unlikely intersections in complex...

Motivation from arithmetic geometryComplex dynamics

Results toward the general conjecture

Unlikely intersections in complex dynamics

Matt Baker(Joint work with L. DeMarco)

Second ERC Research Period on Diophantine GeometryCetraro, ItalyJuly 24, 2014

Baker-DeMarco Unlikely intersections in complex dynamics

Table of contents

1 Motivation from arithmetic geometry

2 Complex dynamics

3 Results toward the general conjecture

The Masser-Zannier theorem

Theorem (Masser–Zannier)

Let Pλ,Qλ be linearly independent points in Eλ(C(λ)), whereEλ : y2 = x(x − 1)(x − λ) is the Legendre family of elliptic curves.Then

{λ ∈ C | Pλ and Qλ are both torsion }

is finite.

Zannier’s question

Motivated by his joint work with Masser, and by the analogybetween torsion points on elliptic curves and preperiodic points fordynamical systems, Zannier posed the following question in 2008:

Question: What can be said about the set of c ∈ C such thatboth 0 and 1 are preperiodic for the map z 7→ z2 + c?

c = 0: 0 7→ 0, 1 7→ 1

c = −1: 1 7→ 0 7→ −1 7→ 0

c = −2: 0 7→ −2 7→ 2 7→ 2, 1 7→ −1 7→ −1.

Open problem: Are there any other such complex parameters c?

c = 0: 0 7→ 0, 1 7→ 1

c = −1: 1 7→ 0 7→ −1 7→ 0

c = −2: 0 7→ −2 7→ 2 7→ 2, 1 7→ −1 7→ −1.

c = 0: 0 7→ 0, 1 7→ 1

c = −1: 1 7→ 0 7→ −1 7→ 0

c = −2: 0 7→ −2 7→ 2 7→ 2, 1 7→ −1 7→ −1.

c = 0: 0 7→ 0, 1 7→ 1

c = −1: 1 7→ 0 7→ −1 7→ 0

c = −2: 0 7→ −2 7→ 2 7→ 2, 1 7→ −1 7→ −1.

Main theorem of [BDM1]

Theorem (B.-DeMarco)

Fix a 6= b in C with a 6= ±b. Then

{c ∈ C | a and b are both preperiodic for z2 + c}

is finite.

Postcritically finite maps

A rational map f ∈ C(t) is called postcritically finite (PCF) if thecritical points of f are all preperiodic.

PCF maps are very important in complex dynamics, roughlyspeaking because of the

Slogan: The dynamics of a rational map f under iteration isgoverned by what happens to the critical points.

Here is a plot of some values of c ∈ C for which z2 + c is PCF:

PCF maps as special points

Let Ratd ⊂ P2d+1 be the space of rational maps of degree d , andlet Md = Ratd/PSL2(C). Let Pd be the subset of Md consistingof (conjugacy classes of) polynomial maps.

Fact: The PCF maps form a countable, Zariski dense subset ofPd . They form a Zariski dense subset of Ratd which is countableoutside of the Lattes locus Latd (which is either empty, if d is nota square, or an algebraic curve).

We view PCF maps in Md as “special points” analogous to CMpoints on Shimura varieties.

Justifying the analogy

There are many reasons to subscribe to the idea that PCF mapsare analogous to CM points; for example:

The set of PCF points in Ratd\Latd has bounded height andthus there are only finitely many non-Lattes PCF maps definedover number fields of bounded degree (analogue of Gauss’sclass number problem). [Benedetto-Ingram-Jones-Levy]

The arboreal Galois representation attached to a PCF maptends to have much smaller image than in the non-PCF case.[Jones et. al.]

Dynamical Andre-Oort

By analogy with the Andre-Oort conjecture, it is natural to askwhich algebraic subvarieties of Md contain a dense set of “special”(PCF) points. Before formulating a conjectural answer, let’s lookat some examples.

Example 1: Consider the parametrized curve in P3 given by

ft(z) = z3 − 3t2z + i .

The (finite) critical points are c1(t) = t and c2(t) = −t, which are“dynamically independent” in a sense to be made precise shortly.It follows from the main theorem of [BDM2] that there are only afinite number of PCF maps in this family of cubic polynomials.

Dynamical Andre-Oort

By analogy with the Andre-Oort conjecture, it is natural to askwhich algebraic subvarieties of Md contain a dense set of “special”(PCF) points. Before formulating a conjectural answer, let’s lookat some examples.

ft(z) = z3 − 3t2z + i .

