Potential Theory on Berkovich SpacesLecture 2: Introduction to Berkovich Analytic

Spaces

Matthew Baker

Georgia Institute of Technology

Arizona Winter School on p-adic GeometryMarch 2007

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Goals

In this lecture, we will:

1 Give an alternative construction of the Berkovich projectiveline.

2 See how to view P1Berk as an inverse limit of finite R-trees.

3 Define the Berkovich analytic space associated to a normedring.

4 Discuss the Berkovich space associated to the ring Z.

5 Briefly discuss global Berkovich spaces and their topologicalstructure.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Goals

In this lecture, we will:

1 Give an alternative construction of the Berkovich projectiveline.

2 See how to view P1Berk as an inverse limit of finite R-trees.

3 Define the Berkovich analytic space associated to a normedring.

4 Discuss the Berkovich space associated to the ring Z.

5 Briefly discuss global Berkovich spaces and their topologicalstructure.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Goals

In this lecture, we will:

1 Give an alternative construction of the Berkovich projectiveline.

2 See how to view P1Berk as an inverse limit of finite R-trees.

3 Define the Berkovich analytic space associated to a normedring.

4 Discuss the Berkovich space associated to the ring Z.

5 Briefly discuss global Berkovich spaces and their topologicalstructure.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Goals

In this lecture, we will:

1 Give an alternative construction of the Berkovich projectiveline.

2 See how to view P1Berk as an inverse limit of finite R-trees.

3 Define the Berkovich analytic space associated to a normedring.

4 Discuss the Berkovich space associated to the ring Z.

5 Briefly discuss global Berkovich spaces and their topologicalstructure.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Goals

In this lecture, we will:

1 Give an alternative construction of the Berkovich projectiveline.

2 See how to view P1Berk as an inverse limit of finite R-trees.

3 Define the Berkovich analytic space associated to a normedring.

4 Discuss the Berkovich space associated to the ring Z.

5 Briefly discuss global Berkovich spaces and their topologicalstructure.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Notation

As in the previous talk, K will denote an algebraically closed fieldwhich is complete with respect to a nontrivial non-archimedeanabsolute value.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Multiplicative seminorms

Recall that a multiplicative seminorm on ring A is a function| | : A → R≥0 with values in the set of nonnegative reals such thatfor every f , g ∈ A, we have:

(S1) |0| = 0, |1| = 1.

(S2) |f · g | = |f | · |g |.(S3) |f + g | ≤ |f |+ |g |.

(S3)′ |f + g | ≤ max{|f |, |g |}.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Multiplicative seminorms

Recall that a multiplicative seminorm on ring A is a function| | : A → R≥0 with values in the set of nonnegative reals such thatfor every f , g ∈ A, we have:

(S1) |0| = 0, |1| = 1.

(S2) |f · g | = |f | · |g |.

(S3) |f + g | ≤ |f |+ |g |.

(S3)′ |f + g | ≤ max{|f |, |g |}.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Multiplicative seminorms

Recall that a multiplicative seminorm on ring A is a function| | : A → R≥0 with values in the set of nonnegative reals such thatfor every f , g ∈ A, we have:

(S1) |0| = 0, |1| = 1.

(S2) |f · g | = |f | · |g |.(S3) |f + g | ≤ |f |+ |g |.

(S3)′ |f + g | ≤ max{|f |, |g |}.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Multiplicative seminorms

Recall that a multiplicative seminorm on ring A is a function| | : A → R≥0 with values in the set of nonnegative reals such thatfor every f , g ∈ A, we have:

(S1) |0| = 0, |1| = 1.

(S2) |f · g | = |f | · |g |.(S3) |f + g | ≤ |f |+ |g |.

The seminorm | | is called non-archimedean if in addition:

(S3)′ |f + g | ≤ max{|f |, |g |}.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Multiplicative seminorms

Recall that a multiplicative seminorm on ring A is a function| | : A → R≥0 with values in the set of nonnegative reals such thatfor every f , g ∈ A, we have:

(S1) |0| = 0, |1| = 1.

(S2) |f · g | = |f | · |g |.(S3) |f + g | ≤ |f |+ |g |.

The seminorm | | is called non-archimedean if in addition:

(S3)′ |f + g | ≤ max{|f |, |g |}.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Definition of A1Berk

Recall that:

As a set, A1Berk,K consists of all multiplicative seminorms on

the polynomial ring K [T ] which extend the usual absolutevalue on K .

The topology on A1Berk,K is the weakest one for which

x 7→ |f |x is continuous for every f ∈ K [T ].

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Definition of A1Berk

Recall that:

As a set, A1Berk,K consists of all multiplicative seminorms on

the polynomial ring K [T ] which extend the usual absolutevalue on K .

The topology on A1Berk,K is the weakest one for which

x 7→ |f |x is continuous for every f ∈ K [T ].

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Definition of A1Berk

Recall that:

As a set, A1Berk,K consists of all multiplicative seminorms on

the polynomial ring K [T ] which extend the usual absolutevalue on K .

The topology on A1Berk,K is the weakest one for which

x 7→ |f |x is continuous for every f ∈ K [T ].

Matthew Baker Lecture 2: Introduction to Berkovich Curves

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Matthew Baker Lecture 2: Introduction to Berkovich Curves

The Berkovich “Proj” construction for P1

In the first lecture, we defined the Berkovich projective lineP1

Berk,K to be the one-point compactification of the locally

compact Hausdorff space A1Berk,K .

A more functorial construction of P1Berk,K proceeds in a

manner analogous to the “Proj” construction in algebraicgeometry, using homogeneous seminorms.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The Berkovich “Proj” construction for P1

In the first lecture, we defined the Berkovich projective lineP1

Berk,K to be the one-point compactification of the locally

compact Hausdorff space A1Berk,K .

A more functorial construction of P1Berk,K proceeds in a

manner analogous to the “Proj” construction in algebraicgeometry, using homogeneous seminorms.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Homogeneous seminorms

Let S denote the set of multiplicative seminorms [ ] on thetwo-variable polynomial ring K [X ,Y ] which extend theabsolute value on K , and which are not identically zero on themaximal ideal (X ,Y ) of K [X ,Y ].

Any such seminorm [ ] is automatically non-archimedean, andtherefore the condition that [ ] is not identically zero on(X ,Y ) is equivalent to saying that [X ] and [Y ] are not bothzero.

We put an equivalence relation on S by declaring that[ ]1 ∼ [ ]2 if and only if there exists a constant C > 0 suchthat [G ]1 = Cd [G ]2 for all homogeneous polynomialsG ∈ K [X ,Y ] of degree d .

