September 29, 2005

Contents

1. A statement of the result 1 2. A strategy of proof 2 3. Obstruction Theory 3 4. The supersingular case 4 5. Ordinary curves (Laures/Hopkins) 4 6. Completion 5 7. Serre-Tate 6 8. Ordinary Curves (at p) 7 9. K(1)-local E_{∞}-rings 8 10. Glueing 9 11. What is a K(2)-local E_{∞} elliptic spectrum, and why is this notion not necessary? 10 12. How do p-divisible groups enter homotopy theory? 10

1. A statement of the result

Let 𝔐_{W} = 𝔐(A, Γ) be the moduli stack of Weierstrass curves and let

𝔐_{W} ⟶ 𝔐_{FG}

be the morphism C ⟼ Ĉ , the formal completion at the identity e.

Theorem (Rezk). 1) This morphism is representable; indeed

Spec(L) ×_{𝔐_{FG}} 𝔐_{W} = Spec(ℤ[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}, e_{4}, e_{5}, ...]) = MUP_{0}tmf

where the a_{i} are the coordinates of the curve and

−\frac{x}{y}_{coord. of C at e} = T + e_{4}T^{4} + ⋯_{universal coord. of F G}

If you mod out by p, v_{1}, v_{2}, v_{3}, this is not a regular sequence unless (p, v_{2}, v_{2}) contains a unit.

2) The induced morphism defined by pull-back

\xymatrix{𝔐_{ell} \ar[d] \ar[r] & 𝔐_{W} \ar[d] \\ 𝔐_{FG}^{(2)} \ar[r] & 𝔐_{FG}}

is representable and flat.

Corollary. Let Spec(R) ⟶^{C} 𝔐_{ell} be an étale morphism. Then there is a 2-periodic elliptic commutative homology theory E(R, C) so that E(R, C)_{0} = C and G_{E(R,C)} = Ĉ.

Proof. Landweber exactness. □

The main theorem is that this can be rigidified to a diagram of E_{∞} ring spectra. First, let us base change over a prime p, fixed now and forever.

Theorem. Let Ét/𝔐_{ell} be the diagram implied above. Then there is a contravariant functor

𝔒_{top} : Ét/𝔐_{ell} ⟶ E_{∞} ring spectra  ‘presheaf’
(𝔒_{top}(Spec(R) ⟶^{C} 𝔐_{ell}))_{∗} = E(R, C)_{∗}  ‘realizing the structure sheaf’

Corollary. Any moduli problem over 𝔐_{ell} has a similar solution.

Definition. / Theorem: tmf is either the homotopy global sections of this presheaf or its connected cover. There is a spectral sequence.

H^{s}(𝔐_{ell}, Ω^{⊗t}) ⟹ π_{2t−s}tmf .

Since E(R, C)_{2t} = Ω_{C}^{⊗t} is the sheaf of invariant differentials on C, the invariants H^{0}(𝔐_{ell}, Ω^{⊗t}) are modular forms of weight t.

2. A strategy of proof

[figure: 𝔐_{FG}^{(1)}; open; ⊆; 𝔐_{FG}^{(2)}; closed; B Aut(\overline{𝔽}_{p}, Γ_{2}); Spec(ℤ_{(p)}); Spec(ℤ_{(p)}); Spec(\overline{𝔽}_{p})]

[figure: B𝔾_{m} ≅ 𝔐_{FG}^{(0)}; ≅; 𝔐_{FG}^{(1)}; ⊇; B Aut(ℤ/p, Γ_{1}) = B(ℤ_{p}^{×}); Spec(ℚ); Spec(ℤ_{(p)}); Spec(ℤ/p)]

By taking pull-backs one gets a corresponding decomposition

[figure: 𝔐^{ord}; 𝔐_{Ell}; ⊇; closed; 𝔐^{ss}; Spec(K(2)_{0}tmf); 𝔐_{FG}^{(1)}; 𝔐_{FG}^{(2)}; ⊇; B Aut(\overline{𝔽}_{p}, Γ_{2}); Spec(\overline{𝔽}_{p})]

𝔐(ℚ[c_{4}, c_{6}], ℚ[c_{4}, c_{6}, λ^{±1}]) = [figure: 𝔐_{ell}^{ℚ}; 𝔐^{ord}; closed; 𝔐_{p}^{ord}; Spec(K(1)_{0}tmf); B𝔾_{m}; 𝔐_{FG}^{(2)}; B Aut(ℤ_{p}^{×}); Spec(𝔽_{p})]

Idea:

• 𝔒_{top}^{ℚ} is easy, using that the category of ℚ − E_{∞} ring spectra is equivalent to rational commutative dgas and Hℚ_{∗}tmf = ℚ[\overline{c}_{4}, \overline{c}_{6}] is a polynomial algebra.

