The Secret Life of Graphs - People | School of …people.math.gatech.edu/~mbaker/pdf/BP2016.pdfModular curves Arakelov theory Dynamical systems Berkovich spaces Tropical geometry The

Modular curvesArakelov theory

Dynamical systemsBerkovich spaces

Tropical geometry

The Secret Life of Graphs

Matt Baker

Georgia Institute of Technology

Benjamin Peirce Centennial ConferenceJune 12, 2016

Matt Baker The Secret Life of Graphs



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Goal of this talk

I will attempt to retrace the rather non-linear development of mymathematical career, which I would characterize as a deterministicsimulation of a random walk.




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Table of contents

1 Modular curves

2 Arakelov theory

3 Dynamical systems

4 Berkovich spaces

5 Tropical geometry




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What to do next?

In my thesis, I studied effective versions of the Manin–Mumfordconjecture for modular curves. Specifically, I computed the intersection ofX0(p)(Q) with J0(p)(Q)tors.

For me the big question was: what to do next?




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What to do next?

In my thesis, I studied effective versions of the Manin–Mumfordconjecture for modular curves. Specifically, I computed the intersection ofX0(p)(Q) with J0(p)(Q)tors.

For me the big question was: what to do next?




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The Bogomolov conjecture

Barry Mazur suggested looking at points of small canonical height onX0(p).

Mazur’s suggestion was motivated by the Bogomolov conjecture, provedby Ullmo and Zhang:

Theorem (Ullmo, 1998)

Let K be a number field and let X/K be a smooth proper algebraic curveof genus at least 2 embedded in its Jacobian J. Let h : J(K )→ R≥0 bethe canonical height associated to the theta divisor. Then there existsε > 0 such that

{P ∈ X (K ) : h(P) < ε}

is finite.




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{P ∈ X (K ) : h(P) < ε}

is finite.




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{P ∈ X (K ) : h(P) < ε}

is finite.




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Equidistribution of small points

The proof of the Bogomolov conjecture is based on an equidistributiontheorem for small points whose proof relies on Arakelov intersectiontheory.

Let A/K be an abelian variety over a number field K , and let {Pn}be a sequence of points in A(K ).

We say {Pn} is small if h(Pn)→ 0 and generic if no subsequence iscontained in a proper subvariety of A.

Let δn denote the discrete probability measure on A(C) supportedequally on the Galois conjugates of Pn.

Theorem (Szpiro-Ullmo-Zhang, 1997)

If the sequence {Pn} is generic and small, then δn converges weakly tothe unit Haar measure on A(C).




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A good excuse to learn some new mathematics

Motivated in part by Mazur’s question (which I never resolved!):

Brian Conrad, Tonghai Yang, and I formed a reading group whichworked through Lang’s book on Arakelov theory.

Tonghai Yang and I read the Gross–Zagier paper together.

I read various papers related to the Bogomolov conjecture, includingShouwu Zhang’s paper ”Admissible Pairing on a Curve”.

I taught a topics course on heights (local decomposition of globalheights, equidistribution theorems, Lehmer’s conjecture, heights overabelian extensions, Call–Silverman dynamical heights, Elkies’ abcimplies Mordell, Mumford’s gap principle,. . . ).




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A smattering of Arakelov theory

In Arakelov intersection theory for algebraic curves over a numberfield, one uses algebraic intersection theory on regular models atfinite places and potential theory on Riemann surfaces at infiniteplaces.

The Archimedean component of the intersection pairing is anormalized Arakelov-Green function g(x , y). It satisfies thedifferential equation

∆yg(x , y) = δx − µ

for some fixed volume form µ on X , together with the normalizationcondition ∫∫

X×Xg(x , y)µ(x)µ(y) = 0.




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∆yg(x , y) = δx − µ


X×Xg(x , y)µ(x)µ(y) = 0.




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∆yg(x , y) = δx − µ


X×Xg(x , y)µ(x)µ(y) = 0.




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∆yg(x , y) = δx − µ


X×Xg(x , y)µ(x)µ(y) = 0.




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Complex, p-adic, and arithmetic dynamics

As I mentioned, thinking about the Bogomolov conjecture got meinterested in various aspects of the theory of canonical heights. Inparticular, I learned during that time about canonical heightsassociated to algebraic dynamical systems.

I started sitting in on Curt McMullen’s seminar on complexdynamics and learned some basic things about Fatou-Julia theoryand the Mandelbrot set.




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Complex, p-adic, and arithmetic dynamics

As I mentioned, thinking about the Bogomolov conjecture got meinterested in various aspects of the theory of canonical heights. Inparticular, I learned during that time about canonical heightsassociated to algebraic dynamical systems.

