# A Lehmer-Type Lower Bound for the Canonical Height on Elliptic Curves Over
Function Fields

Joseph H. Silverman<EMAIL_ADDRESS>Mathematics Department, Box
1917 Brown University, Providence, RI 02912 USA. OrcID 0000-0003-3887-3248. MR
Author ID 162205.

###### Abstract.

Let $\mathbb{F}$ be the function field of a curve over an algebraically closed
field with $\operatorname{char}(\mathbb{F})\neq 2,3$, and let $E/\mathbb{F}$
be an elliptic curve. Then for all finite separable extensions
$\mathbb{K}/\mathbb{F}$ and all non-torsion points $P\in{E(\mathbb{K})}$, the
$\mathbb{F}$-normalized canonical height of $P$ is bounded below by

$\hat{h}_{E}(P)\geq\frac{1}{4570\cdot
h_{\mathbb{F}}(j_{E})^{2}\cdot[\mathbb{K}:\mathbb{F}]^{2}}.$

###### Key words and phrases:

elliptic curve, Lehmer conjecture, function field

###### 2010 Mathematics Subject Classification:

Primary: 11G05; Secondary: 11R58, 14G40

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Local Canonical Heights
  4. 4 Reprise of Material from Previous Papers
  5. 5 Proofs of the Main Theorems
  6. 6 Some Final Remarks
  7. A Proof of an Inequality
  8. B Local Height Formulas via Weierstrass Equations
  9. C Local Height Formulas via Intersection Theory

##  1\. Introduction

The classical Lehmer conjecture for number fields says that there is an
absolute constant $\Cl{KoverQ}>0$ such that for all number fields
$K/{\mathbb{Q}}$ of degree $D=[K:{\mathbb{Q}}]$ and all $\alpha\in{K^{*}}$
that are not roots of unity, the absolute logarithmic Weil height of $\alpha$
satisfies

$h(\alpha)\geq\frac{\Cr{KoverQ}}{D}.$

The best known result [4] gives a lower bound with an additional factor of
$(\log\log D/\log D)^{3}$, while the best known results for the analogous
conjecture for the canonical height on an elliptic curve $E$ and non-torsion
point $P\in{E(K)}$ are as follows:

$\displaystyle{\hat{h}}_{E}(P)$ $\displaystyle\geq\Cl{EoverK1}(E)\cdot
D^{-3}(\log D)^{-2}$ in general [10], $\displaystyle{\hat{h}}_{E}(P)$
$\displaystyle\geq\Cl{EoverK2}(E)\cdot D^{-2}(\log D)^{-2}$ if $j_{E}$ is non-
integral [6], $\displaystyle{\hat{h}}_{E}(P)$
$\displaystyle\geq\Cl{EoverK3}(E)\cdot D^{-1}(\log\log D/\log D)^{3}$ if $E$
has CM [7].

In this paper we consider the function field analogue of Lehmer’s conjecture
for elliptic curves. We set the following notation, which will be used
throughout this article.

$k$ | an algebraically closed field with $\operatorname{char}(k)\neq 2,3$.
---|---
${\mathbb{F}}$ | the function field of a smooth projective curve
| defined over $k$.
$h_{\mathbb{F}}$ | the absolute Weil height $h_{\mathbb{F}}:{\mathbb{P}}^{n}({\overline{\mathbb{F}}})\to[0,\infty)$.
$E/{\mathbb{F}}$ | an elliptic curve defined over ${\mathbb{F}}$ with $j_{E}\notin k$.
${\hat{h}}_{E/{\mathbb{F}}}$ | the absolute logarithmic canonical height on $E({\overline{\mathbb{F}}})$.
(1)

######  Conjecture 1 (Lehmer Conjecture for an Elliptic Curve over a Function
Field).

With notation as in (1), there is a constant $\Cl{lehmer}(E/{\mathbb{F}})>0$
such that for all finite separble extensions ${\mathbb{K}}/{\mathbb{F}}$ of
degree $D=[{\mathbb{K}}:{\mathbb{F}}]\geq 2$, we have

${\hat{h}}_{E/{\mathbb{F}}}(P)\geq\frac{\Cr{lehmer}(E/{\mathbb{F}})}{D}\quad\text{for
all non-torsion $P\in E({\mathbb{K}})$.}$

Our main theorem gives a lower bound of the form
$\Cl{lehmerx}(E/{\mathbb{F}})/D^{2}$, with an explicit value for the constant
$\Cr{lehmerx}(E/{\mathbb{F}})$.

