Variation of canonical height and equidistribution
Laura DeMarco Niki Myrto Mavraki








Let π : E→ B be an elliptic surface defined over a number field K, where B is a smooth projective curve, and let P: B → E be a section defined over K with canonical height ĥ_E(P)≠0.  In this article, we show that the function t ↦ĥ_E_t(P_t) on B() is the height induced from an adelically metrized line bundle with non-negative curvature on B.  Applying theorems of Thuillier and Yuan, we obtain the equidistribution of points t ∈ B() where P_t is torsion, and we give an explicit description of the limiting distribution on B().  Finally, combined with results of Masser and Zannier, we show there is a positive lower bound on the height ĥ_A_t(P_t), after excluding finitely many points t ∈ B, for any “non-special" section P of a family of abelian varieties A → B that split as a product of elliptic curves.   


Variation of canonical height and equidistribution
    Laura De Marco and Niki Myrto Mavraki
    December 30, 2023
==================================================








§ INTRODUCTION


Suppose E → B is an elliptic surface defined over a number field K, so B is a smooth projective curve and all but finitely many fibers E_t, t∈ B(K), are smooth elliptic curves.  We let ĥ_E denote the Néron-Tate canonical height of E viewed as an elliptic curve over the function field k = K(B); we let ĥ_E_t denote the canonical height on the fibers for (all but finitely many) t∈ B().  

Suppose that P → E is a section defined over K for which ĥ_E(P) ≠ 0, so, in particular, the points P_t on the fiber are not torsion in E_t for all t.  Tate showed that the function 
	
    t ↦ĥ_E_t(P_t)

is a Weil height on B(), up to a bounded error  <cit.>.  More precisely, there exists a divisor D_P ∈Pic(B) ⊗ of degree equal to ĥ_E(P) so that 

    ĥ_E_t(P_t) = h_D_P(t) + O(1),

where h_D_P is a Weil height on B() associated to D_P.  In a series of three articles <cit.>, Silverman refined statement (<ref>) by analyzing the Néron decomposition of the canonical height on the fibers 
	
    ĥ_E_t(P_t) = ∑_v ∈ M_K n_v λ̂_E_t, v(P_t)

where M_K denotes the set of places of the number field K, and n_v are the integers appearing in the product formula ∏_v∈ M_K |x|_v^n_v = 1  for all x ∈ K^*.  

In this article, we explain how Silverman's conclusions about the local functions λ̂_E_t, v(P_t) are precisely the input needed to show that t↦ĥ_E_t(P_t) is a “good" height function on the base curve B, from the point of view of equidistribution.  Combining his work with methods from complex dynamics, as in <cit.>, and the inequalities of Zhang on successive minima <cit.>, we prove:  

   
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) satisfying ĥ_E(P) ≠0.  Then 
	
    h_P(t) :=  ĥ_E_t(P_t),

for t with smooth fibers, is the restriction of a height function on B() induced from an adelically metrized ample line bundle , with continuous metrics of non-negative curvature, satisfying 
	
    h_P(B) := c_1(ℒ)^2/(2 c_1(ℒ)) = 0.

  

Theorem <ref> implies that our height function on B satisfies the hypotheses of the equidistribution theorems of Thuillier and Yuan for points of small height on curves <cit.>, and we deduce the following:

   
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) satisfying ĥ_E(P) ≠0.  There is a collection of probability measures μ_P = {μ_P,v: v ∈ M_K} on the Berkovich analytifications B^an_v such that for any infinite, non-repeating sequence of t_n ∈ B() such that 
	
    ĥ_E_t_n(P_t_n) → 0

as n→∞, the discrete measures 
	
    1/|(K/K) · t_n|∑_t ∈(K/K) · t_nδ_t

converge weakly on B^an_v to the measure μ_P,v at each place v of K.   


 
The measures μ_P,v of Corollary <ref> are not difficult to describe, at least at the archimedean places.  At each archimedean place v, there is a canonical positive (1,1)-current T_v on the surface E() (with continuous potentials away from the singular fibers) which restricts to the Haar measure on each smooth fiber E_t().  The measure μ_P,v on B() is just the pull-back of this current by the section P.  Moreover, at every place, the measure μ_P,v is the Laplacian of the local height function λ̂_E_t,v(P_t), away from its singularities.  We give more details about (and a dynamical perpective on) the construction of the current T_v in Section <ref>.



As a consequence of Theorem <ref>, and combined with the work of Masser and Zannier <cit.>, we obtain the so-called Bogomolov extension of their theorems.  Fix integer m≥ 2, and suppose that E_i → B is an elliptic surface over a curve B, defined over , for i = 1, …, m.  We consider sections P of the fiber product A = E_1 ×_B ⋯×_B E_m defined over .  We say that a section P = (P_1, P_2, …, P_m) is special if 


  * for each i = 1, …, m, either P_i is torsion on E_i or ĥ_E_i(P_i)≠0; and

  * for any pair i,j ∈{1, …, m} such that neither P_i nor P_j is torsion, there are an isogeny ϕ: E_i→ E_j and nonzero group endomorphisms a, b of E_j so that a∘ϕ(P_i) = b(P_j). 

If a family of abelian surfaces A → B is isogenous to a fiber product (after performing a base change B' → 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_t = E_1,t×⋯× E_m,t.  That is, there are infinitely many t ∈ B() for which 
	
    ĥ_E_1,t(P_1(t)) = ⋯ = ĥ_E_m,t(P_2(t)) = 0.


For a proof see <cit.> or, for dynamical proofs, see <cit.>.  

The converse statement is also true, but it is much more difficult:  Masser and Zannier proved that if ĥ_E_1,t(P_1(t)) = ⋯ = ĥ_E_m,t(P_2(t)) = 0 for infinitely many t∈ B(), then the section P must be special  <cit.>.  We extend these results of Masser-Zannier from points of height 0 to points of small height:


 
Let B be a quasiprojective smooth algebraic curve defined over .  Suppose A →  B is a family of abelian varieties of relative dimension m ≥ 2 defined over  which is isogeneous to a fibered product of m≥ 2 elliptic surfaces.  Let  be a line bundle on A which restricts to an ample and symmetric line bundle on each fiber A_t, and let ĥ_t be the induced Néron-Tate canonical height on A_t, for each t∈ B().   For each non-special section P: B → A defined over , there is a constant c = c(, P) > 0 so that 
	
    {t ∈ B(): ĥ_t(P_t) < c}
 
is finite.  


If A→ B is isotrivial, then Theorem <ref> is a special case of the Bogomolov Conjecture, proved by Ullmo and Zhang <cit.>.  
  
A key ingredient in their proofs is the equidistribution theorem of Szpiro, Ullmo, and Zhang <cit.>. In his 1998 ICM lecture notes <cit.>, Zhang presented a conjecture about geometrically simple families of abelian varieties, which stated, in its most basic form:

  
Let B be a quasiprojective smooth algebraic curve defined over .  Suppose A →  B is a non-isotrivial family of abelian varieties with fiber dimension > 1, defined over  with a simple generic fiber.  Let  be a line bundle on A which restricts to an ample and symmetric line bundle on each fiber A_t, and let ĥ_t be the induced Néron-Tate canonical height on A_t, for each t∈ B().  For each non-torsion section P: B → A defined over , there is a constant c = c(, P) > 0 so that 
	
    {t ∈ B(): ĥ_t(P_t) < c}
 
is finite.  



