Table Of ContentVARIATION OF CANONICAL HEIGHT AND EQUIDISTRIBUTION
LAURA DE MARCO AND NIKI MYRTO MAVRAKI
7
1
0 Abstract. Let π : E → B be an elliptic surface defined over a number field K, where B
2
is a smooth projective curve, and let P :B →E be a section defined over K with canonical
ar height hˆE(P) (cid:54)= 0. In this article, we show that the function t (cid:55)→ hˆEt(Pt) on B(K) is the
M
height induced from an adelically metrized line bundle with non-negative curvature on B.
ApplyingtheoremsofThuillierandYuan,weobtaintheequidistributionofpointst∈B(K)
2
whereP istorsion,andwegiveanexplicitdescriptionofthelimitingdistributiononB(C).
t
]
T Finally, combined with results of Masser and Zannier, we show there is a positive lower
N bound on the height hˆ (P ), after excluding finitely many points t ∈ B, for any “non-
At t
h. special” section P of a family of abelian varieties A→B that split as a product of elliptic
t curves.
a
m
[
2
v 1. Introduction
7
4
Suppose E → B is an elliptic surface defined over a number field K, so B is a smooth
9
7
projective curve and all but finitely many fibers E , t ∈ B(K), are smooth elliptic curves.
0 t
. We let hˆ denote the N´eron-Tate canonical height of E viewed as an elliptic curve over the
1 E
0 ˆ
function field k = K(B); we let h denote the canonical height on the fibers for (all but
7 Et
1 finitely many) t ∈ B(K).
:
v Suppose that P → E is a section defined over K for which hˆ (P) (cid:54)= 0, so, in particular,
i E
X
the points P on the fiber are not torsion in E for all t. Tate showed that the function
t t
r
a
ˆ
t (cid:55)→ h (P )
Et t
is a Weil height on B(K), up to a bounded error [Ta]. More precisely, there exists a divisor
D ∈ Pic(B)⊗Q of degree equal to hˆ (P) so that
P E
ˆ
(1.1) h (P ) = h (t)+O(1),
Et t DP
where h is a Weil height on B(K) associated to D . In a series of three articles [Si1,
DP P
Si3, Si4], Silverman refined statement (1.1) by analyzing the N´eron decomposition of the
Date: March 3, 2017.
1
VARIATION OF CANONICAL HEIGHT AND EQUIDISTRIBUTION 2
canonical height on the fibers
(cid:88)
ˆ ˆ
h (P ) = n λ (P )
Et t v Et,v t
v∈MK
where M denotes the set of places of the number field K, and n are the integers appearing
K v
in the product formula (cid:81) |x|nv = 1 for all x ∈ K∗.
v∈MK v
ˆ
In this article, we explain how Silverman’s conclusions about the local functions λ (P )
Et,v t
ˆ
are precisely the input needed to show that t (cid:55)→ h (P ) is a “good” height function on the
Et t
base curve B, from the point of view of equidistribution. Combining his work with methods
from complex dynamics, as in [DWY], and the inequalities of Zhang on successive minima
[Zh4, Zh1], we prove:
Theorem 1.1. Let K be a number field and k = K(B) for a smooth projective curve B
defined over K. Fix any elliptic surface E → B defined over K and point P ∈ E(k)
ˆ
satisfying h (P) (cid:54)= 0. Then
E
ˆ
h (t) := h (P ),
P Et t
for t with smooth fibers, is the restriction of a height function on B(K) induced from an
adelically metrized ample line bundle L, with continuous metrics of non-negative curvature,
satisfying
h (B) := c (L)2/(2c (L)) = 0.
