FARGUES–FONTAINE CURVE
NOTES TAKEN BY PAK-HIN LEE
Abstract. These are notes from the ongoing learning seminar on the Fargues–Fontaine
curve at Columbia University during Winter Break 2015, which is organized by Wei Zhang.
The seminar closely follows Sean Howe’s notes for Laurent Fargues’ lectures at the University of Chicago. These notes merely serve to reflect the topics covered at the learning
seminar at Columbia and are not carefully edited; the reader who wishes to learn the materials should instead consult the original papers by Fargues and Fontaine and the excellent
notes by Howe.
Contents
1. Lecture 1 (December 15, 2015): Daniel Gulotta
1.1. Introduction
1.2. Fundamental groups
1.3. The adic Fargues–Fontaine curve
1.4. Connections to p-adic Hodge theory
2. Lecture 2 (December 18, 2015): Qirui Li
2.1. Perfectoid fields
2.2. Tilting
2.3. Geometric picture
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3
4
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Last updated: December 19, 2015. Please send corrections and comments to [email protected].
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1. Lecture 1 (December 15, 2015): Daniel Gulotta
1.1. Introduction. The Fargues–Fontaine curve is a geometric object that can be used to
study local Langlands. Let E and K be local fields (i.e., finite extensions of Qp , or Fp ((t)))
with the same residue characteristic. Local Langlands studies representations1
Gal(K sep /K) → GLn (Q` ).
We can replace GLn by general reductive groups but we will stick to this simpler case. The
Fargues–Fontaine curve X will be a “curve” over E, with a Gal(K sep /K)-action. We will
study vector bundles on X.
Remark. X is not of finite type over E, but it behaves like a curve in many ways: regular,
noetherian, dimension 1, “complete” in the sense that there is a degree function on divisors
so that nonzero effective divisors have degree > 0 and principal divisors have degree 0.
b O
d. What we
Examples of objects that have a Gal(K sep /K)-action are: K sep , OK sep , K,
K
want is roughly
b O )”.
“ Spa(E, O ) × Spa(K,
E
Fp
b
K
This doesn’t literally make sense: E and K could have characteristic 0, so E never contains
Fp . The fundamental group of this object should be the Galois group of E.
1.2. Fundamental groups. For nice connected topological spaces X and Y ,
π1 (X × Y ) ' π1 (X) × π1 (Y ).
This is also true for X, Y varieties over algebraically closed fields of characteristic 0, or in
characteristic p if X or Y is proper. But if X = Y = Spec Fp , then X ×Fp X ∼
= X, so
b but π1 (X) × π1 (Y ) ∼
b × Z.
b
π1 (X ×Fp Y ) ∼
=Z
=Z
The issue is that there is only one Frobenius acting on X × Y .
Theorem 1.1 (Drinfeld). Let X, Y be connected schemes of finite type over Fq . Let (X ×
∼
Y )/p.Fr. be the category of finite étale maps S → X × Y along with an isomorphism S →
FX∗ S, where FX is the Frobenius on X. Then
π1 (X × Y /p.Fr.) ∼
= π1 (X) × π1 (Y ).
b O )
This suggests that we should take the product of the spaces Spa(E, OE ) and Spa(K,
b
K
and quotient out by the Frobenius. More precisely,
• If E and K have characteristic p, then we will take
b O )/φZ
Spa(E, O ) × Spa(K,
E
Fq
b
K
where φ is the Frobenius on K.
• If E has characteristic 0, then we take
b Zp W (O b )/φZ .
OE ⊗
K
1PH:
During the lecture there was some confusion as to what the coefficient field should be. After
discussion with Rahul, we believe the correct treatment is to take Q` -coefficients (where ` 6= p), i.e., we are
studying classical local Langlands, not p-adic local Langlands.
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b with K
b [.
• If K has characteristic 0, we replace K
The Fargues–Fontaine curve satisfies
b [ /p.Fr.
X = Spd E × Spd K
but we won’t be talking about diamonds.