When the dimension of the fibers A_t is equal to 2, the finiteness of {t ∈ B(): ĥ_t(P_t) = 0}  for sections as in Conjecture <ref> was established recently by Masser and Zannier in <cit.>.  It is well known that the conclusion of Conjecture <ref> 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 <cit.> 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 <ref>.  





Theorem <ref>, Corollary <ref>, and Theorem <ref> were obtained in the special case of the Legendre family E_t = {y^2 = x(x-1)(x-t)} over B= ^1 and the abelian variety A_t = E_t× E_t, for sections P with x-coordinates in (t) in <cit.>, using methods 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 ^1(ℚ)), Theorem <ref> 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 <cit.>.  For sections with constant x-coordinate, the hypothesis on P (that ĥ_E(P) ≠0) is equivalent to asking that x(P) ≠ 0,1,∞ <cit.>.














Comments and acknowledgements.  
This project was motivated, in part, by experiments to visualize Silverman's results on the variation of canonical height <cit.> in terms of the measures μ_P,v at archimedean places, and to examine their dependence on P.  In particular, the measure detects the failure of the local height function λ̂_E_t,v(P_t) to be harmonic; compare the comments on non-analyticity preceding Theorem I.0.3 of <cit.>.  The images appearing in  Section <ref> 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 Foundation and the Simons Foundation.






§ SILVERMAN'S WORK




 §.§ Preliminaries
  
Let ℱ be a product formula field of characteristic 0, so there exists a family M_ℱ of non-trivial absolute values on ℱ and a collection of positive integers n_v for v ∈ M_ℱ so that 
	
    ∏_v ∈ M_ℱ |x|_v^n_v = 1

for all x∈ℱ^*.  Let E/ℱ be an elliptic curve with origin O, expressed in Weierstrass form as 
	
    E = { y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6}
 
with discriminant Δ.  Denote by 
	
    ĥ_E:E(ℱ)→[0,∞)

the Néron-Tate canonical height function; it can be defined by 
	
    ĥ_E(P) = 1/2lim_n→∞h(x([n]P))/n^2

where h is the naive Weil height on ^1 and x: E →^1 is the degree 2 projection to the x-coordinate.  

For each v∈ M_ℱ, we let _v denote a minimal, algebraically closed field containing ℱ which is complete with respect to |·|_v.  For each v, we fix an embedding of ℱ into _v.  The canonical height has a decomposition into local heights, as 
	
    ĥ_E(P)= 1/|( ℱ/ℱ)· P| ∑_Q ∈( ℱ/ℱ)· P ∑_v∈ M_ℱ n_v  λ̂_E,v(Q)
 
for P∈ E(ℱ)∖{O}, with the local heights λ̂_E,v characterized by three properties <cit.>:


  * λ̂_E,v is continuous on E(_v)∖{O} and bounded on the complement of any v-adic neighborhood of O;

  * the limit of λ̂_E,v(P) - 1/2log|x(P)|_v exists as P → O in E(_v); and

  * for all P = (x,y) ∈ E(_v) with [2] P ≠ O, 
	
    λ̂([2]P) = 4 λ̂(P) -  log|2y+a_1x+a_3|_v + 1/4log|Δ|_v.


 



 §.§ 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 
	
    ĥ_E(P) ≠0

when viewing P as a point on the elliptic curve E defined over k = (B).  For each t∈ B(K) such that the fiber E_t is non-singular, we have point P_t∈ E_t(K).  We will investigate the function 
	
    t ↦ĥ_E_t(P_t)

which is well defined at all but finitely many t ∈ B(K).  Furthermore, via the embedding of  into _v for each place v∈ M_K, we may view E→ B as defined over _v and consider the Néron local heights λ̂_E_t, v(P_t) on the non-singular fibers E_t as functions of t ∈ B(_v).   


Let D_E(P) be the -divisor on B defined by 

    D_E(P) = ∑_γ∈ B()λ̂_E,_γ (P) · (γ).

Here, λ̂_E,_γ (P) is the local canonical height of the point P on the elliptic curve E over k = (B) at the place ord_γ, for each γ∈ B().  The degree of D_E(P) is equal to ĥ_E(P).  It follows from the definitions that D_E(P) is a subset of the finite set
	
    {t ∈ B(): E_t }∪{t ∈ B(): P_t = O_t}.

By enlarging K, we may assume that the support of D_E(P) is contained in B(K).  



 
That D_E(P) is a -divisor is standard, following from the fact that the numbers λ̂_E,_γ (P) can be viewed as arithmetic intersection numbers on a Néron local model.  See <cit.> for a proof that ĥ_E(P)∈; see <cit.> and <cit.> for proofs that each local function λ̂_E,v also takes values in ; see <cit.> for a dynamical proof.




 §.§ Variation of canonical height: quasi triviality
 
Let h_D_E(P) be an analytic Weil height on B() as defined in <cit.>.  That is, we let g be the genus of B, and for each point γ∈ B(K), we choose an element ξ_γ of K(B) which has a pole of order 2g+1 at γ and no other poles.  For each non-archimedean place v of K, set
	
    λ_D_E(P), v(t) = 1/2g+1 ∑_γ∈ B(K)λ̂_E,_γ (P)  log^+|ξ_γ(t)|_v

for all t ∈ B(_v) ∖ D_E(P).  For archimedean places v, the local height is defined by 
	
    λ_D_E(P), v(t) = 1/2(2g+1) ∑_γ∈ B(K)λ̂_E,_γ (P)  log(1 + |ξ_γ(t)|_v^2).

We set
		
    h_D_E(P)(t) = 1/|(/K)· t| ∑_s ∈(/K)· t ∑_v∈ M_Kλ_D_E(P), v(s)
 
for all t ∈ B().  For fixed choices of ξ_γ, we will call the associated height h_D_E(P) our “reference height" for the divisor D_E(P).  Silverman proved:

  <cit.>
For any choice of reference height h_D_E(P), there is a finite set S of places so that 
 	
    λ̂_E_t, v(P_t) = λ_D_E(P), v(t)

for all t ∈ B() ∖ D_E(P) and all v ∈ M_K ∖ S.  




 §.§ Variation of canonical height: continuity
  
Fix a point t_0∈ B() and a uniformizer u ∈(B) for t_0, and consider the function

    V_P, t_0, v(t):=λ̂_E_t,v(P_t) + λ̂_E,_t_0 (P) log|u(t)|_v,

which is not a priori defined at t_0.  Theorem <ref> implies that 
	
    V_P,t_0,v≡ 0

for all but finitely many places v in a v-adic neighborhood of each t_0.  
Silverman also proved the following:

   <cit.>
Fix t_0 ∈ B() and a uniformizer u at t_0.  For all v ∈ M_K, there exists a neighborhood U ⊂ B(_v) containing t_0 so that the function V_P, t_0,v of (<ref>) extends to a continuous function on U.








§ A DYNAMICAL PERSPECTIVE



Recall that the Néron-Tate height ĥ_E and its local counterparts λ̂_E,v can be defined dynamically.  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 ^1, via the standard quotient identifying a point P with its additive inverse:

    E [d]_π[r]^ϕ    E[d]^π
    ^1   [r]^f_ϕ   ^1

An elementary, but key, observation is that a point is torsion on E if and only if its quotient in ^1 is preperiodic for f_ϕ.  The height ĥ_E on E() satisfies
	
    ĥ_E (P) = lim_n→∞1/4^n h(f_ϕ^n(π P))

where h is the standard logarithmic Weil height on ^1().  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 height functions t↦λ̂_E_t,v(P_t) and the extensions V_P, t_0, v of (<ref>).  We will present this fact as an immediate consequence of now-standard complex-dynamical convergence arguments, at least when the fiber E_t is smooth and the local height λ̂_E_t,v(P_t) is finite.  Near singular fibers, we utilize the maximum principle and standard results on removable singularities for subharmonic functions.  The same reasoning applies in both archimedean and non-archimedean settings.    