P 1 1
Theorem 1.1 implies that our height function on B satisfies the hypotheses of the equidis-
tribution theorems of Thuillier and Yuan for points of small height on curves [CL1, Th, Yu],
and we deduce the following:
Corollary 1.2. Let K be a number field and k = K(B) for a smooth projective curve
B defined over K. Fix any elliptic surface E → B defined over K and point P ∈ E(k)
ˆ
satisfying h (P) (cid:54)= 0. There is a collection of probability measures µ = {µ : v ∈ M }
E P P,v K
on the Berkovich analytifications Ban such that for any infinite, non-repeating sequence of
v
t ∈ B(K) such that
n
ˆ
h (P ) → 0
Etn tn
as n → ∞, the discrete measures
1 (cid:88)
δ
t
|Gal(K/K)·t |
n
t∈Gal(K/K)·tn
converge weakly on Ban to the measure µ at each place v of K.
v P,v
VARIATION OF CANONICAL HEIGHT AND EQUIDISTRIBUTION 3
Remark 1.3. The measures µ of Corollary 1.2 are not difficult to describe, at least at the
P,v
archimedean places. At each archimedean place v, there is a canonical positive (1,1)-current
T on the surface E(C) (with continuous potentials away from the singular fibers) which
v
restricts to the Haar measure on each smooth fiber E (C). The measure µ on B(C) is just
t P,v
the pull-back of this current by the section P. Moreover, at every place, the measure µ
P,v
ˆ
is the Laplacian of the local height function λ (P ), away from its singularities. We give
Et,v t
more details about (and a dynamical perpective on) the construction of the current T in
v
Section 3.
As a consequence of Theorem 1.1, and combined with the work of Masser and Zannier
[MZ1, MZ2, MZ3], we obtain the so-called Bogomolov extension of their theorems. Fix
integer m ≥ 2, and suppose that E → B is an elliptic surface over a curve B, defined over
i
Q, for i = 1,...,m. We consider sections P of the fiber product A = E × ··· × E
1 B B m
defined over Q. We say that a section P = (P ,P ,...,P ) is special if
1 2 m
ˆ
• for each i = 1,...,m, either P is torsion on E or h (P ) (cid:54)= 0; and
i i Ei i
• for any pair i,j ∈ {1,...,m} such that neither P nor P is torsion, there are an
i j
isogeny φ : E → E and nonzero group endomorphisms a,b of E so that a◦φ(P ) =
i j j i
b(P ).
j
If a family of abelian surfaces A → B is isogenous to a fiber product (after performing a
base change B(cid:48) → B if needed), we say that a section of A is special if it is special on the
fiber product.
It is well known that a special section will always pass through infinitely many torsion
points in the fibers A = E ×···×E . That is, there are infinitely many t ∈ B(Q) for
t 1,t m,t
which
ˆ ˆ
h (P (t)) = ··· = h (P (t)) = 0.
E1,t 1 Em,t 2
For a proof see [Za, Chapter 3] or, for dynamical proofs, see [De1].
The converse statement is also true, but it is much more difficult: Masser and Zannier
proved that if hˆ (P (t)) = ··· = hˆ (P (t)) = 0 for infinitely many t ∈ B(Q), then the
E1,t 1 Em,t 2
section P must be special [MZ2, MZ3]. We extend these results of Masser-Zannier from
points of height 0 to points of small height:
Theorem 1.4. Let B be a quasiprojective smooth algebraic curve defined over Q. Suppose
A → B is a family of abelian varieties of relative dimension m ≥ 2 defined over Q which
is isogeneous to a fibered product of m ≥ 2 elliptic surfaces. Let L be a line bundle on A
VARIATION OF CANONICAL HEIGHT AND EQUIDISTRIBUTION 4
ˆ
which restricts to an ample and symmetric line bundle on each fiber A , and let h be the
t t
induced N´eron-Tate canonical height on A , for each t ∈ B(Q). For each non-special section
t
P : B → A defined over Q, there is a constant c = c(L,P) > 0 so that
{t ∈ B(Q) : hˆ (P ) < c}
t t
is finite.