As a set, we define P1Berk to be the equivalence classes of

elements of S .

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Homogeneous seminorms

Let S denote the set of multiplicative seminorms [ ] on thetwo-variable polynomial ring K [X ,Y ] which extend theabsolute value on K , and which are not identically zero on themaximal ideal (X ,Y ) of K [X ,Y ].

Any such seminorm [ ] is automatically non-archimedean, andtherefore the condition that [ ] is not identically zero on(X ,Y ) is equivalent to saying that [X ] and [Y ] are not bothzero.

We put an equivalence relation on S by declaring that[ ]1 ∼ [ ]2 if and only if there exists a constant C > 0 suchthat [G ]1 = Cd [G ]2 for all homogeneous polynomialsG ∈ K [X ,Y ] of degree d .

As a set, we define P1Berk to be the equivalence classes of

elements of S .

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Homogeneous seminorms

Let S denote the set of multiplicative seminorms [ ] on thetwo-variable polynomial ring K [X ,Y ] which extend theabsolute value on K , and which are not identically zero on themaximal ideal (X ,Y ) of K [X ,Y ].

Any such seminorm [ ] is automatically non-archimedean, andtherefore the condition that [ ] is not identically zero on(X ,Y ) is equivalent to saying that [X ] and [Y ] are not bothzero.

We put an equivalence relation on S by declaring that[ ]1 ∼ [ ]2 if and only if there exists a constant C > 0 suchthat [G ]1 = Cd [G ]2 for all homogeneous polynomialsG ∈ K [X ,Y ] of degree d .

As a set, we define P1Berk to be the equivalence classes of

elements of S .

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Homogeneous seminorms

Let S denote the set of multiplicative seminorms [ ] on thetwo-variable polynomial ring K [X ,Y ] which extend theabsolute value on K , and which are not identically zero on themaximal ideal (X ,Y ) of K [X ,Y ].

Any such seminorm [ ] is automatically non-archimedean, andtherefore the condition that [ ] is not identically zero on(X ,Y ) is equivalent to saying that [X ] and [Y ] are not bothzero.

We put an equivalence relation on S by declaring that[ ]1 ∼ [ ]2 if and only if there exists a constant C > 0 suchthat [G ]1 = Cd [G ]2 for all homogeneous polynomialsG ∈ K [X ,Y ] of degree d .

As a set, we define P1Berk to be the equivalence classes of

elements of S .

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Embedding P1(K ) into P1Berk

Define the point ∞ in P1Berk to be the equivalence class of the

seminorm [ ]∞ defined by [G ]∞ = |G (1, 0)|.

More generally, if P ∈ P1(K ) has homogeneous coordinates(a : b), the equivalence class of the evaluation seminorm[G ]P = |G (a, b)| is independent of the choice of homogeneouscoordinates, and therefore [ ]P is a well-defined point of P1

Berk.

This furnishes an embedding of P1(K ) into P1Berk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Embedding P1(K ) into P1Berk

Define the point ∞ in P1Berk to be the equivalence class of the

seminorm [ ]∞ defined by [G ]∞ = |G (1, 0)|.More generally, if P ∈ P1(K ) has homogeneous coordinates(a : b), the equivalence class of the evaluation seminorm[G ]P = |G (a, b)| is independent of the choice of homogeneouscoordinates, and therefore [ ]P is a well-defined point of P1

Berk.

This furnishes an embedding of P1(K ) into P1Berk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Embedding P1(K ) into P1Berk

Define the point ∞ in P1Berk to be the equivalence class of the

seminorm [ ]∞ defined by [G ]∞ = |G (1, 0)|.More generally, if P ∈ P1(K ) has homogeneous coordinates(a : b), the equivalence class of the evaluation seminorm[G ]P = |G (a, b)| is independent of the choice of homogeneouscoordinates, and therefore [ ]P is a well-defined point of P1

Berk.

This furnishes an embedding of P1(K ) into P1Berk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The topology on P1Berk

Definition

We say that a seminorm [ ] in S is normalized ifmax{[X ], [Y ]} = 1.

There is a unique normalized seminorm within eachequivalence class.

We denote by [ ]∗z the normalized seminorm corresponding toa point z ∈ P1

Berk.

Explicitly, if [ ]z is any representative of the equivalence classof z , then

[G ]∗z = [G ]z/ max{[X ]z , [Y ]z}d

for all homogeneous polynomials G ∈ K [X ,Y ] of degree d .

Definition

The topology on P1Berk is defined to be the weakest one such that

z 7→ [G ]∗z is continuous for all G ∈ K [X ,Y ].

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The topology on P1Berk

Definition

We say that a seminorm [ ] in S is normalized ifmax{[X ], [Y ]} = 1.

There is a unique normalized seminorm within eachequivalence class.

We denote by [ ]∗z the normalized seminorm corresponding toa point z ∈ P1

Berk.

Explicitly, if [ ]z is any representative of the equivalence classof z , then

[G ]∗z = [G ]z/ max{[X ]z , [Y ]z}d

for all homogeneous polynomials G ∈ K [X ,Y ] of degree d .

Definition

The topology on P1Berk is defined to be the weakest one such that

z 7→ [G ]∗z is continuous for all G ∈ K [X ,Y ].

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The topology on P1Berk

Definition

We say that a seminorm [ ] in S is normalized ifmax{[X ], [Y ]} = 1.

There is a unique normalized seminorm within eachequivalence class.

We denote by [ ]∗z the normalized seminorm corresponding toa point z ∈ P1

Berk.

Explicitly, if [ ]z is any representative of the equivalence classof z , then

[G ]∗z = [G ]z/ max{[X ]z , [Y ]z}d

for all homogeneous polynomials G ∈ K [X ,Y ] of degree d .

Definition

The topology on P1Berk is defined to be the weakest one such that

z 7→ [G ]∗z is continuous for all G ∈ K [X ,Y ].

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The topology on P1Berk

Definition

We say that a seminorm [ ] in S is normalized ifmax{[X ], [Y ]} = 1.

There is a unique normalized seminorm within eachequivalence class.

We denote by [ ]∗z the normalized seminorm corresponding toa point z ∈ P1

Berk.

Explicitly, if [ ]z is any representative of the equivalence classof z , then

[G ]∗z = [G ]z/ max{[X ]z , [Y ]z}d

for all homogeneous polynomials G ∈ K [X ,Y ] of degree d .

Definition

The topology on P1Berk is defined to be the weakest one such that

z 7→ [G ]∗z is continuous for all G ∈ K [X ,Y ].

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The topology on P1Berk

Definition

We say that a seminorm [ ] in S is normalized ifmax{[X ], [Y ]} = 1.

