Let $p\ge 3$ be a prime, let $n\>m\ge 1$. Let ${\pi }_n$ be the norm of ${\zeta }_{p^n}-1$ under $C_{p-1}$, so that ${\mathbb{Z}}_{\left(p\right)}\left[{\pi }_n\right]|{\mathbb{Z}}_{\left(p\right)}$ is a purely ramified extension of discrete valuation rings of degree $p^{n-1}$. The minimal polynomial of ${\pi }_n$ over $\mathbb{Q}\left({\pi }_m\right)$ is an Eisenstein polynomial; we give lower bounds for its coefficient valuations at ${\pi }_m$. The function field analogue, as introduced by Carlitz and Hayes, is studied as well.

The paper contains an expanded version of the talk delivered by the first author during the conference ALANT3 in Będlewo in June 2014. We survey recent results on independence of systems of Galois representations attached to ℓ-adic cohomology of schemes. Some other topics ranging from the Mumford-Tate conjecture and the Geyer-Jarden conjecture to applications of geometric class field theory are also considered. In addition, we have highlighted a variety of open questions which can lead to interesting...

Let $J_0\left(𝔫\right)$ be the Jacobian variety of the Drinfeld modular curve $X_0\left(𝔫\right)$ over ${𝔽}_q\left(t\right)$, where $𝔫$ is an ideal in ${𝔽}_q\left[t\right]$. Let $0\to B\to J_0\left(𝔫\right)\to A\to 0$ be an exact sequence of abelian varieties. Assume $B$, as a subvariety of $J_0\left(𝔫\right)$ , is stable under the action of the Hecke algebra $𝕋\subset$ End $\left(J_0\left(𝔫\right)\right)$. We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0\to {\Phi }_{B,\infty }\to {\Phi }_{J,\infty }\to {\Phi }_{A,\infty }\to 0$ of the Néron models of these abelian varieties over ${𝔽}_q\left[\left[\frac{1}{t}\right]\right]$. This criterion is always satisfied when either $A$ or $B$ is one-dimensional. Moreover, we prove that the sequence...

We survey methods to compute three-point branched covers of the projective line, also known as Belyĭ maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and $p$-adic methods. Along the way, we pose several questions and provide numerous examples.

Let $\Lambda$ be a maximal $𝔽{}_q\left[T\right]$-order in a division quaternion algebra over $𝔽{}_q\left(T\right)$ which is split at the place $\infty$. The present article gives an algorithm to compute a fundamental domain for the action of the group of units ${\Lambda }^{*}$ on the Bruhat-Tits tree $𝒯$ associated to ${\mathrm{PGL}}_2\left(𝔽{}_q\left((1/T)\right)\right)$. This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group ${\Lambda }^{*}$ in terms of generators and relations. Moreover we determine an upper bound...

Let $\mathbb{Q}\left(\alpha \right)$ be an algebraic number field given by the minimal polynomial $f$ of $\alpha$. We want to determine all subfields $\mathbb{Q}\left(\beta \right)\subset \mathbb{Q}\left(\alpha \right)$ of given degree. It is convenient to describe each subfield by a pair $(g,h)\in \mathbb{Z}\left[t\right]\times \mathbb{Q}\left[t\right]$ such that $g$ is the minimal polynomial of $\beta =h\left(\alpha \right)$. There is a bijection between the block systems of the Galois group of $f$ and the subfields of $\mathbb{Q}\left(\alpha \right)$. These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding...

If conditional independence constraints define a family of positive distributions that is log-convex then this family turns out to be a Markov model over an undirected graph. This is proved for the distributions on products of finite sets and for the regular Gaussian ones. As a consequence, the assertion known as Brook factorization theorem, Hammersley–Clifford theorem or Gibbs–Markov equivalence is obtained.