Arithmetic and Dynamical Degrees on Abelian Varieties

0.19 MB·English

by  Joseph H. Silverman

ARITHMETIC AND DYNAMICAL DEGREES ON
ABELIAN VARIETIES
5
1 JOSEPH H. SILVERMAN
0
2
Abstract. Let φ : X 99K X be a dominant rational map of a
n
a smooth variety and let x ∈ X, all defined over Q¯. The dynamical
J degree δ(φ) measures the geometric complexity of the iterates of

7 φ, andthe arithmeticdegreeα(φ,x) measuresthe arithmeticcom-
1 plexityoftheforwardφ-orbitofx. Itisknownthatα(φ,x)≤δ(φ),

and it is conjectured that if the φ-orbit of x is Zariskidense in X,
]
T then α(φ,x) = δ(φ), i.e., arithmetic complexity equals geometric
N complexity. In this note we prove this conjecture in the case that
. X is an abelian variety, extending earlier work in which the con-
h
jecture was proven for isogenies.
t
a
m
[

1. Introduction
1
v Let K = Q¯, or more generally let K be an algebraically closed field
5
0 on which one has a good theory of height functions as described, for
2
example, in [7, Part B] or [10, Chapters 2–5]. Let X be a smooth
4
0 projective variety of dimension d, let ϕ : X 99K X be a dominant
. rational map, and let H be an ample divisor on X, all defined over K.
1
0 Further let h : X(K) → [1,∞) be a Weil height function associated
X,H
5
to H. We write ϕn for the n’th iterate of ϕ.
1
:
v Definition. The dynanmical degree of ϕ is the quantity
i
X 1/n
δ(ϕ) = lim (ϕn)∗H ·Hd−1 .
r
a n→∞
(cid:16) (cid:17)
Definition. Let x ∈ X be a po(cid:0)int whos(cid:1)e forward ϕ-orbit
O (x) = {ϕn(x) : n ≥ 0}
ϕ
iswell-defined. Thearithmetic degree of x(relative to ϕ)isthequantity
α(ϕ,x) = lim h fn(x) 1/n.
X,H
n→∞
(cid:0) (cid:1)
Date: January 20, 2015.
2010 Mathematics Subject Classification. Primary: 37P30; Secondary: 11G10,
11G50, 37P15.
Key words and phrases. dynamical degree, arithmetic degree, abelian variety.
The author’s research supported by Simons Collaboration Grant #241309.
1
2 JOSEPH H.SILVERMAN
It is known that the limit defining δ(ϕ) exists and is a birational
invariant; see [5, Proposition 1.2(iii)] and [9, Corollary 16]. Bellon and
Viallet [1] conjectured that δ(ϕ) is an algebraic integer. Kawaguchi
and the author [9] proved that α(ϕ,x) ≤ δ(ϕ), and they made the
following conjectures about the arithmetic degree and its relation to
the dynamical degree.
Conjecture 1. (Kawaguchi–Silverman [9, 12])
(a) The limit defining α(ϕ,x) exists.
(b) α(ϕ,x) is an algebraic integer.
(c) α(ϕ,x) : x ∈ X such that O (x) exists is a finite set.
ϕ
(d) If the orbit O (x) is Zariski dense in X, then α(ϕ,x) = δ(ϕ).
ϕ
(cid:8) (cid:9)
Conjecture 1(a,b,c) is known when ϕ is a morphism [8]. But (d),
which is the deepest part of the conjecture, has been proven in only
a few situations, such as group endomorphisms of the torus Gd [12,
m
Theorem 4], group endomorphisms of abelian varieties [8, Theorem 4],
and in a few other special cases; see [9, Section 8]. The goal of this
note is to extend the result in [8] to arbitrary dominant self-maps of
abelian varieties.
Theorem 2. Let A/K be an abelian variety, let ϕ : A → A be a
dominant rational map, and let P ∈ A be a point whose orbit O (P)
ϕ
is Zariski dense in A. Then
α(ϕ,P) = δ(ϕ).