In <ref> we provide the background to justify the explicit description of the limiting distribution μ_P,v at the archimedean places v of K, as mentioned in Remark <ref>.




 §.§ Canonical height and escape rates
  
As in <ref>, 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 ^1 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 (<ref>). In coordinates, we have ϕ(x)=x^4-b_4x^2-2b_6x-b_8 and ψ(x)=4x^3+b_2x^2+2b_4x+b_6=(2y+a_1x+a_3)^2 for P = (x,y).  

By a lift of f, we mean any homogeneous polynomial map F on ^2, defined over , so that τ∘ F = f ∘τ, where τ: ^2∖{(0,0)}→^1 is the tautological projection.  A lift of a point x ∈^1 is a choice of X ∈^2∖{(0,0)} so that τ(X) = x.  


The standard lift of f will be the map F:𝔸^2→𝔸^2 defined by 

    F(z,w) = (w^4ϕ(z/w), w^4ψ(z/w)).

For each v ∈ℳ_, the v-adic escape rate is defined by 

    _F,v(z,w)=lim_n→∞log||F^n(z,w)||_v/4^n

where 
	
    (z,w)_v = max{|z|_v, |w|_v}.
  
Any other lift of f is of the form cF for some c ∈^*; observe that 

    _cF,v = _F,v + 1/3log|c|_v.

Note that 
	
    _F,v(α x, α y) = _F,v(x,y) + log |α|_v

for any choice of lift F. Furthermore, _F,v is continuous on (_v)^2∖{(0,0)}, as proved in the archimedean case by <cit.>.  For non-archimedean absolute values v, _F,v extends continuously to the product of Berkovich affine lines ^1, an_v ×^1,an_v ∖{(0,0)} <cit.>.  


  
For the standard lift F of f, and for each place v of , the local canonical height function satisfies
	
    λ̂_E,v(P) = 1/2_F,v(x,y) - 1/2log|y|_v - 1/12log|Δ|_v

where x(P) = (x:y).


The proof is immediate from the properties of _F,v by checking the three characterizing conditions for λ̂_E,v.  




 §.§ 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 = (B); viewing E as an elliptic curve defined over k, we also fix a point P ∈ E(k).    Recall the function V_P, t_0, v(t) defined in (<ref>).  

  
For every t_0 ∈ B() and uniformizer u in k at t_0, the function 
	
    V_P, t_0, v(t):=λ̂_E_t,v(P_t) + λ̂_E,_t_0 (P) log|u(t)|_v,

extends to a continuous and subharmonic function on a neighborhood of t_0 in the Berkovich analytification B_v^an.


The continuity was already established in Theorem <ref>, though it was not explicitly stated for the Berkovich space.  The argument below takes care of that.  We begin with a lemma. 

 
Fix α∈ k^* and t_0 ∈ B().  Let u ∈ k be a uniformizer at t_0.  For each place v of K, the function 
	
    t↦log|α_t|_v - (_t_0α)  log|u(t)|_v

is harmonic in a neighborhood of t_0 in the Berkovich analytification B_v^an.


This is Silverman's <cit.> plus a removable singularities lemma for harmonic functions.  See also <cit.> for the extension of a harmonic function to a disk in the Berkovich space B_v^an. 


Fix P ∈ E(k).  Let F and X be lifts of f and x(P) to k^2, respectively.  Iterating F, we set
	
    (A_n, B_n) := F^n(X) ∈ k^2

and observe that

    𝒢_F, _t_0(X) = - lim_n→∞min{_t_0 A_n, _t_0 B_n}/4^n

from the definition of the escape rate.  We let F_t and X_t denote the specializations of F and X at a point t ∈ B(); they are well defined for all but finitely many t.  Observe that if F is the standard lift for E then so is F_t for all t.  



  
Fix P ∈ E(k), t_0 ∈ B(), and v∈ M_K.  For any choice of lifts F of f and X of x(P), the function 
	
    G_P(t;v) := 𝒢_F_t, v(X_t) + 𝒢_F, t_0(X) log|u(t)|_v

extends to a continuous and subharmonic function in a neighborhood of t_0 in B_v^an.  


First observe that the conclusion does not depend on the choices of F and X.  Indeed, 

    𝒢_c_t F_t, v(α_t X_t) + 𝒢_c F, t_0(α X) log|u(t)|_v    =   𝒢_F_t, v(X_t) + 𝒢_F, t_0(X) log|u(t)|_v  
          	+  1/3( log|c_t|_v - (_t_0c) log|u(t)|_v) 
          	+  log|α_t|_v - (_t_0α)log|u(t)|_v

for any c, α∈ k^*.  So by Lemma <ref> the function G_P(t; v) is continuous and subharmonic for one choice if and only if it is continuous and subharmonic for all choices.   

Let F be the standard lift of f.  Suppose that P=O.  Since F(1,0) = (1,0), we compute that 
 
    G_O(t;v)=  𝒢_F_t, v(1,0)+ 𝒢_F, t_0(1,0) log|u(t)|_v ≡ 0.


Now suppose that P ≠ O.  Fix t_0∈ B(K) and local uniformizer u at t_0.  Choose a lift F of f so that the coefficients of F have no poles at t_0, with F_t_0≠ (0,0).  Choose lift X of x(P) so that X_t is well defined for all t near t_0 and X_t_0≠ (0,0).  As above, we write 
	
    F^n(X) = (A_n, B_n)

and put 
	
    a_n = min{_t_0 A_n, _t_0 B_n}

so that a_n≥ 0 for all n and a_0 = 0.  Set
      
    F_n(t) = F_t^n(X_t)/u(t)^a_n.

For each place v, we set
	
    h_n,v(t) = logF_n(t)_v/4^n.

By construction, the limit of h_n,v (for t near t_0 with t≠t_0) is exactly the function G_P for these choices.  In fact, for t in a small neighborhood of t_0, but with t≠ t_0, the function f_t on ^1 is a well-defined rational function of degree 4; so the specialization of the homogeneous polynomial map F_t satisfies F_t^-1{(0,0)} = {(0,0)}.  Furthermore, as the coefficients of F_t depend analytically on t, the functions h_n,v converge locally uniformly to the function G_P away from t=t_0.  This can be seen with a standard telescoping sum argument, used often in complex dynamics, as in <cit.>.  In particular, G_P is continuous on a punctured neighborhood of t_0.  

At the archimedean places v, and for each n, the function h_n,v is clearly continuous and subharmonic in a neighborhood of t_0.  At non-archimedean places v, this definition extends to a Berkovich disk around t_0, setting 
	
    h_n,v(t) = 1/4^nmax{log[A_n(T)/T^a_n]_t, log[B_n(T)/T^a_n]_t}

where [·]_t is the seminorm on K[[T]] associated to the point t.  Each of these functions h_n,v is continuous and subharmonic for t in a Berkovich disk around t_0.  (Compare <cit.> Example 8.7, Proposition 8.26(D), and equation (10.9).)  

 
For all v, and by shrinking the radius r if necessary, the functions h_n,v are uniformly bounded from above on the (Berkovich) disk D_r.  


