Potential Theory on Berkovich Spaces Lecture 1: The...

Potential Theory on Berkovich SpacesLecture 1: The Berkovich Projective Line

Matthew Baker

Georgia Institute of Technology

Arizona Winter School on p-adic GeometryMarch 2007

Matthew Baker Lecture 1: The Berkovich Projective Line

In this course, we will describe in detail the structure ofBerkovich analytic curves, with a particular emphasis on theBerkovich projective line.

In particular, we will introduce potential theory (Laplacians,harmonic and subharmonic functions, . . . ) on such spaces.The results obtained closely parallel classical facts over C (e.g.the maximum modulus principle, Poisson formula. . . )

The ultimate goal (which we will unfortunately not say muchabout) is to treat archimedean and non-archimedean analyticspaces in a unified way, and to make precise Arakelov’sanalogy between intersection theory on arithmetic surfacesand potential theory on Riemann surfaces.

Credits

The definitions and results we will be describing have beendeveloped by various people, including (in no particular order):

Vladimir Berkovich

M.B. and Robert Rumely

Antoine Chambert-Loir

Amaury Thuillier

Juan Rivera-Letelier

Charles Favre and Mattias Jonsson

Ernst Kani

Credits

Vladimir Berkovich

Amaury Thuillier

Ernst Kani

Credits

Vladimir Berkovich

Amaury Thuillier

Ernst Kani

Credits

Vladimir Berkovich

Amaury Thuillier

Ernst Kani

Credits

Vladimir Berkovich

Amaury Thuillier

Ernst Kani

Credits

Vladimir Berkovich

Amaury Thuillier

Ernst Kani

Credits

Vladimir Berkovich

Amaury Thuillier

Ernst Kani

Credits

Vladimir Berkovich

Amaury Thuillier

Ernst Kani

Notation

K : an algebraically closed field which is complete with respectto a nontrivial non-archimedean absolute value (e.g. K = Cp)

K : the residue field of K

B(a, r): the closed disk {z ∈ K : |z − a| ≤ r} of radius rabout a in K . Here r is any positive real number, andsometimes we allow the degenerate case r = 0 as well. Ifr ∈ |K ∗| we call the disk rational, and if r 6∈ |K ∗| we call itirrational.

B(a, r)−: the open disk {z ∈ K : |z − a| < r} of radius rabout a in K .

Notation

Motivation

The usual topology on K is totally disconnected and notlocally compact.

Tate dealt with this problem by developing rigid analysis, inwhich one works with a certain Grothendieck topology on K .This gives a satisfactory theory of analytic functions on K ,but the underlying topological space is unchanged.

The Berkovich affine line A1Berk over K is a locally compact,

Hausdorff, and uniquely path-connected topological spacewhich contains K as a dense subspace. The Berkovichprojective line P1

Berk is obtained by adjoining a point ∞ toA1

One can view P1Berk as a profinite R-tree. This allows one to

define a Laplacian operator on P1Berk which comes from the

usual Laplacian on a finite graph. The tree structure alsoleads to a good theory of harmonic and subharmonic functionswhich closely parallels the classical theory over C.

Motivation

Multiplicative seminorms

A multiplicative seminorm on a ring A is a function | |x : A → R≥0

satisfying:

|0|x = 0 and |1|x = 1.

|fg |x = |f |x · |g |x for all f , g ∈ A.

|f + g |x ≤ |f |x + |g |x for all f , g ∈ A.

satisfying:

|0|x = 0 and |1|x = 1.

satisfying:

|0|x = 0 and |1|x = 1.

satisfying:

|0|x = 0 and |1|x = 1.

