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

Note: there is no rigid geometry in this set of notes! That will come next time, when we talk about the period mapping.

References: For starters, the original paper of Lubin-Tate (which involves no rigid geometry, only formal geometry) is: J. Lubin and J. Tate, Formal moduli for one-parameter formal Lie groups, Bulletin de la Soc. Math. de France, 94 (1966), 49–59. (This is not their paper on local class field theory, though of course the two are closely related.) As for “Gross-Hopkins”, there are two such papers. One (here [GH1]) is: M.J. Hopkins and B.H. Gross, The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory, Bulletin of the AMS 30 (1994), 76–86. This paper makes the link between Lubin-Tate spaces and stable homotopy theory; I won’t do that here. It is quite cursory on the geometry side (I’m not capable of judging on the homotopy side), to the point of being barely legible. The second paper (here [GH2]) is: M.J. Hopkins and B.H. Gross, Equivariant vector bundles on the Lubin-Tate moduli space, in Topology and Representation Theory, Contemporary Mathematics 158, AMS, 1994, 23–88. As the page count suggests, this is much more detailed and focuses entirely on the rigid geometry.

Formal groups

Before we do any geometry, here’s a quick review of formal groups. The standard reference is Hazewinkel, Formal Groups, though I think his formulas have some errors in them; use with caution.

A (commutative) formal group of dimension n over a ring R is a (commutative) cogroup structure on R⟦x_{1}, …, x_{n}⟧ with identity 0, i.e., a (commutative) comultiplication satisfying the usual (co)group axioms. It’s of course enough to specify how x_{1}, …, x_{n} behave under the comultiplication; their images form an n-tuple of power series

F (X, Y) = (F_{1}(X, Y), …, F_{n}(X, Y))

(where X is short for x_{1}, …, x_{n} and Y for y_{1}, …, y_{n}), such that

F (X, 0) = X, F (0, Y) = Y
F (X, F (Y, Z)) = F (F (X, Y), Z)
F (X, Y) = F (Y, X).

(Exercise: the existence of inverses is automatic given the other axioms.) A morphism f : F → F′ of formal groups is an n-tuple of n-variate power series such that F′(f(X), f(Y)) = F (X, Y). Let Lie(F) denote the (relative) tangent space of R⟦x_{1}, …, x_{n}⟧, i.e., the trivial Lie algebra over R generated by x_{1}, …, x_{n}. Every endomorphism of F induces a R-linear endomorphism on Lie(F), which is just look at the linear terms in the power series.

Examples: take any algebraic group of dimension n and restrict the group law to the tangent space at the origin, and you get a formal group of dimension n. For 𝔾_{a}, you get

F (x, y) = x + y.

For 𝔾_{m}, you get

F (x, y) = x + y + xy,

or x + y − xy for another choice of coordinates, or crazier things for more bizarre choices of coordinates. (Those two are isomorphic if ℚ ⊆ R, even though the original groups are not, but not in general. I’ll make a stronger statement below.) Any elliptic curve gives a formal group of dimension 1, as constructed in Silverman’s book. Also, abelian varieties and linear groups give you other examples in higher dimension; however, I’m mostly interested here in dimension 1.

If I knew what it was, I would mention here the connection between formal groups of dimension 1 and stable homotopy theory (which as far as I can tell is due more or less entirely to Hopkins). However, I don’t; maybe Mark can enlighten us a bit at some point.

Formal 𝔬-modules

I’m also going to work a bit (following [GH2]) with Drinfeld’s more general notion of formal 𝔬-modules, where 𝔬 is a complete DVR with finite residue field k = 𝔽_{q}. Fix a choice of a uniformizer π of 𝔬, and put K = Frac 𝔬 as usual. Let R be a (commutative) 𝔬-algebra; I’ll call the structure map i : 𝔬 → R if I need to refer to it. A formal 𝔬-module of dimension n over R is a (commutative) formal group F of dimension n equipped with a ring homomorphism θ = θ_{F} : 𝔬 → End_{R}(F), such that

θ(a)(X) ≡ i(a)X (mod (x_{1}, …, x_{n})^{2}).

(That is, the action of θ(a) on Lie(F) is by multiplication by i(a).) A formal group is automatically a formal ℤ_{p}-module as long as ℤ_{p} ⊆ R.

Convention: I’ll write a_{F} instead of θ_{F} (a), as in [GH2].

Example: the first Lubin-Tate paper (Formal complex multiplication in local fields, Ann. Math. 81 (1965), 380–387) show that you can uniquely specify a formal 𝔬-module of dimension 1 by specifying the action of π: it must be given by a series f(x) with f(x) ≡ πx (mod x^{2}) and f(x) ≡ x^{q} (mod π).

