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

Volume 189 / 2019

Marc Hindry, Cecília Salgado Acta Arithmetica 189 (2019), 263-282 MSC: Primary 11G05, 11G10, 11G30, 14D10; Secondary 14H40, 14K15. DOI: 10.4064/aa180411-4-7 Published online: 17 May 2019

Abstract

We consider the following question: given a family $\mathcal{A}$ 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 $\mathcal{A}_t(k)$ 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 $\mathcal{A}(k(B))$. 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.

Authors

  • Marc HindryInstitut de Mathématiques Jussieu
    – Paris Rive Gauche (IMG-PRG)
    Current address:
    UFR de Mathématiques
    Bâtiment Sophie Germain
    Université Paris Diderot
    75013 Paris, France
    e-mail
  • Cecília SalgadoUniversidade Federal do Rio de Janeiro (UFRJ)
    Current address:
    Instituto de Matemática
    Bloco C do CT
    Cidade Universitária
    Ilha do Fundăo
    CEP 21941-909, Rio de Janeiro, RJ, Brasil
    e-mail

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