Table Of ContentSpringer Proceedings in Mathematics & Statistics
Frank Neumann
Sibylle Schroll Editors
Galois Covers,
Grothendieck-Teichmüller
Theory and Dessins
d’Enfants
Interactions between Geometry,
Topology, Number Theory and Algebra,
Leicester, UK, June 2018
Springer Proceedings in Mathematics &
Statistics
Volume 330
Springer Proceedings in Mathematics & Statistics
This book series features volumes composed of selected contributions from
workshops and conferences in all areas of current research in mathematics and
statistics, including operation research and optimization. In addition to an overall
evaluation of the interest, scientific quality, and timeliness of each proposal at the
hands of the publisher, individual contributions are all refereed to the high quality
standards of leading journals in the field. Thus, this series provides the research
community with well-edited, authoritative reports on developments in the most
exciting areas of mathematical and statistical research today.
More information about this series at http://www.springer.com/series/10533
Frank Neumann Sibylle Schroll
(cid:129)
Editors
Galois Covers,
ü
Grothendieck-Teichm ller
Theory and Dessins
’
d Enfants
Interactions between Geometry, Topology,
Number Theory and Algebra, Leicester, UK,
June 2018
123
Editors
FrankNeumann Sibylle Schroll
PureMathematics Group,Schoolof PureMathematics Group,Schoolof
Mathematics andActuarial Science Mathematics andActuarial Science
University of Leicester University of Leicester
Leicester, Leicestershire, UK Leicester, Leicestershire, UK
ISSN 2194-1009 ISSN 2194-1017 (electronic)
SpringerProceedings in Mathematics& Statistics
ISBN978-3-030-51794-6 ISBN978-3-030-51795-3 (eBook)
https://doi.org/10.1007/978-3-030-51795-3
Mathematics Subject Classification: 14J10, 14H10, 14H57, 14K10, 14G32, 14G05, 14J25, 14F35,
14J15,11G32,11D45,20D05
©SpringerNatureSwitzerlandAG2020
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, expressed or implied, with respect to the material contained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard
tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
This proceedings volume presents original peer-reviewed contributions from the
London Mathematical Society Midlands Regional Meeting and Workshop on
‘Galois Covers, Grothendieck-Teichmüller Theory and Dessins d’Enfants’. The
regionalmeetingandworkshopwereheldattheUniversityofLeicesterinLeicester,
United Kingdom from 4 to 7 June 2018 and were sponsored by the London
Mathematical Society (LMS) with additional support from the University of
Leicester, the Anglo-Franco-German Representation Theory Network (REPNET)
and the Engineering and Physical Sciences Research Council (EPSRC). It was
attendedbywellover70participantsfromtheUnitedKingdomandotherEuropean
countries.
The LMS Midlands Regional Meeting took place on 4 June 2018. The invited
speakers were Minhyong Kim (University of Oxford, UK), Leila Schneps (Institut
de Mathématiques de Jussieu Paris, France) and Fabrizio Catanese (University of
Bayreuth, Germany). The meeting started with official society business chaired by
the LMS president Caroline Series (University of Warwick, UK). In the opening
talkentitledDiophantinegeometryandprincipalbundlesMinhyongKimdescribed
fascinatingnewlinksbetweennumbertheory,geometryandphysics.Inthesecond
talk,LeilaSchnepsrelatedGrothendieck-Teichmüllertheorywithnumbertheoryin
her inspirational talk Pro-unipotent Grothendieck-Teichmüller theory: surprising
connections with number theory. In the final talk of the LMS meeting Fabrizio
Catanese gave an inspiring overview entitled Mathematical Mysteries behind the
InterplayofAlgebra andTopologyinModuli Theory onnewconnections between
algebraic and topological aspects of the classification theory of algebraic varieties
with symmetries.
TheregionalmeetingwasfollowedbyaninternationalworkshoponGaloiscovers,
Grothendieck-TeichmüllertheoryandDessinsd'enfantsfrom5to7June2018and
featured 16 invited international speakers, including the aforementioned three
regional meeting speakers. This multidisciplinary workshop brought together
world-leadingexpertsaswellasgraduatestudentsandearlycareerresearchersfrom
geometry,topology,algebraandnumbertheory.Italsoinitiatednewlinksandmany
futurecollaborations.Severalofthetalksfeaturedintroductionsoroverviewsonthe
v
vi Preface
different interdisciplinary themes of the workshop. The invited speakers were:
Fabrizio Catanese (University of Bayreuth, Germany), Anna Felikson (Durham
University,UK),PaolaFrediani(UniversityofPavia,Italy),BenoitFresse(University
of Lille, France), Christian Gleissner (University of Bayreuth, Germany), Pierre
Guillot(IRMA,Strasbourg,France),GarethJones(UniversityofSouthampton,UK),
Minhyong Kim (University of Oxford, UK), Bernhard Koeck (University of
Southampton,UK),MichaelLönne(UniversityofBayreuth,Germany),GoranMalic
(UniversityofManchester,UK),MartaMazzocco(UniversityofBirmingham,UK),
RobertoPignatelli(UniversityofTrento,Italy),MohamedSaïdi(UniversityofExeter,
UK),LeilaSchneps(InstitutdeMathématiquesdeJussieuParis,France)andRichard
Webb(UniversityofCambridge,UK).
