TY - JOUR

T1 - Integral points on quadrics in three variables whose coordinates have few prime factors

AU - Liu, Jianya

AU - Sarnak, Peter

N1 - Funding Information: ∗ Supported by the 973 Program, NSFC Grant #10531060. ∗∗ Supported by Oscar Veblen Fund (IAS) and an NSF Grant. Received February 3, 2008 and in revised form December 14, 2008

PY - 2010

Y1 - 2010

N2 - The main theorem states that if f(x1, x2, x3) is an indefinite anisotropic integral quadratic form with determinant d(f), and t a non-zero integer such that d(f)t is square-free, then as long as there is one integer solution to f(x1, x2, x3) = t there are infinitely many such solutions for which the product x1x2x3 has at most 26 prime factors. The proof relies on the affine linear sieve and in particular automorphic spectral methods to obtain a sharp level of distribution in the associated counting problem. The 26 comes from applying the sharpest known bounds towards Selberg's eigenvalue conjecture. Assuming the latter the number 26 may be reduced to 22.

AB - The main theorem states that if f(x1, x2, x3) is an indefinite anisotropic integral quadratic form with determinant d(f), and t a non-zero integer such that d(f)t is square-free, then as long as there is one integer solution to f(x1, x2, x3) = t there are infinitely many such solutions for which the product x1x2x3 has at most 26 prime factors. The proof relies on the affine linear sieve and in particular automorphic spectral methods to obtain a sharp level of distribution in the associated counting problem. The 26 comes from applying the sharpest known bounds towards Selberg's eigenvalue conjecture. Assuming the latter the number 26 may be reduced to 22.

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U2 - 10.1007/s11856-010-0069-y

DO - 10.1007/s11856-010-0069-y

M3 - Article

SN - 0021-2172

VL - 178

SP - 393

EP - 426

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -