Quadratic Points on Intersections of Two Quadrics over Local Fields

Abstract

By a recent result of Creutz and Viray, a smooth complete intersection of two quadrics in 4-dimensional projective space over a nonarchimedean local field $k$ always has a quadratic point. We extend this result by showing that such an intersection of quadrics must in fact obtain points over all quadratic extensions of $k$ except possibly one. Our approach uses the Theorems of Springer and Amer-Brumer, as well as Tian's theory of semistable models for intersections of two quadrics.