Arithmetik Abelscher Varietäten mit komplexer Multiplikation

109 Pages·1984·1.869 MB·German

by  Claus-Günther Schmidt (auth.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
2801
rehtniLG-sualC Schmidt
Arithmetik
Abelscher net~teiraV tim
rexelpmok Multiplikation
galreV-regnirpS
Berlin Heidelberg New York oykoT 1984
Autor
Claus-G~inther Schmidt
Institut des Hautes Etudes Scientifiques
91440 Bures-sur-Yvette, France
AMS Subject Classification (1980): 14K22, 10D25, 10 D45, 12A35, 14G05
ISBN 3-540-13863-3 Springer-Vertag Berlin Heidelberg New York Tokyo
ISBN 0-387-13863-3 Springer-Verlag New York Heidelberg Berlin Tokyo
CIP-Kurztitelaufnahme der Deutschen Bibliothek Schmidt, Claus-G~inther: Arithmetik
Abelscher Variet&ten mit komplexer Multiplikation / Claus-Gi}nther Schmidt. - Berlin; Heidel-
berg; NewYork; Springer, Tokyo: 1984.
(Lecture notes in mathematics; 1082)
ISBN 3-540-13863-3 (Berlin ...)
ISBN 0-387-13863-3 (New York...)
:EN GT
This work is subject to copyright. HA rights are reserved, whether the whole or part of the material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks. Under
§ 54 of thGee rman Copyright Law where copies armea de for other than private use, a feies
payable to "Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag by Berlin Heidelberg t984
Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach / Bergstr.
2146/3140-543210
V ORW OR T
Mein besonderer Dank gilt Prof. G. Frey fur kritische Durch-
sicht des Manuskripts und Verbesserungsvorschl~ge. Ebenso
mSchte ich hier Prof. Serge Lang danken fur die freundliche
Uber!assung eines Vorabdrucks seines kHrzlich erschienenen
Buchs Hber "Complex multiplication".
Das Tippen des Manuskripts hat dankenswerterweise das Sekre-
tariat des Institut des Hautes Etudes Scientifiques Uber-
nommen.
Bures-sur-Yvette, Mai 198~
Claus-GUnther Schmidt
INHALTSVERZEICHNIS
English Summary VII
Einleitung
.I GrSssencharaktere vom Typ A
o
i. Idelklassencharaktere und zugehSrige Divisorfunktionen 4
.2 Variation des F~hrers unter Wertebereichsvorgaben 12
.3 Die Klassengruppenannullatoren vom Typ A 15
o
.4 Jacobi-Summen 16
Appendix : Stickelberger-Ideale der Maximalordnung 19
II. Abelsche Variet~ten mit komplexer Multiplikation
.i CM-Typ und Dual 24
.2 Der Gr~ssencharakter einer CM-Varietgt 31
.3 Die CM-Varietgten eines Gr~ssencharakters 33
III. Die Halbsysteme Abelscher CM-Typen
.i Primitive Halbsysteme und solche vom Vollrang 39
.2 Der Existenzsatz f~r Vollranghalbsysteme 41
.3 Eine Indexformel 42
IV. Geometrische Annullator-Kriterien
.i Die Automorphieregel 44
.2 Eine Variante zum Satz von Shimura-Casselman 46
.3 Das Zerfallskr~terium 52
.4 Galois-Operation auf dem Kohomologiering 53
V. ModulikSrper und unverzweigte Erweiterungen
.i ModulikSrper und ~-Variet~ten 58
.2 Das Kompositum der ModulikSrper 65
.3 Dualit~ten 69
IV
VI. Die CM-Varietgten der Fermat-Jacobischen
.i Der grobe Zerfall der Fermat-Jacobischen 78
.2 Die Zetafunktion 82
.3 Geometrische Kummer-Jacobi-Relationen 86
Literaturverzeichnis 89
Namen- und Sachverzeichnis 91
Symbolverzeichnis 93
SUMMARY
Since Kronecker, number theorists have kept exploiting the idea of
generating abelian extensions of number fields by means of special
values of appropriately chosen analytic functions. One of the most
beautiful contributions to this program, which is also known as Hil-
bert's 12th problem, is the theory of complex mulitplication on abe-
lian varieties built up by Shimura and Taniyama. It supplies firstly
for these CM-varieties a proof of Weils's conjecture and secondly, by
adjunction of torsion points of abelian varieties to certain number
fields, a tool to construct controlled ramified abelian extensions.
