Construction of period matrices by algebraic techniques (Q2339739)

Let \(S\) be a compact Riemann surface of genus \(g\). It is well known that for a fixed canonical homology basis \(\alpha_j\), \(\beta_j\), \(1\leq j\leq g\), there is the dual basis of holomorphic differentials \(\theta_j\) on \(S\) such that \(\int_{\alpha_j}\theta_i=\delta_{ij}\), \(\int_{\beta_j}\theta_i=\pi_{ij}\). Here \(\delta_{ij}\) is the Kronecker symbol. The numbers \(\pi_{ij}\) form the period matrix \(\Pi\) for \(S\). The authors give a method for constructing the period matrix for a surface admitting a non-trivial conformal automorphism. Every such automorphism induces a change of the canonical homology basis. The change can be defined by a \(2g\times 2g\) matrix with four \(g\times g\) blocks \(D\), \(C\), \(B\), and \(A\), corresponding to the \(\alpha\) and \(\beta\)-components of the initial and the transformed homology bases. The new period matrix \(\Pi'=(A\Pi+B)(C\Pi+D)^{-1}\) coincides with the old one, therefore, we have the relation \(\Pi=(A\Pi+B)(C\Pi+D)^{-1}\) which allows us to determine \(\Pi\) from a system of quadratic equations. Three particular cases are considered for the Riemann surfaces corresponding to the curves \(w^3=z^2(z-\lambda_1)(z-1)\), \(w^3=z^2(z^2-1)\) (a particular case of the previous surface for \(\lambda=-1\)), and \(w^3=z^4-1\).