I started sitting in on Curt McMullen’s seminar on complexdynamics and learned some basic things about Fatou-Julia theoryand the Mandelbrot set.




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Liang-Chung Hsia

Liang-Chung Hsia, a former student of Joe Silverman’s working on p-adicdynamics, was visiting Harvard that same year and we started to worktogether on proving a Szpiro-Ullmo-Zhang type result for polynomialdynamical systems.




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The canonical measure and Brolin’s theorem

One of our motivations was the following classical result of Brolin:

Theorem (Brolin, 1965)

Let φ ∈ C[z ] be a polynomial of degree d ≥ 2. Then there is a canonicalprobability measure µφ on P1(C) with the following property. For anyz0 ∈ C with infinite backward orbit under φ, let δn be the probabilitymeasure supported equally on the points of φ(−n)(z0). Then δn convergesweakly to µφ.

This was later generalized by Lyubich (and independently [FLM]) torational functions.

Brolin’s theorem was one of the very first uses of potential theory incomplex dynamics.




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An equidistribution theorem for polynomial dynamicalsystems

Leth(x) =

∑v∈MK

log max{1, |x |v}

and define

hφ(x) = limn→∞

1

dnh(φn(x)).

Theorem (B.-Hsia)

Let K be a number field, let φ ∈ K [z ] be a polynomial of degree d ≥ 2,and let Pn ∈ P1(K ) be an infinite sequence with hφ(Pn)→ 0. Let δn bethe probability measure on P1(C) supported equally on the Galoisconjugates of Pn. Then δn converges weakly to the canonical measure µφ.




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Leth(x) =

∑v∈MK

log max{1, |x |v}

and define

hφ(x) = limn→∞

1

dnh(φn(x)).

Theorem (B.-Hsia)





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Leth(x) =

∑v∈MK

log max{1, |x |v}

and define

hφ(x) = limn→∞

1

dnh(φn(x)).

Theorem (B.-Hsia)





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Remarks on the equidistribution theorem

The main tool in our proof was the transfinite diameter.

We also proved a complicated non-Archimedean equidistributiontheorem motivated by a result of Bombieri and Zannier.

We were unable to handle the case of rational functions at that time.




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Laura DeMarco’s thesis

Laura was a graduate student of Curt McMullen, and she and I hadbeen students together at Berkeley before Curt moved to Harvard.

In her thesis, she introduced the notion of homogeneous transfinitediameter for a subset of C2, and applied this to the dynamics ofrational functions in one variable over C.

I quickly realized that this would be a very useful tool for extendingmy work with Hsia from polynomial maps to rational functions!




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Dynamical Green functions

Laura’s work motivated the following definition:Let K be a number field, and let v be a place of K . Let φ ∈ K (z) be arational map of degree d ≥ 2.Dehomogenize φ : P1 → P1 as F : A2 → A2, writingF (x1, x2) = (F1(x1, x2),F2(x1, x2)) with each Fi homogeneous of degreed .Define

HF ,v (x) = limd→∞

1

dnlog max{|F (n

1 (x)|v , |F (n)2 (x)|v}.

The v -adic dynamical Arakelov-Green function gφ,v (x , y) onP1(Cv )× P1(Cv ) is

gφ,v ((x1 : x2), (y1 : y2)) = − log |x1y2 − x2y1|v + HF ,v (x) + HF ,v (y)

− 1

d(d − 1)log |Res(F1,F2)|v ,




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1

dnlog max{|F (n

1 (x)|v , |F (n)2 (x)|v}.



− 1





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1

dnlog max{|F (n

1 (x)|v , |F (n)2 (x)|v}.



− 1





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1

dnlog max{|F (n

1 (x)|v , |F (n)2 (x)|v}.



− 1





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1

dnlog max{|F (n

1 (x)|v , |F (n)2 (x)|v}.



− 1





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Two useful observations

For v Archimedean, gφ,v is a normalized Arakelov-Green function forthe canonical measure µφ on P1(C).

By the product formula and the explicit formula for gφ,v (x , y),∑v

gφ,v (x , y) = hφ(x) + hφ(y).




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Two useful observations

For v Archimedean, gφ,v is a normalized Arakelov-Green function forthe canonical measure µφ on P1(C).

By the product formula and the explicit formula for gφ,v (x , y),∑v

gφ,v (x , y) = hφ(x) + hφ(y).




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Enter Berkovich spaces

I first heard about Berkovich spaces in Richard Taylor’s topics courseon the Local Langlands Conjecture for GLn.

