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On the arithmetic of 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 $n=5$.