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If $X$ is a smooth projective curve over $\mathbb C_p$, then its first etale cohomology $\mathrm H^1_{et}(X,\mathbb Q_\ell)$ (with $\ell\neq p$) carries a certain weight filtration $W_\bullet$ -- also known as the monodromy filtration -- which contains information about the reduction type of $X$. For instance, if $\mathfrak X$ denotes a stable model of $X$ over $\mathcal O_{\mathbb C_p}$, then $W_0\mathrm H^1_{et}(X,\mathbb Q_\ell)$ is the cohomology of the dual graph of $\mathfrak X_{\overline{\mathbb F}_p}$, and $W_1\mathrm H^1_{et}(X,\mathbb Q_\ell)/W_0$ is the direct sum of the first etale cohomology of the components of $\mathfrak X_{\overline{\mathbb F}_p}$.

On the other hand, the reduction type of $X$ is also closely related to the structure of its Berkovich analytification $X^{an}$. For instance, $|X^{an}|$ is homotopy equivalent to the dual graph of $\mathfrak X_{\overline{\mathbb F}_p}$, and the fields $\widetilde{\mathscr H(x)}$ for certain type II points of $X^{an}$ are the function fields of the components of $\mathfrak X_{\overline{\mathbb F}_p}$.

If I recall correctly, the above two facts are supposed to be related, but I can't at the moment remember how this is supposed to go.

Question:

  • Can the weight filtration on $\mathrm H^1_{et}(X,\mathbb Q_\ell)=\mathrm H^1_{et}(X^{an},\mathbb Q_\ell)$ be constructed purely in terms of analytic geometry?

  • How does this construction of the weight filtration explain the structure of the weight-graded pieces of $\mathrm H^1_{et}(X,\mathbb Q_\ell)$ (e.g. as noted above)?

If there are good references that set this out clearly and explicitly, they would be particularly welcome.


To illustrate the sort of answer I'm looking for, here is how I think one wants to construct the weight $0$ part of $\mathrm H^1_{et}(X,\mathbb Q_\ell)$. The morphism of sites $\pi\colon X^{an}_{et}\rightarrow|X^{an}|$ gives rise to a spectral sequence computing the etale cohomology of $X^{an}$, whose low-degree terms give an exact sequence$$0\rightarrow\mathrm H^1(|X^{an}|,\mathbb Q_\ell)\rightarrow\mathrm H^1_{et}(X^{an},\mathbb Q_\ell)\rightarrow\mathrm H^0(|X^{an}|,\mathrm R^1\pi_*\mathbb Q_\ell)\rightarrow0$$(with a slight abuse of notation). The image of $\mathrm H^1(|X^{an}|,\mathbb Q_\ell)$ in $\mathrm H^1(X^{an},\mathbb Q_\ell)$ should be exactly the weight $0$ part $W_0$, and we recover the description of $W_0$ as the cohomology of the dual graph from the fact that $|X^{an}|$ is homotopy equivalent to this dual graph.

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I try to answer your question with some ideas.

First, what you said about the weight zero part was proved by Berkovich itself for any dimension in

Berkovich, V. G., An analog of Tate’s conjecture over local and finitely generated fields, Internat. Math. Res. Notices 2000, no. 13, 665–680.

(I understand that what you call $H^1_{ét}(X,\mathbb{Q}_{\ell})$ is what is usually call $H^1(\overline{X}_{ét},\mathbb{Q}_{\ell})$, where $\overline{X}$ is the base change to a (fixed) algebraic closure of $K$, since the cohomology over $K$ has no weight filtration).

Second, using the duality for curves, which gives you that $$H^1(\overline{X}_{ét},\mathbb{Q}_{\ell})^*\cong H^1(\overline{X}_{ét},\mathbb{Q}_{\ell})(1)$$ (where $^*$ means morphisms to $\mathbb{Q}_{\ell}$, with the usual Galois action), you get a (surjective) morphism $$\psi:H^1(\overline{X}_{ét},\mathbb{Q}_{\ell})\to H^1(|X^{an }|,\mathbb{Q}_{\ell})(-1)$$ If you call $$W_1(H^1(\overline{X}_{ét},\mathbb{Q}_{\ell})):=\operatorname{Ker}(\psi)$$ then the image $W_0(H^1(\overline{X}_{ét},\mathbb{Q}_{\ell}))$ of the map $$H^1(|X^{an }|,\mathbb{Q}_{\ell})\to H^1(\overline{X}_{ét},\mathbb{Q}_{\ell})$$ lies inside $\operatorname{Ker}(\psi)$ (since there are no non zero maps between $\mathbb{Q}_{\ell}$ and $\mathbb{Q}_{\ell}(-1)$), and it gives you the weight filtration.

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