• Let 𝔒̂_{top}^{ss} and 𝔒̂_{top}^{ord,p} denote the (putative) presheaves by completing formal neighborhoods of the respective inclusions.

[figure: 𝔐^{ss}; ⊆; 𝔐̂^{ss}; 𝔐_{ell}; 𝔐^{ord}; 𝔐̂^{ord,p}; i; 𝔐^{ord,p}]

We construct these separately, then notice that 𝔒̂_{top}^{ss} is a presheaf on 𝔐_{ell}. So we can form the diagram, a pull-back

\xymatrix{𝔒̂^{ell,p} \ar[d] \ar[r] & 𝔒̂_{top}^{ss} \ar[d] & L_{K(1)∨K(2)}tmf \ar[d] \ar[r] & L_{K(2)}tmf \ar[d] \\ 𝔒̂_{top}^{ord,p} \ar[r] & Ri_{∗}i^{∗}𝔒̂_{top}^{ss} & L_{K(1)}tmf \ar[r] & L_{K(1)}L_{K(2)}tmf}

where 𝔒̂^{ell,p} is completion along

\xymatrix{𝔐_{p} \ar[d] \ar[r]^{⊆} & 𝔐_{ell} \ar[d] \\ Spec(ℤ/p) \ar[r] & Spec(ℤ_{(p)})}

3. Obstruction Theory

Let A be a fixed E_{∞} ring spectrum (e.g. A = S or tmf if it exists) and E_{∗} an Adams type ring theory, E_{∗}E flat over E_{∗} and denote by Alg_{A} the category of E_{∞} ring spectra under A.

(R) Realization problem: Let Λ be an E_{∗}A algebra in E_{∗}E comodules. Compute B(Λ/E_{∗}A), the Dwyer-Kan realization space of all X ∈ Alg_{A} with E_{∗}X ≅ Λ

B(Λ/E_{∗}A) = ∐_{[X]} Bh Aut_{A}(X) .

(M) Mapping problem: Given X_{1}, X_{2}, compute

Alg_{A}(X_{1}, X_{2}) .

• In both cases there is a (fringed) spectral sequence with E_{2}^{0,0} = (R) E_{2}^{s,t} ≅ H^{s+2}(Λ/E_{∗}A, Ω^{t}A) ⇒ π_{t−s}B for s ≥ 1, (M) E_{2}^{s,t} = H^{s}(Λ_{1}/E_{∗}A, Ω^{t}Λ_{2}) ⇒ π_{t}B for s ≥ t ≥ 0, where H is an appropriately defined André-Quillen cohomology.

Vanishing results: Write Λ_{0} for the comodule invariants, i.e., the equalizer of the two maps

\xymatrix{Λ \ar[r] \ar[r] & Λ ⊗_{E_{∗}} E_{∗}E} .

Suppose (1)Λ(resp Λ_{2}) is a relatively injective E_{∗}E comodule

(2)Λ/E_{∗}A(resp Λ_{2}/E_{∗}A) is étale. Then these groups vanish.

Suppose E_{∗} = K_{∗}(−) = p-completed-K-theory and (1)′ holds and (2)′Λ/E_{∗}A is smooth of relative dimension 1. Then these groups vanish in degrees ≥ 3.

Remark. In the case (2′) at p = 2, we will have an example where it doesn’t hold.

4. The supersingular case

Theorem: 𝔐^{ss} = ∐_{j∈\overline{𝔽}_{p}^{ss}} B Aut (\overline{𝔽}_{p}, C_{j}) where j runs over the supersingular values of the j-line. (= zeroes of the Hasse invariant, i.e., v_{1} = 0).