If A → B is isotrivial, then Theorem 1.4 is a special case of the Bogomolov Conjecture,
proved by Ullmo and Zhang [Zh3, Ul].
A key ingredient in their proofs is the equidistribution theorem of Szpiro, Ullmo, and
Zhang [SUZ]. In his 1998 ICM lecture notes [Zh2], Zhang presented a conjecture about
geometrically simple families of abelian varieties, which stated, in its most basic form:
Conjecture 1.5 (Zhang). Let B be a quasiprojective smooth algebraic curve defined over
Q. Suppose A → B is a non-isotrivial family of abelian varieties with fiber dimension > 1,
defined over Q with a simple generic fiber. Let L be a line bundle on A which restricts to
ˆ
an ample and symmetric line bundle on each fiber A , and let h be the induced N´eron-Tate
t t
canonical height on A , for each t ∈ B(Q). For each non-torsion section P : B → A defined
t
over Q, there is a constant c = c(L,P) > 0 so that
{t ∈ B(Q) : hˆ (P ) < c}
t t
is finite.
When the dimension of the fibers A is equal to 2, the finiteness of {t ∈ B(Q) : hˆ (P ) = 0}
t t t
for sections as in Conjecture 1.5 was established recently by Masser and Zannier in [MZ4].
It is well known that the conclusion of Conjecture 1.5 can fail to hold if A is not simple and
certainly fails if it is a family of elliptic curves, as mentioned above. However, the results
of Masser and Zannier in their earlier work [MZ2, MZ3] suggested a formulation of Zhang’s
conjecture for the non-simple case when A splits as a product of elliptic curves; this is what
we proved in our Theorem 1.4.
Remark 1.6. Theorem 1.1, Corollary 1.2, and Theorem 1.4 were obtained in the special
case of the Legendre family E = {y2 = x(x − 1)(x − t)} over B = P1 and the abelian
t
variety A = E × E , for sections P with x-coordinates in Q(t) in [DWY], using methods
t t t
from complex dynamical systems, without appealing to Silverman and Tate’s results on the
height function. Moreover, restricting further to sections P with constant x-coordinate (in
VARIATION OF CANONICAL HEIGHT AND EQUIDISTRIBUTION 5
P1(Q)), Theorem 1.4 was obtained without relying on the theorems of Masser and Zannier
and gave an alternate proof of their result. This includes the special case treated by Masser
and Zannier in their article [MZ1]. For sections with constant x-coordinate, the hypothesis
ˆ
on P (that h (P) (cid:54)= 0) is equivalent to asking that x(P) (cid:54)= 0,1,∞ [DWY, Proposition 1.4].
E
Comments and acknowledgements. This project was motivated, in part, by experi-
ments to visualize Silverman’s results on the variation of canonical height [Si1, Si3, Si4] in
terms of the measures µ at archimedean places, and to examine their dependence on P.
P,v
ˆ
In particular, the measure detects the failure of the local height function λ (P ) to be
Et,v t
harmonic; compare the comments on non-analyticity preceding Theorem I.0.3 of [Si1]. The
images appearing in Section 6 were first presented at the conference in honor of Silverman’s
birthday, August 2015.
We thank Charles Favre, Dragos Ghioca, Robert Rumely, Joseph Silverman, and Amaury
Thuillier for helpful suggestions. Our research was supported by the National Science Foun-
dation and the Simons Foundation.
2. Silverman’s work
2.1. Preliminaries. Let F be a product formula field of characteristic 0, so there exists a
family M of non-trivial absolute values on F and a collection of positive integers n for
F v
v ∈ M so that
F
(cid:89)
|x|nv = 1
v
v∈MF
for all x ∈ F∗. Let E/F be an elliptic curve with origin O, expressed in Weierstrass form as
E = {y2 +a xy +a y = x3 +a x2 +a x+a }
1 3 2 4 6
with discriminant ∆. Denote by
ˆ
h : E(F) → [0,∞)
E
the N´eron-Tate canonical height function; it can be defined by
1 h(x([n]P))
ˆ
h (P) = lim
E 2 n→∞ n2
where h is the naive Weil height on P1 and x : E → P1 is the degree 2 projection to the
x-coordinate.
