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SL2(R) gives the student an introduction to the infinite
dimensional representation theory of semisimple Lie groups by
concentrating on one example - SL2(R). This field is of interest
not only for its own sake, but for its connections with other areas
such as number theory, as brought out, for example, in the work of
Langlands. The rapid development of representation theory over the
past 40 years has made it increasingly difficult for a student to
enter the field. This book makes the theory accessible to a wide
audience, its only prerequisites being a knowledge of real
analysis, and some differential equations.
This book analyzes the response of the Indonesian press to American
foreign policy during the administrations of Presidents Bush and
Obama. Situated in Southeast Asia, Indonesia is the world's fourth
most populous country and the largest Muslim nation, and as such is
a potentially vital economic and strategic partner to the US in the
21st century. Ever since Indonesian independence post World War II,
relations to the US have been marked by ups and downs. The author
argues that the way the Indonesian public perceives the world has
an impact on the national self-image that again heavily influences
national foreign affairs. For both the US and Indonesia, this is a
crucial moment in bilateral relations. This study explores
Indonesian media responses to American foreign policy by analyzing
more than 400 press articles. In the context of President Obama's
declared "pivot to Asia", both countries need to find a way to
foster better relations.
Kummer's work on cyclotomic fields paved the way for the
development of algebraic number theory in general by Dedekind,
Weber, Hensel, Hilbert, Takagi, Artin and others. However, the
success of this general theory has tended to obscure special facts
proved by Kummer about cyclotomic fields which lie deeper than the
general theory. For a long period in the 20th century this aspect
of Kummer's work seems to have been largely forgotten, except for a
few papers, among which are those by Pollaczek [Po], Artin-Hasse
[A-H] and Vandiver [Va]. In the mid 1950's, the theory of
cyclotomic fields was taken up again by Iwasawa and Leopoldt.
Iwasawa viewed cyclotomic fields as being analogues for number
fields of the constant field extensions of algebraic geometry, and
wrote a great sequence of papers investigating towers of cyclotomic
fields, and more generally, Galois extensions of number fields
whose Galois group is isomorphic to the additive group of p-adic
integers. Leopoldt concentrated on a fixed cyclotomic field, and
established various p-adic analogues of the classical complex
analytic class number formulas. In particular, this led him to
introduce, with Kubota, p-adic analogues of the complex L-functions
attached to cyclotomic extensions of the rationals. Finally, in the
late 1960's, Iwasawa [Iw 1 I] . made the fundamental discovery that
there was a close connection between his work on towers of
cyclotomic fields and these p-adic L-functions of Leopoldt-Kubota.
The small book by Shimura-Taniyama on the subject of complex multi
is a classic. It gives the results obtained by them (and some by
Weil) plication in the higher dimensional case, generalizing in a
non-trivial way the method of Deuring for elliptic curves, by
reduction mod p. Partly through the work of Shimura himself (cf.
[Sh 1] [Sh 2], and [Sh 5]), and some others (Serre, Tate, Kubota,
Ribet, Deligne etc.) it is possible today to make a more snappy and
extensive presentation of the fundamental results than was possible
in 1961. Several persons have found my lecture notes on this
subject useful to them, and so I have decided to publish this short
book to make them more widely available. Readers acquainted with
the standard theory of abelian varieties, and who wish to get
rapidly an idea of the fundamental facts of complex multi
plication, are advised to look first at the two main theorems,
Chapter 3, 6 and Chapter 4, 1, as well as the rest of Chapter 4.
The applications of Chapter 6 could also be profitably read early.
I am much indebted to N. Schappacher for a careful reading of the
manu script resulting in a number of useful suggestions. S. LANG
Contents CHAPTER 1 Analytic Complex Multiplication 4 I. Positive
Definite Involutions . . . 6 2. CM Types and Subfields. . . . . 8
3. Application to Abelian Manifolds. 4. Construction of Abelian
Manifolds with CM 14 21 5. Reflex of a CM Type . . . . .
ARCHIVE COpy DO NOT REMOVE The public in industrialized countries
shows a mounting concern about biological effects of electrical and
magnetic fields. As a result, experimental studies on this subject
are being published in increasing numbers throughout the world.
Prof. H. L. Konig, of the Technical University of Munich, West
Germany, a leading expert and pioneer in this field, has written an
authoritative text in a lucid style which makes the material also
accessible to lay readers. The book describes the effects of
natural as well as artificial electromagnetic energies covering the
en tire measurable frequency range from the highest frequencies,
x-rays, through microwaves, radio waves, and finally extremely low
frequency (ELF) waves. Cit ing the evidence from scientific studies
in various countries, Konig also appraises the biologic effects of
microwaves and high tension power lines, which have become
controversial issues in recent years. Other contributions to the
book have been made by Prof. Albert P. Krueger, University of
California, Berkeley, on air ionization effects and by the mete
orologist Walter Sonning on biometeorology, documenting the
influence of atmo spheric electrical currents on health and
disease. Moreover, the late Dr. Siegnot Lang, a former coworker of
Dr. Konig, has contributed to this book."