There is a unique normalized seminorm within eachequivalence class.

We denote by [ ]∗z the normalized seminorm corresponding toa point z ∈ P1

Berk.

Explicitly, if [ ]z is any representative of the equivalence classof z , then

[G ]∗z = [G ]z/ max{[X ]z , [Y ]z}d

for all homogeneous polynomials G ∈ K [X ,Y ] of degree d .

Definition

The topology on P1Berk is defined to be the weakest one such that

z 7→ [G ]∗z is continuous for all G ∈ K [X ,Y ].

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Remarks on the topology on P1Berk

Remark

1 This definition of P1Berk as a topological space agrees with the

previous one.

2 P1(K ) and HBerk := P1Berk\P1(K ) are both dense in P1

Berk.

3 If ϕ ∈ K (T ) is a rational function of degree d ≥ 1, then theinduced map ϕ : P1(K )→ P1(K ) extends to a continuousmap ϕ : P1

Berk → P1Berk, as we explain in the next slide.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Remarks on the topology on P1Berk

Remark

1 This definition of P1Berk as a topological space agrees with the

previous one.

2 P1(K ) and HBerk := P1Berk\P1(K ) are both dense in P1

Berk.

3 If ϕ ∈ K (T ) is a rational function of degree d ≥ 1, then theinduced map ϕ : P1(K )→ P1(K ) extends to a continuousmap ϕ : P1

Berk → P1Berk, as we explain in the next slide.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Remarks on the topology on P1Berk

Remark

1 This definition of P1Berk as a topological space agrees with the

previous one.

2 P1(K ) and HBerk := P1Berk\P1(K ) are both dense in P1

Berk.

3 If ϕ ∈ K (T ) is a rational function of degree d ≥ 1, then theinduced map ϕ : P1(K )→ P1(K ) extends to a continuousmap ϕ : P1

Berk → P1Berk, as we explain in the next slide.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The action of a rational function on P1Berk

Choose a homogeneous lifting F = (F1,F2) of ϕ, whereFi ∈ K [X ,Y ] are homogeneous of degree d and have nocommon zeros in K .

Let G ∈ K [X ,Y ], and define

[G ]ϕ(z) := [G (F1(X ,Y ),F2(X ,Y ))]z .

The right-hand side is independent of the lifting F of ϕ, up toequivalence of seminorms.

Using properties of the resultant, one checks that[X ]ϕ(z) = [F1(X ,Y )]z and [Y ]ϕ(z) = [F2(X ,Y )]z cannot bothbe zero. Therefore we obtain a continuous map from P1

Berk toitself.

One can show that ϕ : P1Berk → P1

Berk is an open surjectivemapping, and that every point z ∈ P1

Berk has at most dpreimages under ϕ.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The action of a rational function on P1Berk

Choose a homogeneous lifting F = (F1,F2) of ϕ, whereFi ∈ K [X ,Y ] are homogeneous of degree d and have nocommon zeros in K .

Let G ∈ K [X ,Y ], and define

[G ]ϕ(z) := [G (F1(X ,Y ),F2(X ,Y ))]z .

The right-hand side is independent of the lifting F of ϕ, up toequivalence of seminorms.

Using properties of the resultant, one checks that[X ]ϕ(z) = [F1(X ,Y )]z and [Y ]ϕ(z) = [F2(X ,Y )]z cannot bothbe zero. Therefore we obtain a continuous map from P1

Berk toitself.

One can show that ϕ : P1Berk → P1

Berk is an open surjectivemapping, and that every point z ∈ P1

Berk has at most dpreimages under ϕ.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The action of a rational function on P1Berk

Choose a homogeneous lifting F = (F1,F2) of ϕ, whereFi ∈ K [X ,Y ] are homogeneous of degree d and have nocommon zeros in K .

Let G ∈ K [X ,Y ], and define

[G ]ϕ(z) := [G (F1(X ,Y ),F2(X ,Y ))]z .

The right-hand side is independent of the lifting F of ϕ, up toequivalence of seminorms.

Using properties of the resultant, one checks that[X ]ϕ(z) = [F1(X ,Y )]z and [Y ]ϕ(z) = [F2(X ,Y )]z cannot bothbe zero. Therefore we obtain a continuous map from P1

Berk toitself.

One can show that ϕ : P1Berk → P1

Berk is an open surjectivemapping, and that every point z ∈ P1

Berk has at most dpreimages under ϕ.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The action of a rational function on P1Berk

Choose a homogeneous lifting F = (F1,F2) of ϕ, whereFi ∈ K [X ,Y ] are homogeneous of degree d and have nocommon zeros in K .

Let G ∈ K [X ,Y ], and define

[G ]ϕ(z) := [G (F1(X ,Y ),F2(X ,Y ))]z .

The right-hand side is independent of the lifting F of ϕ, up toequivalence of seminorms.

Using properties of the resultant, one checks that[X ]ϕ(z) = [F1(X ,Y )]z and [Y ]ϕ(z) = [F2(X ,Y )]z cannot bothbe zero. Therefore we obtain a continuous map from P1

Berk toitself.

One can show that ϕ : P1Berk → P1

Berk is an open surjectivemapping, and that every point z ∈ P1

Berk has at most dpreimages under ϕ.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The action of a rational function on P1Berk

Choose a homogeneous lifting F = (F1,F2) of ϕ, whereFi ∈ K [X ,Y ] are homogeneous of degree d and have nocommon zeros in K .

Let G ∈ K [X ,Y ], and define

[G ]ϕ(z) := [G (F1(X ,Y ),F2(X ,Y ))]z .

The right-hand side is independent of the lifting F of ϕ, up toequivalence of seminorms.

Using properties of the resultant, one checks that[X ]ϕ(z) = [F1(X ,Y )]z and [Y ]ϕ(z) = [F2(X ,Y )]z cannot bothbe zero. Therefore we obtain a continuous map from P1

Berk toitself.

One can show that ϕ : P1Berk → P1

Berk is an open surjectivemapping, and that every point z ∈ P1

Berk has at most dpreimages under ϕ.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The action of a rational function on P1Berk (continued)

If z ∈ HBerk, then ϕ(z) ∈ HBerk as well. More generally, ϕtakes points of type τ to points of type τ for allτ ∈ {I,II,III,IV}.

In particular, the group PGL(2,K ) acts naturally on P1Berk and

on HBerk via automorphisms.

As we saw in Lecture 1, elements of PGL(2,K ) act viaisometries with respect to the path metric on HBerk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The action of a rational function on P1Berk (continued)

If z ∈ HBerk, then ϕ(z) ∈ HBerk as well. More generally, ϕtakes points of type τ to points of type τ for allτ ∈ {I,II,III,IV}.In particular, the group PGL(2,K ) acts naturally on P1

Berk andon HBerk via automorphisms.

