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# Class numbers of orders in quartic fields [Elektronische Ressource] / vorgelegt von Mark Pavey

149 pages
Class Numbers of OrdersinQuartic FieldsDissertationder Fakult˜ at fur˜ Mathematik und Physikder Eberhard-Karls-Universit˜ at Tubingen˜zur Erlangung des Grades eines Doktors der Naturwissenschaftenvorgelegt vonMark Paveyaus Cheltenham Spa, UK2006Mundliche Prufung:˜ 11.05.2006Dekan: Prof.Dr.P.Schmid1.Berichterstatter: Prof.Dr.A.Deitmar2.Berich Prof.Dr.J.HausenDedicated to my parents:Deryk and Frances Pavey,who have always supported me.AcknowledgmentsFirst of all thanks must go to my doctoral supervisor Professor Anton Deit-mar, without whose enthusiasm and unfailing generosity with his time andexpertise this project could not have been completed.The ﬂrst two years of research for this project were undertaken at ExeterUniversity, UK, with funding from the Engineering and Physical Sciences Re-search Council of Great Britain (EPSRC). I would like to thank the membersof the Exeter Maths Department and EPSRC for their support. Likewise Ithank the members of the Mathematisches Institut der Universit˜ at Tubingen,˜where the work was completed.Finally, I wish to thank all the family and friends whose friendship andencouragement have been invaluable to me over the last four years. I haveto mention particularly the Palmers, the Haywards and my brother Phill inSeaton, and those who studied and drank alongside me: Jon, Pete, Dave,Ralph, Thomas and the rest.
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Class Numbers of Orders
in
Quartic Fields
Dissertation
der Fakult˜ at fur˜ Mathematik und Physik
der Eberhard-Karls-Universit˜ at Tubingen˜
zur Erlangung des Grades eines Doktors der Naturwissenschaften
vorgelegt von
Mark Pavey
aus Cheltenham Spa, UK
2006Mundliche Prufung:˜ 11.05.2006
Dekan: Prof.Dr.P.Schmid
1.Berichterstatter: Prof.Dr.A.Deitmar
2.Berich Prof.Dr.J.HausenDedicated to my parents:
Deryk and Frances Pavey,
who have always supported me.Acknowledgments
First of all thanks must go to my doctoral supervisor Professor Anton Deit-
mar, without whose enthusiasm and unfailing generosity with his time and
expertise this project could not have been completed.
The ﬂrst two years of research for this project were undertaken at Exeter
University, UK, with funding from the Engineering and Physical Sciences Re-
search Council of Great Britain (EPSRC). I would like to thank the members
of the Exeter Maths Department and EPSRC for their support. Likewise I
thank the members of the Mathematisches Institut der Universit˜ at Tubingen,˜
where the work was completed.
Finally, I wish to thank all the family and friends whose friendship and
encouragement have been invaluable to me over the last four years. I have
to mention particularly the Palmers, the Haywards and my brother Phill in
Seaton, and those who studied and drank alongside me: Jon, Pete, Dave,
Ralph, Thomas and the rest. Special thanks to Paul Smith for the Online
Chess Club, without which the whole thing might have got done more quickly,
but it wouldn’t have been as much fun.
5Contents
Introduction iii
1 Euler Characteristics and Inﬂnitesimal Characters 1
1.1 The unitary duals of SL (R) and SL (R) . . . . . . . . . . . . 12 4
1.2 Euler characteristics . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Euler-Poincar¶e functions . . . . . . . . . . . . . . . . . . . . . 9
1.4 Euler c in the case of SL (R) . . . . . . . . . . . 154
1.5 The unitary dual of K . . . . . . . . . . . . . . . . . . . . . 19M
1.6 Inﬂnitesimal characters . . . . . . . . . . . . . . . . . . . . . . 21
2 Analysis of the Ruelle Zeta Function 28
2.1 The Selberg trace formula . . . . . . . . . . . . . . . . . . . . 30
2.2 The Selberg zeta function . . . . . . . . . . . . . . . . . . . . 32
2.3 A functional equation for Z (s) . . . . . . . . . . . . . . . . 37P;
2.4 The Ruelle zeta function . . . . . . . . . . . . . . . . . . . . . 43
2.5 Spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6 The Hochschild-Serre spectral sequence . . . . . . . . . . . . . 48
2.7 Contribution of the trivial representation . . . . . . . . . . . . 51
2.8 Con of the othertations . . . . . . . . . . . . 53
3 A Prime Geodesic Theorem for SL (R) 564
3.1 Analytic properties of R (s) . . . . . . . . . . . . . . . . . . 57¡;
