Related papers: Theta functions and algebraic curves with automorp…

A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Theta_s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set…

In algebraic geometry, enumerating or finding superspecial curves in positive characteristic $p$ is important both in theory and in computation. In this paper, we propose feasible algorithms to enumerate or find superspecial hyperelliptic…

We find equations for the higher dimensional analogue of the modular curve X_0(3) using Mumford's algebraic formalism of algebraic theta functions. As a consequence, we derive a method for the construction of genus 2 hyperelliptic curves…

Suppose $X$ is a hyperelliptic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p=2$. We prove that the de Rham cohomology of $X$ decomposes into pieces indexed by the branch points of the hyperelliptic…

We introduce a new approach of computing the automorphism group and the field of moduli of points $\p=[C]$ in the moduli space of hyperelliptic curves $\H_g$. Further, we show that for every moduli point $\p \in \H_g(L)$ such that the…

We study families of superelliptic curves with fixed automorphism groups. Such families are parametrized with invariants expressed in terms of the coefficients of the curves. Algebraic relations among such invariants determine the lattice…

A basis for the space of generalized theta functions of level one for the spin groups, parameterized by the theta characteristics (the even theta characteristcs for the odd spin groups) on a curve, is shown to be projectively flat over the…

To count bundles on curves, we study zetas of elliptic curves and their zeros. There are two types, i.e., the pure non-abelian zetas defined using moduli spaces of semi-stable bundles, and the group zetas defined for special linear groups.…

In this paper we observe that isomorphism classes of certain metrized vector bundles over P^1-{0,infinity} can be parameterized by arithmetic quotients of loop groups. We construct an asymptotic version of theta functions, which are defined…

Recall that a non-singular planar quartic is a canonically embedded non-hyperelliptic curve of genus three. We say such a curve is symmetric if it admits non-trivial automorphisms. The classification of (necessarily finite) groups appearing…

Let the threefold X be a general smooth conic bundle over the projective plane P(2), and let (J(X), Theta) be the intermediate jacobian of X. In this paper we prove the existence of two natural families C(+) and C(-) of curves on X, such…

Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $(X,\omega) \in \mathcal{H}(2)\sqcup\mathcal{H}(1,1)$ a subgraph $\hat{\mathcal{C}}_{\rm cyl}$ of the curve graph of $S$. The vertices…

This article initiates a geometric study of the automorphism groups of general graph products of groups, and investigates the algebraic and geometric structure of automorphism groups of cyclic product of groups. For a cyclic product of at…

We develop the theory of Abelian functions associated with algebraic curves. The growth in computer power and an advancement of efficient symbolic computation techniques has allowed for recent progress in this area. In this paper we focus…

In this paper, we show that there exist families of curves (defined over an algebraically closed field $k$ of characteristic $p >2$) whose Jacobians have interesting $p$-torsion. For example, for every $0 \leq f \leq g$, we find the…

Let C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We prove that if the G-curve C is very general for these properties, then the natural map…

A linear algebraic group $G$ is represented by the linear space of its algebraic functions $F(G)$ endowed with multiplication and comultiplication which turn it into a Hopf algebra. Supplying $G$ with a Poisson structure, we get a quantized…

In this paper, we consider the relation between the simplest types of arithmetic theta series, those associated to the cycles on the moduli space $\Cal C$ of elliptic curves with CM by the ring of integers $\OK$ in an imaginary quadratic…

A smooth complex projective curve is called pseudoreal if it is isomorphic to its conjugate but is not definable over the reals. Such curves, together with real Riemann surfaces, form the real locus of the moduli space $\mathcal M_g$. This…

In our previous paper [math.NT/0408050], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type…