Period matrices of hyperelliptic curves (Q1313500)

Let \({\mathcal C}\) be a curve of genus \(g\) defined over the field of complex numbers. The Jacobian \(\text{Jac} ({\mathcal C})\) is a complex torus \(\mathbb{C}^ g/ \Lambda\). Choosing bases suitably the lattice \(\Lambda\) can be written in the form \((Z, \mathbf{1}_ g) \mathbb{Z}^{2g}\) with \(Z \in {\mathcal H}_ g\), the Siegel upper half space of complex symmetric \((g \times g)\)-matrices with positive definite imaginary part. In general nothing is known about the correspondence \({\mathcal C} \leftrightarrow Z\). In the paper ``On binary sextics with transformations onto themselves'' [Am. J. Math. 10, 47-70 (1887)], \textit{O. Bolza} computed period matrices \(Z\) for hyperelliptic curves \({\mathcal C}\) of genus \(g\) with extra automorphisms. In the paper under consideration Bolza's method is extended to hyperelliptic curves of arbitrary genus \(g\) with reduced automorphism group of order \(\geq 2g\). Moreover period matrices for hyperelliptic curves with reduced automorphism group \(D_{2g + 2}\), \(Z_{2g + 1}\), and \(D_{2g}\) are explicitly computed.