Corollary. 𝔐^{ss} ⟶ B Aut (𝔽_{p}, Γ_{2}) is (formally) étale.

Topologically: K(2)_{0} ⟶ K(2)_{0}tmf is a colimit of étale morphisms.

≅ ∏_{α} \overline{𝔽}_{p} [\frac{Aut(k, Γ)}{Aut(k, C)}] .

Remark. This can be proved by the Hopkins-Miller-Rezk method: the relative Frobenius

[figure: F : 𝔐^{ss}]

has the property that for all super-singular curves C, there is a canonical isomorphism

(F^{2})^{∗}C ⟶^{≅} ?

Now take E_{∗} = (E_{2})_{∗} = E(\overline{𝔽}_{p}, Γ_{2})_{∗}(−) and apply the Hopkins-Miller proof to realize 𝔒̂_{top}^{ss}. Or more directly: 𝔒̂_{top}^{ss} ≃ ∏_{j} E_{2}(\overline{𝔽}_{p}, Ĉ_{j}) [figure: Aut(C_{j}, Γ)] .

Remark. Let \overline{𝔽}_{p} ⟶^{C_{j}} 𝔐_{ell} be a supersingular curve. Then Serre-Tate implies π_{0} Def (\overline{𝔽}_{p}, C_{j}) ≅ π_{0} Def (\overline{𝔽}_{p}, Ĉ_{j}) ≅ Spf (E(\overline{𝔽}_{p}, Ĉ_{j})). When

𝔐̂^{ss} = 𝔐(E(\overline{𝔽}_{p}, Ĉ), E(\overline{𝔽}_{p}, Ĉ_{j})[ Aut (\overline{𝔽}_{p}, Ĉ_{j})]) .

So 𝔒̂_{Top} is as above.

5. Ordinary curves (Laures/Hopkins)

Here the strategy has been to realize Γ𝔒̂^{ord,p} = L_{K(1)}tmf directly, then extend to the entire sheaf by obstruction theory. This is probably necessary at p = 2. Assuming L_{K(1)}tmf exists we have a picture

(1)  \xymatrix{Spec(K_{0}R) \ar[d]_{étale} \ar[r] & Spec(R) \ar[d]^{étale} \\ Spec(K_{0}tmf) \ar[d]_{smooth} \ar[r]_{étale} & 𝔐̂^{over,p} \ar[d]^{smooth} \\ Spec(ℤ_{p}) \ar[r]_{étale} & 𝔐̂^{(2)}}

If E is a K(1)-local E_{∞} ring spectrum, there is a natural ring operation ψ^{p} : E^{∗}(X) ⟶ E^{∗}(X) satisfying ψ^{p}(x) ≡ x^{p}mod p (= x^{p} + pΘ(x)). If E is complex oriented, the case X = ℂP^{∞} yields an isogeny of the formal group ψ^{p}(T) = T^{p} mod p where T is a coordinate. E is K(1)-elliptic if there exists an elliptic curve C/E^{0} and an isogeny ψ : C ⟶ C lifting ψ^{p}.

Theorem (Katz). (For p ≠ 2) Let E be two-periodic K(1) local E_{∞} elliptic. Then

K_{0}E

is an injective K_{0}K = map(ℤ_{p}^{×}, ℤ_{p}) module. (cf. Laures p = 2, KO_{0}K).

Theorem. K_{0} ⟶ K_{0}tmf is smooth of dimension 1 and Spec(K_{0}tmf) ⟶ 𝔐̂^{ord,ss} is étale.

Proof. Follows from (1) above. □

6. Completion

Consider Z ⊆ 𝔐 cut out by an ideal sheaf ℐ. 𝔐_{Z}^{∧} should be defined so that morphisms f : Spec(R) ⟶ 𝔐_{Z}^{∧} correspond to f̃: Spec(R) ⟶ 𝔐 such that f̃^{∗}(ℐ) ⊆ R is a nilpotent ideal.

Example. 𝔐 = 𝔐(A, Γ), Z ⟺ invariant ideal I ⊆ A, then 𝔐_{Z}^{∧} ≈ \underline{colim}_{N} \underbrace{𝔐(A/I^{N}, Γ/I^{N}Γ)}_{𝔐_{N}}.

