arXiv subjects
Publications and source records attributed to María Inés de Frutos-Fernández.
We present the formalization of Carleson's theorem in the proof assistant Lean. This paper describes the mathematical content, organization of the project, the blueprint, and the main design decisions behind the formalization. It is the result of a large collaborative effort, written and developed in public.
Let $K/k$ be a finite Galois extension of global function fields. Let $E$ be a Drinfeld module over $k$. We state and prove an equivariant refinement of Taelman's analogue of the analytic class number formula for $(E,K/k)$, and derive explicit consequences for the Galois structure of the Taelman class group of $E$ over $K$.
The goal of this paper is to present an ongoing formalization, in the framework provided by the Lean/Mathlib mathematical library, of the construction by Roby (1965) of the universal divided power algebra. This is an analogue, in the theory of divided powers, of the classical algebra of polynomials. It is a crucial tool in the development of crystalline cohomology; it is also used in $p$-adic Hodge theory to define the crystalline period ring. As an algebra, this universal divided power algebra has a fairly simple definition that shows that it is a graded algebra. The main difficulty in Roby's theorem lies in constructing a divided power structure on its augmentation ideal. To that aim, Roby identified the graded pieces with another universal structure: homogeneous polynomial laws.We formalize the first steps of the theory of polynomial laws and show how future work will allow to complete the formalization of the above-mentioned divided power structure. We report on various difficulties that appeared in this formalization: taking care of universes, extending to semirings some aspects of the Mathlib library, and coping with several instances of "invisible mathematics".
This paper is the blueprint underlying the Lean formalization of the proof of Carleson's classical result asserting almost everywhere convergence of Fourier series of continuous functions. We break up the proof into two steps, a reduction of the classical result to a new theorem that appears in a sibling communication and a proof of this new theorem, which is also detailed as blueprint in this paper. An early version of this blueprint was used to initiate the Lean formalization. During the formalization, many contributors elaborated the blueprint with minor corrections, modifications and extensions. The final version is presented here as a guide through the accompanying Lean code.
Given an ideal $I$ in a commutative ring $A$, a divided power structure on $I$ is a collection of maps $\{γ_n \colon I \to A\}_{n \in \mathbb{N}}$, subject to axioms that imply that it behaves like the family $\{x \mapsto \frac{x^n}{n!}\}_{n \in \mathbb{N}}$, but which can be defined even when division by factorials is not possible in $A$. Divided power structures have important applications in diverse areas of mathematics, including algebraic topology, number theory and algebraic geometry. In this article we describe a formalization in Lean 4 of the basic theory of divided power structures, including divided power morphisms and sub-divided power ideals, and we provide several fundamental constructions, in particular quotients and sums. This constitutes the first formalization of this theory in any theorem prover. As a prerequisite of general interest, we expand the formalized theory of multivariate power series rings, endowing them with a topology and defining evaluation and substitution of power series.
Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued fields. In particular, we prove that the unit ball with respect to a discrete valuation on a field is a discrete valuation ring and, conversely, that the adic valuation on the field of fractions of a discrete valuation ring is discrete. We define finite extensions of valuations and of discrete valuation rings, and prove some global-to-local results. Building on this general theory, we formalize the abstract definition and some fundamental properties of local fields. As an application, we show that finite extensions of the field $\mathbb{Q}_p$ of $p$-adic numbers and of the field $\mathbb{F}_p(\!(X)\!)$ of Laurent series over $\mathbb{F}_p$ are local fields.
Let $K$ be a field complete with respect to a nonarchimedean real-valued norm, and let $L/K$ be an algebraic extension. We show that there is a unique norm on $L$ extending the given norm on $K$, with an explicit description. As an application, we extend the $p$-adic norm on the field $\mathbb{Q}_p$ of $p$-adic numbers to its algebraic closure $\mathbb{Q}_p^{\text{alg}}$, and we define the field $\mathbb{C}_p$ of $p$-adic complex numbers as the completion of the latter with respect to the $p$-adic norm. Building on the definition of $\mathbb{C}_p$, we formalize the definition of the Fontaine period ring $B_{\text{HT}}$ 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.
The ring of adèles of a global field and its group of units, the group of idèles, are fundamental objects in modern number theory. We discuss a formalization of their definitions in the Lean 3 theorem prover. As a prerequisite, we formalized adic valuations on Dedekind domains. We present some applications, including the statement of the main theorem of global class field theory and a proof that the ideal class group of a number field is isomorphic to an explicit quotient of its idèle class group.
We provide explicit equations for moduli spaces of Drinfeld shtukas over the projective line with $Γ(N)$, $Γ_1(N)$ and $Γ_0(N)$ level structures, where $N$ is an effective divisor on $\mathbb{P}^1$. If the degree of $N$ is high enough, these moduli spaces are relative surfaces. We study some invariants of the moduli space of shtukas with $Γ_0(N)$ level structure for several degree $4$ divisors on $\mathbb{P}^1$.
We compute rational points on genus $3$ odd degree hyperelliptic curves $C$ over $\mathbb{Q}$ that have Jacobians of Mordell-Weil rank $0$. The computation applies the Chabauty-Coleman method to find the zero set of a certain system of $p$-adic integrals, which is known to be finite and include the set of rational points $C(\mathbb{Q})$. We implemented an algorithm in Sage to carry out the Chabauty-Coleman method on a database of $5870$ curves.