An Erd\H{o}s-Kac law for local solubility in families of varieties

TL;DR

This paper investigates the distribution of local obstructions to the existence of p-adic points in families of varieties, demonstrating that under certain conditions, an Erdős-Kac type law governs these probabilities.

Contribution

It establishes an Erdős-Kac law for local solubility in families of varieties, linking number theory and algebraic geometry in a novel probabilistic framework.

Findings

Erdős-Kac law applies to local solubility probabilities

Distribution of local obstructions follows a normal law in certain cases

Provides new insights into the arithmetic of families of varieties

Abstract

We study probability distributions arising from local obstructions to the existence of $p$-adic points in families of varieties. In certain cases we show that an Erd\H{o}s-Kac type law holds.