Convolution and square in abelian groups III

TL;DR

Convolution and square in abelian groups III extends the study of $d$-critical values on odd-order cyclic groups by constructing explicit $\lambda$-critical values $\lambda_0=\sqrt{a}\pm i\sqrt{b}$ via theta-function frameworks and showing these belong to $\mathcal B_d^o$, i.e. arise from functions with $\overline{\widehat{f}}=f$ (up to symmetry). The work leverages Jacobi theta functions, their modular transformation properties, and Fourier-transform identities to realize $f$ as an eigenfunction of a conjugate-FT action, yielding the key relation $f*f(2t)=\lambda f^2(t)$. It systematically analyzes Gaussian functions, Dirichlet characters, and Jacobi sums to map the structure of $\mathcal B_d$, providing explicit small-$d$ lists and a non-Weil example at $d=17$. The results connect harmonic analysis on finite groups with number-theoretic objects from CM theory and modularity of theta-functions, enriching the landscape of spectral properties for convolution-type equations on finite abelian groups.

Abstract

In the first paper we proved 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 an odd algebraic integer of norm d whose both real and imaginary part are square roots of integers. We show here that the function f can be chosen to be equal to the conjugate of its Fourier transform.

Theorems & Definitions (45)

• Proposition 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Theorem 1.8
• Proposition 2.1
• Remark 2.2