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arXiv · 1211.6910

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

Yuuki Tadokoro
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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.

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Yuuki Tadokoro. 2012-11-29. The period matrix of the hyperelliptic curve $w^2=z^{2g+1}-1$. https://arxiv.org/abs/1211.6910

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