Convolution and Square on Abelian Groups

Explore advanced mathematical concepts in this lecture that constructs functions on finite abelian groups whose convolution square is proportional to their square. Learn how to interpret abelian groups as subgroups of abelian varieties with complex multiplication and discover the application of modularity properties of theta functions in this construction. Delve into sophisticated algebraic and analytic techniques that bridge group theory, algebraic geometry, and modular forms. Examine the theoretical foundations underlying the relationship between convolution operations and square functions on finite abelian structures. Gain insights into how complex multiplication properties of abelian varieties provide the framework for understanding these special functions and their remarkable proportionality properties.