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

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

Multiplicative seminorms (continued)

Remarks

1 We will assume throughout that our field K is complete andalgebraically closed.

2 We will identify seminorms | |x with points x ∈ A1Berk,K .

3 We will usually omit explicit reference to the ground field K ,writing A1

Remarks

Topology on A1Berk

Definition

The topology on A1Berk,K is defined to be the weakest one for

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

Topology on A1Berk

Definition

The topology on A1Berk,K is defined to be the weakest one for

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

Explicitly, a fundamental system of open neighborhoods is given byopen sets of the form

{x ∈ A1Berk : αi < |fi |x < βi}

with f1, . . . , fm ∈ K [T ] and αi , βi ∈ R (i = 1, . . . ,m).

Motivation for the definition of A1Berk

The definition of A1Berk can be motivated by the following

observations:

Every multiplicative seminorm on C[T ] which extends theusual absolute value on C is of the form f 7→ |f (z)| for somez ∈ C (by the Gelfand-Mazur theorem), and thecorresponding space A1

Berk,C is homeomorphic to C.When K is non-archimedean, there are many moremultiplicative seminorms on K [T ] than just the ones given byevaluation at a point of K .

Example

Fix a closed disk B(a, r) = {z ∈ K : |z − a| ≤ r} in K , and define| |B(a,r) by

|f |B(a,r) = supz∈B(a,r)

|f (z)|.

Then | |B(a,r) is a multiplicative seminorm on K [T ] (by Gauss’lemma).

observations:

Berk,C is homeomorphic to C.

When K is non-archimedean, there are many moremultiplicative seminorms on K [T ] than just the ones given byevaluation at a point of K .

Example

|f (z)|.

observations:

Example

|f (z)|.

observations:

Example

|f (z)|.

Embedding K into A1Berk,K

The set of all (possibly degenerate) disks B(a, r) thereforeembeds naturally into A1

In particular, K embeds into A1Berk as the set of disks of

radius zero, and is dense in the Berkovich topology.

Similarly, P1(K ) can naturally be embedded as a dense subsetof P1

A1Berk is uniquely path-connected

If a, a′ are distinct points of K , one can visualize the unique pathin A1

Berk from a to a′ as follows:

Start increasing the “radius” of the degenerate disk B(a, 0)until we have a disk B(a, r) which also contains a′.

This disk can also be written as B(a′, s) with r = s = |a− a′|.Now decrease s until the radius reaches zero and we have thedegenerate disk B(a′, 0).

In this way we have “connected up” the totally disconnectedspace K by adding points corresponding to closed disks!

This disk can also be written as B(a′, s) with r = s = |a− a′|.

Now decrease s until the radius reaches zero and we have thedegenerate disk B(a′, 0).

Nested sequences of closed disks

In order to obtain a compact space from this construction, it isusually necessary to add even more points.

For example, the field Cp is not spherically complete: thismeans that there are decreasing sequences of closed disks inCp having empty intersection.

We need to add points corresponding to such sequences inorder to obtain a compact space, since if {B(an, rn)} is anydecreasing nested sequence of closed disks, the map

f 7→ limn→∞

|f |B(an,rn)

defines a multiplicative seminorm on K [T ] extending theusual absolute value on K .

Two such sequences of disks with empty intersection definethe same seminorm if and only if the sequences are cofinal.

f 7→ limn→∞

|f |B(an,rn)

f 7→ limn→∞

|f |B(an,rn)

f 7→ limn→∞

|f |B(an,rn)

Berkovich’s classification theorem

According to a result of Berkovich, we have now described allpoints of A1

Theorem (Berkovich’s Classification Theorem)

Every point x ∈ A1Berk corresponds to a nested sequence

B(a1, r1) ⊇ B(a2, r2) ⊇ B(a3, r3) ⊇ · · · of closed disks, in thesense that

|f |x = limn→∞

|f |B(an,rn).

Two such nested sequences define the same point of A1Berk if and

only if either:

1 each has a nonempty intersection, and their intersections arethe same; or

2 both have empty intersection, and the sequences are cofinal.

|f |x = limn→∞

|f |B(an,rn).

only if either:

|f |x = limn→∞

|f |B(an,rn).

only if either:

|f |x = limn→∞

|f |B(an,rn).

only if either:

|f |x = limn→∞

|f |B(an,rn).

only if either:

Classification into four types

We can categorize the points of A1Berk into four types according to

the nature of B =⋂

B(an, rn):

Type I: B is a point of K .