As we saw in Lecture 1, elements of PGL(2,K ) act viaisometries with respect to the path metric on HBerk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The action of a rational function on P1Berk (continued)

If z ∈ HBerk, then ϕ(z) ∈ HBerk as well. More generally, ϕtakes points of type τ to points of type τ for allτ ∈ {I,II,III,IV}.In particular, the group PGL(2,K ) acts naturally on P1

Berk andon HBerk via automorphisms.

As we saw in Lecture 1, elements of PGL(2,K ) act viaisometries with respect to the path metric on HBerk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Rational functions and the metric on HBerk

The following result is due to Juan Rivera-Letelier:

Theorem

Let ϕ ∈ K (T ) be a nonzero rational function of degree d ≥ 1.Then for all x , y ∈ HBerk, we have

ρ(ϕ(x), ϕ(y)) ≤ d · ρ(x , y).

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Rational functions and the metric on HBerk

The following result is due to Juan Rivera-Letelier:

Theorem

Let ϕ ∈ K (T ) be a nonzero rational function of degree d ≥ 1.Then for all x , y ∈ HBerk, we have

ρ(ϕ(x), ϕ(y)) ≤ d · ρ(x , y).

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Rational functions and the metric on HBerk

The following result is due to Juan Rivera-Letelier:

Theorem

Let ϕ ∈ K (T ) be a nonzero rational function of degree d ≥ 1.Then for all x , y ∈ HBerk, we have

ρ(ϕ(x), ϕ(y)) ≤ d · ρ(x , y).

This result is a consequence of the stronger fact that, locally in thedirection of a tangent vector ~v , a rational function ϕ stretchesdistances in HBerk by an integer factor between 1 and d , called thelocal degree of ~v .

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Generalities on R-trees

Our next goal is to describe P1Berk as a profinite R-tree.

Definition

1 An R-tree is a metric space T such that for each distinct pairof points x , y ∈ T , there is a unique path in T from x to y .

2 A finite R-tree is an R-tree with only finitely many branchpoints.(Intuitively, a finite R-tree is just a finite tree in the usualgraph-theoretic sense, but where the edges are thought of asline segments having definite lengths.)

3 A profinite R-tree is an inverse limit of finite R-trees.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Generalities on R-trees

Our next goal is to describe P1Berk as a profinite R-tree.

Definition

1 An R-tree is a metric space T such that for each distinct pairof points x , y ∈ T , there is a unique path in T from x to y .

2 A finite R-tree is an R-tree with only finitely many branchpoints.(Intuitively, a finite R-tree is just a finite tree in the usualgraph-theoretic sense, but where the edges are thought of asline segments having definite lengths.)

3 A profinite R-tree is an inverse limit of finite R-trees.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Generalities on R-trees

Our next goal is to describe P1Berk as a profinite R-tree.

Definition

1 An R-tree is a metric space T such that for each distinct pairof points x , y ∈ T , there is a unique path in T from x to y .

2 A finite R-tree is an R-tree with only finitely many branchpoints.(Intuitively, a finite R-tree is just a finite tree in the usualgraph-theoretic sense, but where the edges are thought of asline segments having definite lengths.)

3 A profinite R-tree is an inverse limit of finite R-trees.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Generalities on R-trees

Our next goal is to describe P1Berk as a profinite R-tree.

Definition

1 An R-tree is a metric space T such that for each distinct pairof points x , y ∈ T , there is a unique path in T from x to y .

2 A finite R-tree is an R-tree with only finitely many branchpoints.(Intuitively, a finite R-tree is just a finite tree in the usualgraph-theoretic sense, but where the edges are thought of asline segments having definite lengths.)

3 A profinite R-tree is an inverse limit of finite R-trees.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Finite subgraphs of P1Berk

If S ⊂ P1Berk, define the convex hull of S to be the smallest

path-connected subset of P1Berk containing S . (This is the

same as the union of all paths between points of S .)

By a finite subgraph of P1Berk, we will mean the convex hull of

a finite subset of points of type II or III.

Every finite subgraph Γ of P1Berk is in fact a finite R-tree,

where the metric is induced by the path-distance ρ on HBerk.

By construction, a finite subgraph of P1Berk is both finitely

branched and of finite total length with respect to ρ.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Finite subgraphs of P1Berk

If S ⊂ P1Berk, define the convex hull of S to be the smallest

path-connected subset of P1Berk containing S . (This is the

same as the union of all paths between points of S .)

By a finite subgraph of P1Berk, we will mean the convex hull of

a finite subset of points of type II or III.

Every finite subgraph Γ of P1Berk is in fact a finite R-tree,

where the metric is induced by the path-distance ρ on HBerk.

By construction, a finite subgraph of P1Berk is both finitely

branched and of finite total length with respect to ρ.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Finite subgraphs of P1Berk

If S ⊂ P1Berk, define the convex hull of S to be the smallest

path-connected subset of P1Berk containing S . (This is the

same as the union of all paths between points of S .)

By a finite subgraph of P1Berk, we will mean the convex hull of

a finite subset of points of type II or III.

Every finite subgraph Γ of P1Berk is in fact a finite R-tree,

where the metric is induced by the path-distance ρ on HBerk.

By construction, a finite subgraph of P1Berk is both finitely

branched and of finite total length with respect to ρ.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Finite subgraphs of P1Berk

If S ⊂ P1Berk, define the convex hull of S to be the smallest

path-connected subset of P1Berk containing S . (This is the

same as the union of all paths between points of S .)

By a finite subgraph of P1Berk, we will mean the convex hull of

a finite subset of points of type II or III.

Every finite subgraph Γ of P1Berk is in fact a finite R-tree,

where the metric is induced by the path-distance ρ on HBerk.

By construction, a finite subgraph of P1Berk is both finitely

branched and of finite total length with respect to ρ.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Inverse limits of finite subgraphs of P1Berk

The collection of all finite subgraphs of P1Berk forms a directed

system under inclusion.

Moreover, if Γ ≤ Γ′, then by a basic property of R-trees, thereis a continuous retraction map rΓ′,Γ : Γ′ � Γ. So we can alsothink of the finite subgraphs of P1

Berk as forming an inversesystem.

Theorem

1 P1Berk (with its Berkovich topology) is homeomorphic to the

inverse limit lim←− Γ over all finite subgraphs Γ of P1Berk.

2 HRBerk (with its “locally metric” topology) is homeomorphic to

the direct limit lim−→ Γ over all finite subgraphs Γ of P1Berk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Inverse limits of finite subgraphs of P1Berk

The collection of all finite subgraphs of P1Berk forms a directed

system under inclusion.

