A CONJECTURED ANALOGUE OF DEDEKIND'S ETA FUNCTION FOR K3 SURFACES

A fundamental formula in the study of elliptic functions is the product formula for Dedekind's eta function or, equivalently, for the holomorphic cusp form on the upper half plane h which is of weight 12 with respect to the action by PSL(2, Z). A related formula expresses the determinant of the Laplacian which acts on the space of smooth functions on an elliptic curve with a period of the elliptic curve and the Dedekind eta function. In [JT 94a], we constructed a holomorphic function on the moduli space of marked, polarized, algebraic K3 surfaces of fixed degree using determinants of Laplacians. The aim of this article is to state a conjecture which expresses a product formula for this holomorphic form. In addition, we will present speculative relations with the representation theory of the Mathieu group M-24, as well as state many other problems currently under investigation.