Anisotropic quadratic equations in three variables

TL;DR

This paper proves that for certain anisotropic quadratic equations in three variables, there are infinitely many solutions with bounded prime factors, using advanced algebraic and analytic methods, and improves bounds under Selberg's eigenvalue conjecture.

Contribution

It establishes the existence of infinitely many solutions with bounded prime factors for anisotropic quadratic forms, applying the latest bounds from spectral theory.

Findings

Infinitely many solutions with at most 6 prime factors in x_1.

Product x_1 x_2 has at most 16 prime factors.

Bounds can be improved to 5 and 14 under Selberg's conjecture.

Abstract

Let $f(x_1, x_2, x_3)$ be an indefinite anisotropic integral quadratic form with determinant $d(f)$, and $t$ a non-zero integer such that $d(f)t$ is square-free. It is proved in this paper that, as long as there is one integral solution to $f(x_1, x_2, x_3) = t$, there are infinitely many such solutions for which (i) $x_1$ has at most $6$ prime factors, and (ii) the product $x_1 x_2$ has at most $16$ prime factors. Various methods, such as algebraic theory of quadratic forms, harmonic analysis, Jacquet-Langlands theory, as well as combinatorics, interact here, and the above results come from applying the sharpest known bounds towards Selberg's eigenvalue conjecture. Assuming the latter the number $6$ or $16$ may be reduced to $5$ or $14$, respectively.