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Scholze constructed perfectoid modular curve and its canonical and anticanonical part in his paper On torsion in the cohomology of locally symmetric varieties (Annals of Mathematics 182 (2015) pp 945–1066, doi:10.4007/annals.2015.182.3.3, arXiv:1306.2070). According to him, perfectoid modular curves are perfectoid spaces fibered over $\text{Spa}(\mathbb{Q}_p^{\text{cycl}},\mathbb{Z}_p^{\text{cycl}})$. Moreover, using the notion of tilda limit, it's easy to prove that infinite level modular curves parametrize elliptic curves with a complete trivialization of Tate module. Moreover, in 2015, Andreatta, Iovita and Pilloni extended the notion of Coleman's Eigencurve for overconvergent elliptic modular forms, to the boundary of weight space. In order to do that, they construct in Le halo spectral (Ann. Scient. Éc. Norm. Sup. 51 (2018) pp 603–655, doi:10.24033/asens.2362, author pdf) an integral model of the anticanonical part of Scholze's infinite level modular curve, if I understand it well. This object is a formal scheme fibered over the compactified weight space. In particular it is computed as a projective limit along the Frobenius morphism of the tower given by strict neighborhoods of the ordinary locus defined by a suitable equation involving a lifting of Hasse invariant.

Does it provide a moduli interpretation? I guess that an $\text{Spf}(R)$ point of this integral model gives an elliptic curve over $R$ with a trivialization of its Tate module. It seems easy, but I really don't know how to prove it.

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  • $\begingroup$ Thank you for editing! $\endgroup$ – Zariski93 Mar 31 at 13:12

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