Table Of ContentSeries of Congress Reports G Ot S e r i e s o f C o n g r e s s R e p o r t s
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B Miv
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t-Motives: Hodge Structures, Transcendence D io
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and Other Motivic Aspects i Ag Hodge Structures,
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Gebhard Böckle, David Goss, Urs Hartl and Matthew Papanikolas, ss ect Transcendence and
Editors , tr
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H tu Other Motivic Aspects
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This volume contains research and survey articles on Drinfeld modules, Anderson rt re
t-modules and t-motives. Much material that had not been easily accessible in the l s
a
literature is presented here, for example the cohomology theories and Pink’s theory n ,
of Hodge structures attached to Drinfeld modules and t-motives. Also included d T
r Gebhard Böckle
are survey articles on the function field analogue of Fontaine’s theory of p-adic M
a
crystalline Galois representations and on transcendence methods over function a n
t David Goss
fields, encompassing the theories of Frobenius difference equations, automata theory, th s
and Mahler’s method. In addition, this volume contains a small number of research e c
articles on function field Iwasawa theory, 1-t-motifs, and multizeta values. w e Urs Hartl
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This book is a useful source for learning important techniques and an effective p Matthew Papanikolas
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reference for all researchers working in or interested in the area of function field a
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arithmetic, from graduate students to established experts.
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ISBN 978-3-03719-198-9
https://ems.press
SCR Böckle et al. | Egyptienne F | Pantone 116, 287 | RB 31 mm
EMS Series of Congress Reports
EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field
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Previously published:
Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowron´ski (ed.)
K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.)
Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.)
Surveys in Stochastic Processes, Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.)
Representations of Algebras and Related Topics, Andrzej Skowron´ski and Kunio Yamagata (eds.)
Contributions to Algebraic Geometry. Impanga Lecture Notes, Piotr Pragacz (ed.)
Geometry and Arithmetic, Carel Faber, Gavril Farkas and Robin de Jong (eds.)
Derived Categories in Algebraic Geometry. Toyko 2011, Yujiro Kawamata (ed.)
Advances in Representation Theory of Algebras, David J. Benson, Henning Krause and
Andrzej Skowron´ski (eds.)
Valuation Theory in Interaction, Antonio Campillo, Franz-Viktor Kuhlmann and Bernard Teissier (eds.)
Representation Theory – Current Trends and Perspectives, Henning Krause, Peter Littelmann,
Gunter Malle, Karl-Hermann Neeb and Christoph Schweigert (eds.)
Functional Analysis and Operator Theory for Quantum Physics. The Pavel Exner Anniversary Volume,
Jaroslav Dittrich, Hynek Kovarˇík and Ari Laptev (eds.)
Schubert Varieties, Equivariant Cohomology and Characteristic Classes, Jarosław Buczyn´ski, Mateusz
Michałek and Elisa Postinghel (eds.)
Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, Fritz Gesztesy,
Harald Hanche-Olsen, Espen R. Jakobsen, Yurii Lyubarskii, Nils Henrik Risebro and Kristian Seip (eds.)
Spectral Structures and Topological Methods in Mathematics, MIchael Baake, Friedrich Götze and
Werner Hoffmann (eds.)