The interaction which then occurs between the arithmetic of a CM-field
and the geometry of a corresponding CM-variety has already been ex-
ploited repeatedly, either for generating class fields (see Shimura
[st , Kubota and others) or in the opposite direction for in
stance in the analysis of the Mordell-Weil group (Gross, Rohrlich
EG-R~ of certain CM-varieties or in the study of the Hodge con-
jecture (rasp. Tate conjecture) for CM-varieties by Pohlmann, Shioda
and Deligne. That interaction, which is also the crucial feature of
this book, is essentially based on the corresponding GrSBencharakter
of type A o. Under certain conditions this GrSBenchar~.kter supplies
an annihilator of the class group of the CM-field, via the description
of the prime decomposition of its values (as a divisor character) by
its infinity type. In this way one can for instance recove~ inde-
pendently of Stickelberger's theorem, the classical Kummer-Jacobi-re-
lations which describe the annihilation of the class group of a cyclo-
tomic field by so-called Stickelberger elements. However it is not
at all clear, whether by this procedure all annihilators of a certain
type have already been found. Further, thinking of the fundamental
role of the Kummer-Jacobi-relations in the arithmetic of cyclotomic
fields, it would be important to find their analogue for arbitrary
CM-fields. Therefore, to work out systematically the geometric con-
ditions on CM-varieties leading to annihilators of class groups seems
to be premising and is among other things one of the major goals of
this work.
Chapter I begins by giving the basic properties of GrSBencharaktere
of type Ao, analyzing the group of infinity-types and introducing
IIIV
the so-called "test character" with respect to an infinity-type. The
key lemma 2.4 which ensures the existence of the test character, has
consequences of two kinds. First of all it immediately yields lwa-
sawa's theorem ~ on the characterization of annihilators of the
class group as the infinity-types of Gr52encharaktere with prescribed
field of values. Further, applied to infinity-types corresponding to
halfsystems, it lies at the core of ~himura~s construction of polar-
ized OM-varieties CS~ which are already defined over their field of
moduli. The first chapter ends with a comparison of two naturally ari-
sing groups of infinity-types of GrSBencharaktere of an imaginary
abelian number field K. One is ~K, the group of infinity-types of
Jacobi sums and the other one is SK, built up from products of Gauss
sums, which in gener~l is larger than J~K" If we denote by~the maxi-
mal order of the rational group ring of the Galois group G of K/Q, we
show for the ~span ~ J (reap. S )~ of JK (resp. SK) that the quotient
S ~J* is a 2-elementary abelian group with a certain bound for the
2-rank (see Satz A.~).
The second chapter starts with a summary of those parts of the theory
of complex multiplication which are relevant for our purposes, aug-
mented by several recent results such as, for instance, Schappacher's
existence theorem ~ch~ for primitive CM-types or the lower bounds
for the rank of CM-types by Kubota, Ribet ~u, R2~ . Following this,
we determine for a given Gr6Bencharakter@ all CM-varieties whose
corresponding Gr~Bencharakter is equal to~" (Satz 3.4). Furthermore
we obtain s splitting criterion for the associated CM-varieties, in
which the field of values of %& is involved (Satz 3.5) and which sup-
plies a first annihilator criterion (Kor. 3.6).
In the third chapter we consider abstract CM-types of abelian groups
C with respect to an involutionS. The main result here is the exis-
tence theorem for halfsystems of full rank, which for any G non-isomor-
phic to the Kleinian #-group yields the existence of a halfsystem HEG
of maximal rank
I + IGI/2.
"veil
This improves, for abelian G, Schappacher's existence theorem of pri-
mitive CM-types, since every halfsystem of full rank is primitive
but the converse is in general false. Also we show a formula for the
IX
index of the7/ rG)-span of the given halfsystem in the span U(A o) of
all halfsystems. For cyclic G this index formula tells us that there
is even a choice of H such that
7Z[a]'£T -- U(Ao)-
T,=I4
Chapter IV deals with geometric annihilator criteria. First of all we
discuss a sufficient condition for a GrU~encharskter to provide an
annihilator of the class group: This is the automorphy rule, well
known already from Jacobi sums. This rule can be characterized by the
behavior of the isogeny class of a corresponding CM-variety under con-
jugation (Satz 1.4). It is followed by exact criteria which establish
a given infinity-type (not necessarily of halfsystem type) as annihi-
lator of the class group, via the existence of an appropriate CM-vari-
ety or, in the case of halfsystem %ypes, via the splitting behavior
of a given CM-variety. Also among other things the annihilation pro-
perty of a Gr~Sencharakter is interpreted as the diagonalizability of
Frobenius automorphisms under a certain ~-adic representation.
