Xxxxxxxxx lifts Sample Clauses

Xxxxxxxxx lifts. In the classical theory of p-adic modular forms, one usually does not define p-adic modular forms of Γ0(pn)-level, since these can be canonically interpreted as modular forms of tame level by an Xxxxx–Xxxxxx isomorphism. For our considerations, however, it will be important to actually make a difference between the two. This can be understood in terms of a normalisation of q-expansions, as we shall now explain.

Xxxxxxxxx lifts. ‌ Xr,k,5 → Xr−1,k,5, as we shall now discuss. This is our transition map in the variable r. Proposition 6.4.1. Let m ≤ k ≤ k′ ∈ Z≥0, l ∈ p−mZ>0 ∪ {∞} and r ≥ m + 1. 1. The isogeny defined by the canonical subgroup H1,r,k,5 ‹→ Er,k,5 induces a finite and flat morphism φ : Xr,k,5 → Xr−1,k,5 which fits into a Cartesian diagram Er,k,5/H1,r,k,5 Er−1,k,5 Xr,k,5 Xr−1,k,5. 2. If k ≥ r + 1, then φ : X → Xr−1,k,5 reduces mod p/S(p+1)pk−r−1 to the Frobenius relative to Wk,5. If k < r + 1, this is only true after inverting S. 3. The map φr commutes with the transition maps in l and k. Moreover, for k ≥ r + 1, Xr+1,k',5 → Xr,k',5 is the base-change of φ : Xr+1,k,5 → Xr,k,5 along Xr,k',5 → Xr,k,5. Proof. For the proof, we follow [3], Proposition 3.3: The case of k < r follows by applying the Proposition with A0 = Ck,5 and α = Sk and “r”= r − k. The formal scheme denoted there by “Yr” is then precisely our Xr,k,5, and we obtain part 1 of the Proposition. This gives the generic fibre of φ a moduli description, which immediately implies parts 2 and 3. The case of k ≥ r for parts 1 and 3 follows from this by base-change: Since r ≥ m + 1, we may assume after rescaling via rm that l ∈ Z≥0. The desired statements then follow on the generic fibre from the case of k = 0 by base-change along Mk,5 → M0,5. The statements for formal models follow form this by normalisation, using that φ is finite and Xr,k,5 is normal. It remains to prove part 2 in the case of k ≥ r + 1: One way to do this is by slightly modifying the proof of [3], Proposition 3.3. Like in [50], Definition III.2.12, we can regard Xr,k,5 as the formal scheme over Ck,5 representing the functor which sends a Ck,5-algebra R to the set of isomorphism classes of pairs (ƒ, η), where ƒ is a morphism ƒ : Spf(R) → Xk,5 giving rise to a semi-abelian scheme E over R, and η ∈ ω(1−p) is such that Ha(E)η ≡ Spk−r−1 mod p, and where (ƒ, η) and (ƒ′, η′) are equivalent if ƒ = ƒ′ and if there is h ∈ R with η = η′(1 + π · h) where π := p/Spk−r−1 . k−r−1 k k k−r Since the Xxxxx ideal Hdg of X contains Sp , and p = (p/S5p ) · S5p ∈ (Sp ) by l ≥ p−m ≥ p−r, there is on Xr,k,5 a canonical subgroup H1 by [3], Corollaire A.2.1, which reduces to the kernel of Frobenius modulo p/Hdg. Since Spk−r−1 ∈ Hdg, we have p/Hdg ⊆ (p/Spk−r−1 ) = (π), and thus H also reduces to the kernel of Frobenius mod π. We can now define φ as follows: Set E′ = E/H1(E), then E reduces mod π to E(p). We therefore have ωE' ≡ ωE mod π and Ha(E′) ≡ Ha(E)p mod π. But then sin...