A sheaf F on 𝔐_{Z}^{∧} should correspond to family F_{N} over 𝔐_{N}, compatible under pullback.

Example. 𝔐_{F G}^{(n)} ⊇ 𝔐_{F G}^{[n]} cut out by p, v_{1}, …, v_{n−1} [figure: (𝔐_{F G}^{(n)})_{[n]}^{∧}] . Formal completion.

Example (Supersingular locus).

\xymatrix{spf(π_{0}Etmf) \ar[d] \ar[r] & (𝔐_{ell})_{ss}^{∧} \ar[d]_{(⋆)} \ar[r] & 𝔐_{ell} ⊇ 𝔐_{ss} \ar[d] \ar[d] \\ spf(π_{0}E_{2}) \ar[r] & (𝔐_{F G}^{(2)})_{[2]}^{∧} \ar[r] & 𝔐_{F G}^{(2)} ⊇ 𝔐_{F G}^{[2]} \\ Lubin − Tate}

The map (⋆) is étale.

1st proof: It is enough to check that pull back to 𝔐_{F G}^{[2]} is étale.

\xymatrix{𝔐_{ss} \ar[dd]_{(∗)} \ar[d]^{(⋆′)} \ar[r] & (𝔐_{ell})_{ss}^{∧} \ar[d]_{(⋆)} \\ 𝔐_{F G}^{[2]} \ar[d]^{étale} \ar[r] & (𝔐_{F G}^{(2)})_{[2]}^{∧} \\ Spec(𝔽_{p})}

Theorem (Igusa). (∗) is étale. ⟹ (⋆′) étale.

7. Serre-Tate

A p-divisible group G = G[p^{∞}] is a formal direct limit

G[p] ↪ G[p^{2}] ↪ G[p^{3}] ↪ ⋯

of finite (abelian) group schemes, flat over S, such that p^{N} kills G[p^{N}], and multiplication by p^{j} gives a short exact sequence

[figure: 0; G[p^{j}]; G[p^{k}]; ·p^{j}; G[p^{k−j}]; ⊇; 0; G[p^{k}]]

Example. An elliptic curve C has a p-divisible group C[p^{∞}]. Complete at p, get stacks: (𝔐_{ell})_{p}^{∧} ⟶ (𝔐_{p−div})_{p}^{∧}. Let R be a ring with nilpotent ideal I; p nilpotent in R. Then let

Def(R) be the category with objects (C_{0}, G[p^{∞}]; φ : G_{0}[p^{∞}] ⟶^{≅} C_{0}[p^{∞}]) where C_{0} is an elliptic curve over R/I, G[p^{∞}] a p-divisible group over R and φ an isomorphism of p-divisible groups; morphisms are the isogenies.

Theorem (Serre-Tate). The functor

Elliptic curves/R ⟶ Def(R)
C ⟼ (C_{0}, C[p^{∞}], id)

is an equivalence of categories.

Translation into the language of stacks:

[figure: Spec(R/I); { \atop (𝔐_{ell})_{p}^{∧}}; ≡; (𝔐_{ell})_{p}^{∧}; Spec(R); (𝔐_{p−div.})_{p}^{∧}]

Here (𝔐_{ell})_{p}^{∧} is the ‘path space’: objects over R are data (C_{0}, G[p^{∞}], φ : G[p^{∞}] ⟶^{≅} C_{0}[p^{∞}]), all over R. The Serre-Tate theorem says the groupoid of lifts (dotted arrows) is contractible. Thus the morphism (𝔐_{ell})_{p}^{∧} ⟶ (𝔐_{p−div})_{p}^{∧} is ‘formally étale’.

Any p-divisible group G sits in a natural extension

0 ⟶ G^{formal} ⟶ G ⟶ G^{ét} ⟶ 0 .