######  Theorem 2.

With notation as in (1), let ${\mathbb{K}}/{\mathbb{F}}$ be a finite separable
extension of degree $D=[{\mathbb{K}}:{\mathbb{F}}]\geq 2$. Then

${\hat{h}}_{E/{\mathbb{F}}}(P)\geq\frac{1}{4570\cdot
h_{\mathbb{F}}(j_{E})^{2}\cdot D^{2}}\quad\text{for all non-torsion $P\in
E({\mathbb{K}})$.}$

As noted in numerous papers, Theorem 2 is an immediate consequence of the
following sort of counting result:

######  Theorem 3.

With notation as in (1), let ${\mathbb{K}}/{\mathbb{F}}$ be a finite separble
extension of degree $D=[{\mathbb{K}}:{\mathbb{F}}]\geq 2$. Then for all

$0<\delta<1\quad\text{and}\quad 0<\epsilon<1,$

we have

$\\#\left\\{P\in
E({\mathbb{K}}):{\hat{h}}_{E/{\mathbb{F}}}(P)\leq\frac{1-\delta}{2}\left(\frac{1}{2^{5}\cdot(1+\epsilon)\cdot
D}\right)^{2/3}\right\\}\\\
\leq\max\left\\{\frac{12}{\epsilon},\,\frac{h_{\mathbb{F}}(j_{E})}{\delta}\left(2^{5}\cdot(1+\epsilon)\cdot
D\right)^{2/3}\right\\}.$

The proof of Theorem 3 uses Fourier averaging ideas as in [6], with care taken
to optimize the argument and to keep track of how the constants depend on the
curve $E$.

######  Remark 4.

We remark that the classical Lehmer conjecture for ${\mathbb{K}}^{*}$ is
trivial in the function field setting. Philosophically, we view the difficulty
in Lehmer’s conjecture as coming from places of persistant bad reduction. The
group scheme ${\mathbb{G}}_{m}$ has everywhere good reduction over a function
field, whence the triviality of a Lehmer estimate. However, we also
philosophically view ${\mathbb{G}}_{m}$, and indeed all abelian group schemes,
as having persistant bad reduction at all archimedean places, which explains
why Lehmer estimates are non-trivial for ${\mathbb{G}}_{m}$ over number
fields. As we shall see, the challenge in proving a Lehmer-type estimate for
an elliptic curve over a function field comes almost entirely from the places
of multiplicative reduction, i.e., from the places of stable bad reduction.

We briefly describe the contents of this paper. Section 2 sets notation, and
in particular explains how absolute heights are normalized over the base
function field ${\mathbb{F}}$. Section 3 recalls the decomposition of the
canonical height into local heights and gives lower bounds for local heights.
Section 4 gives two useful elliptic curve results from [6], together with a
strengthened version of a numerical inequality appearing in [6] and [8, Lemma
9.4]. The proof of the latter is deferred to Appendix A. Section 5 contains
the proofs of our main results. In Section 6 we illustrate the philosophy
described in Remark 4 with elementary proofs of an exponentially decaying
lower bound for ${\hat{h}}_{E/{\mathbb{F}}}(P)$ and a proof of Lehmer’s
conjecture when the extension ${\mathbb{K}}/{\mathbb{F}}$ is Galois, both in
the function field setting. Finally, in Appendices B and C, we give two
derivations of formulas for the local height in the case of additive,
potentially multiplicative, reduction, i.e., reduction of type $I_{M}^{*}$,
the first via explicit Weierstrass equations and Tate’s algorithm, the second
via intersection theory. We include this material partly because it is not
readily available in the literature, and partly to illustrate and contrast the
two approaches.

