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(A follow-up on this). Consider the modular curve $X_0(N)$. I'm trying to make the jump from understanding the cohomology $H^1(X_0(N), \mathbb{Z})$ to understanding $H^1(X_0(N), \mathcal{O})_\mathfrak{m}$, in the spirit of the modularity theorem: The absolute Galois group $G_\mathbb{Q}$ and a Hecke algebra $\mathbb{T}$ act on $H^1(X_0(N), \mathcal{O})_\mathfrak{m}$, where $\mathcal{O}$ is a DVR with maximal ideal $\mathfrak{m}$, finite flat over the $p$-adic integers. How does $H^1(X_0(N), \mathcal{O})_\mathfrak{m}$ arise intuitively, and why does $\mathcal{O}$ need to have these specific properties? My first (naive) idea is that $X_0(N)$ can be considered an algebraic curve, so the DVR somehow describes rings of functions at its points, and the localization is needed because we need to focus on the maximal ideal, ie those functions that vanish at each specific point. In this spirit, the Hecke operators and diamond operators $\mathbb{T}$ act on each function roughly as they act on the space of modular forms. On the other hand, the universal coefficient theorem gives $H_1 \cong \operatorname{Hom}(H^1, \mathbb{Z})$, and we know that for general ring $R$, we have $H_1(X_0(N),R)=H_1(X_0(N),\mathbb{Z}) \otimes_\mathbb{Z} R$, akin to an extension of scalars. Can someone help me piece this together and point out mistakes in my intuition? Also, can someone briefly describe the action of $G_\mathbb{Q}$?

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    $\begingroup$ This question is so muddled that it's hard to know where to start; I think you're trying to run before you've learned to walk. Just 2 days ago, someone recommended that you read Cornell--Silverman--Stevens. You should keep reading it. $\endgroup$ Feb 13, 2023 at 6:00
  • $\begingroup$ @DavidLoeffler I think you are right; seems I was too eager to understand the whole philosophy in one go. Thank you for the advice! $\endgroup$
    – JBuck
    Feb 13, 2023 at 13:01

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