Graphs and Arithmetic Geometry - Peoplepeople.math.gatech.edu/~mbaker/pdf/JA2009.pdftheory on Berkovich curves Graphs and Arithmetic Geometry Matthew Baker Georgia Institute of Technology

Graphs andArithmeticGeometry

MatthewBaker

The secret lifeof graphs

Tropicalcurves andtheirJacobians

Graphs andarithmeticsurfaces

Harmonicmorphisms ofmetric graphs

Potentialtheory onBerkovichcurves

Graphs and Arithmetic Geometry

Matthew Baker

Georgia Institute of Technology

Journees ArithmetiquesSaint-Etienne, July 2009


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Outline

1 The secret life of graphs

2 Tropical curves and their Jacobians

3 Graphs and arithmetic surfaces

4 Harmonic morphisms of metric graphs

5 Potential theory on Berkovich curves


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References

References:

R. Bacher, P. de la Harpe, T. Nagnibeda, “The lattice ofintegral flows and the lattice of integral cuts on a finitegraph” (1997).

M. Baker and S. Norine, “Riemann-Roch and Abel-Jacobitheory on a finite graph” (2007).


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Graphs

By a graph G , we mean a connected, finite, undirectedmultigraph without loop edges.


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Divisors

The group Div(G ) of divisors on G is the free abeliangroup on V (G ).

We write elements of Div(G ) as formal sums

D =∑

v∈V (G)

av (v)

with av ∈ Z.

A divisor D is effective if av ≥ 0 for all v .

The degree of D =∑

av (v) is deg(D) =∑

av .

We set

Div0(G ) = D ∈ Div(G ) : deg(D) = 0.


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Divisors



D =∑

v∈V (G)

av (v)

with av ∈ Z.




av .

We set

Div0(G ) = D ∈ Div(G ) : deg(D) = 0.


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Divisors



D =∑

v∈V (G)

av (v)

with av ∈ Z.




av .

We set

Div0(G ) = D ∈ Div(G ) : deg(D) = 0.


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Divisors



D =∑

v∈V (G)

av (v)

with av ∈ Z.




av .

We set

Div0(G ) = D ∈ Div(G ) : deg(D) = 0.


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Divisors



D =∑

v∈V (G)

av (v)

with av ∈ Z.




av .

We set

Div0(G ) = D ∈ Div(G ) : deg(D) = 0.


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Rational functions and principal divisors

The group of rational functions on G is

M(G ) = functions f : V (G )→ Z.

The Laplacian operator ∆ :M(G )→ Div0(G ) is definedby

∆f =∑

v∈V (G)

(∑e=vw

(f (v)− f (w))

)(v).

The group of principal divisors on G is the subgroup

Prin(G ) = ∆f : f ∈M(G )

of Div0(G ).


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∆f =∑

v∈V (G)

(∑e=vw

(f (v)− f (w))

)(v).


Prin(G ) = ∆f : f ∈M(G )

of Div0(G ).


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∆f =∑

v∈V (G)

(∑e=vw

(f (v)− f (w))

)(v).


Prin(G ) = ∆f : f ∈M(G )

of Div0(G ).


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Linear equivalence

Divisors D,D ′ ∈ Div(G ) are linearly equivalent, writtenD ∼ D ′, if D − D ′ is principal.

If we think of D as an assignment of Euros to each vertex,then D ∼ D ′ iff one can get from D to D ′ by a sequenceof “legal moves” of the following type:

1 A vertex v lends one Euro across each edge adjacent to v .2 A vertex v borrows one Euro along each edge adjacent to

v .


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Linear equivalence




v .


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Linear equivalence




v .


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Linear equivalence




v .


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The Jacobian

The Jacobian (or Picard group) of G is

Jac(G ) = Div0(G )/ Prin(G ).

This is a finite abelian group whose cardinality is thenumber of spanning trees in G (Kirchhoff’s Matrix-TreeTheorem).


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The Jacobian

The Jacobian (or Picard group) of G is

Jac(G ) = Div0(G )/ Prin(G ).

This is a finite abelian group whose cardinality is thenumber of spanning trees in G (Kirchhoff’s Matrix-TreeTheorem).


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The Riemann-Roch theorem

The canonical divisor on G is

KG =∑

v∈V (G)

(deg(v)− 2) (v).

Its degree is 2g − 2, where g = dimR H1(G , R) is thegenus of G .The complete linear system |D| of a divisor D is

|D| = E ∈ Div(G ) : E ≥ 0, E ∼ D.Define h0(D) by the formula

h0(D) = mindeg(E ) : E ≥ 0, |D − E | = ∅.

