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Transcript of Potential Theory on Berkovich Spaces Lecture 2 ...people.math.gatech.edu/~mbaker/pdf/AWS2.pdf ·...
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