VARIATION OF CANONICAL HEIGHT AND EQUIDISTRIBUTION 6
For each v ∈ M , we let C denote a minimal, algebraically closed field containing F
F v
which is complete with respect to |·| . For each v, we fix an embedding of F into C . The
v v
canonical height has a decomposition into local heights, as
1 (cid:88) (cid:88)
ˆ ˆ
h (P) = n λ (Q)
E v E,v
|Gal(F/F)·P|
Q∈Gal(F/F)·P v∈MF
ˆ
for P ∈ E(F) \ {O}, with the local heights λ characterized by three properties [Si2,
E,v
Chapter 6, Theorem 1.1, page 455]:
(1) λˆ is continuous on E(C ) \ {O} and bounded on the complement of any v-adic
E,v v
neighborhood of O;
(2) the limit of λˆ (P)− 1 log|x(P)| exists as P → O in E(C ); and
E,v 2 v v
(3) for all P = (x,y) ∈ E(C ) with [2]P (cid:54)= O,
v
1
ˆ ˆ
λ([2]P) = 4λ(P)−log|2y +a x+a | + log|∆| .
1 3 v v
4
2.2. Variation of canonical height: the set up. Now let K be a number field and
E → B an elliptic surface defined over a number field K with zero section O : B → E. Let
P : B → E be a non-zero section defined over K, and assume that
ˆ
h (P) (cid:54)= 0
E
when viewing P as a point on the elliptic curve E defined over k = K(B). For each t ∈ B(K)
such that the fiber E is non-singular, we have point P ∈ E (K). We will investigate the
t t t
function
ˆ
t (cid:55)→ h (P )
Et t
which is well defined at all but finitely many t ∈ B(K). Furthermore, via the embedding of
K into C for each place v ∈ M , we may view E → B as defined over C and consider the
v K v
N´eron local heights λˆ (P ) on the non-singular fibers E as functions of t ∈ B(C ).
Et,v t t v
Let D (P) be the Q-divisor on B defined by
E
(cid:88)
ˆ
(2.1) D (P) = λ (P)·(γ).
E E,ordγ
γ∈B(K)
ˆ
Here, λ (P) is the local canonical height of the point P on the elliptic curve E over
E,ordγ
ˆ
k = K(B) at the place ord , for each γ ∈ B(K). The degree of D (P) is equal to h (P). It
γ E E
VARIATION OF CANONICAL HEIGHT AND EQUIDISTRIBUTION 7
follows from the definitions that suppD (P) is a subset of the finite set
E
{t ∈ B(K) : E is singular}∪{t ∈ B(K) : P = O }.
t t t
By enlarging K, we may assume that the support of D (P) is contained in B(K).
E
Remark 2.1. That D (P) is a Q-divisor is standard, following from the fact that the
E
ˆ
numbersλ (P)canbeviewedasarithmeticintersectionnumbersonaN´eronlocalmodel.
E,ordγ
See[Si2, ChapterIII,Theorem9.3]foraproofthathˆ (P) ∈ Q; see[CS, Section6andp.203]
E
ˆ
and [La1, Chapter 11 Theorem 5.1] for proofs that each local function λ also takes values
E,v
in Q; see [DG, Theorem 1.1] for a dynamical proof.
2.3. Variation of canonical height: quasi triviality. Let h be an analytic Weil
DE(P)
height on B(K) as defined in [Si4, §3 Example 1(a)]. That is, we let g be the genus of B,
and for each point γ ∈ B(K), we choose an element ξ of K(B) which has a pole of order
γ
2g +1 at γ and no other poles. For each non-archimedean place v of K, set
1 (cid:88)
λ (t) = λˆ (P) log+|ξ (t)|
DE(P),v 2g +1 E,ordγ γ v
γ∈B(K)
for all t ∈ B(C )\suppD (P). For archimedean places v, the local height is defined by
v E
λ (t) = 1 (cid:88) λˆ (P) log(cid:0)1+|ξ (t)|2(cid:1).
