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The size of the primes obstructing the existence of rational points

Published online by Cambridge University Press:  15 May 2020

E. Sofos*
Affiliation:
The Mathematics and Statistics Building, University of Glasgow, University Place, United Kingdom, G12 8QQ, Scotland ([email protected])

Abstract

The sequence of prime numbers p for which a variety over ℚ has no p-adic point plays a fundamental role in arithmetic geometry. This sequence is deterministic, however, we prove that if we choose a typical variety from a family then the sequence has random behaviour. We furthermore prove that this behaviour is modelled by a random walk in Brownian motion. This has several consequences, one of them being the description of the finer properties of the distribution of the primes in this sequence via the Feynman–Kac formula.

Type
Research Article
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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