Example: in equal characteristic p, Drinfeld stumbled across these as the analogues of the formal group associated to an algebraic group, for what we call “Drinfeld modules”. Briefly, a Drinfeld module is an action of a finite extension of the ring 𝔽_{q}[t] on the additive group of a ring in characteristic p, via “additive polynomials”:

x ↦ c_{0}x + c_{1}x^{p} + ⋯ + c_{n}x^{p^{n}}.

These act like abelian varieties in many ways (e.g., producing Galois representations), but have much simpler moduli and so are useful for things like proving the Langlands correspondence for GL_{2} over function fields (Drinfeld’s original application).

You can speak of the “invariant differentials” of F, i.e., the elements ω of the module of formal differentials (i.e., the free R⟦X⟧ over x_{1}, …, x_{n}) such that ω(F (X, Y)) = ω(X)+ω(Y), and ω(a_{F} (X)) = i(a)ω(X) for a ∈ 𝔬. These form a free R-module of rank n, called ω(F); in fact, the “quotient mod degree 2” map from invariant differentials to R dx_{1} ⊕ ⋯ ⊕ R dx_{n} is a bijection, and all invariant differentials are closed [GH2, Proposition 2.2].

Policy: I’m now going to assume dimension 1 forever after, because Lubin-Tate theory applies only in dimension 1. Also, I may skip the 𝔬-module generalizations of some statements about formal groups, but those are all straightforward to extend (or see [GH2]).

Logarithms

If f : F → 𝔾_{a} is a homomorphism of formal 𝔬-modules, you can take its formal derivative

ω = df(x) = \frac{df}{dx} · dx.

That gives a homomorphism d : Hom(F, 𝔾_{a}) → ω(F). By [GH2, Proposition 3.2], if R is flat (i.e., torsion-free) over 𝔬, then d is injective; if R is a K-algebra, then d is bijective. In particular, in the latter case, there is a unique isomorphism f : F → 𝔾_{a} with df equal to any prescribed generator of ω(F). We call f a logarithm for F.

Height

It’s an easy lemma [GH2, Lemma 4.1; beware that f is used to mean two different things in the same sentence!] that if R is a field and F is a formal group with i(π) = 0, then either π_{F} = 0 or there is an integer h such that

π_{F} (x) = f(x^{q^{h}})

for some series f with f′(0) ≠ 0. In the second case, we say F has height h. If R is a complete local ring whose maximal ideal I contains i(π) (hereafter a local 𝔬-algebra), we say F has height h if the reduction of F has height h over R/I. (Define height for a formal 𝔬-module as the height of the underlying formal group. Oh, and this definition doesn’t depend on dimension 1.)

Deformations

If we fix a formal 𝔬-module F_{0} of some height h over R/I, I will refer to a formal 𝔬-module F over R equipped with an isomorphism of its reduction to F_{0} as a deformation of F_{0} over R. Then the Lubin-Tate-Drinfeld theorem explicitly describes a universal deformation of F_{0}. This comes from the following fact: if R is flat over 𝔬, then any formal 𝔬-module (of dimension 1, as always here) can be presented so that its formal logarithm takes the form

f(x) = x + ∑_{k=1}^{∞} b_{k}x^{q^{k}}

for b_{k} ∈ R ⊗ K, and this presentation is unique. A formal 𝔬-module presented this way is said to be 𝔬-typical. (See [GH2, §5].)

Example: for 𝔾_{m} over ℤ_{p}, b_{k} = p^{−k} and you get the formal logarithm of the Artin-Hasse exponential.

As the previous example shows, it’s better to work with a certain change of variable here. Keeping R flat, define v_{1}, v_{2}, … by

πb_{k} = v_{k} + b_{1}v_{k−1}^{q} + ⋯ + b_{k−1}v_{1}^{q^{k−1}};

then the v_{k} turn out to be integral. In fact, unwinding the construction yields a “universal formal 𝔬-typical module” over the infinite polynomial ring 𝔬[v_{1}, v_{2}, …].

It turns out that you can read off heights easily here: the 𝔬-typical module constructed above has height h if and only if v_{1}, …, v_{h−1} vanish in R/I and v_{h} does not.

The Lubin-Tate-Drinfeld theorem (Lubin-Tate for formal groups, Drinfeld for formal 𝔬-modules) now asserts that if you pull back the universal formal 𝔬-typical module to 𝔬⟦u_{1}, …, u_{h−1}⟧ via

v_{i} ↦ u_{i} (i = 1, …, h − 1)
v_{h} ↦ 1
v_{i} ↦ 0 (i ≥ h + 1),

and call the result F, then F is a universal deformation of its reduction modulo (π, u_{1}, …, u_{h−1}) (which is thus defined over 𝔽_{q}, and which has height h). See [GH2, Proposition 12.10] for the (easy) cohomological computation that verifies this.