Related papers: The place of exceptional covers among all diophant…

Within the Schottky problem, the study of special subvarieties of the Torelli locus has long been of great interest. We describe a representation-theoretic criterion for a Jacobian variety arising from a $G$-Galois cover of $\mathbb{P}^1$…

An abelian cover is a finite morphism $X\to Y$ of varieties which is the quotient map for a generically faithful action of a finite abelian group $G$. Abelian covers with $Y$ smooth and $X$ normal were studied in…

This note presents Galois theory for finite fields. It was written as a handout for the MAT401 course ``Polynomial equations and fields'' taught at the University of Toronto in Spring 2026. We use without proofs some basic properties of…

For any prime power $q$, a polynomial $f(X)\in\F_q[X]$ is ``exceptional'' if it induces bijections of $\F_{q^k}$ for infinitely many $k$; this condition is known to be equivalent to $f(X)$ inducing a bijection of $\F_{q^k}$ for at least one…

Let $(R,\mathfrak{m}, k)$ be a strictly local normal $k$-domain of positive characteristic and $P$ be a prime divisor on $X=\text{Spec } R$. We study the Galois category of finite covers over $X$ that are at worst tamely ramified over $P$…

This paper provides a realization of all classical and most exceptional finite groups of Lie type as Galois groups over function fields over F_q and derives explicit additive polynomials for the extensions. Our unified approach is based on…

We relate two different proposals to extend the \'etale topology into homotopy theory, namely via the notion of finite cover introduced by Mathew and via the notion of separable commutative algebra introduced by Balmer. We show that finite…

The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by…

In this article, we consider weak del Pezzo surfaces defined over a finite field, and their associated, singular, anticanonical models. We first define arithmetic types for such surfaces, by considering the Frobenius actions on their Picard…

We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable fields. In addition, we study properties of…

In this manuscript, we apply patching methods to give a positive answer to the inverse differential Galois problem over function fields over Laurent series fields of characteristic zero. More precisely, we show that any linear algebraic…

Given a Galois cover of curves $f$ over a field of characteristic $p$, the lifting problem asks whether there exists a Galois cover over a complete mixed characteristic discrete valuation ring whose reduction is $f$. In this paper, we…

This paper surveys the methods that have been used to attack the conjecture, still open, that an abelian variety over a characteristic $0$ field with finitely generated Galois group is always of infinite rank.

We introduce a new graph invariant of finite groups that provides a complete characterization of the splitting types of unramified prime ideals in normal number field extensions entirely in terms of the Galois group. In particular, each…

In this paper we produce unconditionally new instances of Galois number field extensions exhibiting strong discrepancies in the distribution of Frobenius elements among conjugacy classes of the Galois group. We first prove an inverse Galois…

Let $F$ be a field of characteristic $0$ containing all roots of unity. We construct a functorial compact Hausdorff space $X_F$ whose profinite fundamental group agrees with the absolute Galois group of $F$, i.e. the category of finite…

In this paper, we prove the uniqueness of certain Fourier-Jacobi models for the split exceptional group $G_2$ over finite fields with odd characteristic. Similar results are also proved for $Sp_4$ and $U_4$.

Inspired by recent work of Farb, Kisin and Wolfson, we develop a method for using actions of finite group schemes over a mixed characteristic dvr R to get lower bounds for the essential dimension of a cover of a variety over K = Frac(R). We…

Covering spaces are a fundamental tool in algebraic topology because of the close relationship they bear with the fundamental groups of spaces. Indeed, they are in correspondence with the subgroups of the fundamental group: this is known as…

Special covers are metacyclic covers of the projective line, with Galois group of order pm, which have a specific type of bad reduction to characteristic p. Such covers arise in the study of the arithmetic of Galois covers of the projective…