Dissertation
Galois structure of holomorphic polydifferentials in positive characteristic
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Summer 2020
DOI: 10.17077/etd.005534
Abstract
This thesis uses a combination of methods from the representation theory of finite groups and the geometry of smooth projective curves in positive characteristic. The goal is to describe the precise structure of the space of holomorphic polydifferentials of such a curve as a representation of a finite group that acts on the curve.
More precisely, suppose X is a smooth projective curve over a perfect field k of positive characteristic p, and m is a positive integer. The space of holomorphic m-differentials, also called polydifferentials, is defined to be the k-vector space of global sections of the m-fold tensor product of the sheaf of relative differentials of X over k. If G is a finite group acting on X, then this space naturally becomes a kG-module. It is a classical problem to determine the decomposition of this space into a direct sum of indecomposable kG-modules, thus providing its precise kG-module structure.
In the main result of this thesis, we solve this problem in the case when G has non-trivial cyclic Sylow p-subgroups. Moreover, our proof provides a precise algorithm in the case when k is algebraically closed and G is a p-hypo-elementary group. We then apply the main result to compute examples in which X is either a hyperelliptic curve or a modular curve.
Details
- Title: Subtitle
- Galois structure of holomorphic polydifferentials in positive characteristic
- Creators
- Adam Wood
- Contributors
- Frauke Bleher (Advisor)Victor Camillo (Committee Member)Charles Frohman (Committee Member)Miodrag Iovanov (Committee Member)Ryan Kinser (Committee Member)Maggy Tomova (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Summer 2020
- DOI
- 10.17077/etd.005534
- Publisher
- University of Iowa
- Number of pages
- ix, 186 pages
- Copyright
- Copyright 2020 Adam Wood
- Language
- English
- Description bibliographic
- Includes bibliographical references (pages 183-186).
- Public Abstract (ETD)
This thesis uses a combination of methods from the representation theory of groups and the geometry of curves. A group is an algebraic object with some additional structure. The representation theory of a group uses linear algebra to understand the structure of the group. A curve is a one-dimensional geometric object. In this thesis, we study a particular representation arising from the action of a finite group of a curve. More specifically, we consider the space of holomorphic polydifferentials of a smooth projective curve, and we determine its decomposition into indecomposable representations in the case when the acting group has cyclic Sylow subgroups. Moreover, we provide an algorithm for this decomosition, and we compute specific examples for certain types of curves.
- Academic Unit
- Mathematics
- Record Identifier
- 9983987895802771
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