# ag.algebraic geometry – Berkovich Integration on algebraic curves

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.