Title: Formal duality in finite cyclic groups,
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.