Type II: B is a closed disk with radius belonging to |K ∗|.Type III: B is an irrational disk with radius not belonging to |K ∗|.Type IV: B = ∅.

B(an, rn):

Type II: B is a closed disk with radius belonging to |K ∗|.

Type III: B is an irrational disk with radius not belonging to |K ∗|.Type IV: B = ∅.

B(an, rn):

Type II: B is a closed disk with radius belonging to |K ∗|.Type III: B is an irrational disk with radius not belonging to |K ∗|.

Type IV: B = ∅.

B(an, rn):

The Gauss norm

We will denote by ζa,r the point of A1Berk of type II or III

corresponding to the closed or irrational disk B(a, r).

Following the terminology introduced by Chambert-Loir, thedistinguished point ζGauss = ζ0,1 in A1

Berk corresponding tothe Gauss norm

aiTi |Gauss = max |ai |

on K [T ] will be called the Gauss point.

The Gauss norm

We will denote by ζa,r the point of A1Berk of type II or III

corresponding to the closed or irrational disk B(a, r).

Following the terminology introduced by Chambert-Loir, thedistinguished point ζGauss = ζ0,1 in A1

Berk corresponding tothe Gauss norm

aiTi |Gauss = max |ai |

on K [T ] will be called the Gauss point.

A visual representation of P1Berk

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Alternate representation of P1Berk

Visualizing P1Berk

Note that:

There is branching only at the points of type II, not those oftype III.

Some of the branches extend all the way to the bottom(terminating in points of type I), while others are “cauterizedoff” earlier and terminate at points of type IV. In any case,every branch terminates either at a point of type I or type IV.

Visualizing P1Berk

Note that:

There is branching only at the points of type II, not those oftype III.

Some of the branches extend all the way to the bottom(terminating in points of type I), while others are “cauterizedoff” earlier and terminate at points of type IV. In any case,every branch terminates either at a point of type I or type IV.

Tangent directions

Definition

Let x ∈ P1Berk. The space Tx of tangent directions at x is the set

of equivalence classes of paths `x ,y emanating from x , where y isany point of P1

Berk not equal to x . Two paths `x ,y1 , `x ,y2 areequivalent if they share a common initial segment.

There is a natural bijection between elements ~v ∈ Tx andconnected components of P1

Berk\{x}.We denote by U(x ;~v) the connected component of P1

Berk\{x}corresponding to ~v ∈ Tx .

The open sets U(x ;~v) for x ∈ P1Berk and ~v ∈ Tx generate the

topology on P1Berk.

Tangent directions

Definition

Berk\{x}.

We denote by U(x ;~v) the connected component of P1Berk\{x}

corresponding to ~v ∈ Tx .

topology on P1Berk.

Tangent directions

Definition

topology on P1Berk.

Tangent directions

Definition

topology on P1Berk.

Berkovich disks

For a ∈ K and r > 0, write

B(a, r)− = {x ∈ A1Berk : |T − a|x < r} ,

B(a, r) = {x ∈ A1Berk : |T − a|x ≤ r} .

We call a set of the form B(a, r)− an open Berkovich disk inA1

Berk, and a set of the form B(a, r) a closed Berkovich disk inA1

Similarly, we can define open and closed Berkovich disks inP1

The intersection of a Berkovich open disk with P1(K ) is a(classical) open disk (and similarly for closed disks).

Berkovich disks

B(a, r)− = {x ∈ A1Berk : |T − a|x < r} ,

B(a, r) = {x ∈ A1Berk : |T − a|x ≤ r} .

Berkovich disks

B(a, r)− = {x ∈ A1Berk : |T − a|x < r} ,

B(a, r) = {x ∈ A1Berk : |T − a|x ≤ r} .

