proceedings of the american mathematical society Volume 123, Number 4, April 1995 SUPERLACUNARYCUSP FORMS KEN ONO AND SINAI ROBINS (Communicated by Dennis A. Hejhal) Dedicated to Basil Gordon on the occasion of his retirement Abstract. Many researchers have studied Euler product identities of weight k = j and k = | , often related to the Jacobi Triple Product identity and the Quintuple Product identity. These identities correspond to theta series of weight k = j and k - | , and they exhibit a behavior which is defined as superlacunary. We show there are no eigen-cusp forms of integral weight which are superlacunary. For half-integral weight forms with k > | , we give a mild condition under which there are no superlacunary eigen-cusp forms. These results suggest the nonexistence of similar Euler-Product identities that arise from eigen-cusp forms with weight k / 5 or | . 1. Introduction The Dedekind //-function is defined by 00 r](x) = x^Yl(l-x"), 71=1 where x lies in the upper half plane ß? = {t| Im(r) > 0} and x —e2nix. We maintain this notation throughout the paper. The Dedekind //-function is a modular form of weight \ on the subgroup T0(242) of Y = SL2(Z), with a multiplier system. An //-product is defined to be any function f(x) of the form (i) /(T)=n^T)fä> S\N where r$ £ Z. This is a modular form of weight k = \ Y^ô\nrs with a multiplier system [9], [16]. An »/-polynomial is defined to be any finite linear combina- tion of //-products. The study of the Fourier coefficients of //-products and //-polynomials are related to many well-known number-theoretic functions, in- cluding partition functions and quadratic form representation numbers. They also arise from representations of the "monster" group [3] and the Mathieu group A/24 [15]. The //-products are building blocks of other modular forms in Received by the editors June 25, 1993. 1991 Mathematics Subject Classification. Primary 11F11,11F12 , 11F37; Secondary 11P81. Key words and phrases. Modular forms, lacunarity, superlacunarity. © 1995 American Mathematical Society 0002-9939/95 $1.00+ $.25 per page 1021 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use