Retour sur l'arithm\'etique des intersections de deux quadriques, avec un appendice par A. Kuznestov
| Autor: | Colliot-Thélène, Jean-Louis |
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| Jazyk: | francouzština |
| Rok vydání: | 2022 |
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| Druh dokumentu: | Working Paper |
| DOI: | 10.1515/crelle-2023-0081 |
| Popis: | 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 $n\geq 5$. 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 $n=5$. Comment: Paper in French, appendix by A. Kuznetsov in English. Version v2 takes into account the remarks of a referee. Final version, to appear in Journal f\"ur die reine und angewandte Mathematik (Crelle) |
| Databáze: | arXiv |
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