##  2\. Notation

We set some additional notation.

  * $M_{\mathbb{F}}$

a complete set of inequivalent non-trivial absolute
---
values on ${\mathbb{F}}$.

  * $\operatorname{ord}_{v}$

the normalized valuation
$\operatorname{ord}_{v}:{\mathbb{F}}^{*}\relbar\joinrel\twoheadrightarrow{\mathbb{Z}}$
for $v\in M_{\mathbb{F}}$.

For the base field ${\mathbb{F}}$, we identify the place
$v\in{M_{\mathbb{F}}}$ with the normalized valuation

$\operatorname{ord}_{v}:{\mathbb{F}}^{*}\relbar\joinrel\twoheadrightarrow{\mathbb{Z}}.$

Then for a finite separable extension ${\mathbb{K}}/{\mathbb{F}}$, we define
$M_{\mathbb{K}}$ to be the set of valuations on ${\mathbb{K}}$ that extend the
valuations in $M_{\mathbb{F}}$, i.e., an element $w\in{M_{\mathbb{K}}}$ is a
valuation

$w:{\mathbb{K}}^{*}\longrightarrow{\mathbb{Q}}\quad\text{satisfying}\quad
w|_{{\mathbb{F}}}\in M_{\mathbb{F}}.$

The valuation $w$ is related to its associated normalized valuation

$\operatorname{ord}_{w}:{\mathbb{K}}^{*}\relbar\joinrel\twoheadrightarrow{\mathbb{Z}}\quad\text{by}\quad\operatorname{ord}_{w}=e(w)w,$

where

$e(w)=[{\mathbb{K}}_{w}:{\mathbb{F}}_{w}],$

is the ramification index of $w$ over $w|_{\mathbb{F}}$. (We recall that the
residue fields are all isomorphic to the algebraically closed field $k$.) We
note that the usual $\sum{e_{i}f_{i}}=n$ formula becomes

$\sum_{\begin{subarray}{c}w\in M_{\mathbb{K}}\\\ w\mid v\\\
\end{subarray}}e(w)=[{\mathbb{K}}:{\mathbb{F}}]\quad\text{for any $v\in
M_{\mathbb{F}}$.}$ (2)

The ${\mathbb{F}}$-normalized height of a point

$[\alpha_{0},\ldots,\alpha_{n}]\in{\mathbb{P}}^{n}({\mathbb{K}})$

is

$h_{\mathbb{F}}(P):=\frac{1}{[{\mathbb{K}}:{\mathbb{F}}]}\sum_{w\in
M_{\mathbb{K}}}e(w)\max_{0\leq i\leq n}w(\alpha_{i}^{-1}).$ (3)

This gives a well-defined function

$h:{\mathbb{P}}^{n}({\overline{\mathbb{F}}})\longrightarrow[0,\infty).$

Let $x$ be the $x$-coordinate on a Weierstrass equation for $E/{\mathbb{F}}$,
i.e., let $x:E\to{\mathbb{P}}^{1}$ be a non-constant global section of
${\mathcal{O}}_{E/{\mathbb{F}}}\bigl{(}2(O)\bigr{)}$. Then the _canonical
height_ on $E({\overline{\mathbb{F}}})$ is defined by the usual limit

${\hat{h}}_{E/{\mathbb{F}}}:E({\overline{\mathbb{F}}})\longrightarrow[0,\infty),\quad{\hat{h}}_{E/{\mathbb{F}}}(P):=\lim_{n\to\infty}4^{-n}h_{\mathbb{F}}\bigl{(}x(2^{n}P)\bigr{)},$

We use the parallelogram law for the canonical height [13, Theorem VIII.9.3]
to derive an inequality that we will use later.

