Evaluating zeta functions of abelian number fields at negative integers

135 Pages·2010·0.57 MB·English

Evaluating Zeta Functions of Abelian Number
Fields at Negative Integers
Dylan Attwell-Duval
Master of Science
Department of Mathematics and Statistics
McGill University
Montreal, Quebec
December, 2009
A thesis submitted to McGill University in partial fulfillment of the
requirements of the degree of Master of Science
Copyright (cid:13)c Dylan Attwell-Duval, 2009

Acknowledgements
First and foremost I would like to thank my supervisor, Professor Eyal
Goren. His guidance and encouragement has been outstanding for my entire
tenure at McGill thus far and I would not have been able to complete this
work without his help. Additionally I would like to thank the following pro-
fessors who had an especially great impact on my undergraduate academic
career and without whom I would not be here today: Prof. Almut Buchard,
Prof. Stephen Kudla and Prof. Dror Bar-Natan.
I would also like to thank all the generous philanthropists who donate
to McGill or indeed any academic institution, for without your support
the pursuit of knowledge for a living would be all but an impossibility. In
particular, I have personally benefitted from the donations of Max Binz and
Lorne Trottier during my stay at McGill.
Finally I would like to thank all four of my parents. My success is
nothing more then the fruition of your labours.
iii
Abstract
In this thesis we study abelian number fields and in particular their
zeta functions at the negative integers. The prototypical examples of abelian
number fields are the oft-studied cyclotomic fields, a topic upon which many
texts have been almost exclusively dedicated to (see for example [26] or
nearly any text on global class field theory).
We begin by building up our understanding of the characters of finite
abelian groups and how they are related to Dedekind zeta functions. We
then use tools from number theory such as the Kronecker-Weber theorem
and Bernoulli numbers to find a simple algorithm for determining the values
of these zeta functions at negative integers. We conclude the thesis by
comparing the relative complexity of our method to two alternative methods
that use completely different theoretical tools to attack the more general
problem of non-abelian number fields.
iv
Abr´eg´e
Dans cette th`ese nous ´etudions les corps de nombres ab´eliens et en
particulier leurs fonctions zeta aux entiers n´egatifs. Les exemples-type de
corps de nombres ab´eliens sont les corps cyclotomiques que l’on ´etudie
fr´equemment, un sujet auquel de nombreux textes ont ´et´e enti`erement
consacr´es (voir par exemple [26] ou presque tous les textes sur la th´eorie
globale des corps de classes).
Nous commen¸cons par construire notre comprehension des caract`eres
des groupes ab´eliens finis et de ce qui les lie aux fonctions zeta de Dedekind.
Ensuite nous utilisons des outils de th´eorie des nombres comme le th´eor`eme
de Kronecker-Weber et les nombres de Bernouilli pour trouver un algorithme
simple pour d´eterminer les valeurs de ces fonctions zeta aux entiers n´egatifs.
Nous concluons la th`ese en comparant la complexit´e relative de notre
m´ethode a` deux m´ethodes alternatives qui utilisent des outils th´eoriques
compl`etement diff´erents pour attaquer le probl`eme plus g´en´eral des corps de
nombres non-ab´eliens.
v
TABLE OF CONTENTS
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Abr´eg´e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Dirichlet Characters and Finite Abelian Extensions . . . . . . . . . 8
1.1 Characters of Finite Abelian Groups . . . . . . . . . . . . . 8
1.2 Dirichlet Characters . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Zeta Functions and L-Functions . . . . . . . . . . . . . . . . 18
1.4 Extending L-Functions . . . . . . . . . . . . . . . . . . . . . 25
1.5 Odds and Ends . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 Tools from Number Theory . . . . . . . . . . . . . . . . . . . . . . 38
2.1 Fractional Ideals . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Differents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Ramification Groups . . . . . . . . . . . . . . . . . . . . . . 46
3 The Kronecker-Weber Theorem . . . . . . . . . . . . . . . . . . . . 55
4 Zeta Functions at Negative Integers for Abelian Number Fields . . 64
4.1 Determining Characters . . . . . . . . . . . . . . . . . . . . 64
4.2 Evaluating Abelian Zeta Functions . . . . . . . . . . . . . . 71
