Theta Functions

Theta Functions

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Hershel Farkas, Hebrew University of Jerusalem
Fine Hall 322

In this talk we consider Weierstrass points for the linear space of meromorphic functions on a compact Riemann surface whose divisors are multiples of $\frac{1}{P_0^{\alpha}P_1..P_{g-1}}$ where the points $P_i$ are points on the surface and $\alpha$ is a positive integer for which there is no holomorphic differential whose divisor is a multiple of $P_0^{\alpha)P_1..P_{g-1}}$. Thus by the Riemann Roch theorem the dimension of the space is precisely $\alpha$. It develops that here are two different ways to define the Weierstrass points for this space. One way is to consider the Wronskian determinant of a basis for the space and to define the Weierstrass points as the zeros of the Wronskian and the weight of the Weierstrass point as the order of the zero. Another way is to consider, for a suitably chosen $ e \in C^g$, the Riemann theta function $ \theta(\alpha \Phi_{P_0}(P)-e)$ as a multivalued function on the surface, to define the Weierstrass points as its zeros and the weight of the Weierstrass point as the order of the zero. In this talk we deal with the question of whether to two definitions agree. We show that the sets of zeros are indeed the same and the problem is to detrmine whether the weights are the same. We give a partial solution to this problem.