Theorem (“Riemann-Roch for graphs”, B.–Norine)

For every D ∈ Div(G ), we have

h0(D)− h0(KG − D) = deg(D) + 1− g .


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KG =∑

v∈V (G)

(deg(v)− 2) (v).



h0(D) = mindeg(E ) : E ≥ 0, |D − E | = ∅.



h0(D)− h0(KG − D) = deg(D) + 1− g .


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KG =∑

v∈V (G)

(deg(v)− 2) (v).



h0(D) = mindeg(E ) : E ≥ 0, |D − E | = ∅.



h0(D)− h0(KG − D) = deg(D) + 1− g .


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KG =∑

v∈V (G)

(deg(v)− 2) (v).



h0(D) = mindeg(E ) : E ≥ 0, |D − E | = ∅.



h0(D)− h0(KG − D) = deg(D) + 1− g .


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A consequence of Riemann-Roch

Corollary (B.-Norine)

If the total amount of money on the graph is at least g , thenone can get every vertex out of debt by a sequence of legalmoves.


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References

References:

G. Mikhalkin and I. Zharkov, ”Tropical curves, theirJacobians, and Theta functions” (2008).

A. Gathmann and M. Kerber, ”A Riemann-Roch theoremin tropical geometry” (2008).


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Tropical geometry

Let K be an algebraically closed field which is completewith respect to a (non-trivial) non-archimedean valuationval.

Examples: K = Cp or K = the Puiseux series field CT.If X is a d-dimensional irreducible algebraic subvariety ofthe torus (K ∗)n, then

Trop(X ) = val(X ) ⊆ Rn

is a connected polyhedral complex of pure dimension d .


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Tropical geometry






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Tropical geometry






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A tropical cubic curve in R2


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Metric graphs

A weighted graph is a graph G together with anassignment of a “length” `(e) > 0 to each edge e ∈ E (G ).A (compact) metric graph Γ is just the “geometricrealization” of a weighted graph: it is obtained from aweighted graph G by identifying each edge e with a linesegment of length `(e). In particular, Γ is a compactmetric space.A weighted graph G whose geometric realization is Γ willbe called a model for Γ.


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Metric graphs



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Metric graphs



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Abstract tropical curves

Following Mikhalkin, an abstract tropical curve is just a“metric graph with a finite number of unbounded ends”.

Convention

We will ignore the unbounded ends and use the terms “tropicalcurve” and “metric graph” interchangeably.


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Abstract tropical curves

Following Mikhalkin, an abstract tropical curve is just a“metric graph with a finite number of unbounded ends”.

Convention

We will ignore the unbounded ends and use the terms “tropicalcurve” and “metric graph” interchangeably.


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Divisors

For a tropical curve Γ, we make the following definitions:

Div(Γ) is the free abelian group on Γ.

M(Γ) consists of all continuous piecewise affine functionsf : Γ→ R with integer slopes.

The Laplacian operator ∆ :M(Γ)→ Div0(Γ) is defined by−∆f =

∑p∈Γ σp(f )(p), where σp(f ) is the sum of the

slopes of f in all tangent directions emanating from p.

Prin(Γ) = ∆f : f ∈M(Γ).Jac(Γ) = Div0(Γ)/ Prin(Γ).


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Divisors









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Divisors









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Divisors









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Divisors









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Tropical Abel theorem

Let Ω1(Γ) be the g -dimensional real vector space of harmonic1-forms on Γ.One can identify Ω1(Γ) with H1(Γ, R) and think of a harmonic1-form ω as a real-valued flow on Γ (net amount into eachvertex equals net amount out).

Theorem (“Tropical Abel Theorem”, Mikhalkin–Zharkov)

There is a canonical isomorphism

Div0(Γ)/ Prin(Γ) ∼= Hom(Ω1(Γ), R)/H1(Γ, Z).

In particular, Jac(Γ) is a real torus of dimension g.


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A tropical curve of genus 2 in its Jacobian


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Tropical Riemann-Roch

As before, define KΓ =∑

p∈Γ (deg(p)− 2) (p) and forD ∈ Div(Γ), define

|D| = E ∈ Div(Γ) : E ≥ 0, E ∼ Dh0(D) = mindeg(E ) : E ≥ 0, |D − E | = ∅.

Theorem (“Riemann-Roch for tropical curves”,Gathmann–Kerber, Mikhalkin–Zharkov)

For every D ∈ Div(Γ), we have

h0(D)− h0(KΓ − D) = deg(D) + 1− g .


