# Sieving rational points on varieties

## TL;DR

This paper develops a sieve method for rational points on varieties, enabling improved counting and analysis of rational solutions, with special results for quadrics using a Selberg sieve adaptation.

## Contribution

It introduces a new sieve framework for rational points on varieties, including applications to thin sets, local solubility, and friable points, with enhanced results for quadrics.

## Key findings

- Effective sieve for rational points on varieties
- Applications to counting in thin sets and local solubility
- Sharper estimates for quadrics using a Selberg sieve

## Abstract

A sieve for rational points on suitable varieties is developed, together with applications to counting rational points in thin sets, the number of varieties in a family which are everywhere locally soluble, and to the notion of friable rational points with respect to divisors. In the special case of quadrics, sharper estimates are obtained by developing a version of the Selberg sieve for rational points.