The CM class number one problem for curves

109 Pages·2017·1.42 MB·English

by  Pinar Kilicer

The CM class number one problem for curves
Pinar Kilicer
To cite this version:
Pinar Kilicer. The CM class number one problem for curves. General Mathematics [math.GM]. Uni-
versité de Bordeaux; Universiteit Leiden (Leyde, Pays-Bas), 2016. English. ￿NNT: 2016BORD0046￿.
￿tel-01383309v2￿
HAL Id: tel-01383309
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`
THESE
pr´esent´ee a`
´
L’UNIVERSITE DE BORDEAUX
´ ´
ECOLE DOCTORALE DE MATHEMATIQUES ET
INFORMATIQUE
par Pınar Kılı¸cer
POUR OBTENIR LE GRADE DE
DOCTEUR
´
SPECIALITE : Math´ematiques Pures
The CM class number one problem
for curves
Soutenue le : 5 juillet 2016 a` Leiden
Devant la commission d’examen form´ee de :
ENGE, Andreas Professeur Universit´e de Bordeaux Directeur
STRENG, Marco Docteur Universiteit Leiden Directeur
TOP, Jaap Professeur Rijksuniversiteit Groningen Rapporteur
HESS, Florian Professeur Universit¨at Oldenburg Rapporteur

Contents
Preface viii
List of Notation x
1 Preliminaries 1
1.1 Global class field theory . . . . . . . . . . . . . . . . . . 1
1.2 CM fields and CM types . . . . . . . . . . . . . . . . . . 3
1.3 Abelian varieties . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Polarizations and the dual variety . . . . . . . . . 6
1.3.2 Complex abelian varieties . . . . . . . . . . . . . 9
1.4 Abelian varieties with complex multiplication . . . . . . 10
1.4.1 Construction of abelian varieties with CM . . . . 10
1.5 Polarized simple abelian varieties with CM . . . . . . . . 14
1.5.1 Classes of polarized simple abelian varieties with CM 16
1.5.2 The first main theorem of CM . . . . . . . . . . . 17
2 The CM class number one problem for curves of genus 2 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 The relative class number . . . . . . . . . . . . . . . . . 23
2.3 Non-normal quartic CM fields . . . . . . . . . . . . . . . 26
2.3.1 An effective bound for CM class number one non-
normal quartic fields . . . . . . . . . . . . . . . . 26
2.3.1.1 Almost all ramified primes are inert in F
and Fr . . . . . . . . . . . . . . . . . . 29
2.3.2 Enumerating the fields . . . . . . . . . . . . . . . 38
2.4 Cyclic quartic CM fields . . . . . . . . . . . . . . . . . . 43
v
3 The CM class number one problem for curves of genus 3 49
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Sextic CM fields containing an imaginary quadratic field 51
3.3 Cyclic sextic CM fields . . . . . . . . . . . . . . . . . . . 53
3.4 Non-normal sextic CM fields . . . . . . . . . . . . . . . . 56
4 Simple CM curves of genus 3 over Q 73
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Polarized CM abelian varieties over Q . . . . . . . . . . 74
4.3 Principally polarized simple CM abelian threefolds . . . 77
4.4 Genus-3 CM curve examples over Q . . . . . . . . . . . . 84
Bibliography 91
Summary 93
Samenvatting 95
R´esum´e 97
Acknowledgement 99
Curriculum Vitae 100
Preface
This thesis has four chapters and is organized as follows.
Chapter 1 is an introduction to abelian varieties and complex multi-
plication theory. It also contains facts from unramified class field theory.
We present facts that we will use in later chapters, including the main
theorem of complex multiplication. The results in this chapter are not
new and most are due to Shimura and Taniyama [40].
Chapter 2 is a joint work with Marco Streng that appeared as The
CM class number one problem for curves of genus 2 [18]. In Sections 2.3
and 2.4, we give a solution to the CM class number one problem for
curves of genus 2 (Theorems 2.3.15 and 2.4.5).
Chapter 3 deals with the CM class number one problem for curves
of genus 3 with a simple Jacobian. We give a partial solution to this
problem. We restrict ourselves to the case where the sextic CM field cor-
responding to such a curve contains an imaginary quadratic subfield. We
givethecompletelistofsuchsexticCMfieldsinTable3.1(unconditional)
and Tables 3.3–3.12 (under GRH).
Chapter4givesthecompletelistofsexticCMfieldsK forwhichthere
existprincipallypolarizedsimpleabelianthreefoldswithCMbyO with
K
rational field of moduli.
vii
List of Notation
O the ring of integers (maximal order) of a number field K
K
I the group of fractional ideals of a number field K
K
P the group of principal fractional ideals of a number
K
field K
Cl the quotient I /P
K K K
h the order of Cl
K K
F the group of totally positive elements of
(cid:29)0
a totally real number field F
P+ the group of principal ideals that are generated
K
by the elements of F
(cid:29)0
W the group of roots of unity of K
K
µ the order of the group of roots of unity W
K K
O× the unit group of O
K K
d the discriminant of K
K
K/F a number field extension of finite degree
Gal(K/F) the Galois group of K over F
f the finite part of the conductor of K/F
K/F
D the different of K over F, an ideal of O
K/F K
N the ideal norm from K to F
K/F
tr the trace from K to F
K/F
viii
List of Notation
t the number of primes in F that are ramified in K
K
h∗ the relative class number h /h of K/F
K K F
D the decomposition group of a prime ideal p⊂O
p K
I the inertia group of a prime ideal p⊂O
p K
(cid:16)K/F(cid:17) the Frobenius automorphism in Gal(K/F)
P
corresponding to a prime ideal P⊂O , page 2
K
Q the Hasse unit index [O× :W O×] of K/F, page 24
K K K F
K K the smallest field in Q that contains
1 2
the fields K and K (for K , K ⊂Q)
1 2 1 2
(K,Φ) a CM pair, where K is a CM field
and Φ is a CM type of K
(Kr,Φr) the reflex of (K,Φ)
Fr the maximal totally real subfield of Kr
N the type norm map from K to Kr, page 5
Φ
I (Φ) an ideal group generated by the elements a of I
0 K
such that N (a)=(α)∈P and αα∈Q, page 17
Φ Fr
A∗ the dual abelian variety of an abelian variety A
End(A) the ring of endomorphisms of A/k over k
End (A) End(A)⊗ Q
0 Z
P a polarized abelian variety of a CM type (K,Φ),
page 17
M the field of moduli of P| =(A,θ| ,C), page 75
J J J
f(P) an ideal in O determined by P, page 15
F
ix

Chapter 1
Preliminaries
ABSTRACT. In this chapter, we give the main ingredients
that we will use in the later chapters. This chapter contains
facts from class field theory, complex multiplication theory
and facts related to abelian varieties.
“All the truths of mathematics
are linked to each other, and
all means of discovering them
are equally admissible.”
Adrien-Marie Legendre
1.1 Global class field theory
In this section, we follow Neukirch [32].
Let K be a number field and O its ring of integers. We denote
K
places by p. A cycle or modulus of K is a formal product
m=Ypn(p),
p
where p runs over all places of K with n(p) ∈ Z such that n(p) = 0
≥0
for almost all places of K. Here n(p) is 0 or 1 if p is real, and 0 if p is
complex.
1