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Published in:
2012 | OriginalPaper | Chapter
Theta Functions Wronskians and Weierstrass Points for Linear Spaces of Meromorphic Functions
Author : Hershel M. Farkas
Published in: The Mathematical Legacy of Leon Ehrenpreis
Publisher: Springer Milan
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Abstract
In this note we consider the Weierstrass points for the linear space of meromorphic functions on a compact Riemann surface whose divisors are multiples of \(\frac{1}{P_{0}^{\alpha}P_{1}\cdots P_{g-1}}\), where P
i
are points of the surface, and α is a positive integer for which there is no holomorphic differential on the surface whose divisor is a multiple of \(P_{0}^{\alpha}P_{1}\cdots P_{g-1} \). Thus the dimension of our linear space is precisely α.
The Weierstrass points for our space are those points Q≠P
i
for which there is a function in the space which vanishes to order at least α at the point Q. Thus the Weierstrass points are all zeros of the Wronskian determinant of a basis for our space, and the weight of the Weierstrass point is the order of the zero.
We show that all the Weierstrass points are zeros of the Riemann theta function \(\theta(\alpha \varPhi _{P_{0}}(P)-e) \) on the surface where \(e=\varPhi _{P_{0}}(P_{1}\cdots P_{g-1})+K_{P_{0}}\). The question we investigate is whether the order of the zero of the theta function agrees with the order of the zero of the Wronskian. We prove that this is so at least in the case of zeros of order k=1,2.