arXiv subjects

Béranger Seguin

Publications and source records attributed to Béranger Seguin.

10 recordsLinked to original sources

arXiv2026

Lifts of unramified twists and local-global principles

We prove that two-step nilpotent $p$-extensions of rational global function fields of characteristic $p$ satisfy a quantitative local-global principle when they are counted according to their largest upper ramification break ("last jump"). We had previously shown this only for $p\neq2$. Compared to our previous proof, this proof is also more self-contained, and may apply to heights other than the last jump. As an application, we describe the distribution of last jumps of $D_4$-extensions of rational global function fields of characteristic $2$. We also exhibit a counterexample to the analogous local-global principle when counting by discriminants.

math.NT↗

arXiv2026

On matrices commuting with their Frobenius

The Frobenius of a matrix $M$ with coefficients in $\bar{\mathbb F}_p$ is the matrix $σ(M)$ obtained by raising each coefficient to the $p$-th power. We consider the question of counting matrices with coefficients in $\mathbb F_q$ which commute with their Frobenius, asymptotically when $q$ is a large power of $p$. We give answers for matrices of size $2$, for diagonalizable matrices, and for matrices whose eigenspaces are defined over $\mathbb F_p$. Moreover, we explain what is needed to solve the case of general matrices. We also solve (for both diagonalizable and general matrices) the corresponding problem when one counts matrices $M$ commuting with all the matrices $σ(M)$, $σ^2(M)$, $\ldots$ in their Frobenius orbit.

math.AG↗

arXiv2025

Flimsy Spaces

We study $n$-flimsy spaces, which are the topological spaces that remain connected when removing fewer than $n$ points but become disconnected when removing exactly $n$ points. We show that no such space exists for $n \geq 3$, and that the compact $2$-flimsy spaces are precisely the dense and order-complete cyclically ordered sets equipped with their order topology. Furthermore, we examine variants of the definition obtained by replacing connectedness by path-connectedness, where paths are either parametrized by $[0,1]$ or by arbitrary compact linear continua.

math.GN↗

arXiv2025

Counting Components of Hurwitz Spaces

For a finite group $G$, we obtain asymptotics for the number of connected components of Hurwitz spaces of marked $G$-covers (of both the affine and projective lines) whose monodromy classes are constrained in a certain way, when the number of branch points grows to infinity. More precisely, we compute both the degree and (in many cases) the coefficient of the leading monomial in the count of components of marked $G$-covers whose monodromy group is a given subgroup of $G$. By the work of Ellenberg, Tran, Venkatesh and Westerland, these asymptotics are related to the distribution of field extensions of $\mathbb{F}_q(T)$ with Galois group $G$.

math.AT↗

arXiv2025

Symmetries of various sets of polynomials

Let $K$ be a field of characteristic $0$, and let $k \geq 2$ be an integer. We prove that every $K$-linear bijection $f \colon K[X] \to K[X]$ strongly preserving the set of $k$-free polynomials (or the set of polynomials with a $k$-fold root in $K$) is a constant multiple of a $K$-algebra automorphism of $K[X]$, i.e., that there are elements $a, c \in K^{\times}$ and $b \in K$ such that $f(P)(X) = c P(a X + b)$. When $K$ is a number field or $K=\mathbb R$, we prove that similar statements hold when $f$ preserves the set of polynomials with a root in $K$.

math.AC↗

arXiv2025

Asymptotics of extensions of simple $\mathbb Q$-algebras

We answer various questions concerning the distribution of extensions of a given central simple algebra $K$ over a number field. Specifically, we give asymptotics for the count of inner Galois extensions $L/K$ of fixed degree and center with bounded discriminant. We also relate the distribution of outer extensions of $K$ to the distribution of field extensions of its center $Z(K)$. This paper generalizes the study of asymptotics of field extensions to the noncommutative case in an analogous manner to the program initiated by Deschamps and Legrand to extend inverse Galois theory to division algebras.

math.NT↗

arXiv2025

Counting two-step nilpotent wildly ramified extensions of function fields

We study the asymptotic distribution of wildly ramified extensions of function fields in characteristic $p > 2$, focusing on (certain) $p$-groups of nilpotency class at most $2$. Rather than the discriminant, we count extensions according to an invariant describing the last jump in the ramification filtration at each place. We prove a local-global principle relating the distribution of extensions over global function fields to their distribution over local fields, leading to an asymptotic formula for the number of extensions with a given global last-jump invariant. A key ingredient is Abrashkin's nilpotent Artin-Schreier theory, which lets us parametrize extensions and obtain bounds on the ramification of local extensions by estimating the number of solutions to certain polynomial equations over finite fields.

math.NT↗

arXiv2024

The Geometry of Rings of Components of Hurwitz Spaces

We consider a variant of the ring of components of Hurwitz spaces introduced by Ellenberg, Venkatesh and Westerland. By focusing on Hurwitz spaces classifying covers of the projective line, the resulting ring of components is commutative, which lets us study it from the point of view of algebraic geometry and relate its geometric properties to numerical invariants involved in our previously obtained asymptotic counts. Specifically, we describe a stratification of the prime spectrum of the ring of components, and we compute the dimensions and degrees of the strata. Using the stratification, we give a complete description of the spectrum in some cases.

math.NT↗

arXiv2023

Fields of Definition of Components of Hurwitz Spaces

For a fixed finite group $G$, we study the fields of definition of geometrically irreducible components of Hurwitz moduli schemes of marked branched $G$-covers of the projective line. The main focus is on determining whether components obtained by "gluing" two other components, both defined over a number field $K$, are also defined over $K$. The article presents a list of situations in which a positive answer is obtained. As an application, when $G$ is a semi-direct product of symmetric groups or the Mathieu group $M_{23}$, components defined over $\mathbb{Q}$ of small dimension ($6$ and $4$, respectively) are shown to exist.

math.NT↗

arXiv2022

Study of a division-like property

We introduce a weak division-like property for noncommutative rings: a nontrivial ring is fadelian if for all nonzero $a,x$ there exist $b,c$ such that $x=ab+ca$. We prove properties of fadelian rings, and construct examples of such rings which are not division rings, as well as non-Noetherian and non-Ore examples. We have also formalized some of these results in the Lean proof assistant.

math.RA↗