If C is a (deformation of) a supersingular curve, then C[p^{∞}] = Ĉ, i.e., the p-divisible group is all formal, so

\xymatrix{(𝔐_{ell})_{ss}^{∧} \ar[d]_{formally étale} \ar@{^{(}->}[r] & 𝔐_{ell} \ar[d] \\ (𝔐_{F G}^{(2)})_{[2]}^{∧} \ar@{^{(}->}[r] & 𝔐_{p-div}}

8. Ordinary Curves (at p)

\xymatrix{(𝔐_{ord})_{p}^{∧} \ar[d] \ar[r] & 𝔐_{ell} \ar[d] \\ (𝔐_{F G}^{(1)})_{[I]}^{∧} \ar[d] \ar[r] & 𝔐_{F G}^{(2)} \\ equals (𝔐_{F G}^{(1)})_{p}^{∧} .}

To construct the sheaf (𝔒̂_{top}^{ord})_{p} over (𝔐_{ord})_{p}^{∧}, we first build

Γ(𝔒_{top}^{ord})_{p}^{∧} = L_{K(1)}tmf ,

then extend to the entire sheaf by obstruction theory

[figure: Spec(K_{0}R); étale; Spec(R); étale; V = (K_{0}tmf)_{p}^{∧}; Spec(V); étale; (𝔐^{ord})_{p}^{∧}; Spec(ℤ_{p}); étale; (𝔐_{F G}^{(I)})_{p}^{∧}]

To build sections over [Spec(R) ⟶_{C} 𝔐_{p}^{ord})^{∧}], we must plug in the map of rings V = K_{0}tmf ⟶ ‘K_{0}R’ defined by pull back above into obstruction machinery. But this map is étale by construction, so all obstructions vanish and we obtain Γ(C, 𝔒_{top}). The same techniques show, for Spec(R), Spec(R′) ⟶^{étale} (𝔐_{ord})_{p}^{∧}, that

map_{E_{∞}−rings/L_{K(1)}tmf} (Γ(C, 𝔒_{top}), Γ(C′, 𝔒_{top})) ⟶^{≃} Hom /𝔐^{∧}(Spec(R), Spec(R′))

is a weak equivalence.

\xymatrix{Spec(V) \ar[d] \ar[r]^{étale} & (𝔐_{ord})_{p}^{∧} \ar[d] \ar[dd]^{(⋆)} \\ Spec(ℤ_{p}) \ar[dr]_{id} \ar[r]^{étale}_{G_{m}} & (𝔐^{(1)})_{p}^{∧} \ar[d] \\  & Spec(ℤ_{p})}

The ring V represents the functor

R ⟼ isomorphism classes of (C, Φ : 𝔾̂_{m} ⟶^{≃} Ĉ)

for rings R with p-nilpotent.

Theorem (Katz). The map (⋆) is smooth of dimension 1, and thus Spec(V) ⟶ Spec(ℤ_{p}) is too.

Again for (p ≠ 2), V is a (relative) injective ℤ_{p}^{×}-module (Aut(𝔾̂_{m})). We want to construct an E_{∞}-ring whose K-theory is V.

9. K(1)-local E_{∞}-rings

If E is an E_{∞}-ring, we have ‘power operations’

π_{0}E ⟶^{B} π_{0}E^{BΣ_{p}^{+}}

Theorem (McClure). The map (proj., transfer) : BΣ_{p}^{+} ⟶ S^{0} ∨ S^{0} is a K(1)−equivalence, so K(1)-local E_{∞}-rings admit operations

Ψ, Θ : π_{0}E ⟶ π_{0}E ,

where Ψ is ring map, and Ψ(x) = x^{p} + pΘ(x) (i.e., π_{0}E is a ‘Θ-ring’).

Note that mod p, Ψ is just the Frobenius map x ⟼ x^{p}.

If E is complex orientable, \xymatrix{π_{0}E^{ℂP_{+}^{∞}} \ar[r]^{Ψ} & π_{0}E^{ℂP_{+}^{∞}} \\ π_{0}E \ar[u] \ar[r]_{Ψ} & π_{0}E \ar[u]}  ⟹  [figure: G; Ψ; G; Ψ^{∗}G; Φ; spf (E_{0}); Ψ; spf (E_{0})]

We get an isogeny Φ : G ⟶ Ψ^{p}G of degree p of f.g./E_{0}, which mod p is the Frobenius.

Example. E = K_{p}^{∧}, Ψ = Ψ^{p}, id on K_{0}, so Φ = [p].

