PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

On local-global divisibility over ${\rm GL}_2$-type varieties

Volume 193 / 2020

Florence Gillibert, Gabriele Ranieri Acta Arithmetica 193 (2020), 339-354 MSC: Primary 11R34; Secondary 11G10. DOI: 10.4064/aa180404-15-3 Published online: 14 February 2020

Abstract

Let $k$ be a number field and let $\mathcal A $ be a ${\rm GL}_2$-type variety of dimension $d$ defined over $k$. We show that for every prime number $p$ satisfying certain conditions (see Theorem 2), if local-global divisibility by a power of $p$ does not hold for $\mathcal A $ over $k$, then there exists a cyclic extension $\widetilde {k}$ of $k$ of degree bounded by a constant depending on $d$ such that $\mathcal A $ is $\widetilde {k}$-isogenous to a ${\rm GL}_2$-type variety defined over $\widetilde {k}$ that admits a $\widetilde {k}$-rational point of order $p$.

Moreover, we explain how our result is related to a question of Cassels on the divisibility of the Tate–Shafarevich group, studied by Çiperiani and Stix and by Creutz.

Authors

  • Florence GillibertInstituto de Matemáticas
    Pontificia Universidad Católica de Valparaíso
    Blanco Viel 596
    2340000 Valparaíso, Chile
    e-mail
  • Gabriele RanieriInstituto de Matemáticas
    Pontificia Universidad Católica de Valparaíso
    Blanco Viel 596
    2340000 Valparaíso, Chile
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image