Berkovich disks

B(a, r)− = {x ∈ A1Berk : |T − a|x < r} ,

B(a, r) = {x ∈ A1Berk : |T − a|x ≤ r} .

A Berkovich open disk

Simple domains

Every open set U(x ;~v) with x of type II or III and ~v ∈ Tx is aBerkovich open disk, and conversely.

Simple domains

Every open set U(x ;~v) with x of type II or III and ~v ∈ Tx is aBerkovich open disk, and conversely.

Finite intersections of Berkovich open disks in P1Berk are called

simple domains, and they form a fundamental system of openneighborhoods for the topology on P1

Ends of a simple domain

A simple domain V in P1Berk has a finite set x1, . . . , xn of boundary

points, and a corresponding finite set ~v1, . . . , ~vn of ends, which arethe inward-pointing tangent directions:

Ends of a simple domain

A simple domain V in P1Berk has a finite set x1, . . . , xn of boundary

points, and a corresponding finite set ~v1, . . . , ~vn of ends, which arethe inward-pointing tangent directions:

Tangent directions at ζGauss

The tangent directions ~v ∈ TζGausscorrespond bijectively to

elements of P1(K ), the projective line over the residue field ofK .

Equivalently, elements of TζGausscorrespond to the open disks

of radius 1 contained in the closed unit disk B(0, 1), togetherwith the open disk

B(∞, 1)− := P1(K )\B(0, 1).

The correspondence between elements of TζGaussand open

disks is given explicitly by ~v 7→ U(ζGauss;~v).

B(∞, 1)− := P1(K )\B(0, 1).

Tangent directions at a general point

More generally, for each point x = ζa,r of type II, the set Tx

of tangent directions at x is (non-canonically) isomorphic toP1(K ): there is one tangent direction going “up” to infinity,and the other tangent directions correspond to open disksB(a′, r)− of radius r contained in B(a, r).

For x ∈ P1Berk, we have:

|Tx | =

|P1(K )| x of type II2 x of type III1 x of type I or type IV.

|Tx | =

|P1(K )| x of type II

2 x of type III1 x of type I or type IV.

|Tx | =

|P1(K )| x of type II2 x of type III

1 x of type I or type IV.

|Tx | =

The Berkovich hyperbolic space HBerk

Following notation introduced by Juan Rivera-Letelier, we writeHBerk for the subset of P1

Berk consisting of all points of type II, III,or IV.

We refer to HBerk as “Berkovich hyperbolic space”.

We write HQBerk for the set of type II points, and HR

Berk for theset of points of type II or III.

The subset HQBerk is dense in P1

The diameter function on A1Berk

Define the diameter function diam : A1Berk → R≥0 by setting

diam(x) = lim ri if x corresponds to the nested sequence{B(ai , ri )}.

If x ∈ HRBerk, then diam(x) is just the diameter (= radius) of

the corresponding closed disk.

In terms of multiplicative seminorms, we have

diam(x) = infa∈K

|T − a|x .

Because K is complete, if x is of type IV, then necessarilydiam(x) > 0. Thus diam(x) = 0 for x ∈ A1

Berk of type I, anddiam(x) > 0 for x ∈ HBerk.

diam(x) = infa∈K

|T − a|x .

diam(x) = infa∈K

|T − a|x .

A partial order on A1Berk

The space A1Berk is endowed with a natural partial order,

defined by saying that

x ≤ y ⇐⇒ |f |x ≤ |f |y ∀ f ∈ K [T ].

In terms of (possibly degenerate) disks, if x , y ∈ A1Berk are

points of type I, II, or III, we have x ≤ y if and only if the diskcorresponding to x is contained in the disk corresponding to y .

For each pair of points x , y ∈ A1Berk, there is a unique least

upper bound x ∨ y in A1Berk with respect to this partial order.