As observed above, the functions h_n,v converge locally uniformly away from t=t_0 to the continuous function G_P(t).  Choose a small radius r, and let 
	
    M_v = sup_n max_|t|_v=r h_n,v(t)

which is finite by the convergence.  Because the functions are subharmonic, the Maximum Principle implies that h_n,v(t) ≤ M_v throughout the disk of radius r, for all n.  For the non-archimedean places, there is also a Maximum Principle on the Berkovich disk, where the role of the circle of radius r is played by the Type II point associated to the disk of radius r (see <cit.> Proposition 8.14).  


We can now complete the proof of Proposition <ref>.  As each h_n,v is subharmonic, and the functions are uniformly bounded from above on the disk by Lemma <ref>, we know that the (upper-semicontinuous regularlization of the) limsup of these functions is subharmonic.  See <cit.> Proposition 8.26(F) for a proof in the non-archimedean case. 


[Proof of Theorem <ref>]
Subharmonicity now follows from Proposition <ref>, Lemma <ref>, and Proposition <ref>.  The continuity at each archimedean place is the content of Theorem <ref>.  The continuity at each non-archimedean place is a combination of the continuity on the punctured Berkovich disk (as in the proof of Proposition <ref>) and the continuity on Type I (classical) points given in Theorem <ref>.  




 §.§ The measures on the base
  
Here we provide more details about the description of the measures appearing in the statement of Corollary <ref>, as discussed in Remark <ref>.  

Fix an archimedean place v and any point t_0 ∈ B().  Choosing a uniformizer u at t_0, recall the definition of V_P, t_0, v from (<ref>).  We define 
	
    μ_P,v := dd^c V_P,t_0, v(t)

on a neighborhood of t_0 in B_v^an; note that this is indepedent of the choice of u.  Note that μ_P,v can be expressed as 
	
    μ_P,v = dd^c λ̂_E_t, v(P_t)

for t outside of the finitely many points in the support of the divisor D_E(P) or where the fiber E_t is singular.  Note, further, that μ_P,v assigns no mass to any individual point t_0, because the potentials are bounded by Theorem <ref>.  The details on the metric and the equidistribution theorem in Section <ref> will show that these are exactly the measures that arise as the distribution of the points of small height in Corollary <ref>.

It is well known that the local height function on a smooth elliptic curve is a potential for the Haar measure.  That is, for fixed t we have
	
    dd^c λ̂_E_t, v(·) = ω_t - δ_o

where ω_t is the normalized Haar measure on E_t and δ_o is a delta-mass supported at the origin of E_t; see, e.g., <cit.>.  We present an alternative proof of this fact related to dynamics as part of Proposition <ref>, as a consequence of Proposition <ref>.  

  
Let E → B be an elliptic surface and P: B → E a section, both defined over a number field K.  Let ⊂ E be the union of the finitely many singular fibers in E.  For each archimedean place v of K, there is a positive, closed (1,1) current T_v on E∖ with locally continuous potentials so that T_v|_E_t is the Haar measure on each smooth fiber, and P^* T_v is equal to the measure μ_P,v.   



As T_v has continuous potentials, the restriction T_v|_E_t and the pullback P^* T_v are well defined.  That is, we have T_v|_E_t = dd^c (u|_E_t) where u is a locally defined potential of T_v, and P^*T_v = dd^c (u∘ P) locally on B.  The measure μ_P,v has no atoms, so it is determined by T_v along the image of P in E∖.  



[Proof of Proposition <ref>]
Let us fix any small neighborhood U in the base curve B() so that all fibers E_t are smooth for t∈ U.  Let f_t be the map on ^1 defined in <ref>; by shrinking U if necessary, we can find lifts F_t of f_t that are holomorphic in t ∈ U.  From <cit.> (or the proof of <cit.>), we know that the escape rate
	
    _F_t,v(z,w)=lim_n→∞log||F_t^n(z,w)||_v/4^n

is continuous and plurisubharmonic as a function of (t, z,w) ∈ U× (^2∖{(0,0)}).  Therefore
	
    dd^c _F_t,v(z,w)

projects to a closed and positive (1,1)-current G_v on the complex surface U×^1, with locally continuous potentials.  This current G_v has the property that, restricted to each fiber ^1, its total mass is 1; and the measure on the fiber is the measure of maximal entropy for the rational map f_t <cit.>.  

The restriction E|_U of the elliptic surface E over U maps with degree 2 to the complex surface U×^1 by the projection π of (<ref>).  The current G_v can be pulled back to E as 1/2  dd^c(g ∘π) where g is a locally-defined continuous potential for G_v.  Covering the base of E∖ by sets of the form U, the local definitions glue to form the closed, positive (1,1)-current T_v on E∖.  


If P: B → E is a section defined over the number field K, then P^* T_v has potential given locally by 
	
    1/2   g∘π∘ P =  1/2 _F_t, v(X_t)

for any lift X_t of π(P_t) ∈^1.  Proposition <ref> yields that P^* T_v must coincide with μ_P,v.  

Finally, to conclude that T_v|_E_t is equal to the normalized Haar measure ω_t, we may use the well-known dynamical fact that for each fixed t in the base, the measure ω_t projects by π to ^1 to the unique measure of maximal entropy for the map f_t; see, e.g., <cit.>.








§ THE ADELIC METRIC AND EQUIDISTRIBUTION



In this section we give the proofs of Theorem <ref> and Corollary <ref>.  

We first outline the proofs.  Let E → B be an elliptic surface defined over a number field K with zero section O:  B → E, and let P:B→ E be a section also defined over K so that ĥ_E(P) ≠0.  Recall from <ref> that we introduced a -divisor 
	
    D_E(P) = ∑_γ∈ B()λ̂_E, _γ(P) · (γ)

on B.  By enlarging K, we may assume that D_E(P) lies in B(K).  We will define an adelic metric on the ample line bundle  associated to the divisor D_E(P), inducing a height function h_ such that
	
    h_(t) = ĥ_E_t(P_t)  t ∈ B()

and
	
    h_(t) ≥ 0  t ∈ B().

Applying Silverman's results on the variation of canonical height, Theorems <ref> and <ref>, we will deduce that the metric is continuous and adelic.  From Theorem <ref>, we will conclude that the metric is also semi-positive in the sense of Zhang <cit.>.  We will use Zhang's inequalities <cit.> to deduce that 
	
    h_(B) = 0.

Consequently, we will be able to apply the equidistribution results of Chambert-Loir, Thuillier, and Yuan <cit.> to complete our proofs.  




 §.§ The metric and its properties
 
Let m∈ℕ be such that 
	
    D = m· D_E(P)

is an integral divisor.  Let _m be the associated line bundle on B.  Note that (_m)=m  ĥ_E(P)>0 so ℒ_m is ample; by replacing m with a multiple, we may assume that _m is very ample.  

Fix a place v of K.  Let U be an open subset of B_v^an.  Each section s ∈_m(U) is identified with a meromorphic function f on U satisfying 
	
    (f) ≥ -D.

We set
	
    s(t)_v = {[ e^-mλ̂_E_t,v(P_t)|f(t)|_v                      f(t);                         0   _t f > - m  λ̂_E, _t(P);        e^-m  V_P, t, v(t)                         , ].

taking the locally-defined uniformizer u = f^1/_t f at t in the definition of V_P, t, v from (<ref>).  

  
The metric · = {·_v}_v ∈ M_K on _m is continuous, semipositive, and adelic.


