An Arithmetic-geometric Reciprocity between Theta Functions Attached to Real and Imaginary Quadratic Fields

We use the theta correspondence to construct classical holomorphic modular forms associated to ideal classes in quadratic number fields. These modular forms are theta functions that were originally introduced by Hecke in the 1920s andhave been investigated by several authors since. Our framework allows us to prove old and new results concerning the periods of these modular forms over certain geometric cycles defined by arithmetic data. In particular, we establish a reciprocity relationship between the periods of theta functions attached to ideal classes in real and imaginary quadratic fields. This provides an analogue of (and context for) Hecke’s discovery that certain periods of his imaginary quadratic theta functions are special values of classical Eisenstein series at CM points.

Department dc:contributor.department

Mathematics

Year dc:date.issued

2023

Author dc:creator

• Al-Faisal, Faisal

Advisor dc:contributor.advisor

• Kudla, Stephen

• modular forms
• number theory
• theta correspondence
• theta functions

Handle dc:identifier.uri

http://hdl.handle.net/1807/130103

OAI identifier oai:identifier

oai:utoronto.scholaris.ca:1807/130103