Enriques Surfaces, Analytic Discriminants, and Borcherds's Φ Function

Abstract

In [Bor 96], Borcherds constructed a non-vanishing weight 4 modular form Φ on the moduli space of marked, polarized Enriques surface of degree 2 by considering the twisted denominator function of the fake monster Lie algebra associated to an automorphism of order 2 of the Leech lattice fixing an 8-dimensional subspace. In [JT 94] and [JT 96], we defined and studied a meromorphic (multi-valued) modular form of weight 2, which we call the K3 analytic discriminant, on the moduli space of marked, polarized, K3 surfaces of degree 2d; in certain cases, including when , where p k are distinct primes, our meromorphic form is actually a holomorphic form. Our construction involves a determinant of the Laplacian on a polarized K3 surface with respect to the Calabi-Yau metric together with the L 2 norm of the image of the period map with respect to a properly scaled holomorphic two form. Since the universal cover of any Enriques surface is a K3 surface, we can restrict the K3 analytic discriminant to the moduli space of degree 2 Enriques surfaces. The main result of this paper is the observation that the square of our degree 2 analytic discriminant, viewed as a function on the moduli space of degree 2 Enriques surfaces, is equal to the Borcherd's Φ function, up to a universal multiplicative constant. This result generalizes known results in the study of generalized Kac-Moody algebras and elliptic curves, and suggests further connections with higher dimensional Calabi-Yau varieties, specifically those which can be realized as complete intersections in some, possibly weighted, projective space.