Remark 3. Every map as in Theorem 2 is a composition of a trans-
lation and an isogeny (see Remark 4), so we can write ϕ : A → A
as
ϕ(P) = f(P)+Q
withf : A → AanisogenyandQ ∈ A. Asalreadynoted, ifQ = 0, then
Theorem 2 was proven in [8], and it may seem that the introduction of
translation by a non-zero Q introduces only a minor complication to
the problem. However, the potential interaction between the points P
and Q may lead to signficiant changes in both the orbit of P and the
value of the arithmetic degree α(ϕ,P) .
To illustrate the extent to which taking Q 6= 0 is significant, consider
the following related question. For which ϕ are there any points P ∈ A
whose ϕ-orbit O (P) is Zariski dense in A? If Q = 0, this question
ϕ
is easy to answer, e.g., by using Poincar´e reducibility [11, Section 19,
Theorem 1]. But if Q 6= 0 and the field K is countable, for exam-
ple K = Q¯, then the problem becomes considerably more difficult.
Indeed, the solution, which only recently appeared in [4], uses Faltings’
ARITHMETIC AND DYNAMICAL DEGREES ON ABELIAN VARIETIES 3
theorem (Mordell–Lang conjecture) on the intersection of subvarieties
of A with finitely generated subgroups of A. So at present it requires
deep tools to determine whether there exist any points P ∈ A(K) to
which Theorem 2 applies.
We briefly outline the contents of this note. We begin in Section 2
by setting notation. Section 3 contains a number of preliminary results
describing how dynamical and arithmetic degrees vary in certain situ-
ations. We then apply these tools and results from earlier work to give
the proof of Theorem 2 in Section 4. Finally, in Section 5 we prove
an auxiliary lemma on pullbacks and pushforwards of divisors that is
needed for one of the proofs in Section 3.
2. Notation
We set the following notation, which will be used for the remainder
of this note.
K an algebraically closed field on which their is a good theory
of height functions. For example, the algebraic closure of Q
or the algebraic closure of a one-dimensional function field.
A/K an abelian variety of dimension d defined over K.
Q a point in A(K).
τ the translation-by-Q map,
Q
τ : A −→ A, τ (P) = P +Q.
Q Q
f an isogeny f : A → A defined over K.
ϕ the finite map ϕ : A → A given by ϕ = τ ◦f, i.e.,
Q
ϕ(P) = f(P)+Q.
h a height function h : A(K) → R associated to an ample
A,H A,H
divisor H ∈ Div(A); see for example [7, 10].
ϕn the n’th iterate of ϕ, i.e., ϕn(P) = ϕ◦ϕ◦···◦ϕ(P).
O (P) the forward ϕ-orbit of P, i.e., the set {ϕn(P) : n ≥ 0}.
ϕ
Remark 4. It is a standard fact that every rational map A 99K A is a
morphism, and that every finite morphism A → A is the composition
of an isogeny and a translation [11, Section 4, Corollary 1]. Hence the
set of dominant rational maps A 99K A is the same as the set of maps
of the form ϕ = τ ◦f as in our notation.
Q
As noted earlier, since ϕ : A → A is a morphism, it is known [9] that
the limit defining α (P) exists (and is an algebraic integer).
ϕ
4 JOSEPH H.SILVERMAN
3. Preliminary material
In this section we collect some basic results that are needed to prove
Theorem 2. We begin with a standard (undoubtedly well-known) de-
composition theorem.
Lemma 5. Let A be an abelian variety, let f : A → A be an isogeny,
and let F(X) ∈ Z[X] be a polynomial such that F(f) = 0 in End(A).
Suppose that F factors as
F(X) = F (X)F (X) with F ,F ∈ Z[X] and gcd(F ,F ) = 1,
1 2 1 2 1 2
where the gcd is computed in Q[X]. Let
A = F (f)A and A = F (f)A,
1 1 2 2
so A and A are abelian subvarieties of A. Then we have:
1 2
(a) A = A +A .
1 2
(b) A ∩A is finite.
1 2
More precisely, if we let ρ = Res(F ,F ), then A ∩A ⊂ A[ρ].
1 2 1 2
Proof. The gcd assumption on F and F implies that their resultant
1 2
is non-zero, so we can find polynomials G ,G ∈ Z[X] so that
1 2
G (X)F (X)+G (X)F (X) = ρ = Res(F ,F ) 6= 0.
