Representations of integers by systems of three quadratic forms

2015/09/30 by Lillian B. Pierce, Damaris Schindler, Melanie Matchett Wood · 3 citations
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Limits and Structures in Graph Theory #math.NT

paper · pdf · doi:10.1112/plms/pdw027

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

arxiv created 2017/07/01 · arxiv updated 2017/07/04

Abstract

It is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers (n₁,…,n_R) by a system of quadratic forms Q₁,…, 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 ≥ 10 for "almost all" tuples, under appropriate nonsingularity assumptions on the forms Q₁,Q₂,Q₃. 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.

Citations

• On the number of points on a complete intersection over a finite field

Cited by

• Sums of three cubes over a function field
• Rational points on complete intersections over \mathbbF_q(t)
• Selmer groups of families of elliptic curves with an ℓ-isogeny

Related

• A Version of the Circle Method for the Representation of Integers by Quadratic Forms
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• On the Number of Representations of Integers by various Quadratic and Higher Forms
• Representation of integers by a family of cubic forms in seven variables II
• Solving quadratic forms in restricted variables with the circle method