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March 2015 The period matrix of the hyperelliptic curve $w^2=z^{2g+1}-1$
Yuuki Tadokoro
Tsukuba J. Math. 38(2): 137-158 (March 2015). DOI: 10.21099/tkbjm/1429103717


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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Yuuki Tadokoro. "The period matrix of the hyperelliptic curve $w^2=z^{2g+1}-1$." Tsukuba J. Math. 38 (2) 137 - 158, March 2015.


Published: March 2015
First available in Project Euclid: 15 April 2015

zbMATH: 1327.14149
MathSciNet: MR3336264
Digital Object Identifier: 10.21099/tkbjm/1429103717

Rights: Copyright © 2015 University of Tsukuba, Institute of Mathematics


Vol.38 • No. 2 • March 2015
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