2/3 Rigid CoCycles & Singular Moduli | Jan Vonk, Leiden University
Channel: IAS | Park City Mathematics Institute | PCMI
Date: 2022-08-04

Lecture series: Singular moduli for real quadratic fields 
Lecture#2/3
Singular moduli are the special values of the modular j-invariant at imaginary quadratic arguments. They are algebraic integers with rich prime factorisations, and play a central role in the theory of complex multiplication. Guided by explicit computations, an analogue for real quadratic fields has recently taken shape, based on the notion of rigid meromorphic cocycles. This course will be an introduction to the theory of rigid cocycles. We will discuss the different arithmetic invariants they produce, and their relations to Hilbert's 12th problem, the Birch and Swinnerton-Dyer conjecture, and the work of Gross and Zagier. The focus will be on how explicit computations with rigid cocycles can be carried out, how they influenced the development of the theory, and the computational questions that lie ahead in the further development of this theory

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