Formalizing Norm Extensions and Applications to Number Theory

Author María Inés de Frutos-Fernández



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María Inés de Frutos-Fernández
  • Imperial College London, UK
  • Universidad Autónoma de Madrid, Spain

Acknowledgements

I would like to thank Kevin Buzzard for many helpful conversations during the completion of this project, Thomas Browning for formalizing normal closures, and Yaël Dillies for the discussions on how best to integrate seminorms in mathlib. I also thank the mathlib community and maintainers for their support and insightful suggestions during the development of this work.

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María Inés de Frutos-Fernández. Formalizing Norm Extensions and Applications to Number Theory. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ITP.2023.13

Abstract

The field ℝ of real numbers is obtained from the rational numbers ℚ by taking the completion with respect to the usual absolute value. We then define the complex numbers ℂ as an algebraic closure of ℝ. The p-adic analogue of the real numbers is the field ℚ_p of p-adic numbers, obtained by completing ℚ with respect to the p-adic norm. In this paper, we formalize in Lean 3 the definition of the p-adic analogue of the complex numbers, which is the field ℂ_p of p-adic complex numbers, a field extension of ℚ_p which is both algebraically closed and complete with respect to the extension of the p-adic norm. More generally, given a field K complete with respect to a nonarchimedean real-valued norm, and an algebraic field extension L/K, we show that there is a unique norm on L extending the given norm on K, with an explicit description. Building on the definition of ℂ_p, we formalize the definition of the Fontaine period ring B_{HT} and discuss some applications to the theory of Galois representations and to p-adic Hodge theory. The results formalized in this paper are a prerequisite to formalize Local Class Field Theory, which is a fundamental ingredient of the proof of Fermat’s Last Theorem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Theory of computation → Type theory
Keywords
  • formal mathematics
  • Lean
  • mathlib
  • algebraic number theory
  • p-adic analysis
  • Galois representations
  • p-adic Hodge theory

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