4.3 Evaluating Running Times . . . . . . . . . . . . . . . . . . . 81
4.4 A Final Result . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 A Functional Equation Approach . . . . . . . . . . . . . . . . . . . 86
6 A Modular Function Approach . . . . . . . . . . . . . . . . . . . . 95
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A Computational Algorithms . . . . . . . . . . . . . . . . . . . . . . . 109
vi
B Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
vii
List of Tables
Table page
4–1 Zeta Values for Real Quadratic Fields . . . . . . . . . . . . . . . 74
4–2 Zeta Values for Maximal Real Subfields of Cyclotomic Fields . . 75
4–3 Zeta Values for Totally Real Fields with Galois Group Z/3Z . . . 76
4–4 Zeta Values for Totally Real Fields with Galois Group Z/4Z . . . 77
4–5 Zeta Values for Totally Real Fields with Galois Group V . . . . 78
4
4–6 Zeta Values for Totally Real Fields with Galois Group Z/5Z or
Z/6Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4–7 Number of Fields for which ζ (1−k) is an Integer . . . . . . . . 80
L
4–8 Condition C(L,p) Verification . . . . . . . . . . . . . . . . . . . 85
viii
Introduction
The end result of this project is the development of an efficient algorithm
that researchers in the field can use to evaluate Dedekind zeta functions of
abelian number fields at negative integers. This should serve the community
well as there appears to be a distinct lack of numerical data regarding zeta
values at negative integers, despite there being great overall interest in these
numbers. Indeed the only known similar work comes from [12] where Goren
uses PARI to calculate these values. Goren notes that problems do arise with
his method as PARI is limited by the precision of its analytic estimations, and
he produces an example where PARI returns an incorrect value for a certain
zeta function at −23. The work of this thesis expands on the work of Goren in
the context of abelian number fields (although there is some discussion regard-
ing the more general case) and approaches the problem from the point of view
of L-functions, which ends up circumventing the issue of analytic estimation
and precision. This allows researchers interested in calculating these values to
be confident in the results for relatively large negative integers and also to be
able to compute these values quickly. Indeed, using the same zeta function
that PARI was unable to evaluate at −23 through analytic approximation,
the author was able to evaluate at −199 in under 1.3 seconds and the fraction
returned had a denominator satisfying the necessary prime divisor conditions
(see Chapter 5) which suggests the legitimacy of the returned value.
The main motivation for the thesis problem is the fact that the values of
these zeta functions at negative integers have a habit of popping up in many
1
different sections of number theory as part of some intrinsic property that
people may wish to know.
One common example, as we shall see in section 6 of this text, is that they
appear in the coefficients of certain normalized Eisenstein series of Hilbert
modular forms. Another place in the theory of modular forms in which these
values occur is calculating volumes of certain orbifolds that arise in the study
of Siegel modular forms. For instance, we have the symplectic group Sp(2g,Z)
acting on the Siegel upper half plane H with volume measured by a metric
g
generalizing the Poincar´e metric. If we normalize the volume to give the Euler
characteristic of this space, we have
vol(Sp(2g,Z)\H ) = ζ(−1)ζ(−3)...ζ(1−2g)
g
where ζ is the usual Riemann-zeta function, the most basic example of a
Dedekind zeta function (see [25]).
Another example arises in the theory of motives. In [4], ζ (1−n) appears
K
in the volume of hyperbolic (2n−1)-simplices defined over K. More generally,
the author studies various covolumes arising from products of 2m hyperbolic
planes modulo the action of certain arithmetic groups. In the formula given
for such volumes, we see ζ (−1)ζ (−3)...ζ (1 − 2m) appearing, similar to
K K K
the result above.
A final motivating example comes from the field of Hilbert modular sur-
faces. If K is a real quadratic field, then [24] shows the volume of Γ \H2
K 1
as 2ζ (−1) where Γ is the Hilbert modular group associated to K. This is
K K
particularly interesting because it is also shown (see pg. 72 of [24]) that the
dimension of certain cusp forms S associated with a Hilbert modular group
k
2