Weierstrass factorizations in compact Riemann surfaces (Q1380506)

Let \(\mathcal V\) be a compact Riemann surface of genus \(g\) , \(\infty\) be a point of \(\mathcal V\) and \(\mathcal V ' = \mathcal V -\{\infty\}\). Let \((p_n)\) be a sequence in \(\mathcal V\) to \(\infty\). The authors characterize the subfields of the field of meromorphic functions in \(\mathcal V '\) containing sufficient functions to verify WF(Weierstrass factorization) property. The following is a typical one which they obtained: Let \(V\) be an open neighborhood of \(\infty\) in \(\mathcal V\) with a coordinate \(z\) such that \(z(\infty) = 0\). Let \(\overline{K_\infty}\) be the field generated by all meromorphic functions in \(\mathcal V '\) of the \(h e^{P(1/2)}\) in \(V\), with a polynomial \(P\). Then, it is known that \(\overline{K_\infty}\) verifies WF property. Let \(K_\infty\) be the subfield of \(\overline{K_\infty}\) corresponding to polynomials \(P\) of degree \(\geq g\) . Then, the authors prove that for a sequence \((p_n)\) , the existence of an associated Weierstrass product with factors in \(K_n\) is equivalent to the absolute convergence of \(\sum_{n=1}^{\infty} z(p_n)^{g+1}\).