The geometric sieve for quadrics

Author:

Browning Tim D.1ORCID,Heath-Brown Roger2ORCID

Affiliation:

1. IST Austria , Am Campus 1, 3400 Klosterneuburg , Austria

2. Mathematical Institute , University of Oxford , Woodstock Road , Oxford , OX2 6GG , United Kingdom

Abstract

Abstract We develop a version of Ekedahl’s geometric sieve for integral quadratic forms of rank at least five. As one ranges over the zeros of such quadratic forms, we use the sieve to compute the density of coprime values of polynomials, and furthermore, to address a question about local solubility in families of varieties parameterised by the zeros.

Funder

Engineering and Physical Sciences Research Council

Austrian Science Fund

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Reference18 articles.

1. M. Bhargava, The geometric sieve and the density of squarefree values of invariant polynomials, preprint (2014), https://arxiv.org/abs/1402.0031.

2. M. Bhargava, A. Shankar and X. Wang, Squarefree values of polynomial discriminants I, preprint (2016), https://arxiv.org/abs/1611.09806; to appear in Invent. Math.

3. B. J. Birch, Forms in many variables, Proc. Roy. Soc. Lond. Ser. A 265 (1961/62), 245–263.

4. M. J. Bright, T. D. Browning and D. Loughran, Failures of weak approximation in families, Compos. Math. 152 (2016), no. 7, 1435–1475.

5. T. Browning and D. Loughran, Sieving rational points on varieties, Trans. Amer. Math. Soc. 371 (2019), no. 8, 5757–5785.

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