Moreover, if Γ ≤ Γ′, then by a basic property of R-trees, thereis a continuous retraction map rΓ′,Γ : Γ′ � Γ. So we can alsothink of the finite subgraphs of P1

Berk as forming an inversesystem.

Theorem

1 P1Berk (with its Berkovich topology) is homeomorphic to the

inverse limit lim←− Γ over all finite subgraphs Γ of P1Berk.

2 HRBerk (with its “locally metric” topology) is homeomorphic to

the direct limit lim−→ Γ over all finite subgraphs Γ of P1Berk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Inverse limits of finite subgraphs of P1Berk

The collection of all finite subgraphs of P1Berk forms a directed

system under inclusion.

Moreover, if Γ ≤ Γ′, then by a basic property of R-trees, thereis a continuous retraction map rΓ′,Γ : Γ′ � Γ. So we can alsothink of the finite subgraphs of P1

Berk as forming an inversesystem.

Theorem

1 P1Berk (with its Berkovich topology) is homeomorphic to the

inverse limit lim←− Γ over all finite subgraphs Γ of P1Berk.

2 HRBerk (with its “locally metric” topology) is homeomorphic to

the direct limit lim−→ Γ over all finite subgraphs Γ of P1Berk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Inverse limits of finite subgraphs of P1Berk

The collection of all finite subgraphs of P1Berk forms a directed

system under inclusion.

Moreover, if Γ ≤ Γ′, then by a basic property of R-trees, thereis a continuous retraction map rΓ′,Γ : Γ′ � Γ. So we can alsothink of the finite subgraphs of P1

Berk as forming an inversesystem.

Theorem

1 P1Berk (with its Berkovich topology) is homeomorphic to the

inverse limit lim←− Γ over all finite subgraphs Γ of P1Berk.

2 HRBerk (with its “locally metric” topology) is homeomorphic to

the direct limit lim−→ Γ over all finite subgraphs Γ of P1Berk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Inverse limits of finite subgraphs of P1Berk

The collection of all finite subgraphs of P1Berk forms a directed

system under inclusion.

Moreover, if Γ ≤ Γ′, then by a basic property of R-trees, thereis a continuous retraction map rΓ′,Γ : Γ′ � Γ. So we can alsothink of the finite subgraphs of P1

Berk as forming an inversesystem.

Theorem

1 P1Berk (with its Berkovich topology) is homeomorphic to the

inverse limit lim←− Γ over all finite subgraphs Γ of P1Berk.

2 HRBerk (with its “locally metric” topology) is homeomorphic to

the direct limit lim−→ Γ over all finite subgraphs Γ of P1Berk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The topology on P1Berk revisited

This description of P1Berk as an inverse limit of finite R-trees

helps us visualize the Berkovich topology: two points are“close” if they retract to the same point on a “large” finitesubgraph of P1

Berk.

Given this description of the topology, the compactness ofP1

Berk is an immediate consequence of Tychonoff’s theorem.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The topology on P1Berk revisited

This description of P1Berk as an inverse limit of finite R-trees

helps us visualize the Berkovich topology: two points are“close” if they retract to the same point on a “large” finitesubgraph of P1

Berk.

Given this description of the topology, the compactness ofP1

Berk is an immediate consequence of Tychonoff’s theorem.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Simple domains

Let rΓ be the natural map from P1Berk to Γ coming from the

universal property of the inverse limit.

A fundamental system of open neighborhoods for thetopology on P1

Berk is given by the simple domains, which aresubsets of the form r−1

Γ (V ) for Γ a finite subgraph of P1Berk

and V a connected open subset of Γ.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Simple domains

Let rΓ be the natural map from P1Berk to Γ coming from the

universal property of the inverse limit.A fundamental system of open neighborhoods for thetopology on P1

Berk is given by the simple domains, which aresubsets of the form r−1

Γ (V ) for Γ a finite subgraph of P1Berk

and V a connected open subset of Γ.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Simple domains

Let rΓ be the natural map from P1Berk to Γ coming from the

universal property of the inverse limit.A fundamental system of open neighborhoods for thetopology on P1

Berk is given by the simple domains, which aresubsets of the form r−1

Γ (V ) for Γ a finite subgraph of P1Berk

and V a connected open subset of Γ.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Characterization of simple domains

Lemma

For a subset U ⊆ P1Berk, the following are equivalent:

1 U is a simple domain.

2 U is a finite intersection of Berkovich open disks.

3 U is a connected open set whose boundary is a finite subset ofHR

Berk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Characterization of simple domains

Lemma

For a subset U ⊆ P1Berk, the following are equivalent:

1 U is a simple domain.

2 U is a finite intersection of Berkovich open disks.

3 U is a connected open set whose boundary is a finite subset ofHR

Berk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Characterization of simple domains

Lemma

For a subset U ⊆ P1Berk, the following are equivalent:

1 U is a simple domain.

2 U is a finite intersection of Berkovich open disks.

3 U is a connected open set whose boundary is a finite subset ofHR

Berk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Normed rings

A norm on a ring A is a function | | : A → R≥0 with values in theset of nonnegative reals such that for every f , g ∈ A, we have:

(N1) |0| = 0, |1| = 1.

(N2) |f + g | ≤ |f |+ |g |.(N3) |f · g | ≤ |f | · |g |.(N4) |f | = 0 implies f = 0.

A normed ring is a pair (A, ‖ ‖) consisting of a ring A and anorm ‖ ‖.It is called a Banach ring if A is complete with respect to thisnorm.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Normed rings

A norm on a ring A is a function | | : A → R≥0 with values in theset of nonnegative reals such that for every f , g ∈ A, we have:

(N1) |0| = 0, |1| = 1.

(N2) |f + g | ≤ |f |+ |g |.(N3) |f · g | ≤ |f | · |g |.(N4) |f | = 0 implies f = 0.

A normed ring is a pair (A, ‖ ‖) consisting of a ring A and anorm ‖ ‖.It is called a Banach ring if A is complete with respect to thisnorm.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Normed rings

A norm on a ring A is a function | | : A → R≥0 with values in theset of nonnegative reals such that for every f , g ∈ A, we have:

(N1) |0| = 0, |1| = 1.

(N2) |f + g | ≤ |f |+ |g |.

(N3) |f · g | ≤ |f | · |g |.(N4) |f | = 0 implies f = 0.

A normed ring is a pair (A, ‖ ‖) consisting of a ring A and anorm ‖ ‖.It is called a Banach ring if A is complete with respect to thisnorm.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Normed rings

A norm on a ring A is a function | | : A → R≥0 with values in theset of nonnegative reals such that for every f , g ∈ A, we have:

(N1) |0| = 0, |1| = 1.