Definition (Hopkins-Laures). A K(1)-local E_{∞} elliptic spectrum is an K(1)-local E_{∞}-ring, complex orientable E, with an elliptic curve C/π_{0}E, Ĝ_{e} ≅ Ĉ, and an isogeny Φ̃ : C ⟶ Ψ^{∗}C compatible with the one on Ĝ_{e}.

Example (Atkin-Lehner operator). An ordinary elliptic curve (over a p-torsion ring) contains a canonical subgroup scheme of degree p, the formal part of p-torsion group C[p]

\xymatrix{Ĉ[p] ⊆ C[p] ⊆ & C \ar@{->>}[r] & C/Ĉ[p] = C′ \\ 𝔾̂_{m}[p] \ar[u]^{ϕ} \ar@{^{(}->}[r] & 𝔾̂_{m} \ar[rr]_{p} \ar[u]_{ϕ} \ar@{->>}[r] & 𝔾̂_{m}/𝔾̂_{m}[p] \ar[u] \ar[r]^{∼} & 𝔾̂_{m} \ar[ul]_{ϕ′}}

C′ is again an ordinary curve, and ϕ′ : 𝔾̂_{m} ⟶ Ĉ′ an isomorphism, so (C, ϕ) ⟼ (C′, ϕ′) defines a map

Ψ : V ⟶ V .

Note that over a ring R of characteristic p, Ĉ[p] ≅ spf (R[x]/x^{p}), and C ⟶ C′ can be identified with the Frobenius, so Ψ(x) ≡ x^{p} mod p.

Now the problem is to realize L_{K(1)}tmf as K(1)-local E_{∞}-ring such that K_{0}^{∧}tmf ≃ V as Θ-ring.

The obstructions lie in H^{s}(V_{∗}/K_{∗}, Σ^{−t}V_{∗}) with t−s = −2, s ≥ 2, where V_{∗} = K_{∗}^{∧}tmf. Cohomology is in the category of ‘K_{∗}K-comodule Θ-rings’ where K_{∗}K = map(ℤ_{p}^{×}, ℤ_{p}). There is a composite functor spectral sequence

Ext_{C}^{∗}(H_{∗}^{AQ}(V_{∗}/K_{∗}), Σ^{−∗}V) ⟹ H^{∗}(V_{∗}/K_{∗}, Σ^{−∗}V_{∗})

where C is the category of K_{∗}K-comodules/Θ-modules. We recall that V/ℤ_{p} is smooth of dimension 1, so

H_{s}^{AQ}(V/ℤ_{p}) = \{ \begin{matrix}0 & s > 0 \\ Ω_{Λ/ℤ_{p}} & s = 0 (a locally free rank 1 Λ-module). \end{matrix}

Ω_{Λ/ℤ_{p}} is a comodule over map(ℤ_{p}^{×}, ℤ_{p}) and a Θ-module:

Definition. A Θ-module M is a V -module together with a map Θ : M ⟶ Ψ^{∗}M of V -modules (i.e., Θ(vm) = Ψ(v)Θ(m)). In other words, V × M is an abelian group object in Θ-rings.

For p ≠ 2, V_{∗} is injective as a K_{∗}K-comodule, and we can compute Ext by a resolution of Ω by free Θ-modules:

0 ⟶ Ω ⊗ ℤ_{p}[Θ] ⟶ Ω ⊗ ℤ_{p}[Θ] ⟶ Ω ⟶ 0

free V ⊗ ℤ[Θ]-modules. So only Ext^{0} and Ext^{1} are non-trivial.

10. Glueing

Now we want to construct the map L_{K(1)}tmf ⟶ L_{K(1)}L_{K(2)}tmf. We constructed L_{K(1)}tmf so K_{0}^{∧}tmf = V. We set W = K_{0}^{∧}L_{K(2)}tmf. We need to show that V ⟶ W can be realized by a map of E_{∞}-ring spectra. (The map comes from the presense of an ordinary curve over W, induced from an elliptic curve associated to L_{K(2)}tmf (if p ≠ 2, 3)). Claim 1: W is (relative) injective as map(ℤ_{p}^{×}, ℤ_{p})-comodule (p ≠ 2). Claim 2: L_{K(1)}(K ∧ L_{K(2)}tmf) is a K(1)-local E_{∞}-elliptic spectrum, i.e., the operator φ on π_{0}L_{K(1)}(K ∧ L_{K(2)}tmf) lifts to an isogeny of curves. In particular, V ⟶ W is a map of θ-rings, plug into obstruction machine...