~3.2 Estimating ˆ(x) and ˆ(x) . . . . . . . . . . . . . . . . . . . . 61
3.3 The Wiener-Ikehara Theorem . . . . . . . . . . . . . . . . . . 66
3.4 The Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Estimating ˆ (x) . . . . . . . . . . . . . . . . . . . . . . . . . 76n
13.6 ˆ (x) . . . . . . . . . . . . . . . . . . . . . . . . . 79
13.7 Estimating …(x), …~(x) and … (x) . . . . . . . . . . . . . . . . . 81
iii
4 Division Algebras of Degree Four 85
4.1 Central simple algebras and orders . . . . . . . . . . . . . . . 85
4.2 Division algebras of degree four . . . . . . . . . . . . . . . . . 87
4.3 Subﬂelds of M(Q) generated by ¡ . . . . . . . . . . . . . . . . 89
4.4 Field and order embeddings . . . . . . . . . . . . . . . . . . . 90
4.5 Counting order emb . . . . . . . . . . . . . . . . . . . 96
5 Comparing Geodesics and Orders 108
5.1 Regular geo . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2 Class numbers of orders in totally complex quartic ﬂelds . . . 117Introduction
In this thesis we present two main results. The ﬂrst is a Prime Geodesic
Theorem for compact symmetric spaces formed as a quotient of the Lie group
SL (R). The second is an application of the Prime Geodesic Theorem to4
prove an asymptotic formula for class numbers of orders in totally complex
quartic ﬂelds with no real quadratic subﬂeld. Before stating our results we
give some background.
Let D be the set of all natural numbers D · 0; 1 mod 4 with D not a
square. Then D is the set of all discriminants of orders in real quadratic
ﬂelds. For D2 D the set ( )p
x +y DO = :x·yD mod 2D
2
p
is an order in the real quadratic ﬂeld Q( D) with discriminant D. As D
varies, O runs through the set of all orders of real quadratic ﬂelds. ForD
D 2 D let h(O ) denote the class number and R(O ) the regulator of theD D
order O . It was conjectured by Gauss ([22]) and proved by Siegel ([55])D
that, as x tends to inﬂnity
2 3=2X … x
h(O )R(O ) = +O(x logx);D D
18‡(3)
D2D
D•x
where ‡ is the Riemann zeta function.
For a long time it was believed to be impossible to separate the class
number and the regulator in the summation. However, in [51], Theorem 3.1,
Sarnak showed, using the Selberg trace formula, that as x!1 we have
2xX e
h(O )» :D
2x
D2D
R(O )•x
D
iiiIntroduction iv
More sharply, X ¡ ¢
3x=2 2h(O ) =L(2x) +O e xD
D2D
R(O )•xD
as x!1, where L(x) is the functionZ x te
L(x) = dt:
t1
Sarnak established this result by identifying the regulators with lengths of
closed geodesics of the modular curve SL (Z)nH, where H denotes the upper2
half-plane, and using the Prime Geodesic Theorem for this Riemannian sur-
face. Actually Sarnak proved not this result but the analogue whereh(O) is
replaced by the class number in the narrower sense and R(O) by a \regula-
tor in the narrower sense". But in Sarnak’s proof the group SL (Z) can be2
replaced by PGL (Z) giving the above result.2
The Prime Geodesic Theorem in this context gives an asymptotic formula
for the number of closed geodesics on the surface SL (Z)nH with length less2
than or equal tox> 0. This formula is analogous to the asymptotic formula
for the number of primes less than x given in the Prime Number Theorem.
The Selberg zeta function (see [52]) is used in the proof of the Prime Geodesic
Theorem in a way analogous to the way the Riemann zeta function is used in
the proof of the Prime Number Thoerem (see [9]). The required properties of
the Selberg zeta function are deduced from the Selberg trace formula ([52]).
It seems that following Sarnak’s result no asymptotic results for class
numbers in ﬂelds of degree greater than two were proven until in [15], Theo-
rem 1.1, Deitmar proved an asymptotic formula for class numbers of orders
in complex cubic ﬂelds, that is, cubic ﬂelds with one real embedding and one
pair of complex conjugate embeddings. Deitmar’s result can be stated as
follows.
Let S be a ﬂnite set of prime numbers containing at least two elements
and let C(S) be the set of all complex cubic ﬂelds F such that all primes
p 2 S are non-decomposed in F. For F 2 C(S) let O (S) be the set ofF
all isomorphism classes of orders in F which are maximal at all p 2 S, ie.
are such that the completionO =O› Z is the order of the ﬂeldp p
F = F › Q for all p 2 S. Let O(S) be the union of all O (S), where Fp p F
ranges over C(S). For a ﬂeld F 2C(S) and an orderO2O (S) deﬂneFY
‚ (O) =‚ (F) = f (F);S S p
p2S