Anisotropic quadratic equations in three variables

TL;DR

The work studies almost-prime representations of integral indefinite anisotropic quadratic forms in three variables by combining algebraic theory of quadratic forms, harmonic analysis, Jacquet–Langlands theory, and combinatorial sieves. It proves that, assuming $td(f)$ is square-free, if there exists an integral solution to $f({\bf x})=t$ there are infinitely many solutions with $x_1$ belonging to $P_6(B)$ (improvable to $P_5(B)$ under Selberg) and with $x_1x_2$ in $P_{16}(B)$ (improvable to $P_{14}(B)$ under Selberg), with strong Zariski-density conclusions. The core technical device is an equidistribution result for orbits modulo $d$ in the anisotropic setting, together with weighted sieve methods (linear and nonlinear) to convert average distribution into almost-prime conclusions; spectral gaps from Selberg’s conjecture (or the Kim–Sarnak bound) play a crucial role. The results extend previous work on almost-primes on quadrics and connect deep arithmetic geometry with analytic sieve techniques, yielding explicit density statements for solutions on the corresponding quadric. The methods have potential implications for Diophantine problems on quadratic forms and related orbits under spin groups.

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.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• Lemma 3.2