(N2) |f + g | ≤ |f |+ |g |.(N3) |f · g | ≤ |f | · |g |.

(N4) |f | = 0 implies f = 0.

A normed ring is a pair (A, ‖ ‖) consisting of a ring A and anorm ‖ ‖.It is called a Banach ring if A is complete with respect to thisnorm.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Normed rings

A norm on a ring A is a function | | : A → R≥0 with values in theset of nonnegative reals such that for every f , g ∈ A, we have:

(N1) |0| = 0, |1| = 1.

(N2) |f + g | ≤ |f |+ |g |.(N3) |f · g | ≤ |f | · |g |.(N4) |f | = 0 implies f = 0.

A normed ring is a pair (A, ‖ ‖) consisting of a ring A and anorm ‖ ‖.It is called a Banach ring if A is complete with respect to thisnorm.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Normed rings

A norm on a ring A is a function | | : A → R≥0 with values in theset of nonnegative reals such that for every f , g ∈ A, we have:

(N1) |0| = 0, |1| = 1.

(N2) |f + g | ≤ |f |+ |g |.(N3) |f · g | ≤ |f | · |g |.(N4) |f | = 0 implies f = 0.

A normed ring is a pair (A, ‖ ‖) consisting of a ring A and anorm ‖ ‖.

It is called a Banach ring if A is complete with respect to thisnorm.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Normed rings

A norm on a ring A is a function | | : A → R≥0 with values in theset of nonnegative reals such that for every f , g ∈ A, we have:

(N1) |0| = 0, |1| = 1.

(N2) |f + g | ≤ |f |+ |g |.(N3) |f · g | ≤ |f | · |g |.(N4) |f | = 0 implies f = 0.

A normed ring is a pair (A, ‖ ‖) consisting of a ring A and anorm ‖ ‖.It is called a Banach ring if A is complete with respect to thisnorm.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Multiplicative seminorms

A bounded multiplicative seminorm on a normed ring (A, ‖ ‖) is afunction | | : A → R≥0 with values in the set of nonnegative realssuch that for every f , g ∈ A, we have:

(S1) |0| = 0, |1| = 1.

(S2) |f + g | ≤ |f |+ |g |.(S3) |f · g | = |f | · |g |.(S4) There exists a constant C > 0 such that |f | ≤ C‖f ‖ for all

f ∈ A.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Multiplicative seminorms

A bounded multiplicative seminorm on a normed ring (A, ‖ ‖) is afunction | | : A → R≥0 with values in the set of nonnegative realssuch that for every f , g ∈ A, we have:

(S1) |0| = 0, |1| = 1.

(S2) |f + g | ≤ |f |+ |g |.

(S3) |f · g | = |f | · |g |.(S4) There exists a constant C > 0 such that |f | ≤ C‖f ‖ for all

f ∈ A.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Multiplicative seminorms

A bounded multiplicative seminorm on a normed ring (A, ‖ ‖) is afunction | | : A → R≥0 with values in the set of nonnegative realssuch that for every f , g ∈ A, we have:

(S1) |0| = 0, |1| = 1.

(S2) |f + g | ≤ |f |+ |g |.(S3) |f · g | = |f | · |g |.

(S4) There exists a constant C > 0 such that |f | ≤ C‖f ‖ for allf ∈ A.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Multiplicative seminorms

A bounded multiplicative seminorm on a normed ring (A, ‖ ‖) is afunction | | : A → R≥0 with values in the set of nonnegative realssuch that for every f , g ∈ A, we have:

(S1) |0| = 0, |1| = 1.

(S2) |f + g | ≤ |f |+ |g |.(S3) |f · g | = |f | · |g |.(S4) There exists a constant C > 0 such that |f | ≤ C‖f ‖ for all

f ∈ A.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The Berkovich spectrum of a normed ring

Let (A, ‖ ‖) be a normed ring. We define a topological spaceM(A), called the Berkovich spectrum of A, as follows.

As a set,M(A) consists of all bounded multiplicativeseminorms on A.

The Berkovich topology onM(A) is the weakest one forwhich all functions of the form | | 7→ |f | for f ∈ A arecontinuous.

A fundamental system of open neighborhoods for thetopology onM(A) is given by open sets of the form

{x ∈M(A) : αi < |fi |x < βi}

with f1, . . . , fm ∈ A and αi , βi ∈ R (i = 1, . . . ,m).

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The Berkovich spectrum of a normed ring

Let (A, ‖ ‖) be a normed ring. We define a topological spaceM(A), called the Berkovich spectrum of A, as follows.

As a set,M(A) consists of all bounded multiplicativeseminorms on A.

The Berkovich topology onM(A) is the weakest one forwhich all functions of the form | | 7→ |f | for f ∈ A arecontinuous.

A fundamental system of open neighborhoods for thetopology onM(A) is given by open sets of the form

{x ∈M(A) : αi < |fi |x < βi}

with f1, . . . , fm ∈ A and αi , βi ∈ R (i = 1, . . . ,m).

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The Berkovich spectrum of a normed ring

Let (A, ‖ ‖) be a normed ring. We define a topological spaceM(A), called the Berkovich spectrum of A, as follows.

As a set,M(A) consists of all bounded multiplicativeseminorms on A.

The Berkovich topology onM(A) is the weakest one forwhich all functions of the form | | 7→ |f | for f ∈ A arecontinuous.

A fundamental system of open neighborhoods for thetopology onM(A) is given by open sets of the form

{x ∈M(A) : αi < |fi |x < βi}

with f1, . . . , fm ∈ A and αi , βi ∈ R (i = 1, . . . ,m).

Matthew Baker Lecture 2: Introduction to Berkovich Curves

The Berkovich spectrum of a normed ring

Let (A, ‖ ‖) be a normed ring. We define a topological spaceM(A), called the Berkovich spectrum of A, as follows.

As a set,M(A) consists of all bounded multiplicativeseminorms on A.

The Berkovich topology onM(A) is the weakest one forwhich all functions of the form | | 7→ |f | for f ∈ A arecontinuous.

A fundamental system of open neighborhoods for thetopology onM(A) is given by open sets of the form

{x ∈M(A) : αi < |fi |x < βi}

with f1, . . . , fm ∈ A and αi , βi ∈ R (i = 1, . . . ,m).

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Compactness of the Berkovich spectrum

Theorem (Berkovich)

If A is a Banach ring, then the spectrum M(A) is a non-emptycompact Hausdorff space.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Compactness of the Berkovich spectrum

Theorem (Berkovich)

If A is a Banach ring, then the spectrum M(A) is a non-emptycompact Hausdorff space.