The (finite) critical points are c1(t) = t and c2(t) = −t, which are“dynamically independent” in a sense to be made precise shortly.It follows from the main theorem of [BDM2] that there are only afinite number of PCF maps in this family of cubic polynomials.

The Lattes curves

Example 2: For the curve in M4 defined by the Lattes family

fλ(z) =(z2 − λ)2

4z(z − 1)(z − λ),

every point is PCF. By a (difficult) theorem of Thurston, Lattescurves are the only examples of positive-dimensional subvarietiesof Md consisting entirely of PCF points (which one might call veryspecial).

Example of a special subvariety

ft(z) = z2(z3 − t3).

One can show that this curve is special, i.e., the above familycontains infinitely many PCF maps. To see this, let β = 3

√2/5 and

let ω be a primitive cube root of unity. Then the critical points arecj(t) = ωjβt for j = 1, 2, 3 and c4(t) = 0.The critical point c4 is persistently periodic, since it’s fixed inevery fiber. And for i , j ∈ {1, 2, 3} and t0 ∈ C, we have

ci (t0) is preperiodic if and only if cj(t0) is preperiodic.

Indeed, these three points satisfy the critical orbit relations

f(2)t (cj(t)) = ω · f (2)(cj−1(t))

and in addition we have f(2)t (ωz) = ωf

(2)t (z).

ft(z) = z2(z3 − t3).

√2/5 and

f(2)t (cj(t)) = ω · f (2)(cj−1(t))

(2)t (z).

ft(z) = z2(z3 − t3).

√2/5 and

f(2)t (cj(t)) = ω · f (2)(cj−1(t))

(2)t (z).

Dynamically dependent orbits

Let V be a irreducible complex algebraic variety of dimension atleast 1, and let

f = {ft : t ∈ V }

be a family of rational maps of degree at least 2 over C.

The dimension of the family is the dimension of the image of V inMd = Ratd/PSL2(C) under the morphism V → Ratd defined byf .

Let a1(t), . . . , am(t) ∈ P1(C(V )) be marked points.We say that ai1 , . . . , ain have dynamically dependent orbits if thepoint (ai1 , . . . , ain) ∈ An(C(V )) is contained in an algebraichypersurface which is invariant under the action of (f , f , . . . , f ).

f = {ft : t ∈ V }

The general conjecture of [BDM2]

Conjecture (B.-DeMarco): Let f = {ft : t ∈ V } be anN-dimensional family of rational maps of degree at least 2 over C,and let a1(t), . . . , am(t) ∈ P1(C(V )) be marked points. Then theset of t ∈ V such that a1(t), . . . , am(t) are simultaneouslypreperiodic is Zariski dense in V if and only if at most N of themarked points a1, . . . , am have dynamically independent orbits.

We call the special case where a1(t), . . . , am(t) are the criticalpoints of f the dynamical Andre-Oort conjecture, since it givesa conjectural characterization of the special subvarieties.

The general conjecture of [BDM2]

Conjecture (B.-DeMarco): Let f = {ft : t ∈ V } be anN-dimensional family of rational maps of degree at least 2 over C,and let a1(t), . . . , am(t) ∈ P1(C(V )) be marked points. Then theset of t ∈ V such that a1(t), . . . , am(t) are simultaneouslypreperiodic is Zariski dense in V if and only if at most N of themarked points a1, . . . , am have dynamically independent orbits.

We call the special case where a1(t), . . . , am(t) are the criticalpoints of f the dynamical Andre-Oort conjecture, since it givesa conjectural characterization of the special subvarieties.

A concrete special case

As a concrete special case, one has:

Conjecture: Let C ⊂ A2C be an algebraic curve containing

infinitely many points (a, b) such that both z2 + a and z2 + b arePCF. Then C is either horizontal, vertical, or diagonal.

A concrete special case

As a concrete special case, one has:

Conjecture: Let C ⊂ A2C be an algebraic curve containing

infinitely many points (a, b) such that both z2 + a and z2 + b arePCF. Then C is either horizontal, vertical, or diagonal.

The one-parameter polynomial case

We proved the following stronger version of the main conjecturefor 1-parameter families of polynomial maps assuming that theai (t) are also polynomials:

Let f = {ft} be a one-parameter family of complex polynomials ofdegree at least 2, and let a1(t), . . . , am(t) ∈ C[t] be polynomials.Then the set of t ∈ V such that a1(t), . . . , am(t) aresimultaneously preperiodic is infinite if and only if every pair ofcritical points which are not persistently periodic satisfies a criticalorbit relation of the form

f (m)(ci ) = h ◦ f (n)(cj)

where h is a polynomial that commutes with some iterate of f .