The continuity and semipositivity follows from Theorem <ref>.  (In <cit.>, semipositivity of a continuously metrized line bundle on a curve is defined terms of subharmonicity of potentials for the curvature form at each archimedean place, and as a uniform limit of “smooth semipositive" metrics at each non-archimedean place.  In <cit.>, it is established that subharmonicity of potentials is a sufficient notion at all places, and he proves in <cit.> that this notion of semipositivity coincides with that of Zhang <cit.>.  See also <cit.> where this same argument is applied in a dynamical context.)  The adelic condition follows from Theorem <ref>.  




 §.§ The associated height function

A height function on B() is defined by setting 
	
    h_P(t) :=  1/m 1/| (/K)· t| ∑_s ∈(/K)· t∑_v∈ M_K -n_v  logϕ(s)_v

where ϕ is any global section of _m which is nonvanishing along the Galois orbit of t, and ·_v is the metric of <ref>.   Recall that D_E(P) ⊂ B(K); we may assume that our sections ϕ are defined over K, and the product formula guarantees our height is independent of the choice of ϕ.  

Our next goal is to prove the following two important facts about this height function h_P.  

 
The height function h_P satisfies
	
    h_P(t)= ĥ_E_t(P_t)

for all t∈ B() such that the fiber E_t is smooth.


  
The height function h_P satisfies
	
    h_P(t) ≥ 0

for all t ∈ B().


[Proof of Proposition <ref>]
First fix t∈ B()∖ D_E(P) with smooth fiber E_t.  Choose a section ϕ defined over K that does not vanish along the Galois orbit of t, and let f be the associated meromorphic function on B.  Then f takes finite and nonzero values along the Galois orbit of t.  We have, 

    h_P(t)    =   1/m 1/| (/K)· t|∑_s ∈(/K)· t ∑_v ∈ M_K  n_v   (m λ̂_E_s, v(P_s) - log|f(s)|_v) 
       =   1/m 1/| (/K)· t|∑_s ∈(/K)· t ∑_v ∈ M_K  m  n_v  λ̂_E_s, v(P_s) 
       =   ĥ_E_t(P_t).

where the second equality follows from the product formula.  

For t_0∈ D_E(P) such that E_t_0 is smooth, it is necessarily the case that P_t_0 = O_t_0, and therefore ĥ_E_t_0(P_t_0) = 0.  To compute h_P(t_0), observe that t_0 ∈ B(K) so its Galois orbit is trivial; fixing a uniformizer u∈ K(B) at t_0, we have 
	
    h_P(t_0) = ∑_v∈ M_K  n_v   V_P, t_0, v(t_0)

where V_P,t_0, v is the function of (<ref>) associated to the uniformizer u. 

We can compute h_P(t_0) using the dynamical interpretation of the local heights, described in Section <ref>.  Fix a Weierstrass equation for E in a neighborhood of t_0 and write P=(x_P,y_P). The assumption that P_t_0=O_t_0 is equivalent to _t_0x_P<0.  After possibly shrinking U, write x_P=(u)^_t_0(x_P)A_0 for the chosen uniformizer u and a function A_0∈ K(B) that does not vanish in U. We choose a lift X of x_P on U defined as X=(A_0,B_0), where B_0:=(u)^-_t_0(x_P). Notice that A_0 and B_0 are regular at t_0.  Let F be the standard lift in these coordinates, defined in (<ref>); it satisfies F_t_0(1,0) = (1,0), and we have _F,_t_0(A_0,B_0)=0.  Since _t_0Δ_E=0, Proposition <ref> implies that
	
    V_P,t_0,v(t)=1/2 _F_t,v(A_0(t),B_0(t))-1/12 log|Δ_E(t)|_v

for all t ∈ U.  Therefore, 

    V_P, t_0, v(t_0)    =   1/2 _F_t_0,v(A_0(t_0),0) -1/12 log|Δ_E(t_0)|_v 
       =   1/2 lim_n→∞1/4^nlogF^n_t_0(A_0(t_0),0)_v - 1/12 log|Δ_E(t_0)|_v 
       =   1/2 lim_n→∞1/4^nlog |A_0(t_0)^4^n|_v - 1/12 log|Δ_E(t_0)|_v 
       =   1/2 log|A_0(t_0)|_v - 1/12 log|Δ_E(t_0)|_v.

The product formula now yields that h_P(t_0)=0, as claimed.



To prove Proposition <ref>, we first reduce to the case that the elliptic surface E→ B has semi-stable reduction; that is all of its fibers are either smooth or have multiplicative reduction.  The next lemma describes how the height associated with the divisor D_E(P) behaves under base extensions of the elliptic surface E→ B. It is adapted from <cit.>. We include it here for completeness.


Let μ: B'→ B be a finite map of smooth projective curves, let E'→ B' be a minimal model for E×_B B', and let P':B'→ E' be the extension of the section P. For each t_0 ∈ B() and t_0'∈μ^-1({t_0})⊂ B'(_v), there is a neighborhood U of t_0' in B'(_v) and a regular non-vanishing function f on U such that 

    V_P,t_0,v(μ(t'))-V_P',t_0',v(t')=log|f(t')|_v

on U∖{t'_0}. 
In particular,

    V_P,t_0,v(t_0)-V_P',t_0',v(t'_0)=log|f(t'_0)|_v.




Let u be a uniformizer at t_0, u' a uniformizer at t_0' and n=_t'_0(μ^*u). Since local heights are invariant under base extension we have 

    λ̂_E', _t'_0(P')=n λ̂_E, _t_0(P).

Notice that for all t' in a punctured neighborhood of t'_0 the fibers E'_t' are smooth. Hence the map E'→ E gives an isomorphism between the fibers E'_t'→ E_μ(t'). Under this isomorphism P'_t'∈ E'_t' is mapped to P_μ(t')∈ E_μ(t'). Invoking now the uniqueness of the Néron local heights, we have 

    λ̂_E_μ(t'),v(P_μ(t'))=λ̂_E'_t',v(P'_t').

Combining (<ref>) and (<ref>) we get that for t' in a punctured neighborhood of t_0', 

    V_P,t_0,v(μ(t'))=V_P',t_0',v(t')+λ̂_E,_t_0(P)log|u(μ(t'))/u'^n(t')|_v.

The definition of n yields that the function f(t')=(u(μ(t'))/u'^n(t'))^λ̂_E, _t_0(P) is regular and non-vanishing at t_0'. The first part of the lemma follows. 

Finally, Theorem <ref> allows us to conclude that 

    V_P,t_0,v(μ(t'_0))-V_P',t_0',v(t'_0)=log|f(t'_0)|_v

at the point t'_0, as claimed.  



The following lemma will allow us to prove Proposition <ref> in the case that a fiber has multiplicative reduction. The proof is lengthy, but it is merely a collection of computations using the explicit formulas for the local height functions, as in <cit.>.

   
Let E→ B be an elliptic surface and let P:B→ E be a non-zero section defined over K. Then there exists a finite extension L of the number field K so that, for each t_0∈ B() such that E_t_0 has multiplicative reduction, there exists an x(t_0) ∈ L^* so that 
	
    V_P,t_0,v(t_0) = log|x(t_0)|_v

at all places v of L.  



We let

    E:y^2=x^3+ax+b,

be a minimal Weierstrass equation for E over an affine subset W⊂ B defined over K with t_0∈ W. Here a,b∈ K(B) are regular functions at t_0. 
Using this Weierstrass equation we write 

    P=(x_P,y_P),

where x_P,y_P∈ K(B).
Since E→ B has multiplicative reduction over t_0∈ B(K), we have 

    N:=_t_0Δ_E≥ 1 and min{_t_0a,_t_0b}=0.

