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Given a $G$-Galois branched cover of the projective line over a number field $K$, we study whether there exists a closed point of $\mathbb{P}^1_K$ with a connected fiber such that the $G$-Galois field extension induced by specialization…

We prove automorphy lifting results for certain essentially conjugate self-dual $p$-adic Galois representations $\rho$ over CM imaginary fields $F$, which satisfy in particular that $p$ splits in $F$, and that the restriction of $\rho$ on…

We introduce a new obstruction to lifting smooth proper varieties in characteristic $p>0$ to characteristic $0$. It is based on Grothendieck's specialization homomorphism and the resulting discrete finiteness properties of \'etale…

In this note we prove an effective characterization of when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations, weakening the…

We examine in detail the stable reduction of Galois covers of the projective line over a complete discrete valuation field of mixed characteristic (0, p), where G has a cyclic p-Sylow subgroup of order p^n. If G is further assumed to be…

Let E be the supersingular elliptic curve defined over k, the algebraic closure of the finite field with two elements, which is unique up to k-isomorphism. Denote by 0 its identity element and let C be the quotient of E-{0} under the action…

Let G be a reductive group over an algebraically closed field of positive characteristic. In this article we show an analogue for Morozov theorem for characteristics that are separably good for G (and under additional hypotheses on the…

Raynaud gave a criterion for a branched $G$-cover of curves defined over a mixed-characteristic discretely valued field $K$ with residue characteristic $p$ to have good reduction in the case of either a three-point cover of $\mathbb{P}^1$…

The Gruenberg-Kegel graph ${\rm GK}(G)=(V_G, E_G)$ of a finite group $G$ is a simple graph with vertex set $V_G=\pi(G)$, the set of all primes dividing the order of $G$, and such that two distinct vertices $p$ and $q$ are joined by an edge,…

Let $G = GL(n)$ and $K = GL(p) \times GL(q)$ with $p+q=n$, where the groups are taken over $\C$. In this paper we study a certain family of $K$-orbit closures on the flag variety $X$ of $G$. The geometry of these orbit closures plays a…

Let $k$ be a field of characteristic zero and ${\bar k}$ an algebraic closure of $k$. For a geometrically integral variety $X$ over $k$, we write ${\bar k}(X)$ for the function field of ${\bar X}=X\times_k{\bar k}$. If $X$ has a smooth…

Let k be an algebraically closed field of characteristic p > 0. Let H be a subgroup of GL(n,k). We are interested in the determination of the vector invariants of H. When the characteristic of k is 0, it is known that the invariants of d…

Let k be an algebraically closed field of characteristic $p>0$, and $G_0$ be a Barsotti-Tate group (or $p$-divisible group) over k. We denote by $S$ the "algebraic" local moduli in characteristic p of $G_0$, by $G$ the universal deformation…

Let G be a semisimple group over an algebraically closed field of characteristic p>0. We give a (partly conjectural) simple, closed formula for the character of many indecomposable tilting rational G-modules, assuming that p is large.

In [2], an exhaustive construction is achieved for the class of all 4-dimensional unital division algebras over finite fields of odd order, whose left nucleus is not minimal and whose automorphism group contains Klein's four-group. We…

Let $K$ be a non-archimedean local field and $\varphi : \mathbb{P}^1 \to \mathbb{P}^1$ a rational endomorphism of degree $d \geq 2$ over $K$. In the tame case ($p \nmid d$), we show that strict good reduction is equivalent to the existence…

Let $X$ be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus $g \ge 2$ defined over an algebraically closed field $K$ of odd characteristic $p$. Let $Aut(X)$ be the group of all automorphisms of $X$ which fix…

Let $K$ be a local field of characteristic $p$ and let $L/K$ be a totally ramified Galois extension such that Gal$(L/K)\cong C_{p^n}$. In this paper we find sufficient conditions for $L/K$ to admit a Galois scaffold. This leads to…

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We prove that if G has isotropic rank >=1 and R is a regular domain…

Let $k$ be an algebraically closed field of characteristic not equal to 2 or 3, let $G$ be an almost simple algebraic group of type $F_4$, $G_2$ or $D_4$ and let $\theta$ be an automorphism of $G$ of finite order, coprime to the…