I had no idea at the time that Berkovich spaces would change mylife! (though I immediately thought they seemed cool).

When I left Harvard for a tenure-track job at UGA, Robert Rumelysuggested that Berkovich spaces should be the right tool to clean upmy messy theory of non-Archimedean pseudo-equidistribution, andto extend my theorem with Hsia to rational functions.

We spent the next couple of years working out the details. . .

Antoine Chambert-Loir had the same idea independently. . .




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Crash course on Berkovich analytic spaces

Let k be a complete and algebraically closed non-Archimedeanvalued field.

If V is an irreducible algebraic variety over k , the Berkovich analyticspace associated to V is a path-connected, locally compactHausdorff space V an containing V (k) as a dense subspace.

The construction V ; V an is functorial.

For an open affine subscheme U = Spec(A) of V , Uan is the spaceof all bounded multiplicative seminorms on A extending the givenabsolute value on k (endowed with the topology of pointwiseconvergence).




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Example: The Berkovich projective line




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Example: A Berkovich elliptic curve




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Example: A Berkovich K3 surface




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Potential theory and dynamics on the Berkovich projectiveline

Rumely and I showed that potential theory on trees could be used todo non-Archimedean potential theory on (P1)an, with results thatclosely parallel the classical theory of harmonic and subharmonicfunctions on P1(C) (Poisson formula, Harnack’s principle,Poincare-Lelong formula, Frostman’s theorem,. . . )

In particular, the Laplacian on metric graphs can be used to define aLaplacian operator on (P1)an. If one thinks of a metric graph as aresistive electrical network, non-Archimedean potential theory isintimately related to Kirchhoff’s laws.

We defined the non-Archimedean canonical measure µφ,v attachedto a rational map φ over Cv by the formula

∆ygφ,v (x , y) = δx − µφ,v .




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∆ygφ,v (x , y) = δx − µφ,v .




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∆ygφ,v (x , y) = δx − µφ,v .




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Adelic equidistribution for rational functions

Theorem (B.–Rumely, Chambert-Loir, Favre–Rivera-Letelier, 2006)

Let K be a number field, and let φ ∈ K (z) be a rational function ofdegree d ≥ 2. For each place v of K , there is a canonical probabilitymeasure µφ,v supported on P1

v with the following property. Let

Pn ∈ P1(K ) be an infinite sequence with hφ(Pn)→ 0. Let δn be theprobability measure supported equally on the Galois conjugates of Pn.Then δn converges weakly to the canonical measure µφ,v on P1

v for allplaces v of K .

The proof uses the product formula∑v

gφ,v (x , y) = hφ(x) + hφ(y)

together with non-Archimedean versions of various classical results ontransfinite diameter, capacities, Laplacians, and subharmonic functions.




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Let K be a number field, and let φ ∈ K (z) be a rational function ofdegree d ≥ 2. For each place v of K, there is a canonical probabilitymeasure µφ,v supported on P1





gφ,v (x , y) = hφ(x) + hφ(y)





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gφ,v (x , y) = hφ(x) + hφ(y)





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gφ,v (x , y) = hφ(x) + hφ(y)





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gφ,v (x , y) = hφ(x) + hφ(y)





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A sample application

The adelic equidistribution theorem and potential theory on the Berkovichprojective line have found many applications in recent years. For example:

Theorem (B.-DeMarco, 2011)

Let a, b ∈ C with a 6= ±b. Then the set of c ∈ C such that both a and bhave finite orbit under z2 + c is finite.

When a, b are algebraic, we apply the equidistribution theorem in thenumber field setting. When they are transcendental, we apply it to thefunction field Q(a, b). In particular, we really need Berkovich spaces tohandle the transcendental case!




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“Dynamical Andre–Oort” theorems

A rational map is post-critically finite if the forward orbit of every criticalpoint is finite.

(B.–DeMarco, 2013) For each complex number λ, let Per1(λ) be thecurve (inside the moduli space of cubic polynomials up toconjugacy) parametrizing polynomials with a fixed point ofmultiplier λ. Then Per1(λ) contains an infinite number ofpost-critically finite polynomials if and only if λ = 0.

(Ghioca–Krieger–Nguyen–Ye, 2015) Let C be an irreducible planecurve over C. If there are infinitely many points (a, b) ∈ C such thatboth z2 + a and z2 + b are post-critically finite, then C is either ahorizontal, vertical, or diagonal line.




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Intersection theory as non-Archimedean potential theory

OK, so what about curves of higher genus?

Amaury Thuillier, a student of Chambert-Loir, developed(independently and at the same time) non-Archimedean potentialtheory for arbitrary Berkovich curves.

