On the application of large deviation estimates to local solubility in families of varieties

TL;DR

This paper applies large deviation theory, specifically the Gartner--Ellis theorem, to analyze local solubility in families of varieties, providing a partial proof of a conjecture related to arithmetic geometry.

Contribution

It introduces a novel application of large deviation estimates to the study of local solubility in algebraic families, advancing understanding of the Loughran--Smeets conjecture.

Findings

Proves a weak version of the Loughran--Smeets conjecture for general fibrations

Demonstrates the effectiveness of large deviation techniques in arithmetic geometry

Provides new insights into the distribution of local solutions in algebraic families

Abstract

We apply the G\"artner--Ellis theorem on large deviations to prove a weak version of the Loughran--Smeets conjecture for general fibrations.