Diophantine problems represent some of the strongest aesthetic
attractions to algebraic geometry. They consist in giving criteria
for the existence of solutions of algebraic equations in rings and
fields, and eventually for the number of such solutions. The
fundamental ring of interest is the ring of ordinary integers Z,
and the fundamental field of interest is the field Q of rational
numbers. One discovers rapidly that to have all the technical
freedom needed in handling general problems, one must consider
rings and fields of finite type over the integers and rationals.
Furthermore, one is led to consider also finite fields, p-adic
fields (including the real and complex numbers) as representing a
localization of the problems under consideration. We shall deal
with global problems, all of which will be of a qualitative nature.
On the one hand we have curves defined over say the rational
numbers. Ifthe curve is affine one may ask for its points in Z, and
thanks to Siegel, one can classify all curves which have infinitely
many integral points. This problem is treated in Chapter VII. One
may ask also for those which have infinitely many rational points,
and for this, there is only Mordell's conjecture that if the genus
is :;;; 2, then there is only a finite number of rational points.
In the present book, we have put together the basic theory of the
units and cuspidal divisor class group in the modular function
fields, developed over the past few years. Let i) be the upper half
plane, and N a positive integer. Let r(N) be the subgroup of SL (Z)
consisting of those matrices == 1 mod N. Then r(N)\i) 2 is complex
analytic isomorphic to an affine curve YeN), whose compactifi
cation is called the modular curve X(N). The affine ring of regular
functions on yeN) over C is the integral closure of C j] in the
function field of X(N) over C. Here j is the classical modular
function. However, for arithmetic applications, one considers the
curve as defined over the cyclotomic field Q(JlN) of N-th roots of
unity, and one takes the integral closure either of Q j] or Z j],
depending on how much arithmetic one wants to throw in. The units
in these rings consist of those modular functions which have no
zeros or poles in the upper half plane. The points of X(N) which
lie at infinity, that is which do not correspond to points on the
above affine set, are called the cusps, because of the way they
look in a fundamental domain in the upper half plane. They generate
a subgroup of the divisor class group, which turns out to be
finite, and is called the cuspidal divisor class group."
It is possible to write endlessly on elliptic curves. (This is not
a threat.) We deal here with diophantine problems, and we lay the
foundations, especially for the theory of integral points. We
review briefly the analytic theory of the Weierstrass function, and
then deal with the arithmetic aspects of the addition formula, over
complete fields and over number fields, giving rise to the theory
of the height and its quadraticity. We apply this to integral
points, covering the inequalities of diophantine approximation both
on the multiplicative group and on the elliptic curve directly.
Thus the book splits naturally in two parts. The first part deals
with the ordinary arithmetic of the elliptic curve: The
transcendental parametrization, the p-adic parametrization, points
of finite order and the group of rational points, and the reduction
of certain diophantine problems by the theory of heights to
diophantine inequalities involving logarithms. The second part
deals with the proofs of selected inequalities, at least strong
enough to obtain the finiteness of integral points.
A belian Varieties has been out of print for a while. Since it was
written, the subject has made some great advances, and Mumford's
book giving a scheme theoretic treatment has appeared (D. Mum-
ford, Abelian Varieties, Tata Lecture Notes, Oxford University
Press, London, 1970). However, some topics covered in my book were
not covered in Mumford's; for instance, the construction of the
Picard variety, the Albanese variety, some formulas concern- ing
numerical questions, the reciprocity law for correspondences and
its application to Kummer theory, Chow's theory for the K/k-trace
and image, and others. Several people have told me they still found
a number of sections of my book useful. There- fore I thank
Springer-Verlag for the opportunity to keep the book in print. S.
LANG v FOREWORD Pour des simplifications plus subs tan- tielles, Ie
developpement futur de la geometrie algebrique ne saurait manquer
sans do ute d' en faire apparaitre. It is with considerable
pleasure that we have seen in recent years the simplifications
expected by Weil realize themselves, and it has seemed timely to
incorporate them into a new book. We treat exclusively abelian
varieties, and do not pretend to write a treatise on algebraic
groups. Hence we have summarized in a first chapter all the general
results on algebraic groups that are used in the sequel. They are
all foundational results.
The small book by Shimura-Taniyama on the subject of complex multi
is a classic. It gives the results obtained by them (and some by
Weil) plication in the higher dimensional case, generalizing in a
non-trivial way the method of Deuring for elliptic curves, by
reduction mod p. Partly through the work of Shimura himself (cf.
