Theta functions play an important role in number theory. For example, they can be used to find meromorphic continuations for zeta and L-functions, or to count the number of representations of a positive integer as a sum of (a given number of) squares. Indeed, (many of) these theta functions appear to be global sections of line bundles on certain complex tori, which enables us to study theta functions from a geometric point of view, e.g., using cohomological methods. In this context, we will see that theta functions give rise to embeddings of (complex) abelian varieties (which we define to be complex tori admitting a positive definite line bundle) into projective space, proving that any abelian variety admits the structure of an algebraic variety. Finally, we shall construct moduli spaces of certain (polarized) abelian varieties and prove that they may be equipped with an algebraic structure as well. If time permits, we could further discuss (representations of) Heisenberg groups and/or Jacobians of curves.

The preliminary discussion takes place on Wednesday, February 10th, between 8-9 o'clock (sharp) via Zoom. Meeting-ID: 848 6577 8547. Password: Leray

A detailed description of the contents of the seminar will soon be available here: https://sites.google.com/view/lukas-prader/teaching

Griffiths, Harris: Principles of algebraic geometry.

Birkenhake, Lange: Complex Abelian Varieties.

Mumford: Abelian Varieties.

Kempf: Complex Abelian Varieties and Theta Functions.

Igusa: Theta functions.

• Organisational meeting/distribution of topics: The preliminary discussion takes place on
  Wednesday, February 10th, between 8-9 o'clock (sharp) via Zoom. Meeting-ID: 848 6577 8547.
  Password: Leray
• If you would like to participate in the seminar, then please let me know! My e-mail address:
  lukas.prader ''at'' ur.de
• Registration for course work/examination/ECTS: FlexNow

• Presentation: Giving a seminar talk of roughly 90 minutes

• Detailed written report of the seminar talk
• active participation

• Benotet:
  • O. g. Studienleistung und o. g. Prüfungsleistung; die Note ergibt sich aus der Prüfungsleistung
  • O. g. Studienleistung und o. g. Prüfungsleistung; die Note ergibt sich aus dem Seminarvortrag
• Unbenotet:
  • O. g. Studienleistung