Hecke algebras, Galois representations, and abelian varieties

158 Pages·2016·1.29 MB·English

Heckealgebras,Galoisrepresentations,andabelianvarieties
Thesiscommittee:
Prof.dr.F.Beukers,UniversiteitUtrecht
Prof.dr.J.Hartmann,UniversityofPennsylvania
Prof.dr.E.Mantovan,CaliforniaInstituteofTechnology
Prof.dr.B.deSmit,UniversiteitLeiden
Dr.M.S.Solleveld,RadboudUniversiteit
ISBN:978-90-393-6563-2
Copyright©2016byValentijnKaremaker. Allrightsreserved.
Hecke algebras, Galois representations,
and abelian varieties
Hecke algebra’s, Galois voorstellingen en abelse variëteiten
(meteensamenvattinginhetNederlands)
Proefschrift
terverkrijgingvandegraadvandoctoraandeUniversiteitUtrechtopgezagvande
rectormagnificus,prof.dr.G.J.vanderZwaan,ingevolgehetbesluitvanhet
collegevoorpromotiesinhetopenbaarteverdedigenopmaandag13juni2016des
middagste2.30uur
door
Valentijn Zoë Karemaker
geborenop13april1990teUtrecht
Promotor: Prof.dr.G.L.M.Cornelissen
This thesis has been financially supported by the Netherlands Organisation for Sci-
entific Research (NWO) under the research project “Quantum statistical mechanics
andanabeliangeometry”.
Contents
1 Introduction 5
1.1 Heckealgebrasandadelicpoints . . . . . . . . . . . . . . . . . . . 5
1.2 Galoisrepresentations . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Abelianvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
I Heckealgebrasandadelicpoints 19
2 Arithmeticequivalenceandlocalisomorphism 21
2.1 Arithmeticequivalenceandlocalisomorphism . . . . . . . . . . . . 21
2.2 Theadditivegroupofadeles . . . . . . . . . . . . . . . . . . . . . 22
2.3 Themultiplicativegroupofadeles . . . . . . . . . . . . . . . . . . 25
3 Adelicpointsonalgebraicgroups 29
3.1 Algebraicgroupsandfertility . . . . . . . . . . . . . . . . . . . . . 29
3.2 Divisibilityandunipotency . . . . . . . . . . . . . . . . . . . . . . 32
3.3 ProofofTheorem3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Heckealgebrasoverglobalandlocalfields 41
4.1 Heckealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 L1-isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Moritaequivalences . . . . . . . . . . . . . . . . . . . . . . . . . . 44
II Galoisrepresentations 55
5 Galoisrepresentationsforabelianvarieties 57
5.1 StructuretheoryofGSp. . . . . . . . . . . . . . . . . . . . . . . . 57
1
2 Contents
5.2 Galoisrepresentationsattachedtothe(cid:96)-torsionofabelianvarieties . 58
5.3 SurjectiveGaloisrepresentations . . . . . . . . . . . . . . . . . . . 60
5.4 SemistablecurvesandtheirgeneralisedJacobians . . . . . . . . . . 64
6 AlgorithmforGaloisrealisationsofGSp (F ) 67
2n (cid:96)
6.1 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Anumericalexampleingenus3 . . . . . . . . . . . . . . . . . . . 69
7 ConstructingJacobianswithlargeGaloisimages 71
7.1 Hyperellipticcurvesandcurvesofgenus3 . . . . . . . . . . . . . . 71
7.2 Statementofmainresult . . . . . . . . . . . . . . . . . . . . . . . 72
7.3 Localconditionsatp . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.4 Localconditionsatq . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.5 Proofofthemaintheorem . . . . . . . . . . . . . . . . . . . . . . 82
7.6 CountingirreducibleWeilpolynomialsofdegree6 . . . . . . . . . 83
III Abelianvarieties 95
8 Abelianvarietiesoverfinitefieldsandtwists 97
8.1 L-polynomialsandsupersingularabelianvarieties . . . . . . . . . . 97
8.2 Twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
9 Theperiodandtheparity 109
9.1 Definitionsandpropertiesofperiodandparity . . . . . . . . . . . . 109
9.2 Analysisinlowgenus . . . . . . . . . . . . . . . . . . . . . . . . . 116
10 Arithmeticstatisticsofdimension 123
11 Samenvatting 131
12 Acknowledgements 137
13 CurriculumVitae 141
Bibliography 143

