We describe an algebra of meromorphic functions on the Siegel domain genus two which contains modular forms for arithmetic index six subgroup symplectic group, is closed under three canonical derivations domain. The main ingredients our study are moduli enhanced curves, Gauss-Manin connection and vector fields living such spaces.

Let A2(n) denote the quotient of the Siegel upper half space of degree two by ?2(n), the principal congruence subgroup of level n in Sp(4; Z). A2(n) is the moduli space of principally polarized abelian varieties of dimension two with a level n structure, and has a compacti-cation A2(n) rst constructed by Igusa. When n 3 this is a smooth projective algebraic variety of dimension three. In this w...

In a series of works [Bo3-5], Borcherds developed a theory of modular forms over domains of type IV which admits an infinite product expansion. Among such modular forms, Borcherds’s Φ-function ([Bo4]) has an interesting geometric background; It is a modular form on the moduli space of Enriques surfaces characterizing the discriminant locus. In his construction, Φ-function is obtained as the den...

Every elliptic curve is isomorphic to the double covering of the projective line P branching at four points. The set of the isomorphism classes of elliptic curves with marking of branching points can be identified with the configuration space M4pts of ordered four points on P. By considering their periods, one obtains a morphism per from M4pts to the set Mct of the isomorphism classes of comple...

We present a new criterion for the complex hyperbolicity of non-compact quotient X bounded symmetric domain. For each $$p \ge 1$$ , this gives precise condition under which subvarieties $$V \subset X$$ with $$\dim V p$$ are general type, and is p-measure hyperbolic. Then, we give several applications related to ball quotients, or Siegel moduli space principally polarized abelian varieties. In c...

Abelian surfaces over C with a polarisation of type (1, t), t a positive integer, are parametrised by a coarse moduli space At which is a quasiprojective variety. In this paper we shall concentrate on the case where t is a prime p ≥ 5, and show that for p sufficiently large (in fact p ≥ 173) any algebraic compactification of Ap is of general type. A few results similar to this are already known...

This paper studies probability density estimation on the Siegel space. The Siegel space is a generalization of the hyperbolic space. Its Riemannian metric provides an interesting structure to the Toeplitz block Toeplitz matrices that appear in the covariance estimation of radar signals. The main techniques of probability density estimation on Riemannian manifolds are reviewed. For computational...

A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. The simi...