### 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\$.