He applies this theory to give a symmetrical version of Arakelovintersection theory for curves in which one is doing potential theoryat all places.

Slogan: Intersection theory is non-Archimedean potential theory.

Remark: The first person to develop this slogan (albeit withoutBerkovich spaces) was Ernst Kani.




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A simple but illustrative example

As a concrete example, the non-Archimedean Arakelov-Green function on(P1)an × (P1)an with respect to a point mass at the Gauss point is

gv (x , y) = − log |x1y2−x2y1|v +log max(|x1|v , |x2|v )+log max(|y1|v , |y2|v ).

This is precisely the intersection product iv (x , y) on the standard modelof P1 over the valuation ring of Cv !




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A simple but illustrative example

As a concrete example, the non-Archimedean Arakelov-Green function on(P1)an × (P1)an with respect to a point mass at the Gauss point is

gv (x , y) = − log |x1y2−x2y1|v +log max(|x1|v , |x2|v )+log max(|y1|v , |y2|v ).

This is precisely the intersection product iv (x , y) on the standard modelof P1 over the valuation ring of Cv !




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The non-Archimedean Poincare-Lelong formula

Theorem (Thuillier)

Let f be a rational function on a curve X over a completenon-Archimedean field k. Then

∆ log |f | = δdiv(f ).




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Reinterpretation in the language of tropical geometry

For a finite metric subgraph Γ of X an containing the skeleton, letTrop(f ) denote the restriction of log |f | to Γ. This is apiecewise-linear function with integer slopes, i.e., a “tropical rationalfunction” on Γ.

For D ∈ Div(X ), let trop(D) denote the retraction of D to Γ. Thisis a “divisor” on Γ.

If F is a tropical rational function on Γ, define the associatedprincipal divisor on Γ to be the Laplacian of F , i.e.,

div(F ) :=∑p∈Γ

∆p(F )(p),

where ∆p(F ) is the sum of the incoming slopes of F at p.

Thuillier’s Poincare-Lelong formula is equivalent to the statementthat for every such Γ, we have

div(Trop(f )) = trop(div(f )).




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div(F ) :=∑p∈Γ

∆p(F )(p),







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div(F ) :=∑p∈Γ

∆p(F )(p),







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div(F ) :=∑p∈Γ

∆p(F )(p),







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Example: retraction of divisors




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The divisor of a tropical rational function




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Zhang’s canonical divisor

In Shouwu Zhang’s Inventiones paper “Admissible pairing on a curve”(an early attempt to prove the Bogomolov conjecture), he encounters(without naming it this) the canonical divisor on a graph:

KΓ =∑p∈Γ

(valence(p)− 2)(v).

The degree of KΓ is 2g − 2, where g is the genus of Γ, i.e. the dimensionof H1(Γ,R).




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KΓ =∑p∈Γ






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KΓ =∑p∈Γ






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The secret life of graphs

We have a notion of principal divisors on a graph, hence of linearequivalence. We also have a canonical divisor and a notion of genus.

Might there be a Riemann-Roch theorem for graphs and/or metricgraphs?




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The secret life of graphs

We have a notion of principal divisors on a graph, hence of linearequivalence. We also have a canonical divisor and a notion of genus.

Might there be a Riemann-Roch theorem for graphs and/or metricgraphs?




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Riemann-Roch for graphs

Yes.

Serguei Norine and I proved a Riemann-Roch theorem for finite graphs,which was quickly extended by Gathmann–Kerber and Mikhalkin–Zharkovto a Riemann-Roch theorem for metric graphs / tropical curves.




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Riemann-Roch for graphs

Yes.

Serguei Norine and I proved a Riemann-Roch theorem for finite graphs,which was quickly extended by Gathmann–Kerber and Mikhalkin–Zharkovto a Riemann-Roch theorem for metric graphs / tropical curves.




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The tropical Riemann-Roch theorem

Let Γ be a metric graph.For D ∈ Div(Γ), define r(D) to be the largest integer k such that D − Eis equivalent to an effective divisor for all effective divisors E of degree k.

Theorem (B.-Norine, Gathmann–Kerber, Mikhalkin–Zharkov, 2007)

For every divisor D on Γ,

r(D)− r(KΓ − D) = deg(D) + 1− g .




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r(D)− r(KΓ − D) = deg(D) + 1− g .




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r(D)− r(KΓ − D) = deg(D) + 1− g .




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Semicontinuity

Shortly after establishing the Riemann-Roch theorem, I noticed that thecombinatorial rank r(D) has the following semicontinuity property:

Lemma (B.)

