Table Of ContentLecture Notes ni
Mathematics
Edited yb .A Dold and .B Eckmann
Subseries: Mathematisches Institut der Universit~.t und Max-Planck-lnstitut
fSr Mathematik, Bonn - vol. 7
Adviser: F. Hirzebruch
5021
B.Z. Moroz
Analytic Arithmetic
ni Algebraic Number Fields
galreV-regnirpS
nilreB grebledieH NewYork London siraP oykoT
Author
B.Z. Moroz
Max-Planck-lnstitut fLir Mathematik, Universit~.t Bonn
Gottfried-Claren-Str. 26, 5300 Bonn 3, Federal Republic of Germany
Mathematics Subject Classification (1980): 11 D57, 11 R39, 11 R42, 11 R44,
11 R45, 22C05
ISBN 3-540-16784-6 Springer-Verlag Berlin Heidelberg New York
ISBN 0-38?-16784-6 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data. Moroz, B.Z. arithmetic Analytic in algebraic
number fields. (Lecture notes in mathematics; "Subseries: 1205) Mathematisches lnstitut der
Universit&t und Max-Planck-lnstitut fur Mathematik, Bonn -vol. ? ". Bibliography: p. Includes index.
.1 Algebrnauimcb er theory. .I Title. .1I Series: Lecture notes ni mathematics (Springer-Verlag; 1205.
QA3.L28 no. 1205 [QA247] 510 [512'.74] 86-20335
ISBN 0-38?-16784-6 (U.S.)
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© Springer-Vertag Berlin Heidelberg 1986
Printed in Germany
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Introduction.
This book is an improved version of our memoir that appeared in Bonner
Mathematische Schriften, [64]. Its purpose is twofold: first, we give
a complete relatively self-contained proof of the theorem concerning
analytic continuation and natural boundary of Euler products (sketched
in Chapter III of [64]) and describe applications of Dirichlet series
represented by Euler products under consideration; secondly, we review
in detail classical methods of analytic number theory in fields of al-
gebraic numbers. Our presentation of these methods (see Chapter )I has
been most influenced by the work of E. Landau, [40], [42], E. Hecke,
[24], and A. Weil, [91] (cf. also [87]). In Chapter II we develop
formalism of Euler products generated by polynomials whose coefficients
lie in the ring of virtual characters of the (absolute) Weil group of
a number field and apply it to study scalar products of Artin-Weil L-
functions. This leads, in particular, to a solution of a long-standing
problem concerning analytic behaviour of the scalar products, or con-
volutions, of L-functions Hecke "mit Gr~ssencharakteren" (cf. [63] for
the history of this problem; one may regard this note as a r~sum~ of
Chapter II, if you like). Chapter III describes applications of those
scalar products to the problem of asymptotic distribution of integral
and prime ideals having equal norms and to a classical problem about
distribution of integral points on a variety defined by a system of
norm-forms. Chapter IV is designed to relate the contents of the book
to the work of other authors and to acknowledge our indebtedness to
these authors.
should like to record here my sincere gratitude to Professor P. Deligne
whose remarks and encouragement helped me to complete this work. This
book, as well as 46[ ,] has been written in the quiet atmosphere of the
Max-Planck-Institut fur Mathematik (Bonn). We are grateful to the
Director of the Institute Professor F. Hirzebruch for his hospitality
and support of our work. The author acknowledges the hospitality of
VI
the Mathematisches Institut Universit~t Z~rich, where parts of the
manuscript have been prepared.
Bonn-am-Rhein, im M~rz 1986.
Table of contents
Chapter I. Classical background.
§I. On the multidimensional arithmetic in the sense of
E. Hecke. p. I
§2. Group theoretic intermission, p. 10
~3. Weil's group and non-abelian L-functions. p. 19
~4. On character sums extended over integral ideals, p. 32
§5. On character sums extended over prime ideals, p. 14
~6. Consequences of the Riemann Hypothesis. p. 50
§7. Equidistribution problems, p. 60
Appendix .I Frobenius classes in Well's groups, p. 69
Appendix .2 Ideal classes and norm-forms, p. 72
Chapter II. Scalar product of L-functions.
§I. Definition and elementary properties of scalar products, p. 78
§2. Digression: virtual characters of compact groups, p. 87
~3. Analytic continuation of Euler products, p. 94
§4. The natural boundary of L(s,H). p. 99
§5. Explicit calculations related to scalar products, p. 107
§6. Proof of the theorem 4.2. p. 125
Chapter III. Ideals with equal norms and integral points
on norm-form varieties.
§I. On character sums extended over ideals having equal norms, p. 141
§2. Equidistribution of ideals with equal norms, p. 151
§3. Equidistribution of integral points in the algebraic
sets defined by a system of norm-forms. p. 160
Chapter IV. Remarks and comments. p. 168
Literature cited. p. 171
Index p. 177
Notations and conventions.
