Convolution and square in abelian groups I

Yves Benoist (1, 2)

1 CNRS - Centre National de la Recherche Scientifique
2 LMO - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: 2020 Math. subject class. Primary 11F03; Secondary 11F27 Functional Equation; Convolution; Cyclic groups; Elliptic curves; Complex multiplication; Theta functions; Modular curve

Domaines: Théorie des nombres [math.NT]

Yves Benoist. Convolution and square in abelian groups I. 2022. ⟨hal-03668240v2⟩