Integral points on quadrics in three variables whose coordinates have few prime factors (Q607841)

Given an integral ternary quadratic form \(f\) and a non-zero integer \(t\) such that the number of integral points on the variety \(V(\mathbb{Z}): f(x_1, x_2, x_3) = t\) is reasonably large, it is interesting to ask if there are solutions in \textit{prime} variables \(x_1, x_2, x_3\). Unless there are obvious reasons why not, this should be the case, but current technology fails to prove such results. However, sieve techniques can produce solutions in almost-primes. If \(f\) is positive definite, this has been established in [\textit{V. Blomer} and \textit{J. Brüdern}, Bull. Lond. Math. Soc. 37, No. 4, 507--513 (2005; Zbl 1084.11056)]. The present paper investigates the situation for indefinite \(f\) which for many reasons is very different from the definite case. If \(f\) is isotropic over \(\mathbb{Q}\), then a suitable subvariety can be parametrized in one variable, and one can produce almost-prime points by standard sieve methods. The present paper proves that if \(f\) is indefinite and anisotropic over \(\mathbb{Q}\) and if in addition \(\det(f) t\) is squarefree, then the there are infinitely many solutions to \(f(x_1, x_2, x_3) = t\) where \(x_1x_2x_3\) has at most 26 prime factors, provided there is at most one integer solution. The general theory of the affine linear sieve of \textit{J. Bourgain, A. Gamburd} and \textit{P. Sarnak} [Invent. Math. 179, No. 1, 559--644 (2010; Zbl 1239.11103)] guarantees almost prime solutions with an unspecified number of prime factors. Here this is worked out in detail and some care has been taken to optimize the constant 26: a weighted sieve is used and the Kim-Sarnak bound for the spectral gap on \(\text{GL}(2)\) which is almost as strong as Selberg's eigenvalue conjecture. The sieve input is an estimate for a (smooth) count of the number of points on \(V\) in a bounded region such that \(x_1x_2x_3\) is divisible by some squarefree integer \(d\). The variety \(V\) is decomposed into finitely many orbits of some subgroup \(\Gamma \subseteq \text{SO}_f(\mathbb{Z})\), where \(\Gamma\) can be realized as a co-compact subgroup of \(\text{SL}_2(\mathbb{Z})\). One is now left with a hyperbolic lattice point count with respect to some congruence subgroup of \(\Gamma\) (depending on \(d\)). Therefore it is crucial to have a spectral gap that is \textit{uniform} in the congruence subgroup.