Retour sur l'arithmétique des intersections de deux quadriques, avec un appendice par A. Kuznestov

Abstract

Lichtenbaum proved that index and period coincide for a curve of genus one over a $p$-adic field. Salberger proved that the Hasse principle holds for a smooth complete intersection of two quadrics $X\subset P^n$ over a number field, if it contains a conic and if $n\ge 5$. Building upon these two results, we extend recent results of Creutz and Viray (2021) on the existence of a quadratic point on intersections of two quadrics over $p$-adic fields and number fields. We then recover Heath-Brown's theorem (2018) that the Hasse principle holds for smooth complete intersections of two quadrics in $P^7$. We also give an alternate proof of a theorem of Iyer and Parimala (2022) on the local-global principle in the case $n=5$.

arXiv commentPaper in French, appendix by A. Kuznetsov in English. Version v2 takes into account the remarks of a referee. Final version, to appear in Journal für die reine und angewandte Mathematik (Crelle)

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Cited by in corpus (2)

• A threefold violating a local-to-global principle for rationality
• Quartic del Pezzo surfaces without quadratic points