1 1 2 2 1 2
We observe that fA ⊂ A and fA ⊂ A and compute
1 1 2 2
A = ρA = G (f)F (f)+G (f)F (f) A
1 1 2 2
= G (f)A +G (f)A
(cid:0) 1 1 2 2 (cid:1)
⊂ A +A ⊂ A.
1 2
HenceA = A +A . Thisproves(a). For(b), supposethatP ∈ A ∩A ,
1 2 1 2
so
P = F (f)P = F (f)P for some P ∈ A and P ∈ A .
1 1 2 2 1 1 2 2
Then
ρP = G (f)F (f)+G (f)F (f) P
1 1 2 2
= G (f)F (f)F (f)P +G (f)F (f)F (f)P
(cid:0) 1 1 2 2 2 (cid:1) 2 1 1
= G (f)F(f)P +G (f)F(f)P since F = F F ,
1 2 2 1 1 2
= 0 since F(f) = 0.
Hence A ∩A ⊂ A[ρ]. (cid:3)
1 2
The next two lemmas relate dynamical and arithmetic degrees. We
statethemsomewhat moregenerallythanneeded inthisnote, since the
proofs are little more difficult and they may be useful for future appli-
cations. The first lemma says that dynamical and arithmetic degrees
ARITHMETIC AND DYNAMICAL DEGREES ON ABELIAN VARIETIES 5
are invariant under finite maps, and the second describes dynamical
and arithmetic degrees on products.
Lemma 6. Let X and Y be non-singular projective varieties, and let
λ
X −−−→ Y
fX fY
 λ 
X −−−→ Y
y y
be a commutative diagram, where f and f are dominant rational
X Y
maps and λ is a finite map, with everything defined over K
(a) Let x ∈ X whose orbit O (x) is well-defined. Then O (x) is
fX fX
Zariski dense in X if and only if O λ(x) is Zariski dense in Y.
fY
(b) The dynamical degrees of f and f are equal,
X Y
(cid:0) (cid:1)
δ(f ) = δ(f )
X Y
(c) Let P ∈ X be a point such that the forward f -orbit of P and the
X
arithmetic degree of P relative to f are well-defined. Then the
X
arithmetic degrees of P and λ(P) satisfy
α(f ,P) = α f ,λ(P) .
X Y
Remark 7. Lemma 6 is a special ((cid:0)relatively(cid:1)easy) case of results of
Dinh–Nguyen [2] and Dinh–Nguyen–Truong [3]. For completeness, we
give an algebraic proof, in the spirit of the present paper, which works
in arbitrary characteristic.
Proof of Lemma 6. (a) We first remark that the f orbit of λ(x) is also
Y
well-defined. To see this, let n ≥ 1 and let U be any Zariski open set on
which fn is well-defined. Then λ◦fn is also well-defined on U, since λ
X X
is a morphism. Also, since λ is a finite map, the image λ(U) is a Zariski
open set, and we note that fn on the set λ(U) agrees with λ◦fn on U.
Y X
Thus fn is defined on λ(U). In particular, since fn is assumed defined
Y X
at x, we see that fn is defined at λ(x).
Y
SupposethatZ = O (x) 6= X. Thenλ(Z)isaproper Zariskiclosed
fX
subset of Y, since finite maps send closed sets to closed sets. Further,
O λ(x) = λ O (x) ⊂ λ(Z).
fY fX
Hence OfY λ(x) is no(cid:0)t Zari(cid:1)ski de(cid:0)nse. Co(cid:1)nversely, suppose that W =
O λ(x) 6= Y. Finite maps (and indeed, morphisms) are continuous
fY (cid:0) (cid:1)
for the Zariski topology, so λ−1(W) is a closed subset of X, and the
(cid:0) (cid:1)
fact that λ is a finite map, hence surjective, implies that λ−1(W) 6= X.
Then
O (x) ⊂ λ−1 O λ(x) ⊂ λ−1(W) ( X,
fX fY
(cid:16) (cid:17)
(cid:0) (cid:1)
6 JOSEPH H.SILVERMAN
so O (x) is not Zariski dense in X.
fX
(b) Let d = dim(X) = dim(Y), and let H be an ample divisor on Y.