Let X be an algebraic curve over a complete non-Archimedean field k.For every finite subgraph Γ of X an,

rΓ(trop(D)) ≥ rX (D).

I began to wonder whether this result might have some applications toclassical algebraic geometry. . .




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Semicontinuity


Lemma (B.)







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Semicontinuity


Lemma (B.)







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Brill-Noether theory

At Harvard, I sat in on Joe Harris’s Second Course in Algebraic Geometryand learned about the proof of the following result known as theBrill-Noether theorem and due originally to Griffiths and Harris:

Theorem (Griffiths-Harris, Eisenbud-Harris, Lazarsfeld,. . . )

Given nonnegative integers g , r , d, define ρ := g − (r + 1)(g − d + r).If X is a nonsingular projective curve of genus g, define W r

d (X ) to be thevariety parametrizing line bundles L of degree d on X with h0(L) ≥ r + 1.Then for a general nonsingular projective curve X of genus g, W r

d (X ) hasdimension ρ if ρ ≥ 0, and is empty if ρ < 0.

The proof, which Joe Harris explained in his class, uses a brilliant ideathat goes back to Castelnuovo:Since dimW r

d (X ) is upper semicontinuous on Mg , to show thestatement for a general smooth curve of genus g it suffices to prove it fora single stable curve of genus g .




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A rational backbone with g elliptic tails




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A tropical approach to degenerating linear series

In 2008 I conjectured a tropical analogue of the Brill-Noethertheorem. The conjecture was motivated by extensive computationalevidence from my summer REU student Adam Tart.

I also proved that this purely combinatorial conjecture would implythe classical Brill-Noether Theorem.

My conjecture was proved by Cools, Draisma, Payne, and Robeva ina very explicit way:

Theorem (CDPR, 2012)

If ρ := g − (r + 1)(g − d + r) < 0, then for the metric graph Γ consistingof a chain of g loops with general edge lengths, there is no divisor ofdegree d and rank at least r on Γ.




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A chain of g loops




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Comparison with limit linear series

Eisenbud and Harris settled a number of longstanding open problemswith their theory of limit linear series.However, their theory only applies to a rather restricted class ofsingular curves, namely those of compact type (i.e., nodal curveswhose dual graph is a tree).

Our approach is more or less orthogonal to the Eisenbud-Harristheory, in that it works best in families which are maximallydegenerate, meaning that the dual graph of the special fiber hasgenus equal to the genus of the generic fiber.




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Other applications of the combinatorics of chains of loops

In just the last year, generic chains of loops have been used to prove thefollowing:

(Jensen–Payne, 2015) [Maximal Rank Conjecture for Quadrics] If Xis a general curve of genus g and L is a general line bundle of degreed and rank r on X , then the natural mapSym2H0(X ,L)→ H0(X ,L⊗2) has maximal rank, i.e., it is eitherinjective or surjective.

(Pflueger, 2016) [Brill–Noether theory for k-gonal curves] For r > 0and g − d + r > 1, a general smooth projective k-gonal curve X ofgenus g has dimW r

d (X ) = ρ if and only if g − k ≤ d − 2r .

(Jensen–Len, 2016) [Surjectivity of Gaussian maps] Fix nonnegativeintegers s and ρ. Then for r sufficiently large, if X is a general curveof genus g := ρ+ rs and L is a general degree d line bundle on Xwith h0(X ,L) = r + 1 and h1(X ,L) = s + 1, the Gaussian–Wahlmap ∧2H0(X ,L)→ H0(KX ⊗L⊗2), f ∧ g 7→ fdg − gdf is surjective.




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Coming full-circle: Applications to number theory

Linear series on metric graphs also play a key role in the proofs of thefollowing two recent breakthroughs in number theory:

Theorem (Katz–Zureick-Brown, 2013)

Let X be a curve of genus g over Q and suppose that the Mordell–Weilrank r of J(Q) is less than g. Then for every prime p > 2r + 2, we have

#X (Q) ≤ #Xsm(Fp) + 2r ,

where X denotes the minimal proper regular model of X over Zp.

Theorem (Katz–Rabinoff–Zureick-Brown, 2015)

There is an explicit bound M(g) = 76g 2 − 82g + 22 such that if X/Q isa curve of genus g with Mordell-Weil rank at most g − 3, then#X (Q) ≤ M(g).




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#X (Q) ≤ #Xsm(Fp) + 2r ,







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#X (Q) ≤ #Xsm(Fp) + 2r ,





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The Secret Life of Graphs - People | School of …people.math.gatech.edu/~mbaker/pdf/BP2016.pdfModular curves Arakelov theory Dynamical systems Berkovich spaces Tropical geometry The