Proof of claim 1 (Sketch). This should be a special case of a more general result: If R is a Landweber exact ring spectrum, then K_{0}^{∧}R is injective. Such R have L_{K(1)}R ≃ (v_{1}^{−1}R)_{p}^{∧} = R′.

\xymatrix{ & Spec(K_{0}K) \ar[r] & Spec((ℤ_{p}) \\ Spec(K_{0}K ∧ R) \ar[ur] \ar[r] \ar[d] & Spec(K_{0}R) \ar[ur] \ar[d]^{étale} \\ Spec((K_{0}R)_{p}^{∧}) \ar[r]_{étale} \ar[d] & Spec((v_{1}^{−1}R)_{p}^{∧}) \ar[d] \\ Spec(ℤ_{p}) \ar[r]_{étale_{(cover)}} & (𝔐_{F G}^{1})_{[1]}^{∧}}

to show injectivity, it is enough to show after base change along π_{0}v_{1}^{−1}R ⟶ K_{0}R; but this is pull back from (ℤ_{p} ⟶ K_{0}K). □

11. What is a K(2)-local E_{∞} elliptic spectrum, and why is this notion not necessary?

Let F ⟶ BΣ_{p}^{+} ⟶^{transfer} S^{0} be a fibration, and assume E is sufficiently ‘nice’ (e.g. Lubin-Tate). Power operations:

[figure: π_{0}E; P; Ψ a ring hom.; π_{0}E^{BΣ_{p}^{+}}; π_{0}E^{F}; i; π_{0}E]

The map i is associated to \xymatrix{F \ar[r] & BΣ_{p}^{+} \ar[r]_{proj} & S^{0}}

Theorem (Strickland). π_{0}E^{F} (as a π_{0}E-algebra by i) classifies subgroup schemes of G = Ĝ_{E} of order p. Equivalently, it classifies isogenies which are deformations of the Frobenius modulo the maximal ideal.

In the following we set S = spf(π_{0}E) and T = spf(π_{0}E^{F}).

[figure: G; i^{∗}G; ψ; φ; φ^{∗}G; G; S; i; T; ψ; S]

Here φ : i^{∗}G ⟶ ψ^{∗}G is an isogeny of degree p over T. Modulo the maximal ideal of π_{0}E^{F}, H is the Frobenius. The kernel of [i^{∗}G ⟶ ψ^{∗}G] is the universal example of a subgroup scheme of rank p. An E_{∞}-elliptic structure would include an elliptic curve C/S, and an isogeny φ̃ : i^{∗}C ⟶ ψ^{∗}C, compatible with φ, which reduces to the Frobenius modulo the maximal ideal. By Serre-Tate, exists a unique such isogeny, once we specify the curve C (with C[p^{∞}] ≃ G). So we never have to specify it!

12. How do p-divisible groups enter homotopy theory?

Let E = E_{2}. We have

E^{ℂP_{+}^{∞}} ≃ holim_{⟵} E^{BC_{p^{k}}^{+}}

since lim_{⟶} BC_{p^{k}} ≃ BC_{p^{∞}}, which after p-completion is equivalent to BS^{1}. We have π_{0}E^{BC_{p+}} ≃

E_{0}[[x]]/[p](x) = \underbrace{(px + v_{1}x^{p} + x^{p^{2}})}_{f(x)} ·(unit) so E^{BC_{p+}} ≃ ∨_{p^{2}}E .

We recall that L_{K(1)}E ≃ (v_{1}^{−1}E)_{p}^{∧}, and consider the map L_{K(1)}(E_{BC_{p+}}) ⟶ (L_{K(1)}E)^{BC_{p+}}.