t-Motives:
Hodge Structures,
Transcendence and
Other Motivic Aspects
Gebhard Böckle
David Goss
Urs Hartl
Matthew Papanikolas
Editors
Editors:
Gebhard Böckle Urs Hartl
Interdisciplinary Center for Scientific Computing Mathematisches Institut
Universität Heidelberg Westfälische Wilhelms-Universität Münster
Im Neuenheimer Feld 368 Einsteinstr. 62
69120 Heidelberg 48149 Münster
Germany Germany
http://www.iwr.uni-heidelberg.de/~Gebhard.Boeckle/ https://www.uni-muenster.de/Arithm/hartl/
Matthew Papanikolas
Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
USA
E-mail: [email protected]
2010 Mathematics Subject Classification (primary; secondary): 11G09; 11J93, 11R58, 13A35
Key words: Drinfeld modules, t-motives, Anderson t-modules, transcendence, Hodge–Pink-structures
ISBN 978-3-03719-198-9
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9 8 7 6 5 4 3 2 1
Preface
In1974Drinfeldrevolutionizedthefieldofarithmeticoverglobalfunctionfields. He
introduceda functionfield analogueof elliptic curvesovernumberfields, which he
calledellipticmodulesbutarenoweponymouslynamedDrinfeldmodules. Forhim
andformanysubsequentdevelopmentstheirmain usewas in the explorationofthe
globalLanglands conjecture for automorphic forms overfunction fields. One of its
predictions is a correspondencebetween automorphic forms and Galois representa-
tions. The deepinsightofDrinfeld was that the modulispacesofDrinfeld modules
canbeassembledinacertaintowersuchthatthecorrespondingdirectlimitoftheas-
sociated `-adic cohomologieswould be an automorphicrepresentationwhich at the
sametimecarriesaGaloisaction.Thiswouldallowhimtorealizethecorrespondence
conjecturedbyLanglandsingeometry. Buildingonthis,Drinfeldhimselfprovedthe
globalLanglands’correspondenceforfunctionfieldsforGL andlaterL.Lafforgue
2
obtainedtheresultforallGL .
n
In a second direction, the analogyof Drinfeld modules with elliptic curves over
number fields made them interesting objects to be studied on their own right. One
couldstudytorsionpointsandGaloisrepresentations,onecoulddefinecohomology
theories such as de Rham or Betti cohomologyand thus investigatetheir periods as
well as transcendence questions. A main advance in this direction is the introduc-
tionoft-motivesbyAnderson. PassingfromDrinfeldmodulestot-motivesmaybe
compared to the passage from elliptic curves to abelian varieties. But more is true.
Thecategoryoft-motivesisalsoasimplefunctionfieldanalogueofGrothendieck’s
conjecturedcategoryofmotivesovernumberfields. Itisthisseconddirectionwhich
constitutesamainthemeofthepresentvolume,includingadvancesonGaloisrepre-
sentations,L-functions,transcendenceresults,Hodgestructuresandperioddomains.
Manyexcitingdevelopmentsinthearithmeticoffunctionsfieldsinrecentdecades
havecenteredaroundthenotionofat-motive. Someofthemostimportantonesare:
• New developments in the transcendence theory over function fields: For in-
stance, it has been shown that the period matrix of a t-motive has transcen-
dencedegreeequalto thedimensionofits motivicGalois group,muchin the
same way that Grothendieck’s conjecture predicts the transcendence degree
of the period matrix of an abelian variety to be equal to the dimension of its
Mumford–Tategroup.
• Hodgestructuresforfunctionfields: DefinedbyPinkin1997,theyallowhim
to define the analogue of the Mumford–Tate group of a t-motive and to for-
mulateaMumford–Tateconjectureforcertaint-motivesandtoprovethecon-
jecture for Drinfeld modules. Also Pink proved the Hodge-conjecturein this
theory, stating, that the Mumford–Tate group is equal to the motivic Galois
group.
• Perioddomains:MorerecentlyPink’sHodgestructureshavebeenusedexten-
sively to lay foundationsfor period domainsoverfunction fields and state an
analogueofFontaine’stheoryofcrystallineGaloisrepresentations.
vi Preface
• Galoisrepresentations:WhiletheTate-conjectureforat-motiveoverafinitely
generatedfieldhasbeenprovedalreadyintheearly90’s,onlyrecentlyresults
on the openness of the image of Galois (l-adically and adelically) have been
obtained.
• Tannakianformalisms:Suchhaverecentlybeendescribedfort-motivesinvar-
ious contexts. They should ultimately link transcendence, Hodge structures
andimagesofGaloisrepresentations.
• L-series: There are now cohomological approaches to L-series attached to
t-motives. Moreover,recentlynewresultsonthezeroesoftheseL-serieshave
emerged.
The above topics are tightly interwoven. The Tannakian formalism is used in
transcendencetheoryaswellasinaformulationofaMumford–Tateconjecturebased
on function field Hodge structures (which is proved for Drinfeld modules). This
in turn spurs the interest in Galois representations over function fields. All of the
abovetopicshavecloserelationstosimilarquestionsinnumbertheory. Thefunction
fieldHodgestructuresandanaloguesofFontaine’stheoryhaveinfluencedquestions
on period spaces for number fields. Other developments such as the transcendence
theoryhavegonefarbeyondcomparableresultsinnumbertheory.
