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The size of the primes obstructing the existence of rational points

Part of: Markov processes Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry Limit theorems Families, fibrations

Published online by Cambridge University Press:  15 May 2020

E. Sofos Open the ORCID record for E. Sofos [Opens in a new window]
E. Sofos*
Affiliation:
The Mathematics and Statistics Building, University of Glasgow, University Place, United Kingdom, G12 8QQ, Scotland ([email protected])
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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.

Keywords

Rational pointsBrownian motion

MSC classification

Primary: 14G05: Rational points
Secondary: 60J65: Brownian motion 11G25: Varieties over finite and local fields 14D10: Arithmetic ground fields (finite, local, global) 60F05: Central limit and other weak theorems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 2 , April 2021 , pp. 652 - 704
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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