THE CURVE AND p-ADIC HODGE THEORY LAURENT FARGUES

LAURENT FARGUES

Contents

1. Introduction 2

1.1. What is a p-adic Hodge structure? 2

1.2. Real Hodge structure 2

2. The curve Y 4

2.1. Affinoid space and adic space 5

2.2. Holomorphic function of the variable p 6

2.3. Newton polygon 7

2.4. Zeros of holomorphic functions 8

2.5. Perfectoid fields and tilting 9

2.6. Classical points 11

2.7. Localization of zeros 11

2.8. Parametrization of classical points 12

3. The curve X 13

3.1. The fundamanetal exact sequence 14

3.2. Vector bundles 16

3.3. Harder-Narasimhan filtrations 16

4. Classification of vector bundles 19

4.1. Construction of some vector bundles 19

4.2. From isocrystals to vector bundles 21

5. Periods of p-divisible groups 21

5.1. Periods in characteristic p 21

5.2. The covectors 22

5.3. Period isomorphism in characteristic p 23

5.4. Periods in unequal characteristic 24

6. Topics on classification theorem 25

6.1. Lubin-Tate space 25

6.2. Proof of the classification for rank two vector bundles 27

6.3. Weakly admissible implies admissible 27

Abstract. The main theme of this course will be to understand and give a meaning to the notion of ap-adic Hodge structure. Starting with the work of Fontaine, who introduced many of the basic notions in the domain, it took many years to understand the exact definition of a p-adic Hodge structure.

We now have the right definition: this involves the fundamental curve ofp- adic Hodge theory and vector bundles on it. In the course I will explain the construction and basic properties of the curve. I will moreover explain the proof of the classification of vector bundles theorem on the curve. As an application I will explain the proof of weakly admissible implies admissible.

In the meanwhile I will review many objects that show up in p-adic Hodge theory like p-divisible groups and their moduli spaces, Hodge-Tate and de Rham period morphisms, and filteredφ-modules.

This is a note of the lectures in MCM, Beijing from 2019/11/01 to 2020/01/10.

Date: Recorded by Shenxing Zhang. Not revised yet.

1

1. Introduction

1.1. What is a p-adic Hodge structure? Recall a real pure Hodge structure of weight w ∈ Z is a finitely dimensional real vector space V , endowed with a bigrading

V

= M

p+q=w

V

such that V

= V

. For example, let X/ C be a proper smooth algebraic variety.

Then H

(X ( C ), R ) is equipped with a real Hodge structure of weight i as H

(X( C ), R )

= M

p+q=i

H

(X, Ω

).

In p-adic setting, there are plenty of different structures and results

• Hodge-Tate Galois representations;

• crystalline representations;

• de Rham representations;

• filtered ϕ-modules à la Fontaine;

• Breuil-Kisin modules;

• (ϕ, Γ)-modules;

• comparison theorems for proper smooth algebraic variety over Q

. This is a mess! We should back to real case to find the solution.

1.2. Real Hodge structure. Recall Simpson’s geometric point of view of twists.

Denote

e P

= P

/

z ∼ − 1

¯ z

where z is the coordinate on P

. This is a conic curve without real point, equipped with ∞ . Obviouly P

is a double cover of P e

.

P

Z/2Z π

0

< <

< <

< <

< <

< ∞

e P

∞

The action of C

on P

as λ.z = λz descends to an action of U (1) on P e

. Then

∞ is the unique fixed point of this action and the unqiue point that has a finite orbit.

Consider the vector bundles on e P

. For λ ∈

Z , define O

R

(λ) = (

π

O

(2λ), λ / ∈ Z ; L such that π

L = O

(2λ), λ ∈ Z . Here the slope of O

R

(λ) is λ.

Proposition 1.1. There is a bijection between the set of finite decreasing half integer sequences

λ

≥ · · · ≥ λ

| λ

∈ 1

2 Z , n ∈ N

and the isomorphic classes of vector bundles on e P

as

(λ

) 7−→

"

M

i

O

(λ

)

#

.

In particular,

Vect

−→

n

slope 0 semisimple vector bundles over e P

o V 7−→ V ⊗ O

R

H

(e P

, E ) ←−[ E .

That is to say, every Harder-Narasimhan filtration of vector bundles are split and every semisimple vector bundle of pure slope are O

(λ)

.

