The leading constant for rational points in families

TL;DR

This paper establishes asymptotic formulas for counting rational points on certain conic families and proposes a new conjecture on the distribution of locally soluble varieties within algebraic families.

Contribution

It provides the first asymptotic results for diagonal planar conics with rational points and introduces a novel conjecture on counting varieties that are locally soluble everywhere.

Findings

Asymptotic formulas for rational points on diagonal planar conics.

A new conjecture on the enumeration of locally soluble varieties.

Insights into the distribution of rational solutions in algebraic families.

Abstract

We prove asymptotics for Serre's problem on the number of diagonal planar conics with a rational point and use this to put forward a new conjecture on counting the number of varieties in a family which are everywhere locally soluble.