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In the paper "Le halo spectral" (http://perso.ens-lyon.fr/vincent.pilloni/halofinal.pdf) and in the following paper "The adic cuspidal, Hilbert eigenvarieties" (http://perso.ens-lyon.fr/vincent.pilloni/Hilbert_adicfinal.pdf), Andreatta, Iovita and Pilloni construct integral models of Igusa tower and anticanonical tower by taking the projective limit along a canonical lifting of Frobenius to characteristic $0$. Those spaces are fibered over a formal scheme which is the connected component containing the trivial character of weight space. Does these spaces satisfy any kind of moduli interpretation? Does the Igusa tower parametrize elliptic curves over a ring with a complete trivialization of the anticanonical subgroups at each level? According to me, a point of the Igusa tower should correspond to a sequence of elliptic curves where the transition morphisms are given by Frobenius. But can this sequence be described as a unique elliptic curve with a trivialization (at least generically) of the anticanonical subgroups? How can I prove this?

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