18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Periods on the Lubin-Tate moduli space

To keep things moving, I’m going to be terse now (as in [GH1]); there are lots of details filled in in [GH2].

References: Jay notes that [GH1] (the terse one) is available online; I’ll put a link on the notes page.

Corrections from last time

Thanks to Jay for these.

page 2: the first Lubin-Tate paper only deals with height 1 (that is all that is needed for local class field theory), and they show uniqueness only over the completion of the maximal unramified extension of K. (Indeed, the fact that they are not isomorphic is key to being able to use them for explicit local reciprocity!)

page 3, top of page: “free R⟦X⟧ over x_{1}, …, x_{n}” should be over dx_{1}, …, dx_{n}. By ω(F (X, Y)), I mean you write ω(X) = f_{1}(X)dx_{1} + ⋯ + f_{n}dx_{n}, then you write F (X, Y) = (F_{1}(X, Y), …, F_{n}(X, Y)), you plug in

ω(F (X, Y)) = f_{1}(F (X, Y))dF_{1}(X, Y) + ... + f_{n}(F (X, Y))dF_{n}(X, Y),

then expand each dF_{i}(X, Y) by the chain rule.

page 3, Height: change “formal group” in the second line to “formal 𝔬-module” and scratch the reference to formal 𝔬-modules in the parenthetical.

A group action

Last time, we built a universal deformation F over A = 𝔬⟦v_{1}, …, v_{h−1}⟧ of a formal 𝔬-module over 𝔽_{q} of height h, which I’ll denote by F_{0}. That means that the group G = Aut(F_{0}) acts on the deformation space Spf A, and on the corresponding rigid analytic space X, which is the open unit polydisc in v_{1}, …, v_{h−1}.

It turns out that D = End(F_{0}) is a division algebra of degree n, G is the group of units in some maximal order therein, and D ⊗ K ≅ M_{n}(K) is split. That means G has a natural n-dimensional linear representation V_{K} over K, as does D^{∗}. In particular, G acts on the hyperplanes of V_{K}, i.e., on ℙ(V_{K}^{∨}); the latter carries a rigid analytic space structure, and the group action is by analytic morphisms.

The crystalline period mapping, to be defined, is a rigid analytic G-equivariant étale morphism Φ : X → ℙ(V_{K}^{∨}) which classifies deformations “up to isogeny” as follows. For A an affinoid algebra over K, let F_{a} and F_{b} be deformations of F_{0} over 𝔬_{A}, corresponding to points a, b ∈ X(A). Then an isogeny of F_{0}, viewed as an element T ∈ D^{∗}, deforms to an isogeny F_{a} → F_{b} if and only if T Φ(a) = Φ(b).

The universal additive extension

In order to specify Φ, I have to give you a G-equivariant line bundle L on X and a K[G]- homomorphism V_{K} → H^{0}(X, L) whose image is basepoint-free (i.e., the images don’t all vanish at a point). For x ∈ X, we then take Φ(x) to be the hyperplane of V_{K} which maps to sections of L vanishing at x.

First, L is the inverse of the analytification of the sheaf ω of invariant differentials, a/k/a the Lie algebra Lie(F). In order to make the map, I must consider the universal additive extension E of F; it sits in an exact sequence

0 → N → E → F → 0

with N = 𝔾_{a} ⊗ Ext(F, 𝔾_{a})^{∨}, and it is universal: if 0 → N′ → E′ → F → 0 is an extension of F by an additive 𝔬-module, then there are unique homomorphisms i : E → E′, j : N → N′ such that

\xymatrix{0 \ar[r] & N \ar[d]^{j} \ar[r] & E \ar[d]^{i} \ar[r] & F \ar[r] \ar[d]^{_{id_{F}}} & 0 \\ 0 \ar[r] & N′ \ar[r] & E \ar[r] & F \ar[r] & 0}

commutes. (This is straightforward, modulo some cohomology arguments which have already been exploited in constructing the universal deformation, namely, that Hom(F, 𝔾_{a}) = 0 and Ext(F, 𝔾_{a}) is free of rank n − 1. See [GH2, Proposition 11.3].) On the level of Lie algebras, we have

0 → Lie(N) → Lie(E) → Lie(F) → 0

and this sequence is G-equivariant.

The bundle Lie(E) turns out to be the covariant Dieudonné module of F_{0}, so it is an “F - crystal”: it carries an integrable connection ∇ : Lie(E) → Lie(E) ⊗ Ω_{A/𝔬}^{1} plus a “Frobenius structure”. The latter can be viewed as an isomorphism σ^{∗} Lie(E) → Lie(E) for any σ : A → A lifting the q-power map on the special fibre. Let M be the analytification of Lie(E), as a rigid vector bundle over X; then by “Dwork’s trick”, M admits a basis of horizontal sections over X. (The idea: by formal integration, you get a basis over a small polydisc. But then you use the Frobenius pullback to “grow” this polydisc.) If you prefer, this can all be described in formulas in this case: see [GH2, Section 22].

We now have our representation V_{K} = H^{0}(X, M)^{∇} on the horizontal sections of M: the surjection M → L → 0 gives the map V_{K} → H^{0}(X, L) whose image is basepoint-free. The verification that Φ is étale and detects isogenies can be found in [GH2, Section 23].

What is Dwork’s trick?

This is worth explaining a bit more, because it also comes up all over the place in p-adic cohomology. Say you have a vector bundle M over the open unit polydisc X over K with coordinates x_{1}, …, x_{n}. (Note: what was n − 1 before is n now for notational simplicity.) It turns out that M is in fact generated freely by global sections. [Correction of 27 Oct 2012: this was previously attributed to Gruson, which is incorrect. Instead see: W. Bartenwerfer, kholomorphe Vektorraumbündel auf offenen Polyzylindern, J. reine angew Math. 326 (1981), 214-240.]

Say you also have an integrable connection ∇ : M → M ⊗ Ω^{1}, i.e., commuting actions of ∂_{i} = \frac{∂}{∂x_{i}} for i = 1, …, n. You can then try to formally solve for the horizontal sections around x_{1} = ⋯ = x_{n} = 0. The resulting sections will converge on some polydisc, but its radius may be much smaller than 1. E.g., if n = 1, the rank is 1, and ∂_{i}v = cv, the horizontal section is exp(− ∫ c)v, which converges on some disc but possibly a small one.

What having a Frobenius structure does is give you an isomorphism between M and its pullback along some map σ lifting the q-power Frobenius on the reduction of Γ(O, X), e.g., one taking K into itself and taking x_{i} to x_{i}^{q}. That pulls back your sections on a tiny polydisc to sections on a larger polydisc (in my example, sections on the disc |x_{i}| ≤ ρ pull back to |x_{i}| ≤ ρ^{1/q}); but in fact the K-vector space of horizontal sections is unique, so these sections actually extend the ones you started with. Repeat ad infinitum.