Related papers: Lifting of curves with automorphisms

We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.

We settle the automorphism groups of curves appearing in a classification list of smooth plane curves with at least two Galois points. One of them is an ordinary curve whose automorphism group exceeds the Hurwitz bound.

Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Its dual group $\hat{G}$ is a split reductive group over $\mathbb{Z}$.…

In this paper we investigate the problem of lifting of Galois covers between algebraic curves from characteristic p>0 to characteristic 0. We prove a refined version of the main result of Garuti concerning this problem in [Ga]. We formulate…

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the…

We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the…

In this paper, an explicit hierarchy of differential equations for the $\tau$-functions defining the moduli space of curves with automorphisms as a subscheme of the Sato Grassmannian is obtained. The Schottky problem for Riemann surfaces…

Let $G$ be a simple simply connected algebraic group over an algebraically closed field $k$ of characteristic $p$, with Frobenius kernel $G_{(1)}$. It is known that when $p\ge 2h-2$, where $h$ is the Coxeter number of $G$, the projective…

We prove potential automorphy results for a single Galois representation $G_F \rightarrow GL_n(\overline{\mathbb{Q}}_l)$ where $F$ is a CM number field. The strategy is to use the $p,q$ switch trick and modify the Dwork motives employed in…

For a prime $p$, a cyclic-by-$p$ group $G$ and a $G$-extension $L|K$ of complete discrete valuation fields of characteristic $p$ with algebraically closed residue field, the local lifting problem asks whether the extension $L|K$ lifts to…

Let $G=D_p$ be the dihedral group of order $2p$, where $p$ is an odd prime. Let $k$ an algebraically closed field of characteristic $p$. We show that any action of $G$ on the ring $k[[y]]$ can be lifted to an action on $R[[y]]$, where $R$…

We study automorphisms of quasi-smooth hypersurfaces in weighted projective spaces, extending classical results for smooth hypersurfaces in projective space to the weighted setting. We establish effective criteria for when a power of a…

We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of $G$-bundles on a smooth projective curve for a reductive algebraic group…

We study the automorphisms of modular curves associated to Cartan subgroups of $\mathrm{GL}_2(\mathbb Z/n\mathbb Z)$ and certain subgroups of their normalizers. We prove that if $n$ is large enough, all the automorphisms are induced by the…

We study the automorphism groups of the reduction $X_0(N) \times \bar{\mathbb{F}}_p$ of a modular curve $X_0(N)$ over primes $ p\nmid N$.

Given $\mathbb P^4_k$, with $k$ algebraically closed field of characteristic $p>0$, and $X\subset \mathbb P^4_k$ integral surface of degree $d$, let $Y=X\cap H$ be the general hyperplane section of $X$. We suppose that $h^0\mathscr…

Let k be a perfect field of characteristic p > 0, and let G be a finite group. We consider the pointed G-curves over k associated by Harbater, Katz, and Gabber to faithful actions of G on k[[t]] over k. We use such "HKG G-curves" to…

For $a,b\geq 3$, we calculate the orders of automorphisms of smooth curves with bidegree $(a,b)$ in the product $\pp$ of the projective line $\mathbb P^1$. We identify smooth curves in $\pp$ which have automorphisms with the largest orders.…

We study the properties of the fundamental group of an affine curve over an algebraically closed field of characteristic $p$, from the point of view of embedding problems. In characteristic zero, the fundamental group is free, but in…

The goal of this paper is to present a number of problems about automorphism groups of nonpositively curved polyhedral complexes and their lattices, meant to highlight possible directions for future research.