Mathematical Research Letters

Volume 2, Issue 3, May 1995 pp. 359-376.

A conjectured analogue of Dedekind's eta function for $K3$ surfaces

Authors: Jay Jorgenson and Andrey Todorov
Author institution: Yale University

Summary: 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 $\bold h$ which is of weight $12$ with respect to the action by $PSL(2, \bold Z)$. 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 \cite{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.

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