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.
Funding source: Engineering and Physical Sciences Research Council
Award Identifier / Grant number: EP/P026710/1
Funding source: Austrian Science Fund
Award Identifier / Grant number: P 32428-N35
Funding statement: During the preparation of this article the first-named author was supported by EPSRC grant EP/P026710/1 and FWF grant P 32428-N35.
Acknowledgements
The authors were inspired to work on this problem following discussions at the AIM workshop “Rational and integral points on higher-dimensional varieties” in May, 2014. They would particularly like to thank David Harari and Olivier Wittenberg for their patient explanations of the issues involved with the geometric sieve for quadrics. The authors are also grateful to Julian Lyczak and Olivier Wittenberg for further helpful remarks. A number of useful comments were also made by the anonymous referee, whose input is gratefully acknowledged.
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