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A criterion for solving embedding problems for the etale fundamental group of curves

Manish Kumar, Poulami Mandal

Abstract

Let $C$ be an affine curve over an algebraically closed field $k$ of characteristic $p>0$. Given an embedding problem $(β:Γ\longrightarrow G, α: π^{et}_1(C)\longrightarrow G)$ for $π_1^{et}(C)$ where $β$ is a surjective homomorphism of finite groups with prime-to-$p$ kernel $H$, we discuss when an $H$-cover of the $G$-cover of $C$ corresponding to $α$ is a solution. When $H$ is abelian and $G$ is a $p$-group, some necessary and sufficient conditions for the solvability of the embedding problem are given in terms of the action of $G$ on certain generalized Picard group.

A criterion for solving embedding problems for the etale fundamental group of curves

Abstract

Let be an affine curve over an algebraically closed field of characteristic . Given an embedding problem for where is a surjective homomorphism of finite groups with prime-to- kernel , we discuss when an -cover of the -cover of corresponding to is a solution. When is abelian and is a -group, some necessary and sufficient conditions for the solvability of the embedding problem are given in terms of the action of on certain generalized Picard group.
Paper Structure (6 sections, 15 theorems, 10 equations)

This paper contains 6 sections, 15 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Galois covers and pullback of coverings
  3. Cyclic covers prime-to-$p$
  4. $G$-action on $P_m$
  5. Effective subgroups
  6. The case when $G$ is a cyclic $p$-group

Key Result

Proposition 2.1

The $H$-covers $W_\sigma\longrightarrow V$ are isomorphic to $W\longrightarrow V$, $\forall \sigma \in$$\mathop{\mathrm{Aut}}\limits(V/Z)$ if and only if the composition $W\xrightarrow{~\psi~} V \xrightarrow{~\pi~} Z$ is also Galois.

Theorems & Definitions (34)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • ...and 24 more