The Harmonic Volumes of Hyperelliptic Curves

We determine the harmonic volumes for all the hyperelliptic curves. This gives a geometric interpretation of a theorem established by A. Tanaka.Comment: 20 pages, 4 figures, to appear in Publ. RIMS, Kyoto Universit

The period matrix of the hyperelliptic curve w2=z2g+1−1w^2=z^{2g+1}-1

A geometric algorithm is introduced for finding a symplectic basis of the first integral homology group of a compact Riemann surface, which is a $p$-cyclic covering of ${\mathbb C} P^1$ branched over 3 points. The algorithm yields a previously unknown symplectic basis of the hyperelliptic curve defined by the affine equation $w^2=z^{2g+1}-1$ for genus $g\geq 2$. We then explicitly obtain the period matrix of this curve, its entries being elements of the $(2g+1)$-st cyclotomic field. In the proof, the details of our algorithm play no significant role.Comment: 18 pages, 10 figure

Nonlinear O(3)O(3) sigma model in discrete complex analysis

We examine a discrete version of the two-dimensional nonlinear $O(3)$ sigma model derived from discrete complex analysis. We adopt two lattices, one rectangular, the other polar. We define a discrete energy $E({f})^{\rm disc.}$ and a discrete area ${\cal{A}}({f})^{\rm disc.}$, where the function $f$ is related to a stereographic projection governed by a unit vector of the model. The discrete energy and area satisfy the inequality $E({f})^{\rm disc.} \ge |{\cal{A}}({f})^{\rm disc.}|$, which is saturated if and only if the function $f$ is discrete (anti-)holomorphic. We show for the rectangular lattice that, except for a factor 2, the discrete energy and the area tend to the usual continuous energy $E({f})$ and the area ${\cal{A}}({f})=4 \pi N, \,\,N\in \pi_2(S^2)$ as the lattice spacings tend to zero. In the polar lattice, we section the plane by $2M$ lines passing through the origin into $2M$ equal sectors and place vertices radially in a geometric progression with a common ratio $q$. For this polar lattice, the Euler--Lagrange equation derived from the discrete energy $E({f})^{\rm disc.}$ yields rotationally symmetric (anti-)holomorphic solutions $f(z)=Cz^{\pm 1}\,\,(C\bar{z}^{\pm 1})$ in the zeroth order of $\kappa:=q^{-1}-q$. We find that the discrete area evaluated by these zeroth-order solutions is expressible as a $q$-integral (the Jackson integral). Moreover, the area tends to $\pm 2\cdot 4\pi$ in the continuum limit ($M \to \infty$ and $q \to 1\!-0$) with fixed discrete conformal structure $\rho_0 =2 \sin{(\pi/M)}/ \kappa$.Comment: v1. 10 pages, 2 figures v2. New title, 19 pages and 3 figures, Sec.2.3 (EL eq. and its continuum limt) and Sec.3 (polar lattice) adde