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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
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