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As the title suggests, my question is devoted to understand (and maybe get some good references) the relation between the Igusa tower for a modular curves and $p$, or maybe $T$-adic modular forms. I post this question since I am reading Andreatta, Iovita and Pilloni's paper "Le Halo Spectral", where, first of all, they construct a sheaf whose global sections should produce a theory of overconvergent $T$-adic modular forms. In chapter $4$ of that paper, the construction goes approximately as follows. First they consider level $N\geq 3$ modular curve $X$ over $\mathbb{F}_p$, $(p,N)=1$ and its ordinary locus $X_{ord}$ defined as the complement of the vanishing locus of the Hasse invariant. Over the ordinary locus it is well defined the Igusa tower, which parametrizes, as a moduli space, the trivializations of the dual of the kernel of Frobenius. Now, they consider, for a given character $\kappa:\mathbb{Z}_p^\ast\rightarrow\mathbb{C}_p^\ast$ the etale $\phi$-module given by Katz, which in this case has the fordddm $\mathcal{O}_{\mathcal{IG}_\infty}[\kappa^{-1}]$, where $\mathcal{IG}_\infty$ is the projective limit of the Igusa tower. This sheaf is essentially the sheaf of functions over the Igusa tower, defined over the base change of the ordinary locus to $\mathbb{F}[[T]]$ which transform via the inverse of $\kappa$ under the action of $\mathbb{Z}_p^\ast$. In the rest of the chapter they show that this construction overconverges in a neighborhood of the ordinary locus, and they define integral overconvergent modular forms as global sections of that sheaf, which is invertible over the suitable neighborhood of the ordinary locus. Now, the point is, how does the Igusa tower relates to the usual definition of overconvergent modular forms, say functions defined over suitable triples of elliptic curves with growth conditions? How is it possible to compute global sections of that sheaf involved in the construction? Say, is it possible to find a suitable basis, as in Katz? Thank you for any suggestion or also reference and sorry if my question is maybe too vague!

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  • $\begingroup$ I add this comment coming from my recent readings. Maybe the answer to my question can be found in Hida's papers, where he uses Igusa tower to trivialize the line bundle of modular forms of a suitable integer weight. Is it the right way to visualize this construction? Can you help me? $\endgroup$ – rime Dec 15 '17 at 8:56

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