Y
The assumption that λ is a finite morphism implies that H := λ∗H
X Y
is an ample divisor on X. This follows from [6, Exercise 5.7(d)], or we
can use the Nakai–Moishezon Criterion [6, Theorem A.5.1] and note
that for every irreducible subvariety W ⊂ X of dimension r we have
λ (H ·Wr) = λ (λ∗H ·Wr) = H ·(λ W)r > 0,
∗ X ∗ Y Y ∗
since λ W is a positive multiple of an r-dimensional irreducible sub-
∗
variety of Y. This means that we can use H to compute δ(f ). In
X X
the following computation we use that fact that since λ is a finite mor-
phism, we have
(fN)∗ ◦λ∗ = (λ◦fN)∗ = (fN ◦λ)∗ = λ∗ ◦(fN)∗. (1)
X X Y Y
We give the justification for this formula at the end of this paper
(Lemma 11), but we note that for the proof of Theorem 2, all of the
relevant maps are morphisms, so (1) is trivially true. We compute
1/n
δ(f ) = lim (fn)∗H ·Hd−1
X X X X
n→∞
(cid:16) (cid:17) 1/n
= lim (fn)∗ ◦λ∗H ·(λ∗H )d−1
X Y Y
n→∞
(cid:16) (cid:17)1/n
= lim λ∗ ◦(fn)∗H ·(λ∗H )d−1 from (1),
Y Y Y
n→∞
(cid:16) (cid:17)1/n
= lim deg(λ) (fn)∗H ·∗ Hd−1
Y Y Y
n→∞
(cid:16) (cid:0) 1/n (cid:1)(cid:17)
= lim (fn)∗H ·Hd−1
Y Y Y
n→∞
= δ(f (cid:16)). (cid:17)
Y
This completes the proof of (b).
(c) We do ananalogous height computation, where the O(1) quantities
depend on X, Y, λ, f , f , and the choice of height functions for H
X Y X
and H , but do not depend of n.
Y
α(f ,P) = lim h fn(P) 1/n
X n→∞ X,HX X
= nl→im∞hX,λ∗H(cid:0)Y fXn(P(cid:1)) 1/n
(cid:0) (cid:1) 1/n
= lim h λ◦fn(P) +O(1)
n→∞ X,HY X
(cid:16) (cid:0) (cid:1) (cid:17)1/n
= lim h fn ◦λ(P) +O(1)
n→∞ X,HY Y
(cid:16) (cid:17)
= α f ,λ(P)(cid:0). (cid:1)
Y
(cid:0) (cid:1)
ARITHMETIC AND DYNAMICAL DEGREES ON ABELIAN VARIETIES 7
(cid:3)
This completes the proof of (c).
Lemma 8. Let Y and Z be non-singular projective varieties, let
f : Y → Y and f : Z → Z
Y Z
be dominant rational maps, and let f := f ×f be the induced map
Y,Z Y Z
on the product Y ×Z, with everything defined over K.
(a) Let y ∈ Y and z ∈ Z be points whose forward orbits via f ,
Y
respectively f , are well-defined, and suppose that O (y,z) is
Z fY,Z
Zariski dense in Y ×Z. Then O (y) is Zariski dense in Y and
fY
O (z) is Zariski dense in Z.
fZ
(b) The dynamical degrees of f , f , and f are related by
Y Z Y,Z
δ(f ) = max δ(f ),δ(f ) .
Y,Z Y Z
(c) Let (PY,PZ) ∈ (Y × Z)(K) b(cid:8)e a point su(cid:9)ch that the arithmetic
degrees α(f ,P ) and α(f ,P ) are well-defined. Then
Y Y Z Z
α f ,(P ,P ) = max α(f ,P ),α(f ,P ) .
Y,Z Y Z Y Y Z Z
Proof of 8. ((cid:0)a ) This elemen(cid:1)tary fact(cid:8)has nothing to do wit(cid:9)h orbits. Let
S ⊂ Y and T ⊂ Z be sets of points. By symmetry, it suffices to prove
that if S×T is Zariski dense in Y ×Z, then S is Zariski dense in Y. We
prove the contrapositive, so assume that S is not Zariski dense in Y.
This means that there is a proper Zariski closed subset W ⊂ Y with
S ⊂ W. Then S ×T ⊂ W ×Z ( Y ×Z, which shows that S ×T is
not Zariski dense in Y ×Z.