4 Contents
1
CHAPTER
Introduction
Oneofthemainobjectsofstudyofmodernalgebraicnumbertheoryistheabsolute
Galois group G = Gal(K/K) of a local or global field K with separable closure
K
K. In particular, one wants to understand the absolute Galois group of the rational
numbers,G = Gal(Q/Q).Thisisaverylargegroup,yetitiscompactwithrespect
Q
totheprofinitetopology.
One way of gaining insight into G is by studying its representations. These
K
mayariseindifferentways. Firstofall,throughtheLanglandscorrespondence,they
cancomefromautomorphicrepresentations,orequivalently,fromrepresentationsof
certainHeckealgebras. Secondly,byconsideringtheactionofG onvectorspaces
K
associated to varieties (e.g. the geometric torsion points of an abelian variety), one
findsgeometricGaloisrepresentations.
This thesis treats three questions related to G and its representations. The
K
first of these, which is anabelian in nature, asks to what extent the representation
theory of a Hecke algebra of a field K (which is either a number field or a local
non-archimedean field of characteristic zero) determines K. The second question
concerns Galois representations attached to abelian varieties, and especially those
with surjective image, which realise symplectic groups as Galois groups. The third
question focuses on supersingular abelian varieties over finite fields, and arithmetic
properties of these varieties which are determined by the characteristic polynomial
oftheFrobeniusendomorphism;thetopologicalFrobeniusmapisageneratorofG
K
forfinitefieldsK.
1.1 Heckealgebrasandadelicpoints
Let K be a field and let G denote its absolute Galois group. First suppose that
K
K is a number field. When K is Galois over Q, Neukirch [78] proved that G
K
∼
determines K, in the sense that any isomorphism G = G (as profinite groups)
K L
∼
induces a unique field isomorphism K = L. Uchida [117] later proved this result
whenK isnotnecessarilyGalois;asNeukirchpointsoutin[79],thesameresultwas
obtainedindependentlybyIwasawa(unpublished),usingresultsbyIkeda[47].
5
6 Introduction
Bycontrast,theabelianisationGab,correspondingtotheone-dimensionalrepre-
K
sentationsofG ,doesnotdetermineK,cf.[83]or[3].
K
Now suppose that K is a non-archimedean local field of characteristic zero. In
this case, Yamagata showed in [129] that the analogous statement of the result by
Neukirch and Uchida is false. Jarden and Ritter [52] prove that G determines the
K
absolutefielddegree[K: Q ]andthemaximalabeliansubextensionofK overQ .
p p
In addition, Mochizuki [76] proved that the absolute Galois group together with its
ramificationfiltrationdoesdeterminealocalfieldofcharacteristic0,andAbrashkin
[1],[2]extendedthisresulttoanycharacteristicp > 0.
A couple of natural questions then arise: when K is a number field, do irre-
ducible two-dimensional (being the “lowest-dimensional non-abelian”) representa-
tionsofG determineK? WhenK isanon-archimedeanlocalfieldofcharacteris-
K
ticzero,towhatextentdoestherepresentationtheoryofG determineK?
K
BythephilosophyoftheLanglandsprogramme, n-dimensionalirreduciblerep-
resentations of G (or, more generally, of the Weil group W ) should be in cor-
K K
respondence with certain automorphic representations of GL , while preserving L-
n
series.
Over non-archimedean local fields of characteristic zero, the local Langlands
correspondence was proven by Harris and Taylor [40] and Henniart [42]. More
precisely, their results state that equivalence classes of admissible irreducible rep-
resentations of GL (K) are in bijection with equivalence classes of n-dimensional
n
Frobenius semisimple representation of the Weil-Deligne group W(cid:48) of K, see e.g.
K
[122]; W(cid:48) is a group extension of the Weil group W , from which there exists a
K K
continuoushomomorphismtoG withdenseimage.
K
When K is a number field, no analogous result is known, although various spe-
cial caseshave beenconsidered. Onebelieves thatirreduciblen-dimensional repre-
sentations of G should correspond to cuspidal representations of GL (A ) “of
K n K
Galois type” [22, p. 244]. Moreover, all cuspidal representations of GL (A )
n K
should be in correspondence with irreducible n-dimensional representations of the
so-called Langlands group, which is the conjectural global analogue of the Weil-
Delignegroup.
Automorphic (admissible) representations of GL (A ) in turn correspond to
n K
(admissible) modules over the Hecke algebra H (K). Therefore, our questions
GLn
inspirethenextquestion.