At the beginning of the fifth chapter we introduce the concept of
"~-varieties", generalizing B. Gross" ~-curves (see ]G[ .) For cy-
clic CM-fields we prove a series of existence statements and we ex-
tend the classification theorem on ~-curves to ~-varieties. There
follows an analysis (which is slightly modified as compared with
Shimura's work ~ ) of the unramified extensions given by fields of
moduli. The field of moduli as well as the field of values of a test
character constitute a measure of how far away a given infinity-type
is from being an annihilator of the class group. This fact is shown
to rely quantitatively on a duality with respect to a certain pairing
and the index of all annihilators of the class group in the group of
all infinity types is described in terms of Galois cohomology. Further¥
the duality between the field of values and the field of moduli is in-
terpreted geometrically, via descent theory, by proving the existence
o£ a certain CM-structure on the restriction of scalars of a corres-
ponding CM-variety.
In Chapter Vl we describe the ~- (resp. ~-) isogenous splitting of
the Jacobian of the Fermat-curve into CM-varieties and we identify
the corresponding Gr~Bencharaktere as Jacobi sums. Thus we obtain, in
particular, a proof of the Kummer-Jacobi-relations which works with-
out using Gauss sums. In conclusion we discuss what kind of new
"Kummer-Jacobi-relations" the geometric method of chapter IV might
eventually yield.
EINLEITUNG
Seit Kronecker zehrt die Zahlentheorie yon der Idee, Abelsche Erweiterungen yon
ZahlkSrpern durch spezielle Werte geeigneter analytischer Funktionen zu erzeugen.
Dieses auch als zwSlftes Hilbertsches Problem bekannte Programm erfuhr eine seiner
schSnsten Best~tigungen durch die Theorie der komplexen Multiplikation auf Abelschen
Variet~ten yon Shimura und Taniyama. Sie liefert einerseits fur jene CM-Variet~ten
einen Beweis der Weilschen Vermutun~en und andererseits durch Adjunktion yon Torsions-
punktender Abelschen Variet~ten an gewisse ZahlkSrper die MSglichkeit, kontrolliert
verzweigte Abelsche Erweiterungen zu konstruieren. Die dabei auftretende Wechselwir-
kung zwischen der Arithmetik eines CM-KSrpers und der Geometrie einer zugehSrigen CM-
Variet~t hat sich schon wiederholt als sehr nHtzlich erwiesen, sei es zur Erzeugung
von KlassenkSrpern (vgl. Shimura [SI], Kubota [Ku] .u ).a oder umgekehrt etwa zur
Untersuchung der Mordell-Weil-Gruppe (Gross, Rohrlich [G-Ro]) gewisser CM-Variet~ten
oder beim Studium der Hodge-Vermutung (bzw. Tate-Vermutung) fur CM-Variet~ten durch
Pohlmann, Shioda und Deligne. Jene Wechselwirkung, die auch der Angelpunkt der vorlie-
genden Arbeit ist, beruht wesentlich auf dem der CM-Variet~t zugeordneten GrSssen-
charakter vom Typ ° , A der unter Umst~nden ~ber die Beschreibung der Primzerlegung
seiner Werte (als Divisorcharakter) durch seinen Unendlichtyp einen Annullator der
Klassengruppe des CM-KSrpes liefert. Man kann etwa die klassisehen Kummer-Jacobi-
Relationen, welche die Annullation der Klassengruppe eines KreiskSrpers durch sog.
Stickelberger-Elemente beschreiben, unabh~ngig vom Stiekelbergerschen Satz auf diesem
Wege finden. Es ist jedoch vSllig unklar, ob man damit schon alle Annullatoren, bzw.
solche eines gewissen Typs gefunden hat. Ferner w~re es sehr wichtig, im Hinblick auf
die fundamentale Bedeutung der Kummer-Jacobi-Relationen fur die Arithmetik yon Kreis-
kSrpern, deren Analogon fHr beliebige CM-KSrper zu finden. Deshalb liegt es nahe, und
dies ist .u .a ein Anliegen dieser Arbeit, systematisch die geometrischen Bedin~ungen
an CM-Variet~ten auszuloten, welche zu Klassengruppenannullatoren f~hren.
In Kapitel I werden zun~chst die wichtigsten Ei~enschaften der GrSssencharaktere vom
Typ ° A bereitgestellt, die Gruppe U(~o) der Unendlichtypen ~nalysiert und der sog.
"Testcharakter" zu ein~n Unendlichtyp eingefNhrt° Des Nchl~ssellemma 2.1~ das die