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

TL;DR

This paper presents a geometric algorithm to compute the period matrix of hyperelliptic curves, specifically for the curve defined by $w^2=z^{2g+1}-1$, revealing new symplectic bases and period matrix entries in cyclotomic fields.

Contribution

The authors introduce a novel geometric algorithm for determining symplectic bases and period matrices of hyperelliptic curves, including explicit computation for the curve $w^2=z^{2g+1}-1$.

Findings

Explicit symplectic basis for the curve's homology group.

Period matrix entries in the $(2g+1)$-st cyclotomic field.

New computational method for hyperelliptic curves.

Abstract

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.