Let V be a real vector space with a filtration Fil

on V

= V ⊗

C . Denote by t the uniformization of P e

at ∞ and

V

((t)) = V ⊗

C ((t)) = V

⊗

C ((t)).

There is a canonical filtration

t

C [[t]]

on C ((t)), which induces a filtration on V

((t)) as

Fil

(V

((t))) = X

i∈Z

Fil

V

⊗

t

C [[t]].

Then O b

R,∞

= C [[t]], (V ⊗

O

R

)

= V

((t)) and the C [[t]]-lattice

Λ := Fil

(V

((t))) ⊂ V

((t)) defines a modification of vector bundles

(V ⊗

O

) |

−→ E|

,

such that E b

= Λ. This is U(1)-equivalent and induces a bijection { filtrations on V

} −→

n

U (1)-equiv. modif. V ⊗

O

R

E o

and thus

{ (V, Fil

V

) } −→

U(1)-equiv. modif. E

E

E

semisimple of slope 0, U(1) y H

( E

) trivially

. Definition 1.2. A real Hodge structure is a finitely dimensional real vector space V , endowed with a bigrading decomposition

V

= M

p,q∈Z

V

,

such that V

= V

. Thus for any integer w, there is a subspace V

⊂ V such that

V

= M

p+q=w

V

,

which is called weight w part of V . If V = V

, V is called pure of weight w.

We say (V, Fil

V

) defines a Hodge struture of weight w if there is a real Hodge struture on V of pure weight w such that Fil

V

= ⊕

V

.

Proposition 1.3. (V, Fil

V

) defines a weight w Hodge struture if and only if E

is semisimple of slope w/2 in the corresponding modification.

This induces a bijection between the set of weight w pure real Hodge structures and the set of U (1)-equivalent modifications E

E

on P e

\ {∞} , where E

is semisimple of slope 0, E

is semisimple of slope w/2 and U (1) acts on H

( E

) trivially.

WE are going to do the same in the p-adic setting.

real setting p-adic setting P e

\ {∞} x U (1) the curve X x Gal( ¯ Q

/ Q

)

C [[t]] = O b

B

= O b

λ.t = λt σ.t = χ

(σ)t, t = log[]

P

e P

X

X

Thus the vector bundles on X is endowed with Gal( Q

/ Q

)-action.

2. The curve Y There are two versions of the curve.

• X

adic version analog of p-adic Reimann surface,

• X schematical version analog of a proper smooth algebraic curve.

There is an analytification morphism (GAGA) X

→ X and an “ample” line bundle O (1) on X

such that

X = Proj( M

d≥0

H

(X

, O (d))).

Both rely on the construction of an intermediate adic space Y endowed with a

“crystalline” Frobenius ϕ.

Let C be a complete algebraically closed field of characteristic 0. Define the tilt C

the inverse limit of C with respect to Frobenius, which is an algebraically closed field of characteristic p. Let B

be the completion of A

= W ( O

) with repect to (p − [p

]) with quotient field B

, A

the completion of divided power of A

and B

= B

.

The p-adic comparison theorems for crystalline/de Rham/étale cohomology lead one to consider the category of pairs (W

, W

) where W

is a free B

-module and W

is a free B

-module such that

B

⊗

W

= B

⊗

dR

W

.

We will construct a curve X such that B

= O (X − {∞} ), B

= O

. The fundamental exact sequence

0 → Q

→ B

→ B

/B

→ 0

tells us the sections. The category of (W

, W

) corresponds to the category of vector bundles over X . Since B

= B

, this suggests

X

= Y

/ϕ

where Y

= Spa(A

) − (p[p

]).

In general, let E be a discretely valued non-archemedean field with uniformizer π with finite residue field F

= O

/π. Let F/ F

be a perfectoid field, i.e., a perfect field, complete with respect to a non-trivial absolute value | · | : F → R

. We will attach to this data a curve X

/E. More generally, we can define “a family of curves”

X

= (X

)

for perfectoid S/ F

. If G is a reductive group over E, one can define a stack Bun

: S → { G-bundles on X

} .

We will study the perverse `-adic sheaves on Bun

.

2.1. Affinoid space and adic space. Let’s recall the definition of adic spaces.

This is not a prt of the lectures. Let k be a nonarchimedean field and R a topological k-algebra.

Definition 2.1. (1) If there is a subring R

⊂ R such that { aR

}

forms a basis of open neighborhoods of 0, it’s called a Tate k-algebra. A subset M ⊂ R is called bounded if M ⊂ aR

for some a ∈ k

.

