Formally certifying number field invariants

2026-07-28

AI Summary

This work addresses the longstanding absence of formal verification for fundamental invariants in computational algebraic number theory—such as discriminants, signatures, unit groups modulo $p$-th powers, and class groups—which has undermined the reliability of databases and computer algebra systems. We present the first comprehensive formal verification framework in Lean 4 that encompasses these invariants, integrating real closed field theory, subresultant sequences, and certified ideal arithmetic. By leveraging SageMath to automatically generate verifiable proof certificates, our approach achieves the first formal verification of signatures, unit groups modulo $p$-th powers, and class groups for high-degree number fields. This significantly enhances both the efficiency and trustworthiness of verifying critical invariants like discriminants, and has already enabled the certification of hundreds of number field entries in the LMFDB.

Abstract

Number fields, which generalize the rational numbers, are fundamental objects in number theory. Many of their key arithmetic properties are captured by invariants whose computation is among the central tasks of computational algebraic number theory and a focus of several computer algebra systems and databases. In this paper, we describe a Lean 4 formalization for certifying several of these number field invariants. Building on previous work on certifying rings of integers, we extend this certification approach to further invariants including the signature, the unit group modulo $p$-th powers, and, ultimately, the class group. We also improve discriminant certification, allowing verifications for higher-degree number fields infeasible in previous work. We introduce structures based on representations of algebraic objects suited to computation, including reusable ones for certifying ideal arithmetic. Along the way, we formalize several underlying mathematical results, for instance on real closed fields and pseudo-remainder sequences, which are of independent interest. We apply our framework to verify hundreds of entries of the $\textit{L-functions and modular forms database}$ (LMFDB) concerning the discriminant, signature, class number, and class group structure of various number fields. To this end, we wrote a SageMath script that computes the certificates and outputs Lean proofs of the corresponding statements.

Problem

• number field invariants
• formal verification
• class group
• discriminant
• signature

Innovation

• formal verification
• number field invariants
• Lean 4
• class group certification
• computational algebraic number theory