As a particular example, we will now consider the Berkovichanalytic space M(Z) associated to the Banach ring (Z, | |∞),where | |∞ denotes the usual archimedean absolute value.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Ostrowski’s theorem

Recall the statement of Ostrowski’s theorem:

Theorem (Ostrowski)

Every non-trivial absolute value on Q is equivalent to either | |∞,or to the standard p-adic absolute value | |p for some primenumber p (normalized so that |p|p = 1

p .)

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Ostrowski’s theorem

Recall the statement of Ostrowski’s theorem:

Theorem (Ostrowski)

Every non-trivial absolute value on Q is equivalent to either | |∞,or to the standard p-adic absolute value | |p for some primenumber p (normalized so that |p|p = 1

p .)

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Ostrowski’s theorem

Recall the statement of Ostrowski’s theorem:

Theorem (Ostrowski)

Every non-trivial absolute value on Q is equivalent to either | |∞,or to the standard p-adic absolute value | |p for some primenumber p (normalized so that |p|p = 1

p .)

Thus, if we let MQ denote the set of places (equivalence classes ofnon-trivial absolute values) of Q, then there is a bijection

MQ ↔ {prime numbers p} ∪ {∞}.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Classification of points ofM(Z)

Following Berkovich, one can classify all multiplicative seminormson Z as follows:

The archimedean absolute values | |∞,ε for 0 < ε ≤ 1 definedby

|n|∞,ε = |n|ε∞.

The p-adic absolute values | |p,ε for 0 < ε <∞ defined by

|n|p,ε = |n|εp.

The trivial absolute value | |0 defined by

|n|0 =

{0 n = 01 n 6= 0.

The p-trivial multiplicative seminorms | |p,∞ defined by

|n|p,∞ =

{0 p | n1 p - n.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Classification of points ofM(Z)

Following Berkovich, one can classify all multiplicative seminormson Z as follows:

The archimedean absolute values | |∞,ε for 0 < ε ≤ 1 definedby

|n|∞,ε = |n|ε∞.

The p-adic absolute values | |p,ε for 0 < ε <∞ defined by

|n|p,ε = |n|εp.

The trivial absolute value | |0 defined by

|n|0 =

{0 n = 01 n 6= 0.

The p-trivial multiplicative seminorms | |p,∞ defined by

|n|p,∞ =

{0 p | n1 p - n.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Classification of points ofM(Z)

Following Berkovich, one can classify all multiplicative seminormson Z as follows:

The archimedean absolute values | |∞,ε for 0 < ε ≤ 1 definedby

|n|∞,ε = |n|ε∞.

The p-adic absolute values | |p,ε for 0 < ε <∞ defined by

|n|p,ε = |n|εp.

The trivial absolute value | |0 defined by

|n|0 =

{0 n = 01 n 6= 0.

The p-trivial multiplicative seminorms | |p,∞ defined by

|n|p,∞ =

{0 p | n1 p - n.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Classification of points ofM(Z)

Following Berkovich, one can classify all multiplicative seminormson Z as follows:

The archimedean absolute values | |∞,ε for 0 < ε ≤ 1 definedby

|n|∞,ε = |n|ε∞.

The p-adic absolute values | |p,ε for 0 < ε <∞ defined by

|n|p,ε = |n|εp.

The trivial absolute value | |0 defined by

|n|0 =

{0 n = 01 n 6= 0.

The p-trivial multiplicative seminorms | |p,∞ defined by

|n|p,∞ =

{0 p | n1 p - n.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Classification of points ofM(Z)

Following Berkovich, one can classify all multiplicative seminormson Z as follows:

The archimedean absolute values | |∞,ε for 0 < ε ≤ 1 definedby

|n|∞,ε = |n|ε∞.

The p-adic absolute values | |p,ε for 0 < ε <∞ defined by

|n|p,ε = |n|εp.

The trivial absolute value | |0 defined by

|n|0 =

{0 n = 01 n 6= 0.

The p-trivial multiplicative seminorms | |p,∞ defined by

|n|p,∞ =

{0 p | n1 p - n.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

VisualizingM(Z)

Remark

Note that the different tangent directions emanating from | |0 arein one-to-one correspondence with the places of Q.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

VisualizingM(Z)

Remark

Note that the different tangent directions emanating from | |0 arein one-to-one correspondence with the places of Q.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Topology onM(Z)

In the topology onM(Z), the interval corresponding to each placeof Q is embedded homeomorphically, and open neighborhoods ofthe point | |0 contain all but finitely many of these intervals.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

MetrizingM(Z)

We identify the segment `∞ = {| |∞,ε}0≤ε≤1 with the realinterval [0, 1] via the parameter ε.

For each prime p, we identify the segment `p = {| |p,ε}0≤ε≤∞with the extended-real interval [0,∞] via the parameter ε.

In this way, the complement HZ inM(Z) of all points of type| |p,∞ becomes a metric space.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

MetrizingM(Z)

We identify the segment `∞ = {| |∞,ε}0≤ε≤1 with the realinterval [0, 1] via the parameter ε.

For each prime p, we identify the segment `p = {| |p,ε}0≤ε≤∞with the extended-real interval [0,∞] via the parameter ε.

In this way, the complement HZ inM(Z) of all points of type| |p,∞ becomes a metric space.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

MetrizingM(Z)

We identify the segment `∞ = {| |∞,ε}0≤ε≤1 with the realinterval [0, 1] via the parameter ε.

For each prime p, we identify the segment `p = {| |p,ε}0≤ε≤∞with the extended-real interval [0,∞] via the parameter ε.

In this way, the complement HZ inM(Z) of all points of type| |p,∞ becomes a metric space.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

MetrizingM(Z)

We identify the segment `∞ = {| |∞,ε}0≤ε≤1 with the realinterval [0, 1] via the parameter ε.

For each prime p, we identify the segment `p = {| |p,ε}0≤ε≤∞with the extended-real interval [0,∞] via the parameter ε.

In this way, the complement HZ inM(Z) of all points of type| |p,∞ becomes a metric space.

The metric topology on HZis not the same as thesubspace topology comingfrom M(Z).

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Remarks onM(Z)

Remark

1 The points ofM(Z) having distance 1 from the trivialseminorm | |0 are precisely the points corresponding to thestandard absolute values | |p = | |p,1 and | |∞ = | |∞,1.