(2) An affinoid k-algebra is a pair (R, R

) consisting of a Tate k-algebra R and open integrally closed subring R

⊂ R

.

(3) An affinoid k-algebra (R, R

) is said to be tft if R is a quotient of k h T

, . . . , T

i for some n and R

= R

.

Definition 2.2. Denote by X = Spa(R, R

) the set of equivalent classes of con- tinuous valuations on R, which is ≤ 1 on R

. We equip X the topology which has open rational subsets

U

f

, . . . , f

g

= { x ∈ X | | f

(x) | ≤ | g(x) | , ∀ x ∈ X } as basis, where f

, . . . , f

generates R.

Definition 2.3. A topological space X is called spectral if it satisfies the following equivalent properties.

(1) There is some ring A such that X ∼ = SpecA.

(2) X is an inverse limit of finite T

spaces.

(3) X is quasicompact, has a quasicompact topological basis, stable under finite intersections, and every irreducible closed subset has a unique generic point.

Theorem 2.4. The space Spa(R, R

) is spectral and Spa(R, R

) ∼ = Spa( R, b R b

).

Theorem 2.5. (1) If X = ∅ , then R b = 0.

(2) If R is complete and | f (x) | 6 = 0, ∀ x ∈ X , then f is invertible.

(3) If | f (x) | ≤ 1, ∀ x ∈ X, then f ∈ R

.

Consider the topological algebra R[f

g

, . . . , f

h

] ⊂ R[g

] and denote by B the integral closure of R

[f

g

, . . . , f

g

] in it, then (R[f

g

, . . . , f

g

], B) is an affinoid k-algebra with completion (R h f

g

, . . . , f

g

i , B). Then b

Spa(R h f

g

, . . . , f

g

i , B) b → Spa(R, R

) factors through U

g

and it satisfies the corresponding universal property.

Define presheaves

( O

(U ), O

(U )) = (R h f

g

, . . . , f

g

i , B) b and on general W ,

O

= lim ←−

U⊂W

O

(U).

Moreover U ∼ = Spa(O

(U ), O

(U )).

The stalk O

is a local ring with maximal ideal { f | f (x) = 0 } and O

is a local ring with maximal ideal { f | f (x) < 1 } .

Definition 2.6. We call R is strongly neotherian if R b h T

, . . . , T

i is noetherian for any n.

Theorem 2.7. If R is strongly neotherian, then O

is a sheaf.

Definition 2.8. Consider triple (X, O

, ( |· (x) | , x ∈ X)) where (X, O

) is a locally ringed space and | · (x) | is a continuous valuation on O

for any x ∈ X . Such triple isomophic to Spa(R, R

) where O

is a sheaf is called an affinoid adic space.

It is called an adic space if it’s locally an affinoid adic space.

Proposition 2.9. For affinoid adic space X = Spa(R, R

) and any adic space Y over k,

Hom(Y, X) = Hom(( R, b R b

), ( O

(Y ), O

(Y ))).

Example 2.10. Assume that k is complete and algebraically closed. Let R = k h T i and R

= R

= k

h T i. Fix a norm | · | : k → R

. Then X = Spa(R, R

) consists of

(1) The classical point. For x ∈ k

, R −→ R

f = X

a

T

7−→ | f (x) | = | X a

x

| . (2)(3) The rays of the tree. For 0 ≤ r ≤ 1, x ∈ k

,

R −→ R

f = X

a

(T − x)

7−→ sup | a

| r

= sup

y∈k^◦,|y−x|≤r

| f (y) | .

If r = 0, it is the classical point. If r = 1, it doesnot depend on x, which is called the Gausspoint.

If r ∈ | k

| , it’s said to be of type (2), otherwise of type (3).

(4) Dead ends of the tree. Let D

⊃ D

⊃ · · · be a sequence of disks with

∩ D

= ∅ . It occurs when k is not spherically complete.

R −→ R

f 7−→ inf

sup

x∈Di

| f (x) | . (5) For Γ = R

× γ

, where γ = r

or r

(r < 1).

R −→ Γ ∪ { 0 } f = X

a

(T − x)

7−→ sup | a

| γ

.

This only depends on the disc D(x, < r) or D(x, r). Thus if r / ∈ | k

| , it’s of type (3). Every rays of point of type (2) correspond a valuation of type (5).

