Related papers: Arithmetic statistics of Galois groups

A self-contained exposition is given of the topological and Galois-theoretic properties of the category of combinatorial 1-complexes, or graphs, very much in the spirit of Stallings. A number of classical, as well as some new results about…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$/ Associated to $E$, there is an adelic Galois representation $\rho_E \colon {\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \to {\rm GL}_2(\hat{\mathbb{Z}})$. In this article, we give…

Shape analysis methods have in the past few years become very popular, both for theoretical exploration as well as from an application point of view. Originally developed for planar curves, these methods have been expanded to higher…

We establish three results dealing with the character varieties of finitely generated groups. The first two are concerned with the behavior of $\dim X_n(\Gamma)$ as a function of $n$, and the third addresses the problem of realizing a…

It is well-known that characters classify linear representations of finite groups, that is if characters of two representations of a finite group are the same, these representations are equivalent. It is also well-known that, in general,…

We study Hopf-Galois extensions with central invariants for a finite dimensional Hopf algebra. We collect general facts about them and discuss some examples arising in the study of restricted Lie algebras and quantum groups at roots of…

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an $F_\sigma$ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to…

The formal group law of an elliptic curve has seen recent applications to computational algebraic geometry in the work of Couveignes to compute the order of an elliptic curve over finite fields of small characteristic. The purpose of this…

Galois theory is developed using elementary polynomial and group algebra. The method follows closely the original prescription of Galois, and has the benefit of making the theory accessible to a wide audience. The theory is illustrated by a…

We develop algorithms to compute the differential Galois group $G$ associated to a parameterized second-order homogeneous linear differential equation of the form \[ \tfrac{\partial^2}{\partial x^2} Y + r_1 \tfrac{\partial}{\partial x} Y +…

A q-analogue of de Finetti's theorem is obtained in terms of a boundary problem for the q-Pascal graph. For q a power of prime this leads to a characterisation of random spaces over the Galois field F_q that are invariant under the natural…

By generalizing Frobenius' polynomial method to good partition algebra, we will develop new character theories for a finite group $G$. A uniform defining equations are derived for these kinds of character theories. The new character…

Topos properties of the category of covering groupoids over a fixed groupoid are discussed. A classification result for connected covering groupoids over a fixed groupoid analogous to the fundamental theorem of Galois theory is given.

Turull has described the fields of values for characters of $SL_n(q)$ in terms of the parametrization of the characters of $GL_n(q)$. In this article, we extend these results to the case of $SU_n(q)$.

In this paper, we prove Lusztig's conjecture for finite special linear groups, i.e., we show that characteristic functions of character sheaves coincide with almost characters up to scalar constants, under the condition that the…

The l-adic parabolic cohomology groups attached to noncongruence subgroups of SL_2(Z) are finite-dimensional representations of Gal(Qbar/F) for some number field F. We exhibit examples (with F=Q) giving rise to Galois representations whose…

We develop cycle index generating functions for orthogonal groups in even characteristic, and give some enumerative applications. A key step is the determination of the values of the complex linear-Weil characters of the finite symplectic…

Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho_E \colon \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{GL}(2, \widehat{\mathbb{Z}})$ be the adelic Galois representation attached to $E$. We describe and…

A new algebraic Cayley graph is constructed using finite fields. Its connectedness and diameter bound are studied via Weil's estimate for character sums. These graphs provide a new source of expander graphs, extending classical results of…

We study Galois points for a hypersurface $X$ with $\dim {\rm Sing}(X) \le \dim X-2$. The purpose of this article is to determine the set $\Delta(X)$ of Galois points in characteristic zero: Indeed, we give a sharp upper bound of the number…