Homological sieve and Manin's conjecture

TL;DR

This survey introduces the homological sieve method and discusses its applications to Manin's conjecture in algebraic number theory.

Contribution

It explains the homological sieve technique and highlights its novel applications to the longstanding problem of Manin's conjecture.

Findings

The homological sieve provides new tools for counting rational points.

Applications to Manin's conjecture show promising progress.

The method offers a new perspective in algebraic number theory.

Abstract

This is a report of the author's talk at RIMS workshop Algebraic Number Theory and Related Topics 2025 which was held at RIMS Kyoto University during December 15th-19th 2025. In this survey paper, we explain the homological sieve method, which is proposed by Das, Lehmann, Tosteson, and the author, and its applications to Manin's conjecture.