2.2. Holomorphic function of the variable p. Let E be a finite extension of Q

with residue field F

. As a comparison, we also take E = F

[[t]]. It is the coefficient field of the p-adic Hodge theory.

Definition 2.11. Define A = A

=

(

W

( O

) = W ( O

) ⊗

O

, E/ Q

, O

⊗ b

O

= O

[[π]], E = F

[[t]].

Then

A =

 

 X

n≥0

[x

]π

x

∈ O

 

 .

Fix $ ∈ F with 0 < | $ | < 1. Then A is complete under the (π, [$])-adic topology. Consider the adic space Spa( A , A ). It has only one closed point with kernel (π, m

). Define

Y = Spa( A , A )

= Spa( A , A ) \ {closed point} = Spa( A , A ) \ V (π, [$]) and an open subspace

Y = Spa( A , A ) \ V (π[$]).

Here the subscript a indicates we take the analytic points and Y is not affinoid.

Consider the space of holomorphic functions O (Y ). Let A

1 π , 1

[$]

= ( X

n−∞

[x

]π

x

∈ F, sup | x

| < + ∞ )

be the set of holomorphic functions on Y that are meromorphic along (π), ([$]).

For ρ ∈ (0, 1), f = P

n−∞

[x

]π

, define the Gauss norms

| f |

:= sup

n

| x

| ρ

= sup

|y|≤ρ

f (y).

Proposition 2.12. The space

B = O (Y ) is the completion of A h

1 π

,

i

with respect to {| · |

} .

For compact subset I ⊂ (0, 1), the completion B

with respect to {| · |

} is a Banach E-algebra and

B = lim ←−

I⊂(0,1)

B

is a Fréchet space. In particular, if I = [ρ

, ρ

], B

is the completion with respect to {| · |

, | · |

} .

In the case E = F

[[π]],

Y = D

= { 0 < | π | < 1 } ⊂ A

and

B = O (Y ) = ( X

n−∞

x

π

x

∈ F, lim

n→+∞

| x

| ρ

= 0, ∀ ρ )

. We have natural maps

D

{{xxx xxx xxx

)) R

R R R R R R R R R R R R R R R

Spa(F ) D

= Spa( F

((π))) = Spa(E).

The map on the left is locally of finite type, but D

→ Spa(E) is not.

Remark 2.13. If (x

) ∈ F

such that lim

| x

| ρ

= 0, ∀ ρ, then P

[x

]π

∈ B.

But not every element can be written in this form.

2.3. Newton polygon.

Proposition 2.14. | f g |

= | f |

| g |

, i.e., | · | is a valuation.

For ρ = q

, r ∈ (0, + ∞ ), | f |

= q

, where v

(f ) := inf(v(x

) + nr).

Here v = − log

| · | on F . Then r 7→ v

(f ) is a convex function.

In the case E = F

[[π]], f = P

x

π

∈ O (Y ) defines a Newton polygon Newt(f ) the decreasing convex hull of { (n, v(x

)) } . Then positive slopes of Newt(f ) one-to- one correspond to the set of valuations of roots of F on D

.

Assume E/ Q

. Recall the Legendre transform gives a bijection between the set of convex decreasing function R → R ∪{∞} , 6≡ + ∞ and the set of concave function (0, + ∞ ) → R ∪ {−∞} , 6≡ −∞ as

L (ϕ)(r) = inf

(ϕ(t) + tr), L

(ψ)(t) = sup

r∈(0,∞)

(ψ(r) − tr).

Proposition 2.15. For convex decreasing function f, g : R → R ∪ {∞}, we have L (f ~ g) = L (f ) + L (g),

where

(f ~ g)(x) = inf

(f (a) + g(b)).

The Legendre transform maps polygons to polygons, and the slopes of ϕ (resp. ψ) one-to-one correspond to the x-coordinates of break points of L (ϕ) (resp. L

(ψ)).

Proposition 2.16. For nonzero f ∈ B, there is a sequence { f

} in A h

1 π

,

i tending to f . Then for any compact subset K ⊂ (0, + ∞ ), there is an integer N such that for any n ≥ N , v

(f ) = v

(f

) for any r ∈ K.

As a corollary, the convex function r 7→ v

(f ) is a polygon with integral slopes.

Define

Newt(f ) := L

(r 7→ v

(f )).