DE(P),v 2(2g +1) E,ordγ γ v
γ∈B(K)
We set
1 (cid:88) (cid:88)
h (t) = λ (s)
DE(P) |Gal(K/K)·t| DE(P),v
s∈Gal(K/K)·t v∈MK
for all t ∈ B(K). For fixed choices of ξ , we will call the associated height h our
γ DE(P)
“reference height” for the divisor D (P). Silverman proved:
E
Theorem 2.2. [Si4, Theorem III.4.1] For any choice of reference height h , there is a
DE(P)
finite set S of places so that
ˆ
λ (P ) = λ (t)
Et,v t DE(P),v
for all t ∈ B(K)\suppD (P) and all v ∈ M \S.
E K
2.4. Variation of canonical height: continuity. Fixapointt ∈ B(K)andauniformizer
0
u ∈ K(B) for t , and consider the function
0
ˆ ˆ
(2.2) V (t) := λ (P )+λ (P)log|u(t)| ,
P,t0,v Et,v t E,ordt0 v
VARIATION OF CANONICAL HEIGHT AND EQUIDISTRIBUTION 8
which is not a priori defined at t . Theorem 2.2 implies that
0
V ≡ 0
P,t0,v
for all but finitely many places v in a v-adic neighborhood of each t . Silverman also proved
0
the following:
Theorem 2.3. [Si3, Theorem II.0.1] Fix t ∈ B(K) and a uniformizer u at t . For all
0 0
v ∈ M , there exists a neighborhood U ⊂ B(C ) containing t so that the function V of
K v 0 P,t0,v
(2.2) extends to a continuous function on U.
3. A dynamical perspective
ˆ ˆ
RecallthattheN´eron-Tateheighth anditslocalcounterpartsλ canbedefineddynam-
E E,v
ically. Letting E be an elliptic curve defined over a number field K, the multiplication-by-2
endomorphism φ on E descends to a rational function of degree 4 on P1, via the standard
quotient identifying a point P with its additive inverse:
φ (cid:47)(cid:47)
(3.1) E E
π π
(cid:15)(cid:15) (cid:15)(cid:15)
f
P1 φ (cid:47)(cid:47) P1
An elementary, but key, observation is that a point is torsion on E if and only if its quotient
in P1 is preperiodic for f . The height hˆ on E(K) satisfies
φ E
1
hˆ (P) = lim h(fn(πP))
E n→∞ 4n φ
where h is the standard logarithmic Weil height on P1(K). Now let E → B be an elliptic
surface defined over a number field K, and let P : B → E be a section, also defined over
K. In this section, we use this perspective to give a proof of subharmonicity of the local
ˆ
height functions t (cid:55)→ λ (P ) and the extensions V of (2.2). We will present this fact as
Et,v t P,t0,v
an immediate consequence of now-standard complex-dynamical convergence arguments, at
ˆ
least when the fiber E is smooth and the local height λ (P ) is finite. Near singular fibers,
t Et,v t
we utilize the maximum principle and standard results on removable singularities for sub-
harmonic functions. The same reasoning applies in both archimedean and non-archimedean
settings.
VARIATION OF CANONICAL HEIGHT AND EQUIDISTRIBUTION 9
In §3.3 we provide the background to justify the explicit description of the limiting distri-
bution µ at the archimedean places v of K, as mentioned in Remark 1.3.