1.3. The adic Fargues–Fontaine curve. Let E be a local field with residue field Fq and
uniformizer π. Let F be a characteristic p perfectoid field over Fq (i.e., perfect, complete
b [ ), and $ be
with respect to a nontrivial valuation F → R ; for example, we can take K
≥0
F
a nonzero nonunit element of OF .
Let A be the unique OE -algebra satisfying:
(1) A is π-adically complete,
(2) A is flat over OE ,
(3) A/πA ∼
= OF .
More explicitly,
b Fp OF (= OF [[π]] since OE ∼
• If E has characteristic p, this is OE ⊗
= Fq [[π]]).
• If E has characteristic 0, this is W (OF ) ⊗W (Fq ) OE . (In classical p-adic Hodge theory,
this is called Ainf .)
Consider the adic space Spa(A, A). Points are equivalence classes of continuous valuations | · | : A → Γ ∪ {0} where Γ is a totally ordered abelian group (which we will write
multiplicatively). Furthermore, we require that |a| ≤ 1 for all a ∈ A.
A rational subset of Spa(A, A) is a subset defined by inequalities
|f1 | ≤ g, |f2 | ≤ g, · · · , |fn | ≤ g, |g| =
6 0
for f1 , · · · , fn , g ∈ A such that (f1 , · · · , fn , g) is open.
There is a “Teichmüller lift” [·] : OF → A, which is multiplicative. The maximal ideal of
∞
A is (π, [$F ]1/p ).
1/p∞
Spa(A, A) has one closed point, corresponding to the maximal ideal m = (π, [$F ]) (the
valuation sends m 7→ 0, and everything else to 1). All remaining points are analytic (kernel
of valuation not open).
Let
Y = Spa(A, A)a = Spa(A, A)\{closed point}.
For all points in Y, either |π| 6= 0 or |[πF ]| 6= 0. We can define a continuous map Y → [0, 1]
such that for any rank 1 valuation | · | : A → R,
log |π|
| · | 7→ q
− log |[$
F ]|
In equal characteristic, this actually is a disc since A ∼
= OF [[π]].
Q
Let I ⊂ [0, 1] be an interval with endpoints in q ∪ {0}, not equal to ∅, {0}, {1} or [0, 1].
Define YI to be the inverse image of I. If I = [q r , q s ] is closed, this is a rational subset
defined by the inequalities
|[$F ]|r ≤ π ≤ |[$F ]|s .
In general, YI is an increasing union of rational subsets.
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For any ρ ∈ [0, 1], define the “Gauss norm” | · |ρ as follows. Let f =
element of A can be represented uniquely in this way.) Define
(
supn |xn |ρn
if ρ > 0,
|f |p =
−n
supn||xn |=1 q
if ρ = 0.
P
n [xn ]π
n
∈ A. (Each
This norm extends:
• to A[π −1 , [$F ]−1 ] if ρ ∈
/ {0, 1};
• to A[π −1 ] if ρ = 1;
• to A[[$F ]−1 ] if ρ = 0.
Then O(YI ) is the completion of A[π −1 , [$F ]−1 ] with respect to | · |ρ for ρ ∈ I when 0, 1 ∈
/ I.
If 0 ∈ I, we take A[[$F ]−1 ] instead. If 1 ∈ I, we take A[π −1 ] instead. If I is closed, we only
need to take the endpoint norms.
Now we are ready to define the adic Fargues–Fontaine curve. The Frobenius action on A
X
X
[xn ]π n 7→
[xqn ]π n
n
n
defines a map φ : Y → Y. This map sends the radius ρ 7→ ρq and fits in the commutative
diagram
/
Y
φ
/
Y
I
ρ7→ρq
I
The fixed points of I are 0 and 1. Y has a point above 0 with |π| = 0, and a point above 1
with |[$F ]| = 0. φ does not fix any other points of Y.
Let Y = Y(0,1) and X = Y /φ. This is an adic space over E.
Definition 1.2. X is the Fargues–Fontaine curve.
