A Formalization of Dedekind Domains and Class Groups of Global Fields

Baanen, Anne; Dahmen, Sander R.; Narayanan, Ashvni; Capriglio, Filippo Alberto Edoardo Nuccio Mortarino Majno Di

doi:10.1007/s10817-022-09644-0

Cited by 7 publications

(7 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For this section, we assume that R is a non-Archimedean normed commutative Q p -algebra, which is complete, nontrivial, and has no zero divisors. The scalar multiplication structure obtained from Q and Q p are compatible, given by is_scalar_tower Q Q_[p] R (see Section 4.2 of [2]). The prime p is odd, and we choose positive natural numbers d and c which are mutually coprime and are also coprime to p. The Dirichlet character χ has level dp m , where m is positive.…”

Section: Integralsmentioning

confidence: 99%

Formalization of $p$-adic $L$-functions in Lean 3

Narayanan¹

2023

Preprint

View full text Add to dashboard Cite

The Euler-Riemann zeta function is a largely studied numbertheoretic object, and the birthplace of several conjectures, such as the Riemann Hypothesis. Different approaches are used to study it, including p-adic analysis : deriving information from p-adic zeta functions. A generalized version of p-adic zeta functions (Riemann zeta function) are p-adic L-functions (resp. Dirichlet L-functions). This paper describes formalization of p-adic L-functions in an interactive theorem prover Lean 3. Kubota-Leopoldt p-adic L-functions are meromorphic functions emerging from the special values they take at negative integers in terms of generalized Bernoulli numbers. They also take twisted values of the Dirichlet L-function at negative integers. This work has never been done before in any theorem prover. Our work is done with the support of mathlib 3, one of Lean's mathematical libraries. It required formalization of a lot of associated topics, such as Dirichlet characters, Bernoulli polynomials etc. We formalize these first, then the definition of a p-adic L-function in terms of an integral with respect to the Bernoulli measure, proving that they take the required values at negative integers. ACM Subject ClassificationMathematics of computing → Mathematical software; Theory of computation → Formal languages and automata theory Keywords and phrases formal math, algebraic number theory, Lean, mathlib Supplementary Material Copies of the source files relevant to this paper are available in a separate repository. Software: https://github.com/laughinggas/p-adic-L-functions Funding Ashvni Narayanan: EPSRC Grant EP/S021590/1 (UK) Acknowledgements The author is supported by EPSRC. The author would like to thank her PhD supervisor Prof Kevin Buzzard for several helpful insights. She would also like to thank Dr Filippo A. E. Nuccio for helpful conversations that helped give shape to the project. She is very grateful to the entire Lean community for their timely help and consistent support.

show abstract

Section: Integralsmentioning

confidence: 99%

Formalization of $p$-adic $L$-functions in Lean 3

Narayanan¹

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…I assume some familiarity with basic ring and field theory, corresponding to a first course in the subject, and found in many undergraduate texts such as [48,76]. This chapter contains material adapted from the paper "A formalization of Dedekind domains and class groups of global fields", specifically the extended version published in the Journal of Automated Reasoning [11] of a paper presented at the ITP 2021 conference [10], and from the paper "Formalized Class Group Computations and Integral Points on Mordell Elliptic Curves", presented at CPP 2023 [9]. All authors contributed to these papers.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

“…This chapter describes the formalization process, noting the idioms we found useful in our development and Mathlib's decentralized collaboration processes involved in this project. This chapter is adapted from the paper "A formalization of Dedekind domains and class groups of global fields", specifically the extended version published in the Journal of Automated Reasoning [11] of a paper presented at the ITP 2021 conference [10]. All authors contributed to the formalization project as well as to both versions of the original paper.…”

Section: A Formalization Of Dedekind Domains and Class Groups Of Glob...mentioning

confidence: 99%

“…• New definitions and results: matrix inverses and Cramer's rule 9 , special linear group 10 , quadratic forms 11 , conversion between bilinear forms and matrices 12 , Lagrange formulas for expanding determinants 13 , Vandermonde matrices 14 , third isomorphism theorem for groups and modules 15 , field of rational functions 16 .…”

Section: Selected Further Contributionsmentioning

confidence: 99%

“…Source "A formalization of Dedekind domains and class groups of global fields". Extended version published in the Journal of Automated Reasoning [11]. Original version presented at ITP 2021 [10].…”

