Convolution and Square in Abelian Groups I

We prove that the functional equation f ? f(2 t)=lambda f(t)(2) , for t in Z/dZ with d odd, admits a nonzero solution f if lambda=root a + i root b with a, b positive integers such that a + b = d and a equivalent to(d+1)(2)/4 mod 4. The proof involves theta functions on elliptic curves with complex multiplication.