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

TL;DR

This paper extends recent results on the existence of rational points on intersections of two quadrics over various fields, providing new proofs and generalizations of the Hasse principle and local-global principles in algebraic geometry.

Contribution

It generalizes previous theorems on quadratic points and the Hasse principle for intersections of two quadrics, including new proofs and broader conditions.

Findings

Extended results on quadratic points over p-adic and number fields.

Provided an alternative proof of Heath-Brown's theorem for P^7.

Generalized local-global principles for intersections of two quadrics.

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 $n\geq 5$. 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 $n=5$.