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Berkovich developed a theory of integrating one-forms on his analytic spaces in his book "Integration of One-forms on $P$-adic analytic spaces". As this book is difficult to digest for me, I am wondering how this theory breaks down, if one just considers analytifications of smooth projective curves. More specifically, I am wondering if his theory can be used to translate some of the result over $\mathbb C$ (which I will recall below) to the $p$-adic world.

In the complex world the moduli space $\Omega M_g$ of pairs $(X, \omega)$, where $X$ is a smooth projective curve of genus $g$ and $0 \neq \omega\in H^0(X, \Omega_X)$ is a holomorphic differential on $X$, is a well studied object. The space $\Omega M_g$ is a complex orbifold and the points are called translation surfaces. One first result is that every translation surface can be represented as a finite union of polygons in the complex plane with edge identifications. One gets this equivalence by integrating (using $\omega$) along paths between the zeros of $\omega$.

$\Omega M_g$ comes equipped with a natural stratification: Let $\kappa$ be a partition of $2g-2$ (the number of zeros of $\omega$ counting multiplicity). Then $\mathcal H(\kappa)$ is the subset of $\Omega M_g$ containing the points $(X,\omega)$ such that the order of zeros of $\omega$ corresponds to the partition $\kappa$. This subset $\mathcal H (\kappa)$ is itself a complex orbifold. Roughly speaking, charts are given by integrating the same paths with respect to different differentials. If you want more details on this topic, I would suggest having a look at a nice overview paper by: Alex Wright

I would like to bring those two results over to the $p$-adic world ($\mathbb C_p$), so let me restate my question:

  • Using Berkovich integration on the analytification of a projective smooth curve $X$, is there a nice geometric description of the pair $(X^{an},\omega)$ (where $\omega$ is a global section of the differentials on $X^{an}$)?
  • On the strata of $\Omega M_g$ (which exists algebraically) can we find some kind of "coordinates" by integrating using the differentials?

I would appreciate any kind of feedback, whether those results are clearly unobtainable or might very well be possible.

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  • $\begingroup$ Just a comment on your first question. Which kind of ``nice geometric description'' do you have in mind? I am not sure what the analogue of a polygon should be in Berkovich geometry and gluing edges can certainly be problematic. Maybe gluing discs along their boundaries? Do you have any examples where you see how things work? $\endgroup$
    – Jérôme Poineau
    Sep 8, 2020 at 14:42
  • $\begingroup$ @JérômePoineau Unfortunately I don't have any examples. One could certainly make something similar as the second definition of translation surface in the paper of Wright: By locally integrating using $\omega$ (as one would do over $\mathbb C$) one can get a $p$-adic manifold, i.e. charts to $\mathbb C_p$. But since $\mathbb C_p$ is totally disconnected this is not really usefull I think... $\endgroup$
    – Fabian Ruoff
    Sep 9, 2020 at 7:34

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