Eventually we will study vector bundles on X. It will turn out that the only semistable
vector bundle of slope 0 is the trivial bundle. For example, if we want to study Gal(K sep /K)representations, we can look at the Galois action on the Fargues–Fontaine curve.
1.4. Connections to p-adic Hodge theory. Let E = Qp , K/Qp finite, F = C[p . A is the
ring denoted Ainf in p-adic Hodge theory. There is a map A → Cp
X
X
n
[xn ]pn 7→
x(0)
n p ,
n
n
which defines a point xC ∈ X by pulling back from Cp . xC is the only Gal(K/K)-fixed
+
. The global sections of Y = Y(0,1) are
point of X. The completed local ring at xC is BdR
+
OY = Brig .
[
There are also connections with (φ, Γ)-modules, by considering F = (Qcyc
p ) . In this case
OY = R.
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2. Lecture 2 (December 18, 2015): Qirui Li
2.1. Perfectoid fields. We want to relate the characteristic p local field Fq ((π)) and the
mixed characteristic local field Qp . The idea is that if we keep adjoining p-power roots of
the uniformizer to Qp
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Qp ⊆ Qp (p1/p ) ⊆ Qp (p1/p ) ⊆ · · ·
∞
∞
these roots “converge” to π and the field Qp (p1/p ) “behaves like” Fp ((π 1/p )). Roughly
∞
∞
speaking, the process of passing from Qp (p1/p ) to Fq ((π 1/p )) will be known as “tilting”,
and the reverse will be “untilting”.
Let K be a complete non-discrete valuation field, with ring of integers OK and uniformizer
$ satisfying 0 < |$| < |p|.
Definition 2.1. K is called a perfectoid field if Frob : OK /$ → OK /$ given by x 7→ xp is
surjective.
Example 2.2. Qp\
(p1/p∞ ) is a perfectoid field.
Example 2.3. If p is odd, then Q\
p (ζp∞ ) is a perfectoid field since the map
\
Z\
p (ζp∞ )/p → Zp (ζp∞ )/p
ζpn+1 − 1 7→ ζpn − 1
is an isomorphism.
\
1/p∞ )) is a perfectoid field.
Example 2.4. Fq ((π
2.2. Tilting. Suppose K is a perfectoid field. Define
K [ = limp K = {(x(0) , x(1) , · · · ) | x(n) = (x(n+1) )p },
x7→x
with multiplication
(x(0) , x(1) , · · · ) × (y (0) , y (1) , · · · ) = (x(0) y (0) , x(1) y (1) , · · · )
and addition
(x(0) , x(1) , · · · ) + (y (0) , y (1) , · · · ) = (c(0) , c(1) , · · · )
where
N
c(n) = lim (x(n+N ) + y (n+N ) )p .
N →∞
These make K [ into a ring. We define a valuation on K [ as follows: if x = (x(0) , x(1) , · · · ) ∈
K [ , then
|x| = |x(0) |.
K [ is a complete valuation field, called the tilting of K.
Proposition 2.5. There exists an isomorphism
∼
=
θ : OK [ → (OK /$)[ .
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Proof. The map is given by
(a(0) , a(1) , · · · ) 7→ (a(0) , a(1) , · · · ).
For the inverse map, given (a(0) , a(1) , · · · ) ∈ (OK /$)[ , we let (e
a(0) , e
a(1) , · · · ) be an arbitrary
lifting, and
N
a(n) = lim (e
a(n+N ) )p .
N →∞
(0)
(1)
Then the sequence (a , a , · · · ) lies in OK [ .
\
1/p∞ )), by tilting the isomorphism
Example 2.6. The tilting of Qp\
(p1/p∞ ) is Fp ((π
∞
∼
∞
Zp [p1/p ]/p → Fp [π 1/p ]/π
and using Proposition 2.5.
\
1/p∞
Example 2.7. The tilting of Q\
)), where π reduces to π = (1 − ζp , 1 − ζp2 , · · · ).
p (ζp∞ ) is Fp ((π
Theorem 2.8. If K is a perfectoid field, and L/K is a field extension of degree n, then
(1) L is a perfectoid field.
