[Paper Review] Zeros of Pairs of Quadratic Forms

D. R. Heath‐Brown|arXiv (Cornell University)|Apr 14, 2013

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper establishes the Hasse principle and weak approximation for nonsingular intersections of two quadratic forms in eight variables over any number field. Using local-global methods and a novel local result on hyperbolic planes in pencils of forms, the authors resolve a key obstruction left open by prior work, completing the classification of such varieties for n ≥ 6.

ABSTRACT

We prove the Hasse principle and weak approximation for varieties defined over number fields by the nonsingular intersection of pairs of quadratic forms in 8 variables. The argument develops work of Colliot-Thelene, Sansuc and Swinnerton-Dyer, and centres on a purely local problem about forms which split off 3 hyperbolic planes.

Motivation & Objective

• To establish the Hasse principle and weak approximation for nonsingular intersections of two quadratic forms in eight variables over a number field.
• To resolve a critical local obstruction in the method of Colliot-Thélène, Sansuc, and Swinnerton-Dyer for the 8-variable case.
• To extend the local-to-global principle from n ≥ 9 to n = 8, confirming a conjecture for this case.
• To demonstrate that the existence of a non-trivial common zero over all completions implies a global solution, under nonsingularity.
• To show that weak approximation holds automatically for such varieties via a result from Colliot-Thélène et al.

Proposed method

• Prove a key local theorem (Theorem 2) stating that over a non-archimedean completion with residue field size ≥ 32, a nonsingular common zero implies a form in the pencil contains at least three hyperbolic planes.
• Use the local result to reduce the global problem to a function field setting via base change to a suitable extension field.
• Construct a degree-q extension k' of k where all prime ideals have norm ≥ 32, ensuring applicability of Theorem 2.
• Apply Springer's theorem on forms over function fields to deduce the existence of a rational point over k(T) from a point over k'(T).
• Invoke the Amer–Brumer Theorem to lift a rational point over k(T) to a simultaneous zero over k.
• Leverage Theorem 3.11 of Colliot-Thélène, Sansuc, and Swinnerton-Dyer to conclude weak approximation holds.

Experimental results

Research questions

• RQ1 Does the Hasse principle hold for nonsingular intersections of two quadratic forms in eight variables over a number field?
• RQ2 Can the local obstruction preventing the extension of the n ≥ 9 results to n = 8 be resolved?
• RQ3 Is weak approximation valid for such intersections when the variety is nonsingular?
• RQ4 Under what conditions does a form in the pencil of two quadratic forms contain multiple hyperbolic planes?
• RQ5 Can the global existence of rational points be deduced from local solubility and a refined local analysis?

Key findings

• The Hasse principle holds for nonsingular intersections of two quadratic forms in eight variables over any number field.
• Weak approximation holds for such varieties, as a consequence of a result by Colliot-Thélène, Sansuc, and Swinnerton-Dyer.
• A key local result (Theorem 2) establishes that if a nonsingular common zero exists over a non-archimedean completion with residue field size ≥ 32, then some form in the pencil contains at least three hyperbolic planes.
• The global result is deduced by base change to a suitable extension field where the local condition is satisfied, followed by application of Springer's theorem and the Amer–Brumer Theorem.
• The method fails for singular intersections, indicating the necessity of the nonsingularity assumption.
• The proof shows that the existence of a rational point over all completions implies a global point, even when the number of variables is reduced to eight.