Let v be a place of K (archimedean or non-archimedean). We denote by j_E the j-invariant of E→ W, given by 
    j_E(t)=1728(4a(t))^3/Δ_E(t).
 
Notice that equation (<ref>) yields that j_E has a pole at t_0. Hence, we can find a v-adic open neighborhood U of t_0 and an analytic map 

    ψ: U→{q∈_v : |q|_v<1},

such that the following holds: If j is the modular j-invariant <cit.>, then we have

    j_E(t)=j(ψ(t)) and _t_0ψ=N.

The function ψ(t) is given as 

    ψ(t)=1/j_E(t)+744/j^2_E(t)+750420/j^3_E(t)+…∈ℤ[[(j_E(t))^-1 ]].

In the following, we choose a uniformizer u ∈ K(B) at t_0, and we identify ψ with its expression ψ(t)∈_v[[u]] and write 

    ψ(t)=β u(t)^N+u(t)^N+1f(t), for t∈ U∖{t_0}.

Equation (<ref>) yields that β∈ K∖{0} and f(t)∈ K[[u]]. 
Following the proof of  <cit.> and after possibly shrinking U we have isomorphisms

    E_t(_v)   _v^*ψ(t)^ C_ψ(t):y^2=4x^3-g_2(ψ(t))x-g_3(ψ(t)),

for t∈ U∖{t_0}.
Under these isomorphisms, we have

    P_t   ↦   w(t)  ↦ (℘(w(t),ψ(t)),℘'(w(t),ψ(t))).

Here g_2,g_3 are the modular invariants, given by their usual q-series 

    g_2(q)=1/12(1+240∑_n=1^∞n^3q^n/1-q^n), g_3(q)=1/216(-1+504∑_n=1^∞n^5q^n/1-q^n)

and ℘ is the Weierstrass ℘-function given by

    ℘(w,q)=1/12+∑_n∈q^nw/(1-q^nw)^2-2∑_n=1^∞nq^n/1-q^n, ℘'(w,q)=∑_n∈q^nw(1+q^nw)/(1-q^nw)^3.


In view of <cit.>, after possibly replacing P by -P, we may assume that w:U→_v is an analytic map satisfying

    0≤_t_0w≤1/2_t_0ψ.

In the following we identify w with its series in _v[[u]] and write 

    w(t)= u^m(t)+u^m+1(t)g(t),

where ∈_v and g(t)∈_v[[u]].

We claim that w(t)∈K[[u]]. To see this, notice that from <cit.> we have that for t∈ U  

    (℘(w(t),ψ(t)),℘'(w(t),ψ(t))=(ν^-2(t)x_P(t),2ν^-3(t)y_P(t)),

where

    ν(t)^12=Δ_E(t)/Δ(ψ(t)).

In the equation above Δ denotes the modular discriminant given by

    Δ(q)=g_2(q)^3-27g_3(q)^3.

Since the functions ψ,Δ_E and Δ are defined over K, we have that Y(t):=2ν^-3(t)y_P(t) is also defined over K.  Since Y(t)=℘'(w(t),ψ(t))∈ K[[u]] and ψ(t)∈ K[[u]] we get that w(t)∈ K[[u]].

Therefore, there are non-zero constants ,β,γ∈K∖{0}, non-negative integers k,m∈ and functions f(t), g(t), h(t)∈K[[u]] such that for all t∈ U 

    ψ(t)=β u^N(t)+f(t)u^N+1(t), w(t)= u^m(t)+g(t)u^m+1, 1-w(t)=γ u^k(t)+h(t)u^k+1(t).


Next, we aim to express x(t_0) (as in the statement of the lemma) in terms of ,β,γ∈K.

Using the isomorphisms in <ref>, the uniqueness of the local canonical heights and the explicit formulas for the local canonical heights <cit.>, we get

    λ̂_E_t,v(P_t)=λ̂(w(t),ψ(t))   =-1/2B_2(log|w(t)|_v/log|ψ(t)|_v)log|ψ(t)|_v-log|1-w(t)|_v
       -∑_n≥ 1log|(1-ψ(t)^nw(t))(1-ψ(t)^nw(t)^-1)|_v,

where B_2(s)=s^2-s+1/6 is the second Bernoulli polynomial.

Since _t_0ψ=N≥ 1 and using (<ref>), we get

    lim_tv→ t_0∑_n≥ 1log|(1-ψ(t)^nw(t))(1-ψ(t)^nw(t)^-1)|_v=0.

In what follows, for F(t)∈_v[[u]] we write

    F(t):=o_v(1), if lim_tv→ t_0F(t)=0.

In view of <cit.>, we have

    B_2(log|w(t)|_v/log|ψ(t)|_v)log|ψ(t)|_v   =log^2|w(t)|_v/log|ψ(t)|_v-log|w(t)|_v+1/6log|ψ(t)|_v
       =m^2/Nlog|u(t)|_v+m/N^2log(||_v^2N/|β|_v^m)-log||_v
       -mlog|u(t)|_v+log|β|_v/6+N/6log|u(t)|_v+o_v(1)

Using equations  (<ref>) and (<ref>), equation (<ref>) yields

    λ̂_E_t,v(P_t)+1/2(m^2/N-m+N/6+2k)log|u(t)|_v   =-1/2(m/N^2log(||_v^2N/|β|_v^m)-log||_v+log|β|_v/6)
       -log|γ|_v+o_v(1).

Finally, notice that <cit.> implies

    λ̂_E,_t_0(P)=_t_0(1-w)+1/2B_2(_t_0 w/_t_0ψ)_t_0ψ=1/2(m^2/N-m+N/6+2k).

Therefore

    V_P,t_0,v(t_0)   =lim_tv→ t_0V_P,t_0,v(t)=-1/2(m/N^2log(||_v^2N/|β|_v^m)-log||_v+log|β|_v/6)-log|γ|_v
       =log|x(t_0)|_v,

where x(t_0)=β^m^2/2N^2-1/2/γ^m/N-1/2 belongs to a finite extension of K, denoted by L.




[Proof of Proposition <ref>]
By <cit.> there is a finite map of smooth projective curves B'→ B such that if E'→ B' is a minimal model for E×_BB', then E' has semi-stable reduction over the singular fibers of E→ B. Moreover, we may choose B' so that everything is defined over K. Thus, by Lemma <ref> and using the product formula, we may assume that the singular fibers of our elliptic surface E → B have multiplicative reduction.  

For all t ∈ B() for which E_t is smooth, we know from Proposition <ref> that h_P(t) = ĥ_E_t(P_t).  The canonical height is always non-negative, so we may conclude that h_P(t)≥ 0 for all such t.  

Assume now that t_0 ∈ B() has a fiber with multiplicative reduction.  Enlarging the number field K if necessary we may assume that t_0∈ B(K) and that its corresponding x(t_0) defined in the statement of Lemma <ref> is in K^*. Then, on using the product formula, Lemma <ref> implies that h_P(t_0) = 0.  This completes the proof.  





 §.§ Proofs of Theorem <ref> and Corollary <ref>


[Proof of Theorem <ref>]
Let _P be the line bundle on B induced from the divisor D_E(P).  From Theorem <ref>, we know that its m-th tensor power can be equipped with a continuous, adelic, semipositive metric, so that the corresponding height function is (a multiple of) the canonical height ĥ_E_t(P_t) on the smooth fibers.  Thus, by pulling back the metric to _P, we obtain a continuous, semipositive, adelic metric on _P inducing the desired height function.  