[Sh 1] [Sh 2], and [Sh 5]), and some others (Serre, Tate, Kubota,
Ribet, Deligne etc.) it is possible today to make a more snappy and
extensive presentation of the fundamental results than was possible
in 1961. Several persons have found my lecture notes on this
subject useful to them, and so I have decided to publish this short
book to make them more widely available. Readers acquainted with
the standard theory of abelian varieties, and who wish to get
rapidly an idea of the fundamental facts of complex multi
plication, are advised to look first at the two main theorems,
Chapter 3, 6 and Chapter 4, 1, as well as the rest of Chapter 4.
The applications of Chapter 6 could also be profitably read early.
I am much indebted to N. Schappacher for a careful reading of the
manu script resulting in a number of useful suggestions. S. LANG
Contents CHAPTER 1 Analytic Complex Multiplication 4 I. Positive
Definite Involutions . . . 6 2. CM Types and Subfields. . . . . 8
3. Application to Abelian Manifolds. 4. Construction of Abelian
Manifolds with CM 14 21 5. Reflex of a CM Type . . . . .
Diophantine problems represent some of the strongest aesthetic
attractions to algebraic geometry. They consist in giving criteria
for the existence of solutions of algebraic equations in rings and
fields, and eventually for the number of such solutions. The
fundamental ring of interest is the ring of ordinary integers Z,
and the fundamental field of interest is the field Q of rational
numbers. One discovers rapidly that to have all the technical
freedom needed in handling general problems, one must consider
rings and fields of finite type over the integers and rationals.
Furthermore, one is led to consider also finite fields, p-adic
fields (including the real and complex numbers) as representing a
localization of the problems under consideration. We shall deal
with global problems, all of which will be of a qualitative nature.
On the one hand we have curves defined over say the rational
numbers. Ifthe curve is affine one may ask for its points in Z, and
thanks to Siegel, one can classify all curves which have infinitely
many integral points. This problem is treated in Chapter VII. One
may ask also for those which have infinitely many rational points,
and for this, there is only Mordell's conjecture that if the genus
is :;;; 2, then there is only a finite number of rational points.
In the present book, we have put together the basic theory of the
units and cuspidal divisor class group in the modular function
fields, developed over the past few years. Let i) be the upper half
plane, and N a positive integer. Let r(N) be the subgroup of SL (Z)
consisting of those matrices == 1 mod N. Then r(N)\i) 2 is complex
analytic isomorphic to an affine curve YeN), whose compactifi
cation is called the modular curve X(N). The affine ring of regular
functions on yeN) over C is the integral closure of C j] in the
function field of X(N) over C. Here j is the classical modular
function. However, for arithmetic applications, one considers the
curve as defined over the cyclotomic field Q(JlN) of N-th roots of
unity, and one takes the integral closure either of Q j] or Z j],
depending on how much arithmetic one wants to throw in. The units
in these rings consist of those modular functions which have no
zeros or poles in the upper half plane. The points of X(N) which
lie at infinity, that is which do not correspond to points on the
above affine set, are called the cusps, because of the way they
look in a fundamental domain in the upper half plane. They generate
a subgroup of the divisor class group, which turns out to be
finite, and is called the cuspidal divisor class group."
It is possible to write endlessly on elliptic curves. (This is not
a threat.) We deal here with diophantine problems, and we lay the
foundations, especially for the theory of integral points. We
review briefly the analytic theory of the Weierstrass function, and
then deal with the arithmetic aspects of the addition formula, over
complete fields and over number fields, giving rise to the theory
of the height and its quadraticity. We apply this to integral
points, covering the inequalities of diophantine approximation both
on the multiplicative group and on the elliptic curve directly.
Thus the book splits naturally in two parts. The first part deals
with the ordinary arithmetic of the elliptic curve: The
transcendental parametrization, the p-adic parametrization, points
of finite order and the group of rational points, and the reduction
of certain diophantine problems by the theory of heights to
diophantine inequalities involving logarithms. The second part
deals with the proofs of selected inequalities, at least strong
enough to obtain the finiteness of integral points.
This book is a riotous, irreverent account of the people and events
that have shaped Britain. Always getting those kings and queens
confused? Never sure what happened when? Then you need this book.
