heights and dynamics over arbitrary fields

Heights and dynamics over arbitrary fields

Alex Carney, University of Rochester

October 15th, 2020

Rigidity of Dynamical systems

Theorem (Yuan-Zhang, C.)

Let f , g : X → X be two polarizable algebraic dynamical systems definedover any field. The following are equivalent:

1 Prep(f ) ⊂ Prep(g).

2 Prep(f ) ∩ Prep(g) is Zariski dense in X .

3 Prep(f ) = Prep(g).

Remarks

Proven by Yuan-Zhang in characteristic zero, by C. for positivecharacteristic.

Prep(f ) is always dense in X (Fakhruddin).

Alex Carney, University of Rochester Heights and dynamics over arbitrary fields 1/21

Proof Outline

1 Define heights over finitely generated fields.

2 Construct canonical heights hf and hg , and equate canonical heightzero points and preperiodic points via Northcott.

3 Relate hf and hg using arithmetic intersection theory (Hodge IndexTheorem) over finitely generated fields.


1. Base k and base Z

Let K be a finitely generated extension of a field k .

We consider two cases

1 Base Zk = Qd = tr.deg(K/k)All models are schemes over SpecZ.

2 Base k

k can be any field (incl. Q)d + 1 = tr.deg(K/k)All models are schemes over Spec k.

Example

K/k = Q(t)/Q can be base Z (d=1) or base k (d=0).


Existing Theory, R-valued heights

Base Z: Moriwaki heights

Base k: Geometric heights (Lang, Serre, others)

But these require a polarization:

1 A model B for K , over Spec k or SpecZ.

2 (Hermitian, in base Z) line bundles H1, . . . ,Hd on B.

This determines a set of absolute values satisfying the product formulavia intersections on B.

Problem

Polarizations are only unique when d = 0.


Models over k and Z

Fix a projective K -variety X .

Open Models

An open arithmetic model for X is a quasi-projective integral schemeover the base (k or Z), whose base change to K is X .

· · · ↪−→W2 ↪−→W1

They form an inverse system via inclusion.

Projective models

A projective model X for an open model W is an open embeddingW ↪→ X into a projective arithmetic model, such that X\W is thesupport of an effective Cartier divisor.

· · · −→ X2 −→ X1

They form an inverse system via dominant morphisms.


Limits of models

Limits of projective models

For an open model U for X , define

Pic(U) := lim−→X

Pic(X )Q

over the inverse system of projective models for UPic(U)cont = completion w/r/t partial ordering by effectivity on X\U .

Limits of open models

Pic(X ) := lim−−−→U→V

Pic(U)cont

over the inverse system of flat maps U → V of open arithmetic modelsfor X → SpecK .

Special case: Pic(K ) := Pic(SpecK ).


A relative intersection pairing

Let X → Y be a morphism of projective K -varieties of pure dimension n.

The vector-valued product

Pic(X )n+1 −→ Pic(Y )

(L0, . . . , Ln) 7→ L0 · · · Ln

Defined via the Deligne pairing.

Important case: Y = SpecK .

The R-valued product

Pic(K )d+1 −→ R

(H0, . . . ,Hd) 7→ H0 · · ·Hd

Pic(K )-valued product + polarization = R-valued product.


Vector valued heights

The Pic(K ) height

Fix L ∈ Pic(X ) and Z a closed subvariety of X . Define

hL(Z ) :=

(L|Z)dim Z+1

(d + 1) (L|Z )dim Z∈ Pic(K )

When x is a K -point, hL(x) = π∗(L|x)/[K (x) : K ].

Relation to existing heights

d = 0: hL is a Q-line bundle on the unique model for K , its degreeis the usual (geometric) height.

d > 0: Fix a polarization H1, . . . ,Hd . The R-producthL(Z ) · H1 · · ·Hd is the Moriwaki/geometric height.


Proof Outline





Canonical heights

Let f : X → X be a dynamical system with a polarizationf ∗L = qL = L⊗q for L ample, q > 1.

Definition

A canonical height hf for f is a height such that

hf (f (x)) = qhf (x).

Tate’s limiting argument

1 Extend L to some L ∈ Pic(X ).

2 Set Lf := limn→∞

1

qn(f n)∗L, show this converges.

3 hf := hLf.


Isotriviality and base k

Suppose we are in the base k setting.

Theorem (Baker)

Suppose f : P1K → P1

K is a rational function. The set{x ∈ P1(K ) : hf (x) = 0} is finite ⇔ k = Fq or f is not isotrivial.

