Related papers: Resultants and the Borcherds Phi-function

We give a formula that relates the difference of the j-invariants with the Borcherds Phi-function, an automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor.

We classify all primitive embeddings of the lattice of numerical equivalence classes of divisors of an Enriques surface with the intersection form multiplied by 2 into an even unimodular hyperbolic lattice of rank 26. These embeddings have…

In his celebrated 1998 Inventiones paper, Borcherds constructed meromorphic automorphic forms Psi(F) for arithmetic subgroups associated to even integral lattices M of signature (n,2). The input to his construction is a vector valued weakly…

This is the abstruct of the revised paper. We study the equivariant analytic torsion for K3 surfaces with an anti-symplectic involution with the invariant lattice M (such a surface is called a 2-elementary K3 surface of type M in this…

We give a short and "classical" proof of Borcherds' theorem that the moduli space of Enriques surfaces is quasi-affine. The use of the Borcherds' product is replaced in our proof by an application of the Grothendieck-Riemann-Roch theorem.

We show that certain spaces of vector valued modular forms are isomorphic to spaces of scalar valued modular forms whose Fourier coefficients are supported on suitable progressions. As an application we give a very explicit description of…

The moduli space of cubic surfaces in complex projective space is known to be isomorphic to the quotient of the complex 4-ball by a certain arithmetic group. We apply Borcherds' techniques to construct automorphic forms for this group and…

A holomorphic torsion invariant of K3 surfaces with involution was introduced by the second-named author. In this paper, we completely determine its structure as an automorphic function on the moduli space of such K3 surfaces. On every…

A general method of finding functional determinants is presented that depends on the asymptotic behaviour of the resolvent. Its application to the case of a bounded trihedral corner for which the eigenvalues are known only implicitly is…

We classify Enriques surfaces with smooth K3 cover and finite automorphism group in arbitrary positive characteristic. The classification is the same as over the complex numbers except that some types are missing in small characteristics.…

We construct the moduli space of Enriques surfaces in positive characteristic and eventually over the integers, and determine its local and global structure. As an application, we show lifting of Enriques surfaces to characteristic zero.…

We calculate the dimension of the locus of elliptic surfaces over P^1 with a section and a given Picard number, in the corresponding moduli space.

We prove that, for an Enriques surface in odd characteristic, the automorphism group is finitely generated and it acts on the effective nef cone with a rational polyhedral fundamental domain. We also construct a smooth projective surface in…

In this paper, we contribute to previously known results on lattices constructed by algebraic function fields, or function field lattices in short. First, motivated by the non-well-roundedness property of certain hyperelliptic function…

We consider an equivariant analogue of a conjecture of Borcherds. Let $Y$ be a real $K3$ surface without real points. Let $g$ be a Ricci-flat Kaehler metric on $Y$ invariant under the complex conjugation. We shall prove that the equivariant…

Let $f$ be an automorphism of a complex Enriques surface $Y$ and let $p_f$ denote the characteristic polynomial of the isometry $f^*$ of the numerical N\'eron-Severi lattice of $Y$ induced by $f$. We apply a modification of McMullen's…

We classify Enriques involutions on a K3 surface, up to conjugation in the automorphism group, in terms of lattice theory. We enumerate such involutions on singular K3 surfaces with transcendental lattice of discriminant smaller than or…

The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of…

We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type $\frac{1}{4}(1,1)$. The moduli space of such log-Enriques surfaces with $k$ singular points is a modular variety…

The $i$-th eigenvalue $\lambda_i$ of the Laplace-Beltrami operator on a surface can be considered as a functional on the space of all Riemannian metrics of unit volume on this surface. Surprisingly only few examples of extremal metrics for…