A dictionary of modular threefolds [Elektronische Ressource] / vorgelegt von Christian Meyer

Sujets

• Mathematics
• Physik

Informations

• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2005
• Nombre de lectures: 7
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

A dictionary of modular threefolds
Dissertation zur Erlangung des Grades
\Doktor der Naturwissenschaften"
am Fachbereich Mathematik und Informatik
der Johannes Gutenberg-Universit at in Mainz,
vorgelegt von
Christian Meyer,
geboren in Mainz.
Mainz, den 22. Februar 2005D77 { Mainzer Dissertation3
Contents
Introduction 6
1 Arithmetic on Calabi{Yau threefolds 9
1.1 Calabi{Yau varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Arithmetic on Calabi{Yau varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Dimension = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Dimension 1: Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.2 2: K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Dimension 3: Calabi{Yau threefolds . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.1 Modularity of rigid Calabi{Yau threefolds . . . . . . . . . . . . . . . . . . 17
1.5.2 Modularity of non-rigid Calabi{Yau threefolds . . . . . . . . . . . . . . . 20
1.6 Construction of Calabi{Yau threefolds . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6.1 Ordinary double points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6.2 Threefolds with many nodes . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6.3 Higher singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7 Correspondences and twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7.1 Correspondences and relatives . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7.2 Relatives by construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.7.3 Twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.8 Computational matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.8.1 Computation of Hodge and Betti numbers . . . . . . . . . . . . . . . . . . 29
1.8.2 Algorithms for counting points . . . . . . . . . . . . . . . . . . . . . . . . 30
1.8.3 Computation of coe cien ts of modular forms . . . . . . . . . . . . . . . . 30
1.8.4 Hard- and software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
64 CONTENTS
2 Fibre products of elliptic surfaces 32
2.1 Examples of Schoen and Schutt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Relatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
43 Quintics in P 43
3.1 Schoen’s quintic and the standard family of quintics . . . . . . . . . . . . . . . . 43
3.2 Equations for the mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Hirzebruch’s quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Van Geemen’s and Werner’s quintics . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5 Consani’s and Scholten’s quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Van Straten’s -symmetric quintics . . . . . . . . . . . . . . . . . . . . . . . . . 566
3.7 The Barth-Nieto quintic and its double cover . . . . . . . . . . . . . . . . . . . . 60
4 Double octics 62
4.1 Cynk’s octic arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Arrangements of eight planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Six planes and a quadric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4 Four planes and two quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5 Four quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.6 Segre’s construction (squaring of coordinates) . . . . . . . . . . . . . . . . . . . . 102
4.7 Application to Kummer surfaces and other quartics . . . . . . . . . . . . . . . . . 103
4.8 Playing with cubic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.9 -symmetric quintics and Barth’s quintic with 15 cusps . . . . . . . . . . . . . . 1125
4.10 octics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175
4.11 Sarti’s Heisenberg-invariant surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 Other examples 131
5.1 A rigid complete intersection with small Euler number . . . . . . . . . . . . . . . 131
5.2 A family of nodal complete intersections . . . . . . . . . . . . . . . . . . . . . . . 134
5.3 Van Geemen’s and Werner’s complete intersections . . . . . . . . . . . . . . . . . 146
5.4 Nygaard’s and van Geemen’s intersection . . . . . . . . . . . . . . . . . 148
5.5 Libgober’s and Teitelbaum’s complete in . . . . . . . . . . . . . . . . . 150
55.6 An intersection of two cubics in P with 108 nodes . . . . . . . . . . . . . . . . . 1525
5.7 Verrill’s threefolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.8 Hulek’s and Verrill’s threefolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.9 Bernadara’s complete intersections . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.10 -symmetric in . . . . . . . . . . . . . . . . . . . . . . . . . 1596
5.11 Rodriguez-Villegas’ hypergeometric threefolds . . . . . . . . . . . . . . . . . . . . 161
6 Tables, correspondences, conclusions 164
6.1 Modular threefolds with small levels . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.1.1 Level 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.1.2 Level 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.1.3 Level 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.1.4 Level 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.1.5 Level 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.1.6 Level 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.1.7 Level 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.2 Modular threefolds with large levels . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.3 Hodge and Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.4 Bad primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.4.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.4.2 Powers of bad primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.4.3 Which newforms do occur? . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.5 Other aspects and questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
A Arrangements of eight planes 186
B Modular double octics 198
C Weight four newforms 225
D Weight two newforms 283
References 2906
Introduction
The proof of the Taniyama-Shimura Conjecture by A. Wiles et al. in the 1990s (cf. [15]), which
implied a proof of Fermat’s Last Theorem, has been met with approval from the mathematical
community and has even aroused great interest in the public (cf. [1], [95]). It connects, in a
very fascinating way, di eren t mathematical subjects, such as algebraic geometry and number
theory.
The two main mathematical theories involved are those of elliptic curves and of modular forms.
The Taniyama-Shimura conjecture relates the numbers of points on elliptic curves over nite
elds to Fourier coe cien ts of certain modular forms of weight two.
An elliptic curve is a special case of a so called Calabi{Yau manifold, namely a Calabi{Yau
manifold of dimension one. Calabi{Yau manifolds are of great importance in string theory,
a main branch of modern theoretical physics. It is a very natural task to try to extend the
results for elliptic curves to Calabi{Yau manifolds of higher dimension. Calabi{Yau manifolds
of dimension two are called K3 surfaces. Their arithmetic, i.e., their properties over nite elds,
has also been studied but we will take one further step forward and concentrate on Calabi{Yau
manifolds of dimension three, the so called Calabi{Yau threefolds.
The arithmetic of Calabi{Yau threefolds de ned over Q is mainly determined by the L-series
of their middle etale cohomology space. The dimension of this space is a positive even number
and can be used to classify Calabi{Yau threefolds. If the dimension is two then the threefold
allows no complex deformations and is therefore called rigid (and non-rigid otherwise). For a
rigid Calabi{Yau threefold X which is de ned over Q there is a precise conjecture about its
connection with modular forms. There should exist a newform of weight four for some Hecke
subgroup (N) the L-series of which agrees with the L-series of the middle cohomology of X.0
In this case X is called modular.
The conjecture has been checked in several examples before and there is also a partial general
result by Dieulefait and Manoharmayum (a modularity proof under mild restrictions concerning
the primes of bad reduction). It is rather di cult to construct rigid Calabi{Yau threefolds.
For non-rigid Calabi{Yau threefolds the situation becomes much more complicated. We expect
that the L-series of their middle cohomology is also determined by modul