Representations of integers by systems of three quadratic forms

Abstract

It is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers $(n_1,\dots ,n_R)$ by a system of quadratic forms $Q_1,\dots ,Q_R$ in $k$ variables, as long as $k$ is sufficiently large; reducing the required number of variables remains a significant open problem. In this work, we consider the case of 3 forms and improve on the classical result by reducing the number of required variables to $k\ge 10$ for "almost all" tuples, under appropriate nonsingularity assumptions on the forms $Q_1,Q_2,Q_3$. To accomplish this, we develop a three-dimensional analogue of Kloosterman's circle method, in particular capitalizing on geometric properties of appropriate systems of three quadratic forms.

64 pages, minor edits to exposition to agree with published version

References in corpus (2)

• Rational points on intersections of cubic and quadric hypersurfaces
• Quadratic forms and systems of forms in many variables

Cited by in corpus (3)

• Rational points on complete intersections over ${\mathbb{F}}_q(t)$
• Quadratic forms and systems of forms in many variables
• Mean values of arithmetic functions and application to sums of powers