Period matrices of some hyperelliptic Riemann surfaces

TL;DR

This paper computes period matrices for a family of hyperelliptic Riemann surfaces defined by specific algebraic equations, using polygonal constructions to explicitly determine their symplectic bases.

Contribution

It provides explicit calculations of period matrices for hyperelliptic curves of genus g using polygonal methods, extending known results to a broader class.

Findings

Explicit period matrices for the given hyperelliptic curves.

Construction of symplectic bases from Euclidean polygons.

Extension of period matrix calculations to arbitrary genus g.

Abstract

In this paper, we calculate period matrices of algebraic curves defined by $$w^2=z(z^2-1)(z^2-a_1^2)(z^2-a_2^2)\cdots (z^2-a_{g-1}^2)$$ for any $g\geq 2$ and $a_1, a_2, \dots, a_{g-1}\in \mathbb{R}$ with $1<a_1<a_2<\cdots <a_{g-1}$. We construct these algebraic curves from Euclidean polygons. A symplectic basis of these curves are given from the polygons.