Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Essential Dimension of Congruence Covers
164
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Consider the algebraic function $Phi_{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Kronecker and Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $Phi_{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties.
rate research
Read More
We discuss two elementary constructions for covers with fixed ramification in positive characteristic. As an application, we compute the number of certain classes of covers between projective lines branched at 4 points and obtain information on the structure of the Hurwitz curve parametrizing these covers.
The goal of this article is to classify unramified covers of a fixed tropical base curve $Gamma$ with an action of a finite abelian group G that preserves and acts transitively on the fibers of the cover. We introduce the notion of dilated cohomology groups for a tropical curve $Gamma$, which generalize simplicial cohomology groups of $Gamma$ with coefficients in G by allowing nontrivial stabilizers at vertices and edges. We show that G-covers of $Gamma$ with a given collection of stabilizers are in natural bijection with the elements of the corresponding first dilated cohomology group of $Gamma$.
We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a hypergeometric differential equation. This generalizes the result of Deligne- Rapoport on the reduction of the modular curve X(p).
We study triple covers of K3 surfaces, following Mirandas theory of triple covers. We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vector bundles on a K3 surface $S$ which determine a non-Galois triple cover of $S$. The examples presented are in any admissible Kodaira dimension and in particular we provide the constructions of irregular covers of K3 surfaces and of surfaces with geometrical genus equal to 2 whose transcendental Hodge structure splits in the sum of two Hodge structures of K3 type.
Let $A$ be the ring of elements in an algebraic function field $K$ over a finite field $F_q$ which are integral outside a fixed place $infty$. In an earlier paper we have shown that the Drinfeld modular group $G=GL_2(A)$ has automorphisms which map congruence subgroups to non-congruence subgroups. Here we prove the existence of (uncountably many) normal genuine non-congruence subgroups, defined to be those which remain non-congruence under the action of every automorphism of $G$. In addition, for all but finitely many cases we evaluate $ngncs(G)$, the smallest index of a normal genuine non-congruence subgroup of $G$, and compare it to the minimal index of an arbitrary normal non-congruence subgroup.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
