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 $0 neq omegain H^0(X, Omega_X)$ 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.