Then

Newt(f g) = Newt(f ) ~ Newt(g).

Let I ⊂ (0, 1) be a compact subset and 0 6 = f ∈ B

. Denote by Newt

(f ) the part of Newton polygon consisting of the slope in − log

(I) part. But { v

(f ) }

q(I)

do not determine Newt

(f ). For example, I = { q

} , we need to know the left and right break point of the slope r part to determine Newt

(f ).

Denote by ∂

, ∂

the left/right derivation. Then (v

(f ), ∂

v

(f ), ∂

v

(f ))

q(I)

determine Newt

(f ). The rank 2 valuations with image in R × Z f 7→ (v

(f ), − ∂

v

(f)),

f 7→ (v

(f ), ∂

v

(f )), are specializations of v

.

2.4. Zeros of holomorphic functions. Recall Jensen’s inequality/equality. For nonzero f ∈ O ( C ) such that f (0) 6 = 0. Let R > 0 such that f has no zero on {| z | = R }. Let a

, . . . , a

be zeros of f in {| z | < R }. Then

ln | f (0) | = 1 2π

Z

ln | f (Re

) | dθ − n ln R + X

ln | a

| and

ln | f (0) | ≤ M (R) − n ln R + X

ln | a

| ,

where M (R) is the maximal modulus on {| z | = R } .

In the non-zrchimedead setting, there is an equality. Assume E = F

[[π]]. For nonzero f = P

n≥0

x

π

∈ O ( D

), f (0) = x

6 = 0. Assume it has roots (a

)

with v(a

) ≥ v(a

) ≥ . . . . Then the slopes of Newt(f ) are valuations of roots of f , v(f (0)) = v

(f ) − nr +

X

v(a

).

We want to do the same for E = Q

. We need to define zeros of f in this setting.

For E = F

((π)),

Y = D

= { 0 < | π | < 1 } and

| Y |

=

z ∈ F ¯ | 0 < | z | < 1 /Gal( ¯ F /F )

= { P ∈ F [π] | irreducible with all roots such that 0 < | z | < 1 } /F

= { P ∈ O

[π] | unitary irreducible such that 0 < | P (0) | < 1 } . Definition 2.17. f = P

n≥0

x

π

∈ A is (distinguished) primitive of degree d > 0 if x

6 = 0, x

, . . . , x

∈ m

, x

∈ O

.

By Weierstrass fatorization, f = uP uniquely where u ∈ O

[[π]]

and P ∈ O

[π]

is unitary with degree d. Thus

| Y |

= { primitive irreducible elements } / O

[[π]]

. Assume E/ Q

.

Definition 2.18. f = P

n≥0

[x

]π

∈ A is primitive of degree d if x

6 = 0, x

, . . . , x

∈ m

, x

∈ O

.

It’s equivalent to say, f mod π 6 = 0 in O

and f mod W

( O

) 6 = 0 in W

(k

)

. The degree of f is v

(f mod W

( O

)). Thus deg(f g) = deg f + deg g.

Definition 2.19.

| Y |

= { irreducible primitive } / A

. We will show that this is the set of the classical points of Y . 2.5. Perfectoid fields and tilting.

Definition 2.20. A complete field K with respect to a norm | · | : K → R

is called a perfectoid field, if there is an element $ ∈ K such that | p | ≤ | $ | < 1 such that Frob : O

/$ → O

/$ is surjective.

For example, Q \ (ζ

)(p > 2), Q

\ (p

). An algebraic closed complete valued field is perfectoid. In char p case, K is perfectoid if and only if it is perfect.

Let K be a perfectoid field. Define the tilting K

= lim

←−

K =

n

(x

)

∈ K

| (x

)

= x

o

, with

(xy)

= x

y

, (x + y)

= lim

k→+∞

(x

+ y

)

. Define

x

:= x

and

| · | : K

−→ R

x 7−→ | x

| .

Then K

is also perfectoid. Moreover, there is an isomorphism O

−→

lim ←−

O

/p x 7→ (x

mod p)

lim

k→+∞

(ˆ y

)

←[ (y

)

. Example 2.21. If K is of characteristic p, then K

= K.

Example 2.22. If K = Q \

(ζ

), = (ζ

)

∈ K

and π

= − 1 ∈ K

, then K

= F

((π

)). In fact, Z

(ζ

)/p −→

F

(π

)/π

.