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h0(D)− h0(KΓ − D) = deg(D) + 1− g .


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h0(D)− h0(KΓ − D) = deg(D) + 1− g .


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h0(D)− h0(KΓ − D) = deg(D) + 1− g .


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References

References:

M. Raynaud, “Specialisation du foncteur de Picard”(1970).

M. Baker, “Specialization of linear systems from curves tographs” (2008).


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Notation and terminology

K : a field which is complete with respect to a (nontrivial)non-archimedean valuation

R: the valuation ring of K

k: the residue field of K (which we assume to bealgebraically closed)

X: an arithmetic surface, i.e., a flat proper scheme over Rwhose generic fiber X is a smooth (proper) geometricallyconnected curve over K . We call X a model for X .


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Semistability and dual graphs

An arithmetic surface X/R is called semistable if itsspecial fiber Xk is reduced and all singularities of Xk areordinary double points. It is called strongly semistable if inaddition every irreducible component of Xk is smooth.

By the Semistable Reduction Theorem, there exists a finiteextension L/K such that XL := X ×K L has a stronglysemistable model.

The dual graph of a strongly semistable arithmetic surfaceX is the weighted graph G = GX whose verticescorrespond to the irreducible components of Xk and whoseedges correspond to the singular points of Xk , with `(e)equal to the thickness of the corresponding singularity.


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Thickness of singularities

If R is a DVR with maximal ideal (π), the thickness ofz ∈ X

singk ⊂ X is the unique natural number k such that z

is cut out etale-locally by an equation of the form xy = πk .

z is a regular point of X iff its thickness is 1.

If R is not a DVR, one can define the thickness using rigidanalytic geometry.


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The reduction graph

The reduction graph Γ = ΓX of X is the metric graphrealization of GX.


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Models and their reductions


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Specialization of divisors

Suppose R is a DVR, and let X/R be a regular stronglysemistable arithmetic surface.

Let Z1, . . . ,Zn be the irreducible components of Xk ,corresponding to the vertices v1, . . . , vn of GX.

Given a Cartier divisor D ∈ Div(X), let L be theassociated line bundle, and define a homomorphismρX : Div(X)→ Div(G ) by

ρX(D) =∑

i

deg(L|Zi)(vi ) ∈ Div(G ).

This extends to a homomorphism ρ : Div(X )→ Div(G ) bytaking Zariski closures.


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ρX(D) =∑

i




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ρX(D) =∑

i




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ρX(D) =∑

i




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Properties of the specialization map

deg(ρ(D)) = deg(D).

If D ∼X D ′ then ρ(D) ∼G ρ(D ′). Thus ρ induces ahomomorphism Jac(X )→ Jac(G ).

(Adjunction formula) Suppose X is totally degenerate,meaning that g(Zi ) = 0 for all i (or equivalently, thatg(X ) = g(GX)). Then ρ(KX ) ∼ KG for any canonicaldivisor KX on X .


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Connection with Neron models

We continue to suppose that R is a DVR.

Let J /R be the Neron model of Jac(X ), and let ΦJ be thegroup of connected components of the special fiber of J .

The following is a reformulation of a theorem of Raynaud:

Theorem (Raynaud)

ΦJ is canonically isomorphic to Jac(GX) for any stronglysemistable regular model X/R for X .


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Theorem (Raynaud)



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Theorem (Raynaud)



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Theorem (Raynaud)



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The specialization inequality

Suppose R is a DVR and X is a strongly semistable regulararithmetic surface over R with dual graph G .

Proposition (“Specialization Inequality”, B.)

For every D ∈ Div(X ), we have h0G (ρ(D)) ≥ h0

X (D).


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The specialization inequality

Suppose R is a DVR and X is a strongly semistable regulararithmetic surface over R with dual graph G .

Proposition (“Specialization Inequality”, B.)

For every D ∈ Div(X ), we have h0G (ρ(D)) ≥ h0

X (D).


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Weierstrass points

If X is a curve, a Weierstrass point on X is a point P ∈ X (K )such that h0(gP) ≥ 2.

It is a classical fact that every curve of genus g ≥ 2 hasWeierstrass points. (In fact, there are exactly g3 − g of them ifwe count with multiplicities.)

If Γ is a tropical curve, we define a Weierstrass point on Γ tobe a point P ∈ Γ such that h0(gP) ≥ 2.

Theorem (B.)

Every tropical curve of genus g ≥ 2 has Weierstrass points.