Definition

Let A be an abelian K -variety. Chow’s K/k-trace is an abelian k-varietyTrK/k(A) with a map τ : TrK/k(A)K → A, universal for all maps from anabelian k-variety to A.

Theorem (Lang-Neron)

Let hNT = h[2] be the Neron-Tate height on A. Then hNT(x) = 0 ⇔ x istorsion or x ∈ TrK/k(A)(k).


Constructible isotriviality

Definition

A constructible map is a composition of rational maps and inversesof totally inseparable rational maps.

A dynamical system f : X → X is constructibly isotrivial providedthere exists a rational map g : Y → Y defined over k such that(X , f ) is constructibly isomorphic to (YK , gK ).

(X , f ) is totally non-isotrivial provided neither (X , f ) itself nor anypositive dimensional periodic closed subvariety of (X , f ) isconstructibly isotrivial.

Example

E = supersingular elliptic curve over k , and Γ = order p subgroupscheme of E [p]× E [p]. Then (EK × EK )/Γ is constructibly isotrivial, butnot isotrivial. τ has a finite kernel.


The Northcott Principle

Definition

For M ∈ Pic(K ), write M ≡ 0 to mean M is numerically trivial.

Theorem (Northcott, base Z, Yuan-Zhang 2013)

Let X be projective over K , L ∈ Pic(X ) ample, and α ∈ Pic(K ). The set

{x ∈ X (K ) : hL(x) ≤ α}

is finite.

Theorem (Northcott, base k , Gauthier-Vigny 2019, C. 2020)

Let f : X → X be polarizable, and suppose k is finite or (X , f ) is totallynon-isotrivial. Then

{x ∈ X (K ) : hf (x) ≡ 0}

is finite.


Proof

Base Z

1 Fix a polarization ⇒ Moriwaki height.

2 Northcott proven by Moriwaki for this R-height.

3 Height approximated uniformly as the polarization varies.

Base k

1 Fix a polarization ⇒ Geometric height.

2 k = Fq: Known for geometric heights.

3 Chatzidakis-Hrushovski (Model theory):limited set dense ⇒ constructibly isotrivial.

4 Induct on dimension.

5 Vary the polarization.


Relative heights in function fields

Let K/k1/k be a tower of finitely generated extensions, and f : X → X apolarizable dynamical system over K .

Two relative heights on K

Height relative to K/k Height relative to K/k1

Values in Pic(K ) = Pic(K/k) Values in Pic(K ) = Pic(K/k1)

Northcott ⇔ non-isotriviality /k Northcott ⇔ non-isotriviality /k1

K/k height intersected with pullbacks from Pic(k1/k) ⇒ K/k1 height.

Example

Let K = C(t).

If (X , f ) totally non-isotrivial over C, use C(t)/C height.

If not, find f.g. subfield of definition K , use K/Q height.


Outline





The Hodge Index Theorem

Theorem (Yuan-Zhang 2013, C. 2020)

Let π : X → SpecK be a projective variety of dimension n. Let

M, L2, . . . , Ln ∈ Pic(X ) with L2, . . . , Ln ample and M · L2 · · · Ln = 0.Then

1 (Inequality)

M2 · L2 · · · Ln ≤ 0.

2 (Equality) Further,

M2 · L2 · · · Ln ≡ 0

if and only if

M ∈ π∗Pic(K )int + TrK/k Pic0(X ).


Proof Sketch

Case d = 0 (I.e. number fields/transcendence degree 1 function fields)proven by Yuan-Zhang/C.

Lemma

Let X be a curve, and M ∈ Pic0(X ). Then M has a unique extension to

M0 ∈ Pic(X ) with M0 = M, for which

M2

0 = −2hNT(M),

where hNT is the vector-valued Neron-Tate canonical height on JacX .

Follows Faltings’ Calculus on arithmetic surfaces/Hriljac’s thesis.

Address M −M0 by studying intersections of fibers.

Induct on the dimension of X .


Proving rigidity of dynamical systems

Preperiodic points always have height zero (doesn’t require Northcott).

The Lefschetz Principle

X , f , g and their polarizations are all defined over some field K which isfinitely generated over either Q or Fq.

Northcott always holds.

Thus canonical height zero points are all preperiodic (entire forwardorbit has height zero).

Proof sketch

1 Construct canonical heights hLfand hMg

.

2 Show Prep(f ) ∩ Prep(g) dense in X ⇒ hLf +Mg(X ) ≡ 0.

3 Apply Hodge Index Theorem to Lf −Mg .


Future directions

Applications to Unlikely Intersections, higher dimensional families.

Better understand constructible isotriviality

Develop vector space structure of Pic(K ) further.


Thank You!


heights and dynamics over arbitrary fields

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