It remains to show that this height h_P satisfies h_P(B) = 0.  This is a consequence of Propositions <ref> and <ref> and Zhang's inequalities on successive minima <cit.>.  Recall that, since ĥ_E(P) ≠0, we know that there are infinitely many t∈ B() for which 
	
    ĥ_E_t(P_t) = 0.

(For a complex-dynamical proof, see <cit.>.)  Therefore, from Proposition <ref>, we may deduce that the essential minimum of h_P on B is equal to 0.  On the other hand, from Proposition <ref>, we know that h_P(t) ≥ 0 for all t ∈ B().  Therefore, from <cit.>, we may conclude that h_P(B) = 0. 


[Proof of Corollary <ref>]
When combined with the equidistribution theorems of Yuan and Thullier <cit.>, we immediately obtain the corollary from Theorem <ref>.  The measures μ_P,v are the curvature distributions associated to the metrics ·_v at each place v.  From the definition of the metric in <ref>, we see that they are given locally by 
	
    μ_P,v = dd^c V_P,t_0, v(t)

in a v-adic neighborhood of any point t_0∈ B(), and for any choice of uniformizer u at t_0.  







§ PROOF OF THEOREM <REF>.




 §.§ Reduction to the case of a fiber product of elliptic surfaces
 

We first show that, to prove the theorem, it suffices to prove the result for sections of the fiber product A = E_1 ×_B⋯×_BE_m of m≥ 2 elliptic surfaces E_i→ B over the same base, and to assume that the line bundle  is generated by the divisor 
    {O_E_1}× E_2×⋯ E_m+E_1×{O_E_2}×⋯× E_m+⋯ + E_1× E_2×⋯×{O_E_m}.


Let B be a quasiprojective smooth algebraic curve defined over .  Suppose A →  B is family of abelian varieties defined over  that is isogenous to a fibered product of m≥ 2 elliptic curves. That is, there is a branched cover B' → B and m≥2 elliptic surfaces E_i → B'  that give rise to an isogeny 
	
    E_1×_B'⋯×_B' E_m → A

over B'.  
Now let  be a line bundle on A which restricts to an ample and symmetric line bundle on each fiber A_t for t∈ B. Then the line bundle  pulls back to a line bundle ' on E_1×_B'⋯×_B' E_m, and it again restricts to an ample and symmetric line bundle on each fiber over t ∈ B'. 

Now suppose that we have a section P: B → A.  The section P pulls back to a section P': B' → A, and this in turn pulls back to a (possibly multi-valued) section of E_1×_B'⋯×_B' E_m.  If multi-valued, we can perform a base change again, passing to a branched cover B”→ B', so that the induced section P”: B”→ E_1×_B”⋯×_B” E_m is well defined.  By definition, the assumption that P is non-special on A means that it is non-special as a section of E_1×_B”⋯×_B” E_m.  

Finally, we observe that the conclusion of Theorem <ref> does not depend on the choice of line bundle.  (We thank Joe Silverman for his help with this argument.)  Recall that, on any abelian variety A defined over , the notion of a “small sequence" of points is independent of the choice of ample and symmetric line bundle.  That is, if we take two ample and symmetric divisors D_1 and D_2, then we know that there exists an integer m_1>0 so that m_1 D_1 - D_2 is ample; similarly there exists m_2>0 so that m_2 D_2 - D_1 is ample.  It follows from properties of the Weil height machine that the heights h_D_1 and h_D_2 will then satisfy
	
    1/m_1 h_D_2 + C_1 ≤ h_D_1≤ m_2   h_D_2 + C_2

for real constants C_1, C_2.   Upon passing to the canonical height, we conclude that 

    1/m_1ĥ_D_2≤ĥ_D_1≤ m_2  ĥ_D_2

on the abelian variety.  In particular, ĥ_D_1(a_i) → 0 for some sequence in A() if and only if ĥ_D_2(a_i) → 0.  Now suppose we have a family of abelian varieties A → B.  Two line bundles _1 and _2 associated to relatively ample and symmetric divisors induce a canonical heights ĥ__1,t and ĥ__2,t on each fiber A_t.  But recalling that amplitude persists on Zariski open sets <cit.>, there exist positive integers m_1 and m_2 so that the line bundles _1^m_1⊗_2^-1 and _2^m_2⊗_1^-1 are relatively ample on a Zariski open subset of the base B.  Passing to the canonical heights once again, we find that the relation (<ref>) holds uniformly over B (after possibly excluding finitely many points).  Therefore, for any section P: B → A, there exists a positive constant c(_1, P) of Theorem <ref> for height ĥ__1 if and only if it there exists such a constant c(_2, P) for ĥ__2.  



 §.§ Proof for a fiber product of elliptic curves

Fix integer m≥ 2, and let E_i → B for i=1,…,m be elliptic surfaces over the same base curve B, defined over .  Let A = E_1×_B⋯×_B E_m, and let  be the line bundle on E_1×_B ⋯×_B E_m associated to the divisor 

    D = {O_E_1}× E_2×⋯ E_m+E_1×{O_E_2}×⋯× E_m+⋯ + E_1× E_2×⋯×{O_E_m}.

For all but finitely many t ∈ B(), the canonical height ĥ__t on the fiber A_t is easily seen to be the sum of canonical heights  (see, e.g., <cit.> for properties of the height functions), so that
	
    ĥ__t = ∑_i=1^mĥ_E_i,t.

	
Now assume that P = (P_1, …, P_m) is a section of A→ B.  Define
	
    ĥ_i(t) := ĥ_E_i,t(P_i(t))

for i = 1,…, m and for all t ∈ B() where all E_i,t are smooth elliptic curves.  Suppose there exists an infinite sequence {t_n}⊂ B() for which

    ĥ_i(t_n) → 0  for all i=1,…,m.

as n→∞.  We will prove that for every pair (i,j), there exists an infinite sequence {s_n}⊂ B() so that 
	
    ĥ_i(s_n) = ĥ_j(s_n) = 0

for all n.  In this way, we reduce our problem to the main results of <cit.> which imply that the pair (P_i, P_j) must be a special section of E_i ×_B E_j.  Finally, we observe that our definition of a special section P = (P_1, P_2, …, P_m) is equivalent to the statement that every pair (P_i, P_j) is special.  Therefore, for any non-special section P, we can conclude that there exists a constant c = c(P) >0 so that the set
	
    {t ∈ B(): ĥ__t(P_t) < c}
 
is finite.

Fix a pair (i,j). First assume that neither E_i nor E_j is isotrivial.  If P_i or P_j is torsion, then the section (P_i, P_j) is special.  Otherwise, we have ĥ_E_i(P_i) ≠0 and ĥ_E_j(P_j) ≠0, and we may apply Theorem <ref> to deduce that the height functions h_i and h_j are “good" on B.  More precisely, we let M_i and M_j be the adelically metrized line bundles on the base curve B associated to the height functions ĥ_i and ĥ_j, from Theorem <ref>.  They are both equipped with continuous adelic metrics of non-negative curvature.  By assumption, we have 

    ĥ_i(t_n) → 0     ĥ_j(t_n) → 0

as n→∞.  Therefore, we may apply the observation of Chambert-Loir <cit.>, which builds upon on Zhang's inequalities <cit.>, to conclude that there exist integers n_i and n_j so that M_i^n_i and M_j^n_j are isomorphic as line bundles on B and their metrics are scalar multiples of one another.  It follows that the height functions ĥ_i and ĥ_j are the same, up to scale, and in particular they have the same zero sets.  In other words, P_i(t) is a torsion point on E_i,t if and only if P_j(t) is a torsion point on E_j,t (for all but finitely many t in B), and there are infinitely many such parameters t∈ B().