The first part of this volume consists of survey articles on central topics in the
arithmeticoffunctionfields,thefirstthreeofwhichfocusonpropertiesoft-motives
and Andersont-modules. There is a briefintroductoryarticle on Drinfeld modules,
t-modules, andt-motivesby BrownawellandPapanikolas. The article byHartland
JuschkaonPink’stheoryofHodgestructuresprovidesanextensiveviewoftheinter-
connectednessbetweencohomologytheories,Hodge–Pinkstructures,t-motives,and
Anderson t-modules. The article by Hartl and Kim investigates local shtukas con-
nectedtoHodge–PinkstructuresandGaloisrepresentationsandprovidesa function
field analogue of Fontaine’s theory of p-adic crystalline Galois representations and
Kisin’stheoryofcrystallineGaloisdeformationrings.
There are three further survey articles on transcendence methods over function
fields. Changhasprovidedanoverviewoftechniquesintranscendencetheoryarising
from solutionsofFrobeniusdifferenceequationsandtheirt-motivicinterpretations.
Pellarin’s article gives an overviewof Mahler’s method for deducingtranscendence
and algebraic independence in the context of function fields. Finally, Thakur has
writtenasurveyofhowautomatatheorycanbeappliedtotranscendenceproblems.
The remaining three articles of the volume are research articles. The article by
Bandini, Bars, and Longhi focuses on Iwasawa theory over function fields, and in
particulartheyformulateaMainConjectureforabelianvarietiesinZN-extensionsof
p
function fields. Taelman constructs and proves fundamental results for 1-t-motifs,
which can be viewed as function field analogues of 1-motives over number fields.
ThearticlebyThakurreviewsthetheoryofmultizetavaluesoverfunctionfieldsand
presentsrecentresultsonlinearrelationsamongthemandtheirperiodinterpretations.
The present volume grew out of the workshop, “t-motives: Hodge structures,
transcendence and other motivic aspects,” held at the Banff International Research
StationonSeptember27–October2,2009,whichbroughttogetherresearchersfrom
Preface vii
acrosstheglobetodiscussprogressinfunctionfieldarithmeticandrelatedtopics.The
workshoppagehttps://www.birs.ca/workshops/2009/09w5094/files/containsfurther
materialfordownloadthatisnotcoveredinthepresentvolume.
Duetothelongtimeittookforthiscollectiontoappear,italsoseemsappropriate
todiscusssomeimportantfurtherdevelopmentssincethetimeoftheworkshop.
• Having introduced good notions of class module and unit group shortly be-
forethe Banffmeeting, thesearchofTaelmanfora classnumberformulafor
special values of L-functions proved to be successful. We refer to [3] for a
first decisive theorem by Taelman in this direction, where some crucial steps
are inspired by a trace formula of V. Lafforgue. This spurred much research
afterwardsbyTaelmanandmanyothers. RecentlyM.Mornevhasgivenaco-
homological reformulation of some of Taelman’s work. There is also recent
workonspecialL-valuesbyB.Anglès,C.-Y.Chang,C.Debry,F. Demeslay,
A.El-Guindy,J.Fang,T.NgoDac,M.Papanikolas,F.Pellarin,andF.Tavares
Ribeiro.
• In[1],F.PellarinintroducedanewkindofDrinfeldmodularformalongwith
anewkindofL-function. TheformsarenowcalledvectorialDrinfeldmodu-
larforms overthe Tate algebra, andthe L-functionsare namedafterPellarin.
Bothconstructionshavenotyetfoundanaloguesovernumberfieldsbutproved
extremely useful in function field arithmetic. Recent research is also due to
B. Anglès, Q. Gazda, O. Gezmis¸, D. Goss, N. Green, A. Maurischat, T. Ngo
Dac,M.Papanikolas,R.Perkins,andF.TavaresRibeiro.
