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
AN - SCOPUS:78650245674
SN - 0021-2172
VL - 178
SP - 393
EP - 426
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -