# Weierstrass points on rigid-analytic surfaces

Does a rigid-analytic surface defined over a nonarchimedean complete field have Weierstrass points (if its genus is big enough let's say)? Is there a good reference that (ideally) lists theorems for rigid-analytic spaces that are the analog of commonly known theorems about complex analytic spaces?

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## 3 Answers

Quick note: I am going to assume you want to talk about complete curves. One can, of course, have a curve with punctures in algebraic geometry, and I'm not sure how you'd want to define a Weierstrass point on it. In rigid geometry, you have even more freedom: you can have the analogue of a Riemann surface with holes of positive area, and I think (not sure) you can also build the analogue of a Riemann surface of infinite genus. I'm going to assume you are not thinking about these issues.

What you want is the rigid GAGA theorem. I'm not sure what the best reference is; I refreshed my memory from Coleman's lectures, numbers 23-25. Rigid GAGA says:

Let $\mathcal{X}$ be a projective rigid analytic variety. Then

(1) $\mathcal{X}$ is the analytification of an algebraic variety $X$.

(2) The analytificiation functor from coherent sheaves on $X$ to coherent sheaves on $\mathcal{X}$ is an equivalence of categories.

(3) The cohomology of a coherent sheaf is naturally isomorphic to that of its analytification.

Thus, if we define Weierstrass points by the condition that the dimension of $H^0(\mathcal{O}(kp), X)$ is higher than expected, we will get the same points whether we work algebraically or analytically. It shouldn't be too hard to show that your favorite definition is equivalent to this.

Of course, all I've done is tell you how to translate between analysis and algebra. The algebra itself may be very difficult, as Felipe Voloch points out.

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I did mean a complete curve. And you were right in assuming that I was not thinking about any of the tricky or out-of-the-ordinary cases :) – Christelle Vincent Dec 13 '09 at 0:10

Do you mean a curve, instead of a surface? The notion of Weierstrass point is usually defined for algebraic curves (= Riemann surfaces, if the ground field is the complex numbers) and is a purely algebraic notion. For non-algebraically closed fields, there can be curves that have no Weierstrass point defined over the ground field.

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I did mean a curve, thanks for pointing this out! I was getting slightly mixed up between the curve over the complex numbers and the Riemann surface. – Christelle Vincent Dec 13 '09 at 0:11

About references I think Brian Conrad's lecture notes (I'd check mainly his references) from his course in AWS07 might be a good place to look at. http://swc.math.arizona.edu/aws/07/ConradNotes11Mar.pdf

From Conrad's notes I remember that two references were helpful,
http://wwwmath.uni-muenster.de/sfb/about/publ/heft378.pdf and "Non-Archimedean analysis" by S. Bosch, U. Güntzer, R. Remmert

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