# Formal duality in finite cyclic groups

## TL;DR

This paper proves a conjecture about the absence of non-trivial formally dual pairs in most finite cyclic groups, using advanced algebraic and number-theoretic methods.

## Contribution

It confirms the conjecture for a broad class of cyclic groups, extending previous results and employing diverse mathematical tools.

## Key findings

- Confirmed the conjecture for cyclic groups of order a product of two prime powers.
- Extended the validity of the conjecture to groups with orders satisfying specific prime divisibility conditions.
- Utilized field descent, polynomial methods, and cyclotomic field number theory in proofs.

## Abstract

The notion of formal duality in finite Abelian groups appeared recently in relation to spherical designs, tight sphere packings, and energy minimizing configurations in Euclidean spaces. For finite cyclic groups it is conjectured that there are no primitive formally dual pairs besides the trivial one and the TITO configuration. This conjecture has been verified for cyclic groups of prime power order, as well as of square-free order. In this paper, we will confirm the conjecture for other classes of cyclic groups, namely almost all cyclic groups of order a product of two prime powers, with finitely many exceptions for each pair of primes, or whose order $N$ satisfies $p\mid\!\mid N$, where $p$ a prime satisfying the so-called self-conjugacy property with respect to $N$. For the above proofs, various tools were needed: the field descent method, used chiefly for the circulant Hadamard conjecture, the techniques of Coven & Meyerowitz for sets that tile $\mathbb{Z}$ or $\mathbb{Z}_N$ by translations, dubbed herein as the polynomial method, as well as basic number theory of cyclotomic fields, especially the splitting of primes in a given cyclotomic extension.