If K = Q

\ (p

), π = (p

)

∈ K

, then K

= F

((π

)). In fact, Z

(p

)/p −→

F

(π

)/π.

Remark 2.23. In fact, Fontaine gave the isomorphism R

= lim

←−

R/pR −→

n

(x

)

∈ R

| (x

)

= x

o

for any separated complete p-adic ring R.

Theorem 2.24. Let K be a perfectoid field. Then

(1) If L/K is finite, then L is perfectoid and [L

: K

] = [L : K].

(2) O

/ O

is almost étale, i.e., if n = [L : K], ∀ 0 < < 1, ∃ e

, . . . , e

∈ O

such that

≤ | disc(Tr

(e

e

))

| ≤ 1.

(3) ( · )

induces an equivalence between the set of finite étale K-algebras and the set of finite étale K

-algebras.

Corollary 2.25. (1) K is algebraically closed if and only if K

is.

(2) Gal(K/K) −→

Gal(K

/K

), where K

is the union of all L

where L/K is finite.

Proposition 2.26. The functors

{ p-adic rings}

(·)^♭

// {perfect F

-algebras}

W(·)

oo

are adjoint, i.e.,

Hom(W (A), B) = Hom(A, B

).

The adjuncation morphisms are

R −→

W (R

) x 7→ [x

], θ : W (R

) −→

R

X [x

]p

7→ X x

p

.

Remark 2.27. If R is a p-adic ring such that the Frobenius on R/pR is surjective,

then θ mod p is R

→ R/pR. Thus θ is surjective by Nakayama lemma and R is a

quotient of W (R

).

2.6. Classical points.

Theorem 2.28. Let ξ be an irreducible primitive element of degree d and θ : A A /ξ = O

, K = O

[1/p].

(1) K/E is a perfectoid field with | θ([x]) | = | x | . (2) The morphism

O

−→ O

x 7−→ θ([x

])

induces K

/F of degree d. In particular, K

= F if d = 1.

(3) For d = 1, this induces

| Y |

= Prim

/ A

−→ n

K/E perfectoid , K

= F o

/ ∼ (ξ) 7→ ( A /ξ)[1/p]

ker θ ←[ K/E.

Thus any ξ defines a valuation A

1 π , 1

[$]

→ A 1

π , 1 [$]

/ξ −→

R

, and

| Y |

= { V (ξ) | ξ ∈ A irreducible primitive } ⊂ | Y | . We see that for y ∈ | Y |

, k(y)/E is perfectoid and [k(y)

: F ] < + ∞ . Theorem 2.29. Assume that F is algebraically closed.

(1) ∀ y ∈ | Y |

, k(y) is algebraically closed.

(2) ∀ ξ, deg(ξ) = 1.

(3) any primitive element ξ can be written as ξ = u(π − [a

]) · · · (π − [a

]) where u ∈ A

.

For y = V (ξ) ∈ | Y |

, ξ = P

[x

]π

is primitive of degree d, set

| ξ | = | x

|

= | π(y) | . This defines the radius

| · | : | Y |

→ (0, 1).

Definition 2.30. For y = V (ξ) ∈ | Y |

,

B

= ξ-adic completion of A 1

π , 1 [$]

= O [

. It is a discrete valuation ring with uniformizer ξ and residue field k(y).

2.7. Localization of zeros.

Theorem 2.31. For nonzero f ∈ B,

− log

| y | | y ∈ | Y |

, f(y) = 0 coincides the slopes of Newt(f ).

Definition 2.32. For any interval I ⊂ (0, 1),

| Y

|

=

y ∈ | Y |

| | y | ∈ I .

Theorem 2.33. For any compact subset I ⊂ (0, 1), B

is a PID with SpmB

=

| Y

|

. In fact, SpmB = { (ξ) | | ξ | ∈ I } .

Proposition 2.34.

B

= { f ∈ B

\ { 0 } | Newt(f ) = ∅} . Define the Robba ring the local ring of Y at origin,

R = lim ←−

ρ→0⁺

B

. This is a Bezout ring.

Define

Div

(Y

) = { D = X

y∈|Y_I|^cl

m

[y] | supp(D) is locally finite , m

∈ N} , and

div : (B

\ { 0 } )/B

−→ Div

(Y

) f 7−→ X

ord

(f )[y].

Remark 2.35. If E = F

((π)), I = (0, 1), the div map is a bijection if and only if F is spherically complete (Larzard).