Inside you'll find rip-roaring stories of power-mad kings,
executions, invasions, high treason, global empire building, and
forbidden love - not bad for a nation of stiff upper lips! *
Revised and expanded to include the historical parliamentary
elections of 2010 and the British mission in Afghanistan *
Accompanied by access to a timeline and 'Who's Who in British
History' section on dummies.com * This new edition contains an
8-page color insert so you can see who, what and where the ensuing
historical action takes place
The rich variety of Europe's history rolled into one thrilling
account. This book takes you on a fascinating journey through the
disasters, triumphs, people, power and politics that have shaped
the Europe we know today - and you'll meet some incredible
characters along the way! From Roman relics to Renaissance, World
Wars and Eurovision, European History For Dummies packs in the
facts alongside the fun and brings the past alive. * Accompanied by
access to a value-add timeline and 'Who's Who in European History'
section on dummies.com * This new edition contains an 8-page colour
insert so you can see who, what, and where the ensuing historical
action takes place.
This book analyzes the response of the Indonesian press to American
foreign policy during the administrations of Presidents Bush and
Obama. Situated in Southeast Asia, Indonesia is the world's fourth
most populous country and the largest Muslim nation, and as such is
a potentially vital economic and strategic partner to the US in the
21st century. Ever since Indonesian independence post World War II,
relations to the US have been marked by ups and downs. The author
argues that the way the Indonesian public perceives the world has
an impact on the national self-image that again heavily influences
national foreign affairs. For both the US and Indonesia, this is a
crucial moment in bilateral relations. This study explores
Indonesian media responses to American foreign policy by analyzing
more than 400 press articles. In the context of President Obama's
declared "pivot to Asia", both countries need to find a way to
foster better relations.
Die Auseinandersetzung im Umweltschutz tiber groBtechnische
Risiken, z. B. bei der friedlichen Nutzung der Kernenergie oder
neuerdings auch bei der Kohlenutzung (Saurer Regen) sind Ausdruck
einer veranderten BewuBtseinslage in der GeselI- schaft. Die
Technoiogie-Folgen mtissen starker beriicksichtigt werden.
Technische GroBschaden und Umweltkatastrophen find en erhOhte
Aufmerksamkeit in der Of- fentlichkeit. Sie werden als von Menschen
verursachte Gefahren nicht mehr schick- salhaft wie
Naturkatastrophen hingenommen. Verschiedene GroBtechnologien
zeichnen sich zwar durch eine hoch entwickelte, qualifizierte
Sicherheitstechnologie aus; das Gefahrdungspotential
groBtechnischer Systeme hat he ute jedoch GroBen- ordnungen
erreicht, daB Storfalle mit katastrophalen Konsequenzen flir Mensch
und Umwelt trotz aller Sicherheitsvorkehrungen als entfernte
Moglichkeiten prinzi- piell nicht ausgeschlossen werden konnen. Die
technische Entwicklung hat uns in neue Dimensionen des Umgangs mit
Energie, gefahrlichen Stoffen und anderen mit Gefahren verbundenen
System en geflihrt. Lange Zeit wurde kaum wahrgenommen, daB die
damit verbundene quantitative VergroBerung der Gefahren zu einer
neuen QualiHit technischer Risiken geflihrt hat. Ich sehe daher in
der Untersuchung von Risiko-und Sicherheitsfragen konventio- neller
und neuer Technologien im Rahmen der F + E-Politik der
Bundesregierung eine wichtige Aufgabe. Staatliche
ForschungsfOrderung muB dazu beitragen, technologische Entwick-
lungen in ihren Zusammenhangen und Auswirkungen zu erkennen, ihre
Chancen und Risiken abzuwagen und Entscheidungen tiber die Nutzung
von Technologien zu begriinden. Dieser Zieivorstellung dient der
Forderschwerpunkt "Risiko-und Sicherheitsfor- schung" beim
Bundesminister flir Forschung und Technologie. Forderungsziel ist
die Minderung der groBtechnischen Gefahren und die rationale
Analyse der ver- bleibenden Risiken. Der Schwerpunkt untersttitzt
damit die unter der Zielrichtung "Technologiefolgenabschatzung"
laufenden Arbeiten.
"Statistiken sind merkwurdige Dinge ...," dies wird so mancher
Mediziner denken, wenn er sich mit der Biometrie befasst. Sei es im
Rahmen seiner Ausbildung oder im Zuge wissenschaftlicher oder
klinischer Studien, Kenntnisse der Statistik und Mathematik sind
unentbehrlich fur die tagliche Arbeit des Mediziners. Ziel dieses
Lehrbuches ist es, den Mediziner systematisch an biometrische
Terminologie und Arbeitsmethoden heranzufuhren, um ihn schliesslich
mit den Grundlagen der Wahrscheinlichkeitsrechung vertraut zu
machen. Nach der Lekture dieses Buches halt der Leser ein Werkzeug
in den Handen, das ihm bei der Losung medizinscher Fragestellungen
hilft ebenso wie bei der Beschreibung von Ergebnissen
wissenschaftlicher Studien und naturlich bei der Doktorarbeit "
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