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Weierstrass points




Theorem (B.)



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Weierstrass points




Theorem (B.)



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Weierstrass points




Theorem (B.)



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Sketch of proof

By a semicontinuity argument on the moduli space ofmetric graphs of genus g , we may assume without loss ofgenerality that Γ has rational edge lengths.

Rescaling, we may assume that Γ has a “regular model” G(i.e., a model with all edge lengths equal to 1).

By deformation theory, there exists a regular, totallydegenerate, strongly semistable arithmetic surface X withdual graph G .

By the specialization inequality, Weierstrass points inX (K ) reduce to Weierstrass points on Γ.


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Sketch of proof






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Sketch of proof






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Sketch of proof






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Remarks

1 There exist tropical curves Γ with infinitely manyWeierstrass points.

2 There are graphs G with no Weierstrass points, forexample:


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Remarks

1 There exist tropical curves Γ with infinitely manyWeierstrass points.

2 There are graphs G with no Weierstrass points, forexample:


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Graphs without Weierstrass points

If G has no Weierstrass points, then a curve X/K having atotally degenerate regular semistable model with dual graph Gcannot have any K -rational Weierstrass points.

More generally, if P ∈ X (K ) is a Weierstrass point, then ρ(P)must be a Weierstrass point on G .

This observation gives a new proof and generalization of thefollowing theorem of Ogg:

Theorem (Ogg)

The cusps 0,∞ are not Weierstrass points on the modularcurve X0(p) for p ≥ 23 a prime number.


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Theorem (Ogg)



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Theorem (Ogg)



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Theorem (Ogg)



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Tropical Brill-Noether theory

One can use similar methods to prove:

Theorem (“Tropical Brill-Noether theorem”, B.)

If g − (r + 1)(g − d + r) ≥ 0, then every tropical curve Γ ofgenus g has a divisor D with deg(D) ≤ d and h0(D) ≥ r + 1.

Our deformation/specialization technique thus allows one todeduce certain theorems in tropical geometry from theirclassical counterparts.


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References

H. Urakawa, “A discrete analogue of the harmonicmorphism and Green kernel comparison theorems” (2000).

M. Baker and S. Norine, “Harmonic morphisms andhyperelliptic graphs” (2009).


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Harmonic morphisms

A continuous map φ : Γ→ Γ′ between metric graphs is called aharmonic morphism if:

φ is piecewise affine with integer slopes.

If p ∈ Γ and f : Γ′ → R is harmonic at φ(p), then f φ isharmonic at p.


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Harmonic morphisms

A continuous map φ : Γ→ Γ′ between metric graphs is called aharmonic morphism if:

φ is piecewise affine with integer slopes.

If p ∈ Γ and f : Γ′ → R is harmonic at φ(p), then f φ isharmonic at p.


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A degree 3 harmonic morphism


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Multiplicities

If φ : Γ→ Γ′ is a harmonic morphism of metric graphs, one candefine a local multiplicity (or ramification index) mφ(x) atevery x ∈ Γ.

The quantity

deg(φ) :=∑

φ(x)=x ′

mφ(x)

is the same for every x ′ ∈ Γ′, and is called the degree of φ.


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Multiplicities

If φ : Γ→ Γ′ is a harmonic morphism of metric graphs, one candefine a local multiplicity (or ramification index) mφ(x) atevery x ∈ Γ.

The quantity

deg(φ) :=∑

φ(x)=x ′

mφ(x)

is the same for every x ′ ∈ Γ′, and is called the degree of φ.


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A degree 4 harmonic morphism


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The tropical Riemann-Hurwitz formula

As with algebraic curves, one can use the multiplicities mφ(x)to define functorial pushforward and pullback maps on divisors,harmonic 1-forms, Jacobians, etc.

Proposition (Tropical Riemann-Hurwitz formula)

Assume (for simplicity) that φ : Γ→ Γ′ is a harmonic morphismof metric graphs having finite fibers. Then

KΓ = φ∗KΓ′ +∑x∈Γ

(2mφ(x)− 2−

∑e3x

(mφ(e)− 1)

)(x).


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(2mφ(x)− 2−

∑e3x

(mφ(e)− 1)

)(x).


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(2mφ(x)− 2−

∑e3x

(mφ(e)− 1)

)(x).


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Connections with arithmetic geometry

Theorem (B., Thuillier)

If f : X→ X′ is a morphism of semistable arithmetic surfacesover R, then f induces in a functorial way a harmonicmorphism φ : ΓX → ΓX′ of reduction graphs.

