×

zbMATH — the first resource for mathematics

On the \(\wp\)-zero value function and \(\wp\)-zero division value functions. (English) Zbl 0766.33018
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 M. Eichler and 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.
MSC:
33E05 Elliptic functions and integrals
14H52 Elliptic curves
32D15 Continuation of analytic objects in several complex variables
30B40 Analytic continuation of functions of one complex variable
PDF BibTeX XML Cite
Full Text: DOI