The size of the primes obstructing the existence of rational points

TL;DR

This paper demonstrates that for typical varieties over b, the primes obstructing rational points behave randomly and can be modeled by Brownian motion, revealing new probabilistic insights into their distribution.

Contribution

It introduces a probabilistic model for the sequence of primes obstructing rational points, showing their behavior aligns with a Brownian motion model.

Findings

Primes obstructing rational points exhibit random behavior for typical varieties.

The sequence's distribution is modeled by a Brownian motion process.

Feynman-Kac formula describes finer properties of the prime distribution.

Abstract

The sequence of the primes $p$ for which a variety over $\mathbb{Q}$ 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 behavior. We furthermore prove that this behavior 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.