Remark

It is necessary to work with metric graphs: f does notnecessarily induce maps on the level of dual graphs, just ongeometric realizations.


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Connections with arithmetic geometry

Theorem (B., Thuillier)

If f : X→ X′ is a morphism of semistable arithmetic surfacesover R, then f induces in a functorial way a harmonicmorphism φ : ΓX → ΓX′ of reduction graphs.

Remark

It is necessary to work with metric graphs: f does notnecessarily induce maps on the level of dual graphs, just ongeometric realizations.


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Induced maps between component groups of Neronmodels

One can use harmonic morphisms of metric graphs to provenew results about maps between component groups of Neronmodels induced by a morphism of algebraic curves over K .(Ongoing joint work with S. Norine and F. Shokrieh.)

Induced maps between component groups of Jacobians ofmodular curves play a key role in Ribet’s proof thatShimura-Taniyama implies Fermat’s Last Theorem.


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Induced maps between component groups of Neronmodels

One can use harmonic morphisms of metric graphs to provenew results about maps between component groups of Neronmodels induced by a morphism of algebraic curves over K .(Ongoing joint work with S. Norine and F. Shokrieh.)

Induced maps between component groups of Jacobians ofmodular curves play a key role in Ribet’s proof thatShimura-Taniyama implies Fermat’s Last Theorem.


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References

References:

M. Baker and R. Rumely, “Potential Theory and Dynamicson the Berkovich Projective Line”, to appear in the AMSSurveys and Monographs series (2009).

A. Thuillier, “Theorie du potentiel sur les courbes engeometrie analytique non archimedienne. Applications a latheorie d’Arakelov”, Ph.D. thesis (2005).


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Analytic spaces

From now on, we assume that K is algebraically closed.

If V is a smooth proper algebraic variety over K , theBerkovich analytic space associated to V is apath-connected, locally contractible compact Hausdorffspace V an containing V (K ) as a dense subspace.

The construction V ; V an is functorial.


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Analytic spaces





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Analytic spaces





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One-dimensional analytic spaces

If X/K is a curve and X is a semistable model for X , then thereduction graph ΓX naturally embeds into X an, and moreoverthere is a canonical deformation retraction τ : X an ΓX.

Theorem

X an is canonically homeomorphic to the inverse limit lim←−XΓX

over all semistable models X for X .

The induced map τ∗ : Div(X )→ Div(ΓX) on divisors coincides(up to linear equivalence) with the specialization mapρ : Div(X )→ Div(ΓX) described earlier.


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Theorem





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Theorem





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Example: Tate elliptic curves

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


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Another picture


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Laplacian on a Berkovich curve

The graph-theoretic Laplacian operators on each ΓX

cohere to give a (measure-valued or distribution-valued)Laplacian operator ∆ on the inverse limit space X an.

A function h : U → R on a domain U ⊆ X an is calledharmonic if its Laplacian is zero on U.

Using the Laplacian on X an, one can develop a theory ofharmonic and subharmonic functions, Green’s functions,etc. closely paralleling the classical theory over C.


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Maps between Berkovich curves

A morphism f : X → Y between curves over K induces amorphism between the corresponding Berkovich analytic spaces.

Theorem (Thuillier)

The induced map f : X an → Y an is a harmonic morphism, i.e.,it pulls back germs of harmonic functions on Y an to germs ofharmonic functions on X an.

Harmonic functions on X an behave much like harmonicfunctions on a Riemann surface: for example, they obey themaximum modulus principle, and there is a Poisson formula forrecovering the values of a harmonic function on a domain fromits boundary values.


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Theorem (Thuillier)




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Theorem (Thuillier)




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Some arithmetic applications of potential theory onBerkovich curves

Fatou-Julia theory on (P1)an for a rational map ϕ ∈ K (T )of degree d ≥ 2. (Rivera-Letelier, Favre–Rivera-Letelier,B.–Rumely)

Non-archimedean Arakelov intersection theory (Thuillier)

Non-archimedean equidistribution of small points(Chambert-Loir, B.–Rumely, B.–Petsche,Favre–Rivera-Letelier, Yuan, Faber, Gubler)

The Bogomolov Conjecture over function fields (Gubler,Zhang, Cinkir)


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Graphs and Arithmetic Geometry - Peoplepeople.math.gatech.edu/~mbaker/pdf/JA2009.pdftheory on Berkovich curves Graphs and Arithmetic Geometry Matthew Baker Georgia Institute of Technology