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I am starting studying the theory of (algebraic) modular forms, and I have some trouble in understanding completely the construction of the Tate curve. My problem is the following: as far as I know the Tate curve $T(q)$ is an elliptic curve over $\mathbb Z((q))$ with an affine equation given by $y^2+xy=x^3+b(q)x+c(q)$ for some $b(q),c(q)\in \mathbb Z[[q]]$ and an invariant differential $\omega_{can}=dx/(2y+x)$. But often I read the equality $\omega_{can}=dx/(2y+x)=dt/t$ and I can't really understand what does it mean formally. For example, say that I consider the curve $T(q)$ base changed to $\mathbb F_p((q))$ (which is the case I am interested in). If I look at the affine part of the curve described by the equation above, an invariant differential should be a basis of the $R$-module $H^0(T(q),\Omega^1_{T(q)/\mathbb F_p((q))})$ ($R$ the coordinate ring of $T(q)$) which I know being free with basis $dx/(2y+x)$. So how should I interpret this $dt/t$ in such a setting? I know that if $E=\mathbb C/\Lambda$ is an elliptic curve over $\mathbb C$ for some lattice $\Lambda$, then using the exponential map we can view it as a quotient of $\mathbb C^*$ by a discrete subgroup $q^{\mathbb Z}$ for some $q$ with $|q|<1$ and in such a case we get an isomorphic complex manifold $\mathbb C^*/q^{\mathbb Z}$ which has a canonical differential $dt/t$. But what does this "$dt/t$" mean when we deal with a scheme over $\mathbb Z[[q]]$? I'm sorry if it is a silly question, thank you in advance to everybody willing to help me!

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The precise meaning with which this is actually useful to prove things rigorously over $\mathbf{Z}(\!(q)\!)$ is via Raynaud's construction over $\mathbf{Z}[\![q]\!]$ (deformation from $q=0$!) via algebraization of formal schemes. It provides a canonical identification of the formal completion along the identity section with $\widehat{\mathbf{G}}_m$ (intrinsically characterized too) such that the formal $\mu_N$'s globalize. The invariant 1-form ${\rm{d}}t/t$ on that formal group comes from a global 1-form on the algebraized Tate curve, namely ${\rm{d}}x/(2y+x)$. That is the link. –  user61789 Jul 3 '13 at 1:11

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You can view it in (at least) two ways.

You can view $\mathbb{F}_p((q))^*/q^{\mathbb{Z}}$ as an analytic manifold over the valued field $\mathbb{F}_p((q))$, analogously to $\mathbb{C}^*/q^{\mathbb{Z}}$ or $\mathbb{Q}_p^*/q^{\mathbb{Z}}$, the latter two for element $q$ in the respective field with $|q|<1$. You can imagine a theory of manifolds over a valued field and differentials. This is better done with the theory of rigid analytic spaces and was in fact the motivation for them.

But another more direct way of viewing this is to look at the ring of power series in two variables $\mathbb{F}_p[[q,t]]$, which has a differentiation $\partial/\partial t$. Then you have two power series $x(q,t),y(q,t)$ that satisfy the equation of the elliptic curve and $t\partial x/\partial t = 2y+x$ is an identity that you can verify.

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