Section: Chaptermentioning

confidence: 99%

See 2 more Smart Citations

Formalizing Fundamental Algebraic Number Theory

Baanen

View full text Add to dashboard Cite

The research described in my thesis involves the extension and maintenance of the library of formal mathematics Mathlib for the interactive theorem prover Lean. The process of mechanization that results in formal mathematics rests on a detailed, uniform and unambiguous translation of mathematical knowledge. Thus, it requires suitable design choices, which needs a thorough understanding of the functionality provided by Lean. After giving a general overview of Lean and Mathlib, the thesis extensively covers a specific mechanism of Lean: the instance parameter. This makes possible the design pattern of typeclasses. Related mechanisms for organizing mathematical structures can be found in many other interactive theorem provers. Based on characteristic practical examples, I describe a variety of design choices that help in applying this functionality. Based on this research, changes and additions were made to Mathlib, also on fundamental levels. Many of the design choices are also applied in the subsequent chapters. After this follows a part of the thesis with the emphasis on formal mathematics, beginning with an overview of the mathematics that is covered. As part of a dynamically changing group of contributors, I worked on formalizing subjects that lie at the basis of algebraic number theory, and these formalizations have since been added to Mathlib. An important theme of this chapter is that it becomes possible to treat different occurences of the same situation in an equal manner, if the right formulation is chosen. Next, we apply the previous work to the solution of concrete problems, namely the formal computation of class numbers and a formal description of all integer solutions to certain Diophantine equations. This required us to face the challenge of writing down definitions that are useful both in the case of high abstraction, and when very specific objects are involved. Computing the desired properties for now remains a matter of manually applying calculation steps, although computer algebra systems can supply the outcome (without a formal proof) within mere moments. Finally, I describe proof tactics I developed, that make it possible to automatically prove certain classes of equalities in the study of rings. By applying these, many manual and tedious calculation steps can be left to the computer. The outcome of this research was not merely to extend the list of theorems available in Mathlib, but to enrich the possibilities of effectively formalizing mathematics.

show abstract

The Design of Mathematical Language

Avigad

2023

Handbook of the History and Philosophy of Mathematical Practice

View full text Add to dashboard Cite

As idealized descriptions of mathematical language, there is a sense in which formal systems specify too little, and there is a sense in which they specify too much. They are silent with respect to a number of features of mathematical language that are essential to the communicative and inferential goals of the subject, while many of these features are independent of a specific choice of foundation. This chapter begins to map out the design features of mathematical language without descending to the level of formal implementation, drawing on examples from the mathematical literature and insights from the design of computational proof assistants. Perspectives Philosophical orientationPhilosophy of mathematics has traditionally been concerned with the nature of mathematical objects, knowledge, and thought. Talking about these in a rigorous way requires some sort of conception of mathematics itself, and some sort of understanding the features of mathematical practice that fall under the scope of the analysis. Toward forming such a conception, what we have the most direct access to is the mathematical literature: the historical record of statements, questions, arguments, definitions, and and other textual artifacts that are constitutive of the subject. These artifacts, rooted in language, form the starting point for philosophical study.However we ultimately try to characterize the goals and methods of mathematics, we have to start with language. Whether we view mathematics as the practice of solving problems, abstracting from experience, or getting at a certain type of truth, what we say about that practice has to fit with what we see in the mathematical literature. We need to understand how mathematical language enables us to carry out those activities and how those activities are manifested in language.One central thesis of this chapter is that, when we study mathematical language, it is important to understand not just what is allowed, but also what is desired. Formal systems specify rules that tell us when a formula is well formed and when an inference is justified, but it doesn't tell us which definitions are good definitions, or which among the myriad inferential steps that can be taken at any given point are most worthy of our attention. Mathematics calls upon its practitioners to carry out complex tasks, and to do so creatively, efficiently, and reliably. Reflection on mathematical practice should help explain how it helps us manage complexity and carry out fruitful exploration.Another central thesis of this chapter is that it is helpful to view mathematical language as the object of design. Mathematical language and method have evolved over the centuries, presumably for good reasons. Some features of mathematical language and method have remained remarkably stable, again, we may presume, for good reasons. Mathematics provides powerful means for abstraction and for managing information, making data salient when it is needed and suppressing it when it is a distraction. Recognizing that it is effective in that ...

show abstract

A Formalization of Dedekind Domains and Class Groups of Global Fields

Cited by 7 publications

References 21 publications

Formalization of $p$-adic $L$-functions in Lean 3

Formalization of $p$-adic $L$-functions in Lean 3

Formalizing Fundamental Algebraic Number Theory

The Design of Mathematical Language

Contact Info

Product

Resources

About