Now suppose that E_i is isotrivial.  The existence of the small sequence t_n in (<ref>) implies that either ĥ_E_i(P_i) ≠0 or P_i is torsion on E_i, and furthermore, if P_i is torsion, then it follows that (P_i, P_j) is a special section of E_i×_B E_j.  Similarly if E_j is isotrivial.  In other words, the existence of the sequence t_n in (<ref>) allows us to conclude that either (P_i,P_j) is a special pair, or we have that both ĥ_E_i(P_i) ≠0 and ĥ_E_j(P_j) ≠0.  Therefore, we may proceed as above in the nonisotrivial case, applying Theorem <ref> to deduce that the heights ĥ_i and ĥ_j coincide, up to scale, and in particular there are infinitely many parameters s∈ B() where 
	
    ĥ_i(s) = ĥ_j(s) = 0.

This concludes the proof of Theorem <ref>.
















§ VARIATION OF CANONICAL HEIGHT, ILLUSTRATED



In this final section, we provide a few illustrations of the distributions μ_P,v for an archimedean place v, arising in Corollary <ref>.  In Proposition <ref>, we present a complex-dynamical proof that the archimedean measures μ_P,v will have support equal to all of B.



 §.§ Images
  Given E→ B and section P, we plot the parameters t where P_t is a torsion point on the fiber E_t of specified order.  As proved in Corollary <ref>, the local height function at each place
	
    t ↦λ̂_E_t,v(P_t)

determines the distribution of the torsion parameters; it is a potential for the measure μ_P,v (away from the singularities).  Recall that if we have two sections P and Q that are linearly related on E, then the distributions of their torsion parameters in B will be the same.

Figure <ref>, top, illustrates the example of Silverman from <cit.>.  Here, we have  
	
    E_t = {y^2 + xy/t + y/t = x^3 + 2 x^2/t}
 
with B = ^1 and P_t = (0,0) in (x,y)-coordinates.  Plotted are the torsion parameters of orders 2^n for all n≤ 8; that is, the points t in the base B where P_t is torsion of order 2^n on the fiber E_t.  Roughly, a smaller yellow dot corresponds to higher order of torsion.  Figure <ref>, bottom, is another section of the same family, where the x-coordinate of P_t is constant and equal to -1/4.  (Strictly speaking, this second P is not a section of our given E→^1, because the y-coordinate will not lie in (B) ≃ℚ(t) but in an extension; however, the property of being torsion and the determination of its order is independent of which point in the fiber we choose.)  Observe the distinctly different pattern of yellow dots in the first and second pictures, especially in the left half of the two pictures, illustrating the linear independence in E(k) of the two sections.  

Figure <ref> illustrates the torsion parameters for two independent sections of the Legendre family, 
	
    E_t = {y^2 = x(x-1)(x-t)}

over B = ^1, studied in <cit.>.   The chosen sections are P_2, with constant x-coordinate equal to 2, and P_5, with constant x-coordinate equal to 5.  As in Figure <ref>, we plot the torsion parameters of orders 2^n for all n≤ 8; generally, a smaller yellow dot signifies higher order of torsion.  It was proved in <cit.> that the limiting distributions for sections with constant x-coordinate satisfy μ_P_x,∞ = μ_P_x', ∞ (at an archimedean place) if and only if x=x'.  It was proved in <cit.> and <cit.> that there are no t ∈^1() for which both (P_2)_t and (P_5)_t are torsion on E_t.   Again, observe the difference in the geometry of the yellow dots for the two independent sections.   

Figure <ref> illustrates our equidistribution result, Corollary <ref>, for the example of the Legendre family with the section P_5. Plotted are the torsion parameters of orders 2^n with (a) n≤ 6, (b) n≤ 8, and (c) n≤ 10.  Observe how the yellow dots fill in the “grid structure" in the base curve B, exactly as do the torsion points for one elliptic curve.  








As mentioned above, the smaller yellow dots in the illustrations correspond, roughly, to higher orders of torsion.  These images are produced with a standard escape-rate algorithm.  We use the dynamical system f_t on ^1, induced from multiplication by 2 on the elliptic curve E_t from Section <ref>, line (<ref>).  The coordinates on ^1 are chosen so that ∞ is the image of the 0 of E_t.  We mark t yellow if |f^n_t(π(P_t))| ≥ 10000 for some n≤ 8.  




 §.§ Density of torsion parameters

In all of these examples, the yellow dots will fill in the picture as the order of torsion grows, and the support of the measures μ_P,v is equal to B().  In fact, this will always be the case, for any (nontrivial) section of a complex elliptic surface, as our final result, Proposition <ref>, shows.  

Let E→ B be an elliptic surface over a projective curve B, defined over , and let P: B→ E a section for which ĥ_E(P) ≠0 (over the function field k = (B)).  Let μ_P be the measure on B defined as in Proposition <ref>, as the pullback of the current T that restricts to Haar measure on each smooth fiber.  In other words, μ_P is locally defined as the Laplacian of the function G_P(t) introduced in Proposition <ref>, which is well defined when working over .

  
Let E→ B be an elliptic surface over a projective curve B, defined over , and let P: B→ E be a section for which ĥ_E(P) ≠0 (over the function field k = (B)).  Then the set 
	
    {t ∈ B:  P_t  E_t}

is dense in B() and 
	
    μ_P = B().



We give a complex-dynamical proof, viewing Proposition <ref> as a consequence of the main result of <cit.>.  (We do not use the equidistribution result, Corollary <ref>.)  An analytic proof is also presented in <cit.>.

Let B^*⊂ B be a finitely-punctured Riemann surface such that the fiber E_t is smooth for all t ∈ B^*.  Let π_t: E_t→^1 be the degree-two projection and f_t: ^1 →^1 be the rational map induced by multiplication-by-2 on E_t, as defined in the introduction to Section <ref>.  It is well known that the holomorphic family {f_t:  t∈ B^*} is structurally stable; see, e.g., <cit.>.  Thus, over any simply-connected subset U of B^*, there is a holomorphic motion of the periodic points of f_t which extends uniquely to a holomorphic motion of all of ^1, conjugating the dynamics.  

The key observation is that μ_P is precisely the “bifurcation measure" of the pair (f,P) on B^*.  See <cit.> and <cit.> for definitions.  The support of μ_P is equal to the bifurcation locus of (f,P); in particular, the parameters t∈ B^* for which π_t(P_t) is preperiodic for f_t are dense in μ_P.  Therefore, it suffices to show that μ_P = B.

Suppose to the contrary that there is an open disk U ⊂ B^* for which μ_P(U) = 0.  Then the pair (f,P) is stable on U, and therefore π_t(P_t) cannot be a repelling periodic point for any t ∈ U.  From the uniqueness of the holomorphic motion, it follows that t↦π_t(P_t) is part of the holomorphic motion on U.  By analytic continuity, then, we deduce that π_t(P_t) must follow the motion of a point over all of B^*.  This implies that the pair (f,P) is stable throughout B^* and the measure μ_P is 0.  But this is absurd by the assumption that ĥ_E(P) ≠0; see <cit.>.  






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