Triangle Fuchsian differential equations with apparent singularities (Q1763045)

Suppose that the Fuchsian differential equation \[ w''+\left(\frac{1-\nu_0}{z}+\frac{1-\nu_1}{z-1} -\sum^n_{i=1}\frac{1}{z-t_i}\right)w'+\frac 1 {z(z-1)}\left(\mu'\mu''+\sum^n_{i=1}\frac{A_i} {z-t_i}\right)w=0 \] \((\,'\,=d/dz)\) admits the Riemann scheme \[ \begin{pmatrix} 0 & 1 & t_1 & \cdots & t_n & \infty \\ 0 & 0 & 0 & \cdots & 0 & \mu' \\ \nu_0 & \nu_1 & 2 & \cdots & 2 & \mu'' \end{pmatrix} . \] This is possible when the accessory parameters \(A_1, \dots, A_n\) satisfy a certain system of quadratic equations. Furthermore, suppose that \[ \nu_0, \nu_1, \nu_{\infty} \in\mathbb {Q}\setminus \mathbb{Z}, \quad \nu_0\pm \nu_1\pm \nu_{\infty} \not\in \mathbb{Z}, \quad \nu_{\infty}:=\mu''-\mu', \] and that \(t_1,..., t_n\) are algebraic numbers. Then, a basis of solutions \((w_1,w_2)\) corresponding to the Riemann scheme above is expressible in terms of periods along \(1\)-cycles of the hypergeometric curve \(X(k,z):\) \(y^k= u^{k(a-c)}(u-1)^{k(c+b-1)}(u-z)^{ka},\) \((a,b,c)= (\mu',\mu'', 1-n-\nu_0),\) where \(k\) is the least common denominator of \(a,b,c\). These solutions define the Schwarz map \(D(z)=w_1(z)/w_2(z).\) This paper discusses the algebraicity of the value \(D(\tau)\) for algebraic numbers \(\tau\), in particular for \(\tau=t_j,\) and its relations to the Prym variety of \(X(k,z).\)