Taylorentwicklungen automorpher Formen und ein Transzendenzproblem aus der Uniformisierungstheorie (Q761487)

This paper consists of two theorems. Theorem 1 asserts that \(\tau\) being an imaginary quadratic number with Im\(\tau\) \(>0\), for every weight k, there exists a basis composed of elliptic modular forms f(z) having the following Taylor expansion: \(f(z)=(z-{\bar\tau})^{-k} \sum^{\infty}_{n=0} r_ nb^ n((z-\tau) / (z-{\bar\tau}))^ n\) with \(r_ n\in {\bar{\mathbb{Q}}}\) and transcendental b. In fact, \(b=\omega^ 2/\pi\), where \(\omega\) is the period of the algebraic curve with period-ratio \(\tau\) and algebraic invariants \(g_ 2\), \(g_ 3.\) In Theorem 2, the author positively answers the question raised by S. Lang on the transcendency of r of the universal covering map of an algebraic curve K: \[ \phi: \{\zeta \in {\mathbb{C}}_ 2| | \zeta | <r\}\to K \] when K is defined by \(y^ 2=u^ 2+v^ 2\), \(x^ 3=u\cdot v^ 2\) in \({\mathbb{P}}^ 3({\mathbb{C}})\) under certain normalization conditions on \(\phi\).