P,v
3.1. Canonical height and escape rates. As in §2.1, we let E be an elliptic curve in
Weierstrass form, defined over a product-formula field F of characteristic 0. We define a
rational function f = φ/ψ on P1 by the formula
f(x(P)) = x([2]P)
for all P ∈ E. Here x(P) is the x−coordinate for a point P ∈ E; this function x plays
the role of π in (3.1). In coordinates, we have φ(x) = x4 − b x2 − 2b x − b and ψ(x) =
4 6 8
4x3 +b x2 +2b x+b = (2y +a x+a )2 for P = (x,y).
2 4 6 1 3
By a lift of f, we mean any homogeneous polynomial map F on A2, defined over F, so
that τ ◦ F = f ◦ τ, where τ : A2 \ {(0,0)} → P1 is the tautological projection. A lift of a
point x ∈ P1 is a choice of X ∈ A2 \{(0,0)} so that τ(X) = x.
The standard lift of f will be the map F : A2 → A2 defined by
(cid:0) (cid:1)
(3.2) F(z,w) = w4φ(z/w),w4ψ(z/w) .
For each v ∈ M , the v-adic escape rate is defined by
F
log||Fn(z,w)||
v
G (z,w) = lim
F,v n→∞ 4n
where
(cid:107)(z,w)(cid:107) = max{|z| ,|w| }.
v v v
Any other lift of f is of the form cF for some c ∈ F∗; observe that
1
G = G + log|c| .
cF,v F,v v
3
Note that
G (αx,αy) = G (x,y)+log|α|
F,v F,v v
for any choice of lift F. Furthermore, G is continuous on (C )2 \ {(0,0)}, as proved in
F,v v
the archimedean case by [HP, FS]. For non-archimedean absolute values v, G extends
F,v
continuously to the product of Berkovich affine lines A1,an × A1,an \ {(0,0)} [BR, Chapter
v v
10].
VARIATION OF CANONICAL HEIGHT AND EQUIDISTRIBUTION 10
Proposition 3.1. For the standard lift F of f, and for each place v of F, the local canonical
height function satisfies
1 1 1
ˆ
λ (P) = G (x,y)− log|y| − log|∆|
E,v F,v v v
2 2 12
where x(P) = (x : y).
Proof. TheproofisimmediatefromthepropertiesofG bycheckingthethreecharacterizing
F,v
conditions for λˆ . (cid:3)
E,v
3.2. Variation of canonical height: subharmonicity. Now let K be a number field and
E → B an elliptic surface defined over a number field K with zero section O : B → E.
Let k = K(B); viewing E as an elliptic curve defined over k, we also fix a point P ∈ E(k).
Recall the function V (t) defined in (2.2).
P,t0,v
Theorem 3.2. For every t ∈ B(K) and uniformizer u in k at t , the function
0 0
ˆ ˆ
V (t) := λ (P )+λ (P)log|u(t)| ,
P,t0,v Et,v t E,ordt0 v
extends to a continuous and subharmonic function on a neighborhood of t in the Berkovich
0
analytification Ban.
v
The continuity was already established in Theorem 2.3, though it was not explicitly stated
for the Berkovich space. The argument below takes care of that. We begin with a lemma.
Lemma 3.3. Fix α ∈ k∗ and t ∈ B(K). Let u ∈ k be a uniformizer at t . For each place
0 0
v of K, the function
t (cid:55)→ log|α | −(ord α) log|u(t)|
t v t0 v
is harmonic in a neighborhood of t in the Berkovich analytification Ban.
0 v
Proof. This is Silverman’s [Si3, Lemma II.1.1(c)] plus a removable singularities lemma for
harmonic functions. See also [BR, Proposition 7.19] for the extension of a harmonic function
to a disk in the Berkovich space Ban. (cid:3)
v
Fix P ∈ E(k). Let F and X be lifts of f and x(P) to k2, respectively. Iterating F, we set
(A ,B ) := Fn(X) ∈ k2
n n
and observe that
min{ord A ,ord B }
(3.3) G (X) = − lim t0 n t0 n
F,ordt0 n→∞ 4n