2 The points | |p,∞ should be thought of as “ideal boundarypoints” which are infinitely far from the points of HZ.

3 Like P1Berk, the space M(Z) can be viewed as an inverse limit

of finite R-trees, and HZ (with its locally metric topology) isthe corresponding direct limit.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Remarks onM(Z)

Remark

1 The points ofM(Z) having distance 1 from the trivialseminorm | |0 are precisely the points corresponding to thestandard absolute values | |p = | |p,1 and | |∞ = | |∞,1.

2 The points | |p,∞ should be thought of as “ideal boundarypoints” which are infinitely far from the points of HZ.

3 Like P1Berk, the space M(Z) can be viewed as an inverse limit

of finite R-trees, and HZ (with its locally metric topology) isthe corresponding direct limit.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Remarks onM(Z)

Remark

1 The points ofM(Z) having distance 1 from the trivialseminorm | |0 are precisely the points corresponding to thestandard absolute values | |p = | |p,1 and | |∞ = | |∞,1.

2 The points | |p,∞ should be thought of as “ideal boundarypoints” which are infinitely far from the points of HZ.

3 Like P1Berk, the space M(Z) can be viewed as an inverse limit

of finite R-trees, and HZ (with its locally metric topology) isthe corresponding direct limit.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Visualizing the metric onM(Z)

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Analytification of an algebraic variety

One can associate in a functorial way to every algebraicvariety X/K a locally ringed topological space XBerk calledthe Berkovich K -analytic space associated to X .

When X = Spec(A) is affine, the underlying topological spaceof XBerk is the set of multiplicative seminorms on A whichextend the given absolute value on K , equipped with theweakest topology for which all functions of the form | | 7→ |f |for f ∈ A are continuous.

When X = A1 (resp. P1), we recover the definition of A1Berk

(resp. P1Berk) given above.

The space XBerk is locally compact and Hausdorff. If X isproper then XBerk is compact. If X is Zariski-connected thenXBerk is path-connected.

There is a canonical embedding of X (K ) (endowed with itstotally disconnected analytic topology) as a dense subspace ofXBerk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Analytification of an algebraic variety

One can associate in a functorial way to every algebraicvariety X/K a locally ringed topological space XBerk calledthe Berkovich K -analytic space associated to X .

When X = Spec(A) is affine, the underlying topological spaceof XBerk is the set of multiplicative seminorms on A whichextend the given absolute value on K , equipped with theweakest topology for which all functions of the form | | 7→ |f |for f ∈ A are continuous.

When X = A1 (resp. P1), we recover the definition of A1Berk

(resp. P1Berk) given above.

The space XBerk is locally compact and Hausdorff. If X isproper then XBerk is compact. If X is Zariski-connected thenXBerk is path-connected.

There is a canonical embedding of X (K ) (endowed with itstotally disconnected analytic topology) as a dense subspace ofXBerk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Analytification of an algebraic variety

One can associate in a functorial way to every algebraicvariety X/K a locally ringed topological space XBerk calledthe Berkovich K -analytic space associated to X .

When X = Spec(A) is affine, the underlying topological spaceof XBerk is the set of multiplicative seminorms on A whichextend the given absolute value on K , equipped with theweakest topology for which all functions of the form | | 7→ |f |for f ∈ A are continuous.

When X = A1 (resp. P1), we recover the definition of A1Berk

(resp. P1Berk) given above.

The space XBerk is locally compact and Hausdorff. If X isproper then XBerk is compact. If X is Zariski-connected thenXBerk is path-connected.

There is a canonical embedding of X (K ) (endowed with itstotally disconnected analytic topology) as a dense subspace ofXBerk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Analytification of an algebraic variety

One can associate in a functorial way to every algebraicvariety X/K a locally ringed topological space XBerk calledthe Berkovich K -analytic space associated to X .

When X = Spec(A) is affine, the underlying topological spaceof XBerk is the set of multiplicative seminorms on A whichextend the given absolute value on K , equipped with theweakest topology for which all functions of the form | | 7→ |f |for f ∈ A are continuous.

When X = A1 (resp. P1), we recover the definition of A1Berk

(resp. P1Berk) given above.

The space XBerk is locally compact and Hausdorff. If X isproper then XBerk is compact. If X is Zariski-connected thenXBerk is path-connected.

There is a canonical embedding of X (K ) (endowed with itstotally disconnected analytic topology) as a dense subspace ofXBerk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Analytification of an algebraic variety

One can associate in a functorial way to every algebraicvariety X/K a locally ringed topological space XBerk calledthe Berkovich K -analytic space associated to X .

When X = Spec(A) is affine, the underlying topological spaceof XBerk is the set of multiplicative seminorms on A whichextend the given absolute value on K , equipped with theweakest topology for which all functions of the form | | 7→ |f |for f ∈ A are continuous.

When X = A1 (resp. P1), we recover the definition of A1Berk

(resp. P1Berk) given above.

The space XBerk is locally compact and Hausdorff. If X isproper then XBerk is compact. If X is Zariski-connected thenXBerk is path-connected.

There is a canonical embedding of X (K ) (endowed with itstotally disconnected analytic topology) as a dense subspace ofXBerk.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Analytification of an algebraic variety (continued)

A theorem of Berkovich implies that the analytification of asmooth projective variety over K is locally contractible.(This is a very difficult result which relies, among otherthings, on de Jong’s theory of alterations.)

As a concrete example, the analytification XBerk of a smoothprojective curve X/K admits a deformation retraction onto afinite metrized graph called the skeleton of XBerk.

If X has good reduction, then the skeleton of X is a point andthe entire space XBerk is contractible.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Analytification of an algebraic variety (continued)

A theorem of Berkovich implies that the analytification of asmooth projective variety over K is locally contractible.(This is a very difficult result which relies, among otherthings, on de Jong’s theory of alterations.)

As a concrete example, the analytification XBerk of a smoothprojective curve X/K admits a deformation retraction onto afinite metrized graph called the skeleton of XBerk.

If X has good reduction, then the skeleton of X is a point andthe entire space XBerk is contractible.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Analytification of an algebraic variety (continued)

A theorem of Berkovich implies that the analytification of asmooth projective variety over K is locally contractible.(This is a very difficult result which relies, among otherthings, on de Jong’s theory of alterations.)

As a concrete example, the analytification XBerk of a smoothprojective curve X/K admits a deformation retraction onto afinite metrized graph called the skeleton of XBerk.

If X has good reduction, then the skeleton of X is a point andthe entire space XBerk is contractible.

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Example: Tate elliptic curves

Here is a picture of the analytic space associated to an ellipticcurve with multiplicative reduction:

Matthew Baker Lecture 2: Introduction to Berkovich Curves

Another picture

Matthew Baker Lecture 2: Introduction to Berkovich Curves

And one more!

Matthew Baker Lecture 2: Introduction to Berkovich Curves