Tsukuba Journal of Mathematics

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

Yuuki Tadokoro

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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.

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

First available in Project Euclid: 15 April 2015

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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

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