Arithmetic intersections on modular curves

161 Pages·2017·1.2 MB·English

Arithmetic intersections on modular curves
Dissertation
zur Erlangung des akademischen Grades
Doctor rerum naturalium
(Dr. rer. nat.)
im Fach Mathematik
eingereicht an der
Mathematisch-Naturwissenschaftlichen Fakultät
Humboldt-Universität zu Berlin
von
M.Sc. Miguel Daygoro Grados Fukuda
Präsidentin der Humboldt-Universität zu Berlin:
Prof. Dr.-Ing. Dr. Sabine Kunst
Dekan der Mathematisch-Naturwissenschaftlichen Fakultät:
Prof. Dr. Elmar Kulke
Gutachter:
1. Prof. Dr. Jürg Kramer (Humboldt-Universität zu Berlin)
2. Prof. Dr. Anna-Maria von Pippich (Technische Universität Darmstadt)
3. Prof. Dr. Bas Edixhoven (Universiteit Leiden)
Tag der mündlichen Prüfung: 07. Februar 2017

To my beloved wife
and my little angel

Acknowledgements
This thesis would not have been possible without the stimulating and constant
support of my research advisors and co-advisor. First of all, I would like to
express my sincere gratitude to Prof. Dr. Anna-Maria von Pippich, for her
immense patience, her insightful feedbacks, and for the tremendous amount of
time she has devoted in guiding me in many aspects through my PhD years.
I extend my deep gratitude to Prof. Dr. Jürg Kramer. I have benefited from
his experience and his clarity in many interesting mathematical discussions.
Thanks to the funding he coordinated for me I was able to work in an excellent
and inspiring atmosphere in HU Berlin and also I could travel to many summer
schools and conferences. Last but not least, I would like to thank my co-advisor
Prof. Dr. Bas Edixhoven whose help was fundamental to develop the geometric
part of this work. I am deeply indebted to all of them not only for the many
stimulating and fruitful mathematical discussions we had, but also for their
inquisitive view and enthusiasm in doing and writing mathematics which I have
found very inspiring.
I would like to thank the Algebra group at TU Darmstadt where part of this
research was done during the SoSe 2015. I want to thank S. Zemel, F. Völz,
Y. Li, and M. Schwagenscheidt for the helpful discussions and for the great
time we spent in Darmstadt.
I also thank Universiteit Leiden for hosting me for one semester. In particular,
I would like to express my gratitude to Prof. Dr. Peter Bruin for the stimulating
and productive mathematical discussions.
A warm thanks to my friends of the Arithmetic Geometry working group Ana
María Botero, Anilatmaja Aryasomayajula, Antareep Mandal, Barbara Jung,
and Giovanni De Gaetano; for having been a family to me during this time
away from my family.
Special thanks go to the secretary of the Institut für Mathematik in HU Berlin,
Frau Marion Thomma, for her invaluable help on innumerable occasions.
I am specially grateful to Gregor Bruns for carefully reading parts of the
manuscript and offering detailed comments and suggestions. His contribution
improved the final presentation of this work.
I also want to acknowledge Jaime Cuadros and Richard Gonzales for their
supportive friendship and appreciation towards me.
I heartily thank my loving and devoted parents Oscar and Elsa, and my sister
Carolina for their constant love, encouragement, and support.
Finally, a very special thank you goes to my wife Lucy Hirota. She has been
my strength and motivation. It is a pleasure to dedicate this thesis to her.

Abstract
An important invariant of modular curves is the arithmetic self-intersection
of the relative dualizing sheaf. On the minimal regular model of X(N) this
self-intersection is completely described by the usual intersection of some
distinguished vertical divisors (geometric contribution) and the evaluation of
the canonical Green’s function at certain cusps (analytic contribution). The
aim of this thesis is to determine each of these contributions in terms of the
level N and study the asymptotic behaviour of the self-intersection as N tends
to infinity.
Keywords:
Arakelov theory, moduli problems, modular curves, automorphic forms, canoni-
cal Green’s function, zeta functions.
i
ii
Zusammenfassung
EinewichtigeInvariantevonModulkurvenistdiearithmetischeSelbstschnittzahl
der relativ dualisierenden Garbe. Auf dem minimalen regulären Modell von
X(N) ist diese Selbstschnittzahl durch den gewöhnlichen Schnitt einiger aus-
gezeichneter vertikaler Divisoren (dem geometrischen Beitrag) und durch die
Auswertung der kanonischen Greenschen Funktion an einigen Spitzen (dem
analytischen Beitrag) vollständig festgelegt. Das Ziel dieser Arbeit ist es, je-
den dieser Beiträge in Abhängigkeit von der Stufe N zu bestimmen und das
asymptotische Verhalten der Selbstschnittzahl zu studieren, wenn die Stufe N
gegen unendlich geht.
Schlagwörter:
Arakelovtheorie, Modulkurven, automorphe Formen, kanonische Greensche
Funktion, Zetafunktionen.
iii
iv