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Counting three-point G-covers with a given special G-deformation datum
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
2011 (English)In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 134, no 3, 513-532 p.Article in journal (Refereed) Published
Abstract [en]

We correct an erroneous description of an automorphism group from 'Three point covers with bad reduction' by Wewers. The cardinality of the automorphism group is used in a counting formula for the number of three-point covers in characteristic zero with the same fibre in characteristic p. We consider the consequences for the counting formula resulting from the new description of the automorphism group, and also consider two families of examples in order to illustrate how the counting formula is related to the action of the inertia group at p on three-point covers of order strictly divisible by p.

Place, publisher, year, edition, pages
2011. Vol. 134, no 3, 513-532 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-95457DOI: 10.1007/s00229-010-0411-zISI: 000286396700012OAI: oai:DiVA.org:uu-95457DiVA: diva2:169669
Available from: 2007-03-08 Created: 2007-03-08 Last updated: 2017-12-14Bibliographically approved
In thesis
1. Arithmetic of three-point covers
Open this publication in new window or tab >>Arithmetic of three-point covers
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Any cover of the Riemann sphere with rational branch points is known to be defined over the algebraic numbers. Hence the Galois group of the rationals acts on the category of such branched covers. Particulars about this action are still scarce, even in the simplest non-abelian case, the case with just three branch points.

The first paper in this thesis describes a new algorithm, which uses modular form techniques in order to compute the equations for a cover of the Riemann sphere which is hyperelliptic as a curve. Given such equations one may easily determine the Galois orbit to which the cover belongs. We compute and discuss all covers of degree 6 and genus 2, and complete the case of covers of degree 7 and genus 1 as well.

The second paper gives a proof of a formula for the number of three-point G-covers with a fixed special G-deformation datum (here G is a finite group which is strictly divisible by a prime number p). Since such a datum is an invariant for the action of the inertia group at p, this gives partial information about the action of this inertia group.

Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen, 2007. 30 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049
Keyword
Algebra and geometry, Algebra och geometri
Identifiers
urn:nbn:se:uu:diva-7497 (URN)978-91-506-1916-4 (ISBN)
Public defence
2007-03-29, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Available from: 2007-03-08 Created: 2007-03-08 Last updated: 2009-08-25Bibliographically approved

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