Siegel modular forms: some geometric applications
Sara Perna
Advisor:Riccardo Salvati Manni
Corso di Dottorato in Matematica Dipartimento di Matematica ‘’Guido Castelnuovo“ Università degli Studi di Roma ‘’La Sapienza“
INTRODUCTION
This thesis is devoted to the investigation of some aspects of the connection between the theory of Siegel modular forms and the study of the geometry of Siegel modular varieties. We will show how one can use this connection to study polarized abelian varieties and their moduli spaces. In order to understand abelian varieties one has to understand first complex tori since abelian varieties turns out to be complex tori that admit an immersion in some projective space. The simpler example of an abelian variety is an elliptic curve. We will present an introduction to the basic theory of complex tori and complex abelian varieties (see Chapter 1) in order to highlight the deep relationship between this subject and the theory of Siegel modular varieties. Indeed these varieties arise naturally as compactifications of moduli spaces of complex abelian varieties. We will mostly talk about Siegel modular forms as tools for the study of complex abelian varieties and their moduli spaces, but they also represent an interesting and rich subject in the theory of automorphic forms. We will develop the theory of Siegel modular forms in Chapter 2 where we will also give many examples of Siegel modular forms. These modular forms will have a prominent role in the exposition of the original results of the thesis which are mostly based on my papers [39], [8], [40]. Let H denote the Siegel space of degree g. This is the space of g g symmetric com- g ⇥ plex matrices with positive definite imaginary part. The group of integral symplectic matrices g := Sp(2g, Z) acts properly discontinuously on Hg as follows: