Moduli stacks of (ϕ, Γ)-modules
Toby Gee (joint with Matthew Emerton)

This project has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. 884596)

Galois representations

K/Qp finite extension.
GK := Gal(K/K).
R a topological ring.
Is there a moduli space of representations ρ : GK → GLd (R)?

Moduli spaces of Galois representations
There is a good theory over Zl , l 6= p, or even Z[1/p].
Basic point: the action of wild inertia doesn’t deform, and tameinertia is “easy”. Can work with Weil–Deligne representations andget finite type lci moduli spaces (Helm,. . . )Over Zp : doesn’t work.
Mazur: fix ρ : GK → GLd (F), F/Fp finite, and consider lifts of ρto ρ : GK → GLd (R), where R is Artin local with residue field F.
Moduli space: Spf Rρ .
How can we let ρ vary?

Existence of the stack
K/Qp finite extension, GK := Gal(K/K).

Theorem
There is a Noetherian formal algebraic stack Xd over Spf Zp , suchthat Xd (Spf Zp ) is naturally equivalent to the groupoid ofcontinuous representations GK → GLd (Zp ).
The underlying reduced substack Xd,red is an algebraic stack offinite type over Fp , and is equidimensional of dimension[K : Qp ]d(d − 1)/2.
Similarly for Xd (Fp ). Not true for general p-adically complete
Zp -algebras, e.g. Fp [x].
n.b. Xd is not a p-adic formal algebraic stack. It is probably[K : Qp ]d2 -dimensional.

The 1-dimensional case

For simplicity from now, unless otherwise stated: K = Qp .
×Characters GQp → Fp are of the form λa εi , 0 ≤ i < p − 1, where
λa is the unramified character taking Frob 7→ a.
X1 has (p − 1) irreducible components, indexed by i.
X1,red is 0-dimensional: one dimension for a, but automorphisms
are Gm .

Definition of the stack

A a finite type Z/pa Z-algebra for some a ≥ 1.

Definition
Xd (A) := rank d projective étale (ϕ, Γ)-modules with
A-coefficients.
Extend to general p-adically complete A by taking limits.
Identification of Xd (Zp ) with GK → GLd (Zp ) is due to Fontaine.

étale (ϕ, Γ)-modules
A a finite type Z/pa Z-algebra for some a ≥ 1.
A rank d projective (ϕ, Γ)-module with A-coefficients is: a rank dprojective A((T ))-module M with commuting semilinear actionsof ϕ and Γ.
ϕ : A((T )) → A((T )) is A-linear, ϕ(1 + T ) = (1 + T )p .
Γ = Gal(Qp (ζp∞ )/Qp ), ε : Γ → Z×p cyclotomic character.
γ : A((T )) → A((T )) is A-linear, γ(1 + T ) = (1 + T )ε(γ) ,étale: M = A((T )) · ϕ(M ).

Closed points and specializations

Irreducible representations GQp → GL2 (Fp ) are again discrete up
to unramified twist, and give 0-dimensional substacks of the1-dimensional algebraic stack X2,red .


λa εi
∗
Reducible indecomposable representations
with
0
λb εj
∗=
6 0 can specialize to ∗ = 0, so these are not closed points.
Fact: closed points of Xd = semisimple GK → GLd (Fp ).

Closed points and specializations
Are there other specializations?
Consider the family of étale ϕ-modules over Fp given by
ap −1
ϕ=Ti 0with 1 ≤ i ≤ p − 2 and ap ∈ Fp . (This can be equipped with anaction of Γ).
If ap 6= 0, corresponding GQp → GL2 (Fp ) is reducible, but if
ap = 0 it is irreducible.
This cannot happen for a literal family of representations!

Irreducible components
Example from the previous slide is a feature, not a bug.
∗
λ a εiwith a, b varying.
For each 0 ≤ i, j < p − 1 consider0
λb ε jThe closure of this is a 1-dimensional substack.
In fact if i ≡ j + 1 (mod p − 1) we have an extra irreducible
component with a = b.

Theorem(K = Qp ) X2 has p(p − 1) irreducible components, indexed by i, jas above.
More generally, the irreducible components of Xd are indexed by(k1 , . . . , kn ) with 0 ≤ ki − ki+1 ≤ p − 1, 0 ≤ kn < p − 1.

A coarse moduli space for GL2 (Qp )In general the specialisation relations are complicated and noobvious coarse moduli space.
For GL2 (Qp ): the only interesting specialisations are those wewrote down before.
Fixing determinants, have a 1-dimensional scheme, in fact a chainof P1 s.

A coarse moduli space for GL2 (Qp )sa Indw Eoane
EE Indwzp

2Ada ti e iIndwz

P'tE I Indwz

The crystalline/potentially semistable substacks
Fix λ = (λ1 ≥ · · · ≥ λd ).

Theoremcrys,λThere is a unique closed substack Xdof Xd which is flatover Spf Zp , and such that if A/Zp is finite flat, thencrys,λXd= {ρ : GQp → GLd (A) |ρ ⊗ Qp is crystalline with Hodge–Tate weights λ}.

crys,λXd

is a p-adic formal algebraic stack.
crys,λIf λ1 > · · · > λd then (Xdequal to dim Xd,red .

)Fp is equidimensional of dimension

Analogous result holds for (potentially) semistable representations,for any K.

The geometric Breuil–Mézard conjecture Icrys,λIf λ1 > · · · > λd then (Xddim Xd,red .

)Fp is equidimensional of dimension
crys,λNot necessarily reduced: write Z(λ) = Z((Xdsum of irreducible components of Xd,red .

)Fp ), a formalQuestion (Breuil–Mézard): Which components? What are themultiplicities?

p-adic local Langlands
Expectation/hope: there is a sheaf M of GLd (K)-representations
on Xd which:satisfies local-global compatibility for completed cohomology
of locally symmetric spaces.
encodes the weight part of Serre’s conjecture.
answers the question of Breuil–Mézard.
...
(p-adic analogue of Hellmann/Ben-Zvi–Chen–Helm–Nadler/Zhu.)
Dotto–Emerton–G. (in progress): holds for GL2 (Qp ).
Idea: Colmez’s construction of (D  P1 )/(D\  P1 ) makes senseon X2 .

The weight part of Serre’s conjecture
K = Qp . Let σ be an irreducible Fp -representation of GLd (Fp ).
M(σ) := HomGLd (Zp ) (M, σ ∨ )∨ .
Expectation: support of M(σ) is a union of irreducible
components of Xd,red .
Z(σ) := Z(M(σ))).
If local-global compatibility holds, the support of Z(σ) exactly
determines the weight part of Serre’s conjecture.
True for modular curves.

The geometric Breuil–Mézard conjecture IIFor λ = (λ1 > · · · > λd ), set M(λ) := HomGLd (Zp ) (M, πλ∨ )∨ ,where πλ = irreducible algebraic GLd (Zp )-representation ofhighest weight (λ1 − (d − 1), . . . , λd−1 − 1, λd ).
crys,λExpectations imply: Z(M(λ)) = Z(λ) = Z((Xd)Fp ).

Then weP
have the geometric Breuil–Mézard conjectureZ(λ) = σ nσ (λ)Z(σ), where nσ (λ) = multiplicity of σin πλ ⊗Zp Fp .
Known for GL2 (Qp ) (Kisin, Paškūnas,. . . )Extends to potentially crystalline/semistable case, using inertial
local Langlands correspondence.
This version can sometimes be proved if λ is small, e.g. G.–Kisin,Le–Le Hung–Levin–Morra.

Patched modules and deformation rings
No construction is known of M other than for GL1 or GL2 (Qp ).
However there is a candidate after pulling back to the versal ringat a fixed ρ : GK → GLd (Fp ): the patched module M∞ of
Caraiani–Emerton–G.–Geraghty–Paškūnas–Shin.
Construction via globalisation and Taylor–Wiles patching of
cohomology of unitary Shimura varieties.
The pullback
P of the geometric Breuil–Mézard conjectureZ(λ) = σ nσ (λ)Z(σ) to the versal ring is equivalent toautomorphy lifting theorems (Kisin).
e.g. can use solvable base change to reduce the potentiallycrystalline case to the crystalline case (G.–Kisin).