## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 38, Number 2 (2015), 137-158.

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

#### 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 $\mathbf{C}P_1$ 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 $g \ge 2$. 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.

#### Article information

**Source**

Tsukuba J. Math., Volume 38, Number 2 (2015), 137-158.

**Dates**

First available in Project Euclid: 15 April 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1429103717

**Digital Object Identifier**

doi:10.21099/tkbjm/1429103717

**Mathematical Reviews number (MathSciNet)**

MR3336264

**Zentralblatt MATH identifier**

1327.14149

#### Citation

Tadokoro, Yuuki. The period matrix of the hyperelliptic curve $w^2=z^{2g+1}-1$. Tsukuba J. Math. 38 (2015), no. 2, 137--158. doi:10.21099/tkbjm/1429103717. https://projecteuclid.org/euclid.tkbjm/1429103717