The distribution of Weierstrass points on a compact Riemann surface. (English) Zbl 0598.14016

The following beautiful result is proved: Let C be a compact Riemann surface of genus \(g\geq 2\) and let \(h:\quad C\to {\mathbb{R}}\) be a continuous, real valued function. Denote by \(J_{g-1+n}\) the jacobian of line bundles of degree \(g-1+n\) on C. Define a function \(Av^ nh\) on \(J_{g- 1+n}\) by the following formula: \[ (Av^ nh)(z)=\frac{1}{gn^ 2}\sum_{x\quad a\quad Weierstrass\quad point\quad of\quad z}h(x). \] Then as \(n\to \infty\), the \(Av^ nh\) converge uniformly to the constant function \(\int_{C}h db\) where b is the Bergman measure on C.
The result strengthtens an unpublished result of D. Mumford which states that if L is a line bundle on C of positive degree, then the Cesaro limit \( Lim_{m\to \infty}(Av^*h)(L^ n)=\int_{C}h db.\) The author’s proof is elegant and uses, in addition to the method of Mumford, several new and interesting techniques like studying \(Av^ nh\) via its Fourier coefficients and estimating high derivatives of the same function by pulling translation invariant vector fields back to the product of C with the theta divisor.
Reviewer: D.Laksov


14H55 Riemann surfaces; Weierstrass points; gap sequences
14H40 Jacobians, Prym varieties
30F10 Compact Riemann surfaces and uniformization
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