# arithmetic geometry – Integrated models of perfectoidal modular curves

Scholze constructed the perfectoid modular curve and its canonical and anticanonical part in his article "On torsion in the cohomology of locally symmetrical manifolds". According to him, perfectoid modular curves are perfectoid spaces with $$text {Spa} ( mathbb {Q} _p ^ { text {cycl}}, mathbb {Z} _p ^ { text {cycl}}$$. Moreover, by using the notion of tilda limit, it is easy to prove that the infinite level modular curves parameterize the elliptic curves with a complete trivialization of the Tate module. Moreover, in 2015, Andreatta, Iovita and Pilloni extended Coleman's Eigencurve concept to elliptic modular shapes that were too convergent, at the limit of weight space. To do this, they build in "The Spectral Halo" an integral model of the anticanonical part of Scholze's infinite-level modular curve, as I understand it. This object is a formal fiber pattern on the compacted weight space. In particular, it is calculated as a projective boundary along the Frobenius morphism of the tower given by strict neighborhoods of the ordinary locus defined by an appropriate equation involving a Hasse invariant lift. Does it provide an interpretation of modules? I guess that's one $$text {Spf} (R)$$ The point of this integral model gives an elliptic curve on $$R$$ with a trivialization of its Tate module. It sounds easy, but I really do not know how to prove it.