Convolution and square in abelian groups I

TL;DR

This paper demonstrates the existence of special functions on cyclic groups of odd order with convolution properties linked to imaginary quadratic integers, using advanced techniques involving theta functions on elliptic curves.

Contribution

It establishes a novel connection between convolution squares on cyclic groups and imaginary quadratic integers via elliptic curve theta functions.

Findings

Existence of non-zero functions with specific convolution properties on cyclic groups.

Connection between convolution squares and imaginary quadratic integers of norm d.

Use of theta functions on elliptic curves with complex multiplication.

Abstract

We prove that on the cyclic groups of odd order d, there exist non zero functions whose convolution square f*f(2t) is proportional to their square f(t)^2 when the proportionality constant is given by an imaginary quadratic integer of norm d which is equal to 1 modulo 2. The proof involves theta functions on elliptic curves with complex multiplication.