We shall use the following notations and abbreviations:
empty set
:= "is defined as"
A\B the set theoretic difference
the set of natural numbers. (including zero)
the ring of natural integers
the field of rational numbers
the field of real numbers
5+ the set of positive real numbers
the field of complex numbers
A the group of invertible elements in a ring A
the set of all the simple (continuous) characters
of a (topological) group G
a fixed algebraic closure of the field k
I denotes the unit element in any of the multiplicative
groups to be considered
{xIP(x)} is the set of objects x satisfying the property P(x)
card S, or simply IsI, stands for the cardinality of a finite set S;
is an extension of number fields: E ~ F
[E:F] denotes the degree of ElF
G(E[F) denotes the Galois group of a finite extension EIF
(a) is a principal ideal generated by
means divisor ~ divides
is the absolute norm, that is NE/~, of a divisor
in a number field E
lxI is the absolute value of a complex number x
,×,k stand for finite sequences (of a fixed length) of divisors,
characters, fields, etc.
Im is the image of the map
Vil
Ker is the kernel of the homomorphism
Re s is the real part of s in
Im s is the imaginary part of s in
fog denotes the composition of two maps, so that
(feg) )a( = f(g(a))
r(s) is the Euler's gamma-function
l.c.m. is the least common multiple
g.c.d. is the greatest common divisor
ol H denotes the restriction of a map P to the set H
c G denotes sometimes (the closure of) the commutator subgroup
of a (topological) group G
A@B is the tensor product of A and B
A~ B is the direct sum of A and B
References of the form : theorem 1.2.3, lemma 1.1, proposition 2,
corollary I.A2.1 mean theorem 3 in ~2 of Chapter I, lemma I in §I of
the same chapter, proposition 2 in the same paragraph and corollary I
in Appendix 2 of Chapter I, respectively; the same system is used for
references to numbered formulae. Relations proved under the assumption
that Riemann Hypothesis, Artin-Weil conjecture or Lindel~f Hypothesis
are valid shall be marked by the letters R, AW, L, respectively,
before their number.
Every paragraph is regarded as a distinct unit, a brief relatively self-
contained article; thus we try to be consistent in our notations through-
out a paragraph but not necessarily over the whole chapter. In the
first three chapters we avoid bibliographical and historical references
which are collected in the Chapter IV.
Chapter I. Classical background.
§I. On the multidimensional arithmetic in the sense of E. Hecke.
Let k be an algebraic number field of degree n = [k:~]. Consider
the following objects:
v is the ring of integers of k;
I S and 2 S are the sets of real and complex places of k respectively,
S I = U S $2;
O S is the set of prime divisors of k identified with the set of
non-archimedean valuations;
S:= O U S S is the set of all primes in k;
r j:= ISjl, j = 1,2, so that n I = + r 2r2;
kp is the completion of k at p for p 6 S;
Up is the group of units of kp for p 6 So;
w is the valuation function on k normalised by the condition
P P
w k( )s = w k( )~ = ~ p E So;
P P P
Io(k) is the monoid of integral ideals of k;
I(k) is the group of fractional ideals of k;
(~) = )~(P~p'" is the principal ideal generated by in k ,
P6S o
0~= ~ pWp (~)
we extend the valuation function w to I and write
P
PCS o
for RE ;I
Jk is the id~le group of k;
Ck:=
Jk/k~ is the id~le-class group of k, where * k is embedded
diagonally in Jk;
X:= K k is regarded as a n-dimensional ~-algebra. The group of
p~s P
units ~ v acts freely as a discrete group of transformations on the
multiplicative group X ,~ the action being given by
X +~ EX r X X E ,@ C C Vef
where k is embedded diagonally in X. Obviously,
I r r1+r 2
X ~ (2Z/22Z) x T r2 JR+ × , (I)
where T = {exp(2zi~) I0 < ~ < I} denotes the unit circle in ~. Let
m be the order of the maximal finite subgroup of k ; by a theorem
of Dirichlet,
~ r1+r2-1
v = ~ x ~/m~ (2)
One can show that, in accordance with (I) and )2( F
r
X*/v* ~ (m/2m) o JR+ x x ~ , )3(
where ° _ r < max {0,ri-I} and ~ is a real (n-1)-dimensional torus.
The diagonal embedding of k into X gives rise to a monomorphism
f : k~/v ~ ÷ X~/v ~
o
of the group of principal ideals of k into the group (3). Let
g: X~/v~ + 7 denote the natural projection map of X~/v ~ on the torus
~. The composition of these maps gof O can be continued to a homomor-
phism
f: I(k) + ~ , fip k g'fo ' )4(
where Pk := k~/v~. Let 4~ 6 Io(k) and let ~ c 1. S One defines a
subgroup
),444(I = {0Z 0~ 6 I I(k), Wp(0~) = I for p[4~, } p O 6 S
of I(k) and a subgroup
P(4~) = {Ca)]0~ 6 k ~, ~ --- I (4~), ~p(~) > 0 for p 6 ~ }
of Pk' where ~ denotes the natural embedding of k in k for
P P
p 6 S, and ~ := (4~,~). The ray class group
H(~) := I (4~)/P(4~)
is a finite group of order
[H(4~) I = h~(4~},
where
]H(I,~)I
h = , H(I,~) = I(k)/P k
are the class number and the class group of k respectively, and
• (~) := card (I(44~)NPk)/P(~).
For a smooth subset T of T and a ray class A in H(~) one
defines two functions
I(-;A,~) JR+ : + ~ ,
~(.;A,Y): 2+ ÷ NI ,
by letting
I(x;A,Y) = card {~I~6 A N Io, f(~) 6 T,[~ I < x}