(2) L[ is a field extension of degree n of K [ .
(3) OL is almost étale over OK , i.e., for any > 0, there exists xi ∈ OL for all 1 ≤ i ≤ n
such that
1 − < | det(Tr(xi xj ))| < 1
(equivalently, ΩOL /OK is killed by arbitrarily small powers of p, or by the maximal
ideal m).
(4) There is an equivalence of categories
{finite étale extensions of K} ↔ {finite étale extensions of K [ }.
Corollary 2.9.
(1) A perfectoid field K is algebraically closed if and only if K [ is algebraically closed.
(2) Gal(K/K) ∼
= Gal(K [ /K [ ).
\
1/p∞ )) are isomorphic.
Example 2.10. The absolute Galois groups of Qp\
(p1/p∞ ) and Fp ((π
2.3. Geometric picture. Recall the definition from last time. Let E be a discrete valuation
field with ring of integers OE and residue field Fq , and F be a perfectoid field over Fq with
pseudo-uniformizer $ of OE . We want to study perfectoid fields over E that tilt to F .
Define
A = WOE (OF ) = W (OF ) ⊗W (Fq ) OE
and
Y = Spa(A)\V (π, $).
There is a Teichmüller lifting [·] : OF → A. Define
Y = Spa(A)\V (π[$]) = Y(0,1) .
P
i
Next we will define the classical points |Y |cl . An f = ∞
i=0 [xi ]π ∈ A is called primitive if
×
x0 6= 0 and there exists d ≥ 0 such that xd ∈ OF . The minimum such d is called the degree
of f .
It is clear that deg(f g) = deg(f ) + deg(g).
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Example 2.11. {Primitive degree 0 elements} ' A× .
Example 2.12. If a ∈
/ OF× , then deg(π − a) = 1.
Definition 2.13.
|Y |cl = {irreducible primitive elements}/A× = Irr/ ∼ .
Recall that B = O(Y ) = O(Y(0,1) ). If E = Fq ((π)), then
(
)
X
B=
[ai ]π i | lim |ai |ρi = 0 for any ρ ∈ (0, 1) .
x→∞
i∈Z
Theorem 2.14. Let y = (f ) ∈ |Y |cl where f is primitive of degree d. Let k(y) = B/(f ) and
θ : B k(y). Then
(1) k(y) is a perfectoid field over E.
N
(2) There is a map F → k(y)[ given by a 7→ θ([a1/p ]), and k(y)[ is a degree d extension
of F .
(3) There is a 1-1 correspondence
∼
=
Irrdeg=1 / ∼↔ {(K, i) | K is perfectoid over E, i : F → K [ }/ ∼
(4) If F is algebraically closed, then every primitive degree 1 element f is (f ) = (π − [a])
for some 0 < |a| < 1.
Warning: The choice of a is not unique, i.e., it might be the case that
(π − [a1 ]) = (π − [a2 ]).
By considering the diagram
/
BO
B/(π − [a1 ])
O
[a]7→[a]
?
[F ]
[a1 ]7→π
/
O
k(y)
O
(x(0) ,x(1) ,··· )7→x(0)
a7→[a]
F
∼
=
/
k(y)[
we see that the ambiguity is the Zp (1)-torsor
{(x(0) , x(1) , · · · ) ∈ k(y)[ | x(0) = π}.
For any y = (f ) ∈ |Y |cl , k(y) = B/(f ), we have the map θ : A[ π1 ] → k(y) with ker θ = f .
The completion of the local ring at y is
1
d
O
/(ker θ)n .
Y,y = lim A
n→∞
π
We define
+
BdR
(k(y))
1
= lim A
/(f )n .
n→∞
π
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For y = α, the local ring is
(∞
)
X
lim OY[0,ρ] =
[xi ]π i | there exists > 0 such that lim |xi |i = 0
ρ→0
i=0
[
and the completion is O
Y,α = WOE (F ). These are also denoted as OE † and OE respectively,
where † means “overconvergent”.
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