Period matrices of some hyperelliptic Riemann surfaces

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 $g\ge 2$ 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.

22 pages, 7 figures