The one-parameter polynomial case

We proved the following stronger version of the main conjecturefor 1-parameter families of polynomial maps assuming that theai (t) are also polynomials:

Let f = {ft} be a one-parameter family of complex polynomials ofdegree at least 2, and let a1(t), . . . , am(t) ∈ C[t] be polynomials.Then the set of t ∈ V such that a1(t), . . . , am(t) aresimultaneously preperiodic is infinite if and only if every pair ofcritical points which are not persistently periodic satisfies a criticalorbit relation of the form

f (m)(ci ) = h ◦ f (n)(cj)

where h is a polynomial that commutes with some iterate of f .

Ingredients in the proof

The proof involves several different kinds of ingredients, notably:

An adelic equidistribution theorem for Galois orbits of pointsof small dynamical height

Potential theory on both P1(C) and its non-ArchimedeanBerkovich space analogue

Complex analysis, especially univalent function theory and thetheory of normal families

The recent work of Medvedev-Scanlon, who use the classicalmethods of Ritt to classify the invariant subvarieties for acertain class of polynomial dynamical systems.

The Milnor curves

For some non-polynomial examples, consider the curve Per1(λ) inP3 (resp. M2) consisting of (conjugacy classes of) rational mapswith a marked fixed point of multiplier λ.

Theorem (B.-DeMarco for P3, DeMarco-Wan-Ye for M2)

The curve Per1(λ) contains infinitely many PCF points if and onlyif λ = 0.

Note that even the case of P3 is not a consequence of the previoustheorem, since the critical points in this family are notparametrized by polynomials.

The Milnor curves

The Masser-Zannier theorem revisited

Recently, DeMarco, Wan, and Ye have given a new proof of theMasser-Zannier theorem using the methods of proof from [BDM1]and [BDM2], as applied to the degree-four Lattes family. Themethod of proof in fact yields a Bogomolov-type strengthening ofthe Masser-Zannier theorem which applies to points of small heightand not just torsion points:

Theorem (DeMarco-Wan-Ye)

For a 6= b in Q\{0, 1}, there exists ε > 0 such that the set ofλ ∈ Q\{0, 1} for which the points Pλ = (a,

√a(a− 1)(a− λ)) and

Qλ = (b,√

b(b − 1)(b − λ)) both have Neron-Tate canonicalheight less than ε is finite.

The Masser-Zannier theorem revisited

Recently, DeMarco, Wan, and Ye have given a new proof of theMasser-Zannier theorem using the methods of proof from [BDM1]and [BDM2], as applied to the degree-four Lattes family. Themethod of proof in fact yields a Bogomolov-type strengthening ofthe Masser-Zannier theorem which applies to points of small heightand not just torsion points:

Theorem (DeMarco-Wan-Ye)

For a 6= b in Q\{0, 1}, there exists ε > 0 such that the set ofλ ∈ Q\{0, 1} for which the points Pλ = (a,

√a(a− 1)(a− λ)) and

Qλ = (b,√

b(b − 1)(b − λ)) both have Neron-Tate canonicalheight less than ε is finite.

A theorem of Ghioca-Krieger-Nguyen

Theorem (Ghioca-Krieger-Nguyen)

Let f (z) ∈ C[z ] be a non-constant polynomial. There are infinitelymany t ∈ C such that both zd + t and zd + f (t) are PCF if andonly if f (z) = ζz for some (d − 1)st root of unity ζ.

For d = 2, this is the special case y = f (x) of the conjecture thanan algebraic curve in A2 containing infinitely many points (a, b)such that both z2 + a and z2 + b are PCF if and only if C is eitherhorizontal, vertical, or diagonal.

A theorem of Ghioca-Krieger-Nguyen

Theorem (Ghioca-Krieger-Nguyen)

Let f (z) ∈ C[z ] be a non-constant polynomial. There are infinitelymany t ∈ C such that both zd + t and zd + f (t) are PCF if andonly if f (z) = ζz for some (d − 1)st root of unity ζ.

For d = 2, this is the special case y = f (x) of the conjecture thanan algebraic curve in A2 containing infinitely many points (a, b)such that both z2 + a and z2 + b are PCF if and only if C is eitherhorizontal, vertical, or diagonal.

Thank you!

And a special thanks to the ERC for funding the conferencethrough ERC Diophantine Problems GA n. 267273.

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