Simultaneous Integer Values of Pairs of Quadratic Forms

TL;DR

This paper proves that pairs of integral quadratic forms in five or more variables can simultaneously represent almost all integer pairs meeting local conditions, including prime values, using advanced circle method techniques.

Contribution

It introduces a two-dimensional Kloosterman refinement within the circle method to analyze simultaneous representations by quadratic forms.

Findings

Pairs of quadratic forms in 5+ variables represent almost all local-satisfying integer pairs.

Forms can attain prime values under local conditions.

The method advances the circle method with a novel two-dimensional Kloosterman refinement.

Abstract

We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.