Canonical lifts of Igusa curves Sample Clauses

Canonical lifts of Igusa curves. ‌ In this section, we relate the canonical lift of the perfectoid Igusa scheme IG∞,∞,∞,∞ over Fp[[t1/p∞ ]] to the integral family of perfectoid modular curves IG∞,∞,∞,5 for any l ∈ Z[1/p]. We recall that for the sake of overall notational consistency we denote by t the image of T ∈ Zp[[T ]] under the reduction mod p. As discussed in §6.1, there is a canonical isomorphism W (Fp[[t1/p∞ ]]) = Zp[[T ]]⟨(1 + T )1/p∞ ⟩ which identifies the perfected weight space W∞ with the canonical Xxxx lift of the point Spf(Fp[[t1/p∞ ]]) that the modular curves in characteristic p are defined over. The goal of this section is to show that a similar identification is possible for the whole overconvergent Igusa tower of modular curves.
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