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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITP.2023.13
URN: urn:nbn:de:0030-drops-183880
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18388/
de Frutos-Fernández, María Inés
Formalizing Norm Extensions and Applications to Number Theory
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.
BibTeX - Entry
@InProceedings{defrutosfernandez:LIPIcs.ITP.2023.13,
author = {de Frutos-Fern\'{a}ndez, Mar{\'\i}a In\'{e}s},
title = {{Formalizing Norm Extensions and Applications to Number Theory}},
booktitle = {14th International Conference on Interactive Theorem Proving (ITP 2023)},
pages = {13:1--13:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-284-6},
ISSN = {1868-8969},
year = {2023},
volume = {268},
editor = {Naumowicz, Adam and Thiemann, Ren\'{e}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18388},
URN = {urn:nbn:de:0030-drops-183880},
doi = {10.4230/LIPIcs.ITP.2023.13},
annote = {Keywords: formal mathematics, Lean, mathlib, algebraic number theory, p-adic analysis, Galois representations, p-adic Hodge theory}
}
