A continuous function f(x+ iy) = u(x, y) + iv(x, y) defined in a domain D ⊂ C (Complex plane) is harmonic in D if u and v are real harmonic in D. Clunie and Shiel-Small [3] showed that in a simply connected domain such functions can be written in the form f = h+g, where both h and g are analytic. We call h the analytic part and g, the co-analytic part of f . Let w(z) = g ′(z) h′(z) be the dilat...

This paper characterizes those pseudo-Anosov mappings whose entropy can be detected homologically by taking a limit over finite covers. The proof is via complex-analytic methods. The same methods show the natural map Mg → Ah, which sends a Riemann surface to the Jacobians of all of its finite covers, is a contraction in most directions.

In this paper we obtain conditions on the divisors of the group order of the Jacobian of a hyperelliptic genus 2 curve, generated by the complex multiplication method described by Weng (2003) and Gaudry et al (2005). Examples, where these conditions imply that the Jacobian has a large cyclic subgroup, are given.

We give a lower bound on the rank of the Cartier operator of Jacobian varieties of hyperelliptic and superelliptic curves in terms of their genus.

The main purpose of this paper is to provide a new approach to the study of generalized theta linear series on moduli spaces of vector bundles on a smooth complex projective curve X, based on vector bundle techniques on abelian varieties. To this effect we naturally associate to these linear series what will be called the Verlinde vector bundles on the Jacobian of X and make use of some special...

We consider an equivariant analogue of a conjecture of Borcherds. Let (Y, σ) be a real K3 surface without real points. We shall prove that the equivariant determinant of the Laplacian of (Y, σ) with respect to a σ-invariant Ricci-flat Kähler metric is expressed as the norm of the Borcherds Φ-function at the “period point”. Here the period of (Y, σ) is not the one in algebraic geometry.

We discuss arithmetic in the Jacobian of a hyperelliptic curve C of genus g. The traditional approach is to fix a point P∞ ∈ C and represent divisor classes in the form E − d(P∞) where E is effective and 0 ≤ d ≤ g. We propose a different representation which is balanced at infinity. The resulting arithmetic is more efficient than previous approaches when there are 2 points at infinity.

We give a new proof of a determinant evaluation due to Andrews, which has been used to enumerate cyclically symmetric and descending plane partitions. We also prove some related results, including a q-analogue of Andrews’s determinant.

Consider a finite groupG acting on a Riemann surface S, and the associated branched Galois cover πG : S → Y = S/G. We introduce the concept of geometric signature for the action of G, and we show that it captures much information: the geometric structure of the lattice of intermediate covers, the isotypical decomposition of the rational representation of the group G acting on the Jacobian varie...

A finite group G acting on an abelian variety A induces a decomposition of A up to isogeny. In this paper we investigate several aspects of this decomposition. We apply the results to the decomposition of the Jacobian variety of a smooth projective curve with G-action. The aim is to express the decomposition in terms of the G-action of the curve. As examples of the results we work out the decom...