## 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The result strengthtens an unpublished result of D. Mumford which states that if L is a line bundle on C of positive degree, then the Cesaro limit $$Lim_{m\to \infty}(Av^*h)(L^ n)=\int_{C}h db.$$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

### MSC:

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

### Keywords:

Weierstrass point; jacobian of line bundles; theta divisor
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