On the \(\wp\)-zero value function and \(\wp\)-zero division value functions (Q1176120)

For \(\tau\) in the upper half plane \({H}\), the Weierstrass elliptic \(\wp\)-function with fundamental periods \(\b{Z}+\b{Z}\tau\) is denoted by \(\wp(z,\tau)\). Let \(D_ 1\), \(D_ 2\) be two domains in \({H}\) and \(u_ 1\), \(u_ 2\) be two analytic functions on \(D_ 1\), \(D_ 2\) respectively. Assume \(\wp\bigl(u_ i(\tau,\tau\bigr)=0\) for \(\tau\in D_ i\) (\(i=1\) and \(i=2\)). Then there exists a curve \(\gamma\) in \({H}\) such that \((u_ 2,D_ 2)\) is an analytic continuation of \((u_ 1,D_ 1)\) along \(\gamma\). This yields what the author calls a many-valued modular form. The proof uses results from \textit{M. Eichler} and \textit{D. Zagier} [Math. Ann. 258, 399-407 (1982; Zbl 0491.33004)]. A similar result holds, for each \(N\geq 1\), for the \(N\)-th division values of the \(\wp\)-function.