ENRIQUES SURFACES, ANALYTIC DISCRIMINANTS, AND BORCHERDSS PHI-FUNCTION

In [Bor 96], Borcherds constructed a non-vanishing weight 4 modular fo rm Phi on the moduli space of marked, polarized Enriques surface of de gree 2 by considering the twisted denominator function of the fake mon ster 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 weigh t 2, which we call the K3 analytic discriminant, on the moduli space o f marked, polarized, K3 surfaces of degree 2d; in certain cases, inclu ding when d = Pi p(k), where p(k) are distinct primes, our meromorphic form is actually a holomorphic form. Our construction involves a dete rminant 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 peri od map with respect to a properly scaled holomorphic two form. Since t he universal cover of any Enriques surface is a K3 surface, we can res trict the K3 analytic discriminant to the moduli space of degree 2 Enr iques 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 Bo rcherd's Phi function, up to a universal multiplicative constant. This result generalizes known results in the study of generalized Kac-Mood y 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, proj ective space.