We would like to thank the authors of these proceedings for their important
contributions. We also thank the many referees for their work and suggestions to
improve this volume. Furthermore, we thank all the speakers and participants who
made this LMS regional meeting and workshop such a great success. Last but not
least,weliketothanktheLondonMathematicalSociety,REPNET,EPSRCandthe
University of Leicester for their generous financial support. Finally, we thank
Springer for the suggestion to include this proceedings volume in the Springer
Proceedings in Mathematics & Statistics series. In particular, we would like to
thank Banu Dhayalan and Remi Lodh from Springer who guided us smoothly and
with expertise through the editing and publishing process.
Leicester, UK Frank Neumann
April 2020 Sibylle Schroll
Contents
Galois Covers, Grothendieck-Teichmüller Theory and Dessins
d’Enfants - An Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Frank Neumann and Sibylle Schroll
OntheNumberofRationalPointsofClassifyingStacksforChevalley
Group Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Scott Balchin and Frank Neumann
Cyclic Symmetry on Complex Tori and Bagnera–De Franchis
Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Fabrizio Catanese
Second Fundamental Form of the Prym Map in the Ramified Case . . . 55
Elisabetta Colombo and Paola Frediani
Strongly Real Beauville Groups III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Ben Fairbairn
The Pluricanonical Systems of a Product-Quotient Variety . . . . . . . . . . 89
Filippo F. Favale, Christian Gleissner, and Roberto Pignatelli
Joining Dessins Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Gareth A. Jones
Arithmetic Chern–Simons Theory I. . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Minhyong Kim
Branch Stabilisation for the Components of Hurwitz Moduli Spaces
of Galois Covers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Michael Lönne
vii
viii Contents
Dessins d’Enfants and Brauer Configuration Algebras . . . . . . . . . . . . . 205
Goran Malić and Sibylle Schroll
On the Elliptic Kashiwara–Vergne Lie Algebra. . . . . . . . . . . . . . . . . . . 227
Élise Raphael and Leila Schneps
Galois Covers,
Grothendieck-Teichmüller Theory and
Dessins d’Enfants - An Introduction
FrankNeumannandSibylleSchroll
Abstract Inthisintroduction,wewillgiveabriefoverviewofthethemesandtopics
ofthearticlesinthisproceedingsvolumeandsummariseeachindividualcontribution
basedontheabstractsandintroduction.
· · ·
Keywords Galoiscovers Grothendieck-Teichmüllertheory Dessinsd’enfants
·
Groupactions Representationtheory
The LMS workshop Galois covers, Grothendieck–Teichmüller theory and Dessins
d’enfants brought together many experts from the United Kingdom and abroad as
well as graduate students and early career researchers centred around three main
themes: Galois covers, Grothendieck–Teichmüller theory and dessins d’enfants.
Thesethemesbringaboutunexpectedlinksbetweenalgebraicgeometry,representa-
tiontheory,numbertheoryandalgebraictopology.Theproceedingsofthisworkshop
reflect the topics and scope of the lectures presented and explore through surveys
andoriginalresearcharticlesthemanyfacetsofthesefascinatingthemes.
1 OverviewofThemes
The absolute Galois group Gal(Q¯/Q) is one of the most fundamental objects of
study in number theory. In his groundbreaking work, Grothendieck [20] proposed
acombinatorialapproachtoinvestigateGal(Q¯/Q)basedonbipartiteribbongraphs
which,becauseoftheirapparentsimplicity,hecalled‘dessinsd’enfants’.Theyarise
asgraphsembeddedintoRiemannsurfacesandarerelatedtoalgebraiccurvesdefined
overalgebraicnumberfields.Thereexistsabijectionbetweendessinsd’enfantsand
B
F.Neumann( )·S.Schroll
SchoolofMathematicsandActuarialScience,PureMathematicsGroup,
UniversityofLeicester,UniversityRoad,LeicesterLE17RH,England,UK
e-mail:[email protected]
S.Schroll
e-mail:[email protected]
©SpringerNatureSwitzerlandAG2020 1
F.NeumannandS.Schroll(eds.),GaloisCovers,Grothendieck-TeichmüllerTheory
andDessinsd’Enfants,SpringerProceedingsinMathematics&Statistics330,
https://doi.org/10.1007/978-3-030-51795-3_1