(b) Let
π : Y ×Z → Y and π : Y ×Z → Z
Y Z
denote the projection maps. Let H and H be, respectively, ample
Y Z
divisors on Y and Z. Then
H := (H ×Z)+(Y ×H ) = π∗H +π∗H
Y,Z Y Z Y Y Z Z
is an ample divisor on Y ×Z. We compute
(fn )∗H = (fn ×fn)∗(π∗H +π∗H )
Y,Z Y,Z Y Z Y Y Z Z
= π∗ ◦(fn)∗H +π∗ ◦(fn)∗H .
Y Y Y Z Z Z
We let
d = dim(Y), d = dim(Z), so dim(Y ×Z) = d +d .
Y Z Y Z
We compute
(fn )∗H ·HdY+dZ−1
Y,Z Y,Z Y,Z
= π∗ ◦(fn)∗H +π∗ ◦(fn)∗H · π∗H +π∗H dY+dZ−1
Y Y Y Z Z Z Y Y Z Z
(cid:16) (cid:16) (cid:17)
(cid:1)
8 JOSEPH H.SILVERMAN
d +d −1
= y Z ((fn)∗H ·HdY−1)(HdZ)
d Y Y Y Z
(cid:18) Z (cid:19)
d +d −1
+ y Z ((fn)∗H ·HdZ−1)(HdY). (2)
d Z Z Z Y
(cid:18) Y (cid:19)
For any dominant rational self-map f : X → X of a non-singular
projective variety of dimension d and any ample divisor H on X, the
dynamical degree of f is, by definition, the number δ(f) satisfying
δ(f)n = (fn)∗H ·Hd−1 ·2o(n) as n → ∞.
Using this formula three times in (2) yields
δ(f )n ·2o(n) = δ(f )n ·2o(n) ·HdZ +δ(f )n ·2o(n) ·HdY.
Y,Z Y Z Z Y
The quantities HdY and HdZ are positive, since H and H are ample.
Y Z Y Z
Now taking the n’th root of both sides and letting n → ∞ gives the
desired result, which completes the proof of (b).
(c) We do a similar computation. Thus
h fn (P ,P )
Y×Z,HY,Z Y,Z Y Z
= hY×Z,(cid:0)πY∗HY fYn,Z(PY(cid:1),PZ) +hY×Z,πZ∗HZ fYn,Z(PY,PZ) +O(1)
= h π ◦fn (P ,P ) +h π ◦fn (P ,P ) +O(1)
Y,HY Y (cid:0) Y,Z Y Z (cid:1) Z,HZ Z (cid:0) Y,Z Y Z (cid:1)
= h fn(P ) +h fn(P ) +O(1).
Y,HY(cid:0) Y Y Z,HZ (cid:1)Z Z (cid:0) (cid:1)
(cid:0) (cid:1) (cid:0) (cid:1)
For any dominant rational self-map f : X → X of a non-singular
projective variety defined over K, any ample divisor H on X, and
any P ∈ X(K) whose f-orbit is well-defined, the arithmetic degree is
the limit (if it exists)
α(f,P) := lim h+ fn(P) 1/n.
X,H
n→∞
(cid:0) (cid:1)
(Here h+ = max{h,1}.) Hence
α f ,(P ,P ) = lim h+ fn (P ,P ) 1/n
Y,Z Y Z n→∞ Y×Z,HY,Z Y,Z Y Z
(cid:0) (cid:1) (cid:0) (cid:1) 1/n
= lim h+ fn(P ) +h+ fn(P ) +O(1)
n→∞ Y,HY Y Y Z,HZ Z Z
(cid:16) (cid:17)
= max α(f ,(cid:0)P ),α(f(cid:1),P ) , (cid:0) (cid:1)
Y Y Z Z
(cid:8) (cid:9) (cid:3)
which completes the proof of (c).
ARITHMETIC AND DYNAMICAL DEGREES ON ABELIAN VARIETIES 9
4. Proof of Theorem 2
Proof of Theorem 2. The translationmapτ induces theidentity map1
Q
τ∗ = id : NS(A) → NS(A),
Q
from which we deduce that
ϕ∗ = f∗ and δ = δ . (3)
ϕ f
We begin by proving Theorem 2 under the assumption that a non-
zero multiple of the point Q is in the image of the map f −1, say
mQ = (f −1)(Q′) for some m 6= 0 and Q′ ∈ A.
