Polynomial identities between Hecke eigenforms

Abstract

In this paper, we study solutions to h = af 2 + bf g + g 2 , where f, g, h are Hecke newforms with respect to Γ 1 (N ) of weight k > 2 and a, b = 0. We show that the number of solutions is finite for all N . Assuming Maeda's conjecture, we prove that the Petersson inner product f 2 , g is nonzero, where f and g are any nonzero cusp eigenforms for SL 2 (Z) of weight k and 2k, respectively. As a corollary, we obtain that, assuming Maeda's conjecture, identities between cusp eigenforms for SL 2 (Z) of the form X 2 + n i=1 α i Y i = 0 all are forced by dimension considerations. We also give a proof using polynomial identities between eigenforms that the j-function is algebraic on zeros of Eisenstein series of weight 12k.