Related papers: Cutting out arithmetic Teichmueller curves in genu…

We discuss Weber's formula which gives the quotient of two Thetanullwerte for a plane smooth quartic in terms of the bitangents. In particular, we show how it can easily be derived from the Riemann-Jacobi formula.

Algebraic curves in Hilbert modular surfaces that are totally geodesic for the Kobayashi metric have very interesting geometric and arithmetic properties, e.g. they are rigid. There are very few methods known to construct such algebraic…

We give a method for finding rational equations of genus 2 curves whose jacobians are abelian varieties $A_f$ attached by Shimura to normalized newforms $f \in S_2( \Gamma_0(N))$. We present all the curves corresponding to principally…

I give a conjectural generating function for the numbers of $\delta$-nodal curves in a linear system of dimension $\delta$ on an algebraic surface. It reproduces the results of Vainsencher for the case $\delta\le 6$ and Kleiman-Piene for…

The existence of theta function solutions of genus two for the ILW equation is established. A numerical example is also presented. The method basically goes along with the Krichever's construction of theta function solutions for soliton…

In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta…

The goal of this work is to give new quantitative results about the distribution of semi-arithmetic hyperbolic surfaces in the moduli space of closed hyperbolic surfaces. We show that two coverings of genus $g$ of a fixed arithmetic surface…

We study the moduli space of stable sheaves of Euler characteristic 1, supported on curves of arithmetic genus 2 contained in a smooth quadric surface. We show that this moduli space is rational. We give a classification of the stable…

Using Galois-Stiefel-Whitney classes of theta characteristics we show that over a totally real base field the moduli stack of smooth genus $g$ curves and the moduli stack of principally polarized abelian varieties of dimension $g$ have…

In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for…

Genus 2 curves are useful in cryptography for both discrete-log based and pairing-based systems, but a method is required to compute genus 2 curves such that the Jacobian has a given number of points. Currently, all known methods involve…

We study arithmetic intersections on twisted (quaternionic) Hilbert modular surfaces and Shimura curves over a real quadratic field. Our first main result is the determination of the degree of the top arithmetic Todd class of an arithmetic…

The explicit computation of the field of moduli of a closed Riemann surface is, in general, a difficult task. In this paper, for each even integer $k \geq 2$, we consider a suitable $2$-real parameter family of non-hyperelliptic pseudo-real…

Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves correspond to rational points on the Hilbert modular surfaces via their…

We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann…

Under some assumptions, we compute the Picard group of moduli of stable sheaves on Abelian surfaces. Considering relative moduli spaces, it is sufficient to consider the moduli of stable sheaves on the product of elliptic curves. By using…

For $4 \nmid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures. In particular, we determine the divisibility properties of the…

We outline a method to compute rational models for the Hilbert modular surfaces Y_{-}(D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in Q(sqrt{D}), via moduli…

The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein…

The theta characteristics on a Riemann surface are permuted by the induced action of the automorphism group, with the orbit structure being important for the geometry of the curve and associated manifolds. We describe two new methods for…