Strong approximation for the intersection of two quadrics

TL;DR

The paper addresses strong approximation for the intersection of two quadrics in projective space, focusing on the rank-four fibration scenario over number fields. It develops a generalized fibration lemma and employs open descent to transfer fiberwise approximation to global strong approximation, applied to punctured affine cones over intersections of two quadrics. The main result shows that for a pure geometrically integral intersection $V\subset \mathbb P^n_k$ with $n\ge 5$ not a cone, the punctured affine cone $\widetilde U$ satisfies strong approximation off a place $v_0$, with the Brauer-Manin obstruction behaving differently depending on $n$ (trivial for $n\ge 6$, finite for $n=5$; SA with BA obstruction for $n=5,6$). The work leverages descent theory, the fibration method, and recent frameworks (HWW22) to extend strong approximation results to higher-rank quadric intersections and yield implications for quadric surface bundles.

Abstract

We study strong approximation for the intersection of two affine quadrics. As its application, we prove the fibration method for weak approximation over number fields of rank four with nonsplit fibers split by quadratic extensions.

Theorems & Definitions (28)

• Theorem 1.2: Har94HWBS19HWW22
• Theorem 1.3: HWbrowningmatthiesen
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Corollary 1.7
• Lemma 2.1
• proof
• Lemma 2.2
• proof