Then we have
mϕn(P) = m fn(P)+(fn−1 +fn−2 +···+f +1)(Q)
= fn(mP)+(fn−1 +fn−2 +···+f +1)(mQ)
(cid:0) (cid:1)
= fn(mP)+(fn−1 +fn−2 +···+f +1)◦(f −1)(Q′)
= fn(mP)+fn(Q′)−Q′
= fn(mP +Q′)−Q′. (4)
In particular, the ϕ orbit of P and the f orbit of mP + Q′ differ by
translation by −Q′, so the assumption that O (P) is Zariski dense and
ϕ
the fact that translation is an automorphism imply that O (mP +Q′)
f
is also Zariski dense. We will also use the standard formula
h ◦m = m2h +O(1). (5)
A,H A,H
We now compute (with additional explanation for steps (6) and (7)
following the computation)
α (P) = lim h ϕn(P) 1/n by definition,
ϕ A,H
n→∞
= lim h (cid:0)mϕn(P(cid:1)) 1/n from (5),
A,H
n→∞
= lim h (cid:0)fn(mP +(cid:1) Q′)−Q′ 1/n from (4),
A,H
n→∞
= nl→im∞hA,τ−∗(cid:0)Q′H fn(mP +Q′) 1(cid:1)/n functoriality,
= α (mP +Q′)(cid:0) (cid:1) by definition, (6)
f
= δ from [8, Theorem 4], (7)
f
= δ from (3).
ϕ
1Letµ:A×A→Abeµ(x,y)=x+y,andletD ∈Div(A). ThenforanyP ∈A,
thedivisorµ∗Dhasthepropertythatµ∗D| =τ∗D×{P}. HenceasP varies,
A×{P} P
the divisors τ∗D are algebraically equivalent, so in particular τ∗D ≡ τ∗D ≡ D,
P Q 0
which shows that τ∗ is the identity map on NS(A).
Q
10 JOSEPH H.SILVERMAN
We note that (6) follows from [9, Proposition 12], which says that the
arithmetic degree may be computed using the height relative to any
ample divisor. (The map τ−Q′ is an isomorphism, so τ−∗Q′H is ample.)
For (7), we have applied [8, Theorem 4] to the isogeny f and the
point mP +Q′, since we’ve already noted that O (mP +Q′) is Zariski
f
dense. This completes the proof of Theorem 2 if Q ∈ (f −1)(A).
We now commence the proof in the general case. The Tate mod-
ule T (A) group of A has rank 2d, and an isogeny is zero if and only
ℓ
if it induces the trivial map on the Tate module, from which we see
that f satisfies a monic integral polynomial equation of degree 2d, say
F(f) = 0 with F(X) ∈ Z[X] monic.
We factor F(X) as
F(X) = F (X)F (X)
1 2
with
F (X) = (X −1)r, F (X) ∈ Z[X], and F (1) 6= 0.
1 2 2
We first deal with the case that r = 0. This means that F(1) 6= 0.
Writing F(X) = (X −1)G(X)+F(1), we have
0 = F(f)Q = (f −1)G(f)Q+F(1)Q,
so
F(1)Q = −(f −1)G(f)Q ∈ (f −1)A.
Thus a non-zero multiple of Q is in (f−1)A, which is the case that we
handled earlier.
We now assume that r ≥ 1, and we define abelian subvarieties of A
by
A = F (f)A and A = F (f)A
1 1 2 2
and consider the map
λ : A ×A −→ A, λ(P ,P ) = P +P .
1 2 1 2 1 2
Lemma 5 tells us that λ is an isogeny. More precisely, Lemma 5(a)
says that λ is surjective, while Lemma 5(b) tells us that
∼
ker(λ) = (P,−P) : P ∈ A ∩A = A ∩A
1 2 1 2
is finite. (cid:8) (cid:9)
We recall the the map ϕ : A → A has the form ϕ(P) = f(P)+Q for
some fixed Q ∈ A. The map λ is onto, so we can find a pair
(Q ,Q ) ∈ A ×A satisfying λ(Q ,Q ) = Q, i.e., Q +Q = Q.
1 2 1 2 1 2 1 2