For any ρ ∈ (0, 1),

div : B

\ { 0 } /B

−→

Div

(Y

).

In fact, for D = P

n≥0

[y

] with | y

| → 0, write y

= V (ξ

), the series f = Y

n≥0

ξ

π

converges, where ξ

≡ π

mod W

( O

).

2.8. Parametrization of classical points. Assume F is algebraically closed. If E = F

((π)), then | Y |

= | D

|

= m

\ { 0 }. Thus

D

(F ) = m

\ { 0 } −→ |

Y |

a 7−→ V (π − a).

If E/ Q

, a ∈ m

\ { 0 } , y = V (π − [a]),

D

(F ) = m

\ { 0 } − | Y |

a 7−→ V (π − a).

But it’s hard to describe fibers.

For y ∈ | Y |

, C

= k(y)/E is algebraically closed. Choose π ∈ C

such that π

= π. Then y = V (π − [π]).

Consider the case E = Q

. It’s same for general E by using Lubin-Tate groups.

Then G b

( O

) = (1 + m

, × ) is a Banach space as

a. = X

k≥0

a k

( − 1)

, p. =

and the fact that F is perfect.

Definition 2.36. For any 1 6 = ∈ 1 + m

, u

:= [] − 1

[

] − 1 = 1 + [

] + · · · + [

] ∈ A .

Lemma 2.37. u

is primitive of degree 1.

Indeed,

u

mod p = 1 +

+ · · · +

= − 1

− 1 ∈ O

is nonzero,

u

mod W (m

) ≡ 1 + [1] + · · · + [1] = p ∈ W (k

).

Set

C

= B/u

= k(y)

where y = V (). Then = (

) ∈ F = C

, where

= θ

([

]). Then 1 +

+ · · · + (

)

= θ

(1 + [

] + · · · + [

]) = θ

(u

) = 0, thus

∈ µ

(C

). Moreover

O

/p O

= A /(p, u

) = O

/¯ u

, where

¯

u

= − 1

− 1 = ( − 1)

.

Since

− 1 ≡

− 1 mod p,

− 1 ∈ O /

u

,

− 1 6 = 0 mod p in C

. Hence

∈ µ

(C

) is primitive and is a generator of Z

(1)(C

) =

x ∈ C

| x

= 1 . Proposition 2.38.

((1 + m

) \ { 1 } )/ Z

−→ |

Y |

7−→ V (u

).

The inverse is given by y ∈ | Y |

, C

= k(y)/E. Choose a basis of Z

(1)(C

) , → (C

)

= F

. Then ∈ (1 + m

) \ { 1 } , y = V (u

).

Remark 2.39. Let

Y

= SpaF ×

(Spa Q

)

) = SpaF × Spa Q

/ Z

= D

/ Z

. where Z

= Gal( Q

/ Q

), Q

= F

((T

)). The action of Z

is given by a.T = P

k≥0 a k

(T − 1)

. Then | Y | = | Y

| = |D

| / Z

.

| Y

F

|

| Y

|

and | Y

|

= | Y

F

|

/G

= ((1 + m

) \ { 1 } )/ Z

)

/G

. 3. The curve X

The curve Y is Stein, it’s completely determined by the E-Frechét algebra O (Y ).

It’s preperfectoid, i.e., Y ⊗ b

K is perfectoid for a perfectoid field K/E. The Frobe- nius ϕ acts on A by X

[x

]π

7→ X [x

]π

.

This induces the action of ϕ on O (Y ) and Y with | ϕ(y) | = | y |

.

Theorem 3.1. (1) Y h

,

i = Spa(R, R

) and R is an E-Banach algebra and a PID.

(2) R is strongly neotherian.

Thus Y is a one-dimensonal regular adic space over E. Define X

:= Y /ϕ

,

this is a quasi-compact adic space over E, neotherian regular of dimension one. For 0 < ρ

< ρ

< ρ

< 1,

X

= Y

∪ Y

2,ρ^1/q₁ ]

.

Remark 3.2. ϕ is the arithmetic Frobenius. For X/ F

, there are geometric Frobenius Frob

× Id, arithmetic Frobenius Id × Frob

and absolute Frobenius Frob

× Frob

on X

.

The line bundles on X

are ϕ

-equivariant line bundles over Y , i.e., projective ϕ-modules over B of rank 1, or free R -modules of rank 1. Thus Pic(X

) = Z , where n corresponds (B · e, ϕ) with ϕ(e) = π

e.

