Multiplicative series, modular forms, and Mandelbrot polynomials

Abstract

We say a power series ∑ n = 0 ∞ a n q n \sum _{n=0}^\infty a_n q^n is multiplicative if the sequence 1 , a 2 / a 1 , … , a n / a 1 , … 1,a_2/a_1,\ldots ,a_n/a_1,\ldots is so. In this paper, we consider multiplicative power series f f such that f 2 f^2 is also multiplicative. We find a number of examples for which f f is a rational function or a theta series and prove that the complete set of solutions is the locus of a (probably reducible) affine variety over C \mathbb {C} . The precise determination of this variety turns out to be a finite computational problem, but it seems to be beyond the reach of current computer algebra systems. The proof of the theorem depends on a bound on the logarithmic capacity of the Mandelbrot set.