Concretely, if x = ζa,r and y = ζb,s are points of type I, II orIII, then x ∨ y is the point of A1

Berk corresponding to thesmallest disk containing both B(a, r) and B(b, s).

x ≤ y ⇐⇒ |f |x ≤ |f |y ∀ f ∈ K [T ].

The path metric on HBerk

If x , y ∈ HBerk with x ≤ y , define the path metric

ρ(x , y) = logvdiam(y)

diam(x),

where logv denotes the logarithm to the base qv , with qv > 1a suitable constant.

For example, if K = Cp and |p|p = 1/p, we would set qv = pin order to have {logv |x |p : x ∈ C∗p} = Q.

More generally, for x , y ∈ HBerk arbitrary, we define

ρ(x , y) = ρ(x , x ∨ y) + ρ(y , x ∨ y) .

This gives an metric on HBerk.

diam(x),

ρ(x , y) = ρ(x , x ∨ y) + ρ(y , x ∨ y) .

diam(x),

ρ(x , y) = ρ(x , x ∨ y) + ρ(y , x ∨ y) .

Remarks on the path metric on HBerk

For closed disks B(a, r) ⊆ B(a,R), we have

ρ(ζa,r , ζa,R) = logv R − logv r .

The points of type I should be thought of as infinitely faraway from the points of HBerk.

The topology on HBerk defined by the metric ρ is not thesubspace topology induced from the Berkovich topology onP1

Berk. However, the inclusion map HBerk ↪→ P1Berk is

continuous with respect to these topologies.

The group PGL(2,K ) of Mobius transformations actscontinuously on P1

Berk in a natural way compatible with theusual action on P1(K ), and this action preserves HBerk. Onecan show that PGL(2,K ) acts via isometries on HBerk, i.e.,

ρ(M(x),M(y)) = ρ(x , y)

for all x , y ∈ HBerk and all M ∈ PGL(2,K ). (This shows thatthe metric ρ is canonical).

ρ(M(x),M(y)) = ρ(x , y)

The canonical distance on A1Berk

The diameter function can be used to extend the usualdistance function |x − y | on K to A1

Berk in a natural way bysetting

[x , y ]∞ = diam(x ∨ y)

for x , y ∈ A1Berk.

We call this extension the canonical distance (or Hsia kernel)on A1

Berk (relative to infinity).

If x , y ∈ K , then [x , y ]∞ = |x − y |.

More generally:

If x = ζa,r and y = ζb,s are points of Type I,II, or III, then

[x , y ]∞ = supx0∈B(a,r)y0∈B(b,s)

|x0 − y0|.

[x , y ]∞ = diam(x ∨ y)

If x , y ∈ K , then [x , y ]∞ = |x − y |.

More generally:

[x , y ]∞ = supx0∈B(a,r)y0∈B(b,s)

|x0 − y0|.

[x , y ]∞ = diam(x ∨ y)

If x , y ∈ K , then [x , y ]∞ = |x − y |.

More generally:

[x , y ]∞ = supx0∈B(a,r)y0∈B(b,s)

|x0 − y0|.

[x , y ]∞ = diam(x ∨ y)

If x , y ∈ K , then [x , y ]∞ = |x − y |.

More generally:

[x , y ]∞ = supx0∈B(a,r)y0∈B(b,s)

|x0 − y0|.

Properties of [x , y ]∞

1 For y fixed, [x , y ]∞ is continuous in x .

2 As a function of two variables, [x , y ]∞ is merely uppersemicontinuous.

3 For all x , y , z ∈ A1Berk, we have the ultrametric inequality

[x , y ]∞ ≤ max([x , z ]∞, [y , z ]∞),

with equality if [x , z ]∞ 6= [y , z ]∞.

4 [x , y ]∞ satisfies all of the axioms for an ultrametric except wehave [x , x ]∞ > 0 for x ∈ HBerk.

[x , y ]∞ ≤ max([x , z ]∞, [y , z ]∞),

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