‎We introduce a higher dimensional Atkin-Lehner theory for‎ ‎Siegel-Parahoric congruence subgroups of $GSp(2g)$‎. ‎Old‎ ‎Siegel forms are induced by geometric correspondences on Siegel‎ ‎moduli spaces which commute with almost all local Hecke algebras‎. ‎We also introduce an algorithm to get equations for moduli spaces of‎ ‎Siegel-Parahoric level structures‎, ‎once we have equations for prime l...

An anti-holomorphic multiplication by the integers Od of a quadratic imaginary number field, on a principally polarized complex abelian variety AC is an action of Od on AC such that the purely imaginary elements act in an anti-holomorphic manner. The coarse moduli space XR of such A (with appropriate level structure) is shown to consist of finitely many isomorphic connected components, each of ...

Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders, whose number under a given maximal length generically has quadratic asymptotics in this length. Siegel–Veech constants are coefficients of these quadratic growth rates, and coincide for almost all su...

Consider the Hurwitz space parameterizing covers of P branched at four points. We study its intersection with divisor classes on the moduli space of curves. As applications, we calculate the slope of Teichmüller curves parameterizing square-tiled cyclic covers. In addition, we come up with a relation among the slope of Teichmüller curves, the sum of Lyapunov exponents and the Siegel-Veech const...

We show that the higher genus 4-point superstring amplitude is strongly constrained by the geometry of moduli space of Riemann surfaces. A detailed analysis leads to a natural proposal which satisfies several conditions. The result is based on the recently derived Siegel induced metric on the moduli space of Riemann surfaces and on combinatorial products of determinants of holomorphic abelian d...

Hilbert-Siegel varieties are moduli spaces for abelian varieties equipped with an action by an order OK in a fixed, totally real field K. As such, they include both the Siegel moduli spaces (use K = Q and the action is the standard one) and Hilbert-Blumenthal varieties (where the dimension of K is the same as that of the abelian varieties in question). In this paper we study certain phenomena a...

The moduli space of principally polarized abelian surfaces is parametrized by three Igusa functions. In this article we investigate a new way to evaluate these functions by using Siegel Eisenstein series. We explain how to compute the Fourier coefficients of certain Siegel modular forms using classical modular forms of halfintegral weight. One of the results in this paper is an explicit algorit...

We give a new method for generating genus 2 curves over a finite field with a given number of points on the Jacobian of the curve. We define two new invariants for genus 2 curves as values of modular functions on the Hilbert moduli space and show how to compute them. We relate them to the usual three Igusa invariants on the Siegel moduli space and give an algorithm to construct curves using the...

We derive the (g − 2)(g − 3)/2 linearly independent relations among the products of pairs in a basis of holomorphic abelian differentials in the case of compact non-hyperelliptic Riemann surfaces of genus g ≥ 4. By the Kodaira-Spencer map this leads to the modular invariant metric on the moduli space induced by the Siegel metric.