: Contribution to journalArticleAcademicpeer-review

5 Citations (Scopus)

Abstract

We consider the following question: given a family A of abelian varieties over a curve B defined over a number field k, how does the rank of the Mordell–Weil group of the fibres At(k) vary? A specialisation theorem of Silverman guarantees that, for almost all t in B(k), the rank of the fibre is at least the generic rank, i.e. the rank of A(k(B)). When the base curve B is rational, we give geometric conditions which ensure that for infinitely many fibres the rank jumps up. Examining the case of Jacobian fibrations, we show that in certain cases we get infinitely many fibres where the rank jumps by at least two units.
Original languageEnglish
Pages (from-to)263-282
Number of pages20
JournalActa Arithmetica
Volume189
DOIs
Publication statusPublished - 2019
Externally publishedYes

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