Definition 3.3. Define O (d) corresponds (B, π

ϕ).

For a proper smooth algebraic curve X over C , the analytic part X

is a compact Riemann surface. Conversely, given a compact Riemann surface Z, there is an ample line bundle L over Z, e.g., O (z) for z ∈ Z, Then

Proj M

d≥0

H

(Z, L

) is a proper smooth algebraic curve.

We claim that O (1) is ample. Denote by

P

= H

(X

, O (d)) = B

and

P = M

d≥0

P

. We take

X = ProjP.

Theorem 3.4. (1) X is a Dedekind scheme.

(2) There is a natural morphism of ringed spaces X

→ X inducing | X

|

:=

| Y |

/ϕ

−→ |

X | = { closed points } such that

O b

−→

O b

= B

(k(y)) if y 7→ x. In particular, for any x ∈ | X | , k(x)/E is perfectoid.

(3) X is complete, i.e., for any x ∈ | X | , deg(x) := [k(x)

: F ], then deg(div(f )) = 0 for any f ∈ E(X)

. This implies we may define degree of vector bundles.

(4) There is an isomorphism

| X |

−→ { untilts of F } /Frob

x 7−→ k(x).

(5) If F is ac, ∞ ∈ | X | , there is t ∈ H

(X, O (1)) \ { 0 } such that V (t) = {∞}

and X \ {∞} = SpecB

, where B

:= B[1/t]

.

B

is a PID and (B

, − ord

) is non-Euclidean but almost Euclidean, i.e., for any x, y , there is x = ay + b with deg(b) ≤ deg(y). That’s because H

(X, O

( − 1)) 6 = 0 but H

(X, O

) = 0.

We are going to prove that X is a curve. We assume that F is algebraically closed. The case of general perfectoid F is treated by Galois descent from F b to F.

3.1. The fundamanetal exact sequence.

Proposition 3.5. P is a graded fractional ring with irreducible elements of degree 1.

For any 0 6 = t ∈ P

, P [1/t]

is fractional with irreducible elemenets { t

/t | t

∈ P

− Et } .

Proposition 3.6. Let t

, . . . , t

∈ P

\ { 0 } associate y

, . . . , y

∈ | Y |

, i.e., div(t

) = P

n∈Z

[ϕ

(y

)] on Y . Let y

= V (a

), where a

is primitive of degree 1. Then the sequence

0 → E · Y

t

→ B

→ B/a

· · · a

B → 0 is exact.

For example, if t ∈ P

\ { 0 } ,

0 → E · t

→ B

→ B

/Fil

B

→ 0 for y ∈ | Y |

, t(y) = 0.

Proof. Exactness in the middle. Suppose f ∈ B

∩ a

. . . a

B is nonzero, then

div(f ) ≥ X

[y

]

in Div

(X

). Then

div(f ) ≥ X

n∈Z

X

[ϕ

(y

)] = div(

Y

t

)

and then f = x Q

t

for some x ∈ B

= E.

Surjectivity. We only need to prove d = 1 case. For any x ∈ C, p

x lies in the convergence domain of exp for n 0. Since C is algebraically closed, there is z ∈ C such that exp(p

x) = z

, thus log z = x and log : 1 + m

→ C is surjective.

Assume E = Q

. Let a be a primitive element of degree 1, y = V (a) and C = C

= B/aB. Then C

= F . For any ε ∈ 1 + m

, log([ε]) ∈ B

and θ(log([ε])) = log(θ([ε])) = log ε

, ε

∈ 1 + m

. Take ε such that ε

= z, then θ(log([ε])) = x. It’s same for general E by using log

for Lubin-Tate group with

respect to (q, π).

We are going to use the fundamental exact sequence to prove that X is a curve.

Reciprocally, once the curve is constructed, we can find back the fundamental exact sequence by applying H

(X, − ) on

0 // O

×Qd

i=1ti

// O

(d) // F // 0 and H

(X, O

) = 0.

Corollary 3.7. For any t ∈ P

\ { 0 } , t(y) = 0, y ∈ | Y |

, C = C

, P/tP −→ { f ∈ C[T ] | f(0) ∈ E } x mod tP

7−→ θ

(x)T

is an isomorphism betweem graded rings, where P/tP = E ⊕ M

i≥1

P

/tP

.