Béranger Seguin

The Frobenius of a matrix \(M\) with coefficients in \(\overline{\mathbb F}_p\) is the matrix \(\sigma(M)\) 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 \(\sigma(M)\), \(\sigma^2(M)\), \(\ldots\) in their Frobenius orbit.

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.

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 skew fields.

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.

For a finite group G, we describe the asymptotical growth of 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, as the number of branch points grows to infinity. More precisely, we compute both the exponent and (in many cases) the coefficient of the leading monomial in the count of components containing covers whose monodromy group is a given subgroup of G. By the work of Ellenberg, Tran, Venkatesh and Westerland, this asymptotical behavior is related to the distribution of field extensions of 𝔽q(T) with Galois group G.

Let K be a field of characteristic 0 and k≥2 be an integer. We prove that every K-linear bijection f:K[X]→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., there are elements a,c∈Kˣ, b∈K such that f(P)(X) = cP(aX+b). When K is a number field or K=ℝ, we prove that similar statements hold when f preserves the set of polynomials with a root in K.

We study a weak divisibility property for noncommutative rings: a nontrivial ring is fadelian if for all nonzero a and x there exist b, c such that x=ab+ca. We prove properties of fadelian rings and construct examples thereof which are not division rings, as well as non-Noetherian and non-Ore examples.

Hurwitz spaces are moduli spaces that classify ramified covers of the projective line on which a fixed group G acts. Their geometric and arithmetic properties are related to number theoretical questions, particularly the inverse Galois problem. In this thesis, we study the connected components of these spaces. Firstly, we prove results concerning the asymptotical behaviour of the count of connected components of Hurwitz spaces as the number of branch points of the covers they classify grows. Secondly, we establish stability results for fields of definitions of connected components of Hurwitz spaces under the gluing operation. These results relate topological and arithmetical properties of covers. Three expository chapters, devoid of original statements, present the various objects. In an appendix, we summarize the thesis for the general public.

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. Applications are given.