Convolution and square in abelian groups II

TL;DR

This paper investigates critical values on abelian groups of odd order, constructing many such values using abelian varieties with complex multiplication, and explores solutions to a specific functional equation.

Contribution

It introduces a method to construct critical values on abelian groups using abelian varieties with complex multiplication, expanding understanding of the functional equation.

Findings

Constructed numerous critical values for abelian groups of odd order.

Linked critical values to properties of abelian varieties with complex multiplication.

Provided new solutions to the functional equation on abelian groups.

Abstract

A critical value on an abelian group G of odd order d is a value $\lambda$ such that the functional equation f$\star$f (2 t) = $\lambda$f (t)^2 on G has a nonzero solution f. We construct many critical values by using abelian varieties with complex multiplication.