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

Jianya Liu, Peter Sarnak

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)393-426
Number of pages34
JournalIsrael Journal of Mathematics
Volume178
Issue number1
DOIs
StatePublished - Dec 1 2010

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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