• The work started by Thakur [4] on multizeta values, on which his article in
the present volume reports, proved to be extremely influential; perhaps also
because of the motivic interpretation given jointly by him and Anderson. In
particularmany transcendenceresults on algebraic independencefor the infi-
nite or v-adic places have been proved in the meanwhile and many relations
among such values are now understood. There has been much work in this
area, particularly by B. Anglès, C.-Y. Chang, H.-J. Chen, N. Green, W.-C.
Huang,Y.-L.Kuan,J.A.LaraRodríguez,Y.-H.Lin,Y.Mishiba,T.NgoDac,
M.Papanikolas,S.Shi,F.TavaresRibeiro,G.Todd,andJ.Yu.
• BreuerandPinkstartedaprogramtodevelopfoundationsofhigherrankDrin-
feldmodularforms. WorkofPink[2]andofsomeofhisstudents,ledtoagood
understandingoftheSatakecompactificationofDrinfeldmodularvarietiesof
higher rank, algebraically and analytically. Very recent work on this and on
higherrankDrinfeldmodularformsisalsoduetoD.Basson,F.Breuer,E.-U.
Gekeler,S.Häberli,M.Papikian,S.Schieder,andF.-T.Wei.
The above list is certainly not complete and we apologize for not mentioning the
manyotherdevelopmentsthattookplaceinthelastyearsinfunctionfieldarithmetic
andthefuturedirectionsthathavebeenstartedrecently.
G.Böckle,D.Goss,U.Hartl,M.Papanikolas
23May2019
viii Preface
N.B. It is with great sorrow that we observe that David Goss passed away during
the compilation of this volume. David was one of the early pioneers in modern ap-
proaches to function field arithmetic, and throughoutrecent decades he enthusiasti-
callychampionednewdevelopmentsinthesubjectandcontinuallyencouragedboth
juniorandestablishedresearcherstoreachfornewdiscoveries. Hewasawonderful
colleagueandfriend.
References
[1] F.Pellarin,ValuesofcertainL-seriesinpositivecharacteristic.Ann.Math.176(3)(2012),
2055–2093.
[2] R.Pink,CompactificationofDrinfeldmodularvarietiesandDrinfeldmodularformsof
arbitraryrank.ManuscriptaMath.140(3–4)(2013),333–361.
[3] L.Taelman,SpecialL-valuesofDrinfeldmodules.Ann.Math.175(1)(2012),369–391.
[4] D.S.Thakur,FunctionFieldArithmetic,WorldScientific,RiverEdge,NJ,2004.
Contents
PartA.Surveyarticles
1 ArapidintroductiontoDrinfeldmodules,t-modules,andt-motives. . 3
byW.DaleBrownawell,MatthewA.Papanikolas
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Exponentialfunctionsofalgebraicgroups . . . . . . . . . . . . . . . . . 3
1.3 Drinfeldmodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 t-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 t-Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Pink’stheoryofHodgestructuresandtheHodgeconjecture
overfunctionfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
byUrsHartl,Ann-KristinJuschka
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Hodge–Pinkstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 MixedA-motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4 MixeddualA-motives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.5 AndersonA-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
2.7 (cid:2)-Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
3 Localshtukas,Hodge–PinkstructuresandGaloisrepresentations . . . 183
byUrsHartl,WansuKim
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
3.2 Localshtukas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
3.3 DivisiblelocalAndersonmodules . . . . . . . . . . . . . . . . . . . . . . . 191
3.4 Tatemodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
3.5 Hodge–Pinkstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
3.6 Admissibilityandweakadmissibility. . . . . . . . . . . . . . . . . . . . . 222
3.7 TorsionlocalshtukasandtorsionGaloisrepresentations . . . . . . . . 230
3.8 DeformationtheoryofGaloisrepresentations . . . . . . . . . . . . . . . 245
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
4 FrobeniusdifferenceequationsanddifferenceGaloisgroups . . . . . . . 261
byChieh-YuChang
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
4.2 t-Motivictranscendencetheory. . . . . . . . . . . . . . . . . . . . . . . . . 262
4.3 Carlitzpolylogarithmsandspecial(cid:3)-values . . . . . . . . . . . . . . . . 268