• L_{K(1)}(E^{BC_{p+}}) ≃ ⋁_{p^{2}} L_{K(1)}E, π_{0}L_{K(1)}(E^{BC_{p+}}) ≃ [v_{1}^{−1}(E_{0}[x]/f(x))]_{p}^{∧} mod p, f(x) ≡

g(x^{p}), where g(x) = v_{1}x + x^{p}.

• (L_{K(1)}E)^{BC_{p}} ≃ ⋁_{p} L_{K(1)}E, π_{0}(L_{K(1)}E)^{BC_{p+}} ≃ (v_{1}^{−1}E_{0})_{p}^{∧}[[x]]/f(x), since f(x) = (px +

v_{1}x^{p})· (unit).

[figure: spf(π_{0}(L_{K(1)}E)^{BC_{p^{k}+}}); spf(π_{0}L_{K(1)}(E^{BC_{p^{k}+}})); spf(π_{0}E^{BC_{p^{k}+}}); G_{(1)}^{formal}[p^{k}]; G_{(1)}[p^{k}]; G[p^{k}]; S_{(1)}; ≃; S_{(1)}; S; spf(π_{0}L_{K(1)}E); spf(π_{0}E)]

i.e., the π_{0}L_{K(1)}(E^{BC_{p^{k}+}}) fit together to give a p-divisible group G_{(1)} := lim_{⟶} G_{(1)}[p^{k}] over S_{(1)}, while the π_{0}(L_{K(1)}E)^{BC_{p^{k}+}} give its formal part

G_{(1)}^{formal} := lim_{⟶} G_{(1)}^{formal} [p^{k}] .

Let Ẽ^{ℂP_{+}^{∞}} := holim_{⟵}L_{K(1)}(E^{BC_{p^{k}+}}), a K(1)-local E_{∞}-ring. (Maybe I really want an inverse system …) This gives a diagram of E_{∞}-rings

(2)  \xymatrix{E^{ℂP_{+}^{∞}} \ar[r] & ∼ E^{ℂP_{+}^{∞}} \ar[r] & (L_{K(1)}E)^{ℂP_{+}^{∞}} \\ E \ar[u] \ar[r] & L_{K(1)}E \ar[u] \ar[r]^{≃} & L_{K(1)}E \ar[u]}

and a corresponding diagram of schemes [figure: G; G(1); G_{(1)}^{formal}; S; S_{(1)}; ≃; S_{(1)}] G_{(1)} = p-divisible group.

The left-hand square appears to be a pull-back. (3) The Power operation π_{0}A ⟶^{ψ} π_{0}A^{F} is natural in the E_{∞}-ring A; so we have commutative diagrams

(3)

[figure: π_{0}E; i; π_{0}E^{F}; ψ; π_{0}E; π_{0}L_{K(1)}E; i; ≃; π_{0}(L_{K(1)}E)^{F}; ψ; π_{0}L_{K(1)}E; S; i; T; φ; S; S_{(1)}; i; ≃; S_{(1)}; can; φ; S_{(1)}]

middle map represents “canonical subgroup”.

Applying the same to the diagram (2) gives

[figure: G_{(1)}; G_{(1)}^{formal}; i^{∗}G; φ; i^{∗}G_{(1)}; φ_{(1)}; i^{∗}G_{(1)}^{formal}; φ_{(1)}^{formal}; ψ^{∗}G; ψ^{∗}(G_{(1)}); ψ^{∗}G_{(1)}^{formal}; T; can; S^{(1)}; ≃; S^{(1)}; isogenies]

I claim that φ_{(1)} is the base change of φ along can. Given this, suppose C is an elliptic curve over π_{0}E attached to G. Then C_{(1)} := S_{(1)} ×_{S} C has G_{(1)} as its p-divisible group. Since we have an isogeny on G_{(1)} (compatible with the one on the formal group G_{(1)}^{formal}), Serre-Tate says this lifts uniquely to one on the curve C_{(1)} [which reduces to the Frobenius mod p]. This gives the K(1)-local E_{∞}-elliptic spectrum structure we need. To show the claim, I need that

[figure: i^{∗}G; G_{(1)}; T; can; S^{(1)}; and; ψ^{∗}G; ψ^{∗}(G_{(1)}); T; can; S]

are pull backs. This appears to follow from (3) and the commutativity of ☺.