######  Lemma 5.

Let $P_{1},\ldots,P_{N}\in{E({\overline{\mathbb{F}}})}$. Then

$\max_{0\leq i\leq
N}{\hat{h}}_{E/{\mathbb{F}}}(P_{i})\geq\frac{1}{2(N+1)^{2}}\sum_{\begin{subarray}{c}j,k=0\\\
j\neq k\\\ \end{subarray}}^{N}{\hat{h}}_{E/{\mathbb{F}}}(P_{j}-P_{k}).$

###### Proof.

We denote the bilinear form associated to the canonical height by

$\langle
P,Q\rangle_{E/{\mathbb{F}}}:=\frac{1}{2}\Bigl{(}{\hat{h}}_{E/{\mathbb{F}}}(P+Q)-{\hat{h}}_{E/{\mathbb{F}}}(P)-{\hat{h}}_{E/{\mathbb{F}}}(Q)\Bigr{)}.$

We compute

$\displaystyle\smash[b]{\sum_{\begin{subarray}{c}j,k=0\\\ j\neq k\\\
\end{subarray}}^{N}{\hat{h}}_{E/{\mathbb{F}}}(P_{j}-P_{k})}$
$\displaystyle=\sum_{j,k=0}^{N}{\hat{h}}_{E/{\mathbb{F}}}(P_{j}-P_{k})\quad\text{since
${\hat{h}}_{E/{\mathbb{F}}}(O)=0$,}$
$\displaystyle=\sum_{j,k=0}^{N}\Bigl{(}{\hat{h}}_{E/{\mathbb{F}}}(P_{j})-2\bigl{\langle}P_{j},P_{k}\bigr{\rangle}_{E/{\mathbb{F}}}+{\hat{h}}_{E/{\mathbb{F}}}(P_{k})\Bigr{)}$
$\displaystyle=\sum_{j,k=0}^{N}\Bigl{(}{\hat{h}}_{E/{\mathbb{F}}}(P_{j})+{\hat{h}}_{E/{\mathbb{F}}}(P_{k})\Bigr{)}-2\left\langle\sum_{j=0}^{N}P_{j},\sum_{k=0}^{N}P_{k}\right>_{\\!\\!E/{\mathbb{F}}}$
$\displaystyle=2(N+1)\sum_{i=0}^{N}{\hat{h}}_{E/{\mathbb{F}}}(P_{i})-2{\hat{h}}_{E/{\mathbb{F}}}\biggl{(}\sum_{i=0}^{N}P_{i}\biggr{)}$
$\displaystyle\leq
2(N+1)\sum_{i=0}^{N}{\hat{h}}_{E/{\mathbb{F}}}(P_{i})\quad\text{since
${\hat{h}}_{E/{\mathbb{F}}}\geq 0$,}$ $\displaystyle\leq 2(N+1)^{2}\max_{0\leq
i\leq N}{\hat{h}}_{E/{\mathbb{F}}}(P_{i}).\qed$

##  3\. Local Canonical Heights

We recall the principle properties of local canonical heights on elliptic
curves.

######  Theorem 6.

(Néron–Tate) With notation as in (1), let ${\mathbb{K}}/{\mathbb{F}}$ be a
finite separable extension.

  * (a)

For each $w\in{M_{\mathbb{K}}}$, there is a unique local canonical
height111The elliptic curve and base field are implicit in the notation, but
should we need to specify them, we will write
$\hat{\lambda}_{E/{\mathbb{F}},w}$.

$\hat{\lambda}_{w}:E({\mathbb{K}}_{w})\smallsetminus
0\longrightarrow{\mathbb{R}}$

having the following three properties:

    * (i)

$\hat{\lambda}_{w}$ is continuous and bounded on the complement of any
$w$-adic neighborhood of $0$.

    * (ii)

Let

$E:y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$ (4)

be a Weierstrass equation for $E/{\mathbb{F}}$, and let $\Delta$ be the
associated discriminant [13, III §1]. Then

$\lim_{\begin{subarray}{c}P\to 0\\\ \textup{$w$-adic}\\\
\end{subarray}}\left[\hat{\lambda}_{w}(P)+\frac{1}{12}w\left(\frac{x(P)^{6}}{\Delta}\right)\right]=0.$

    * (iii)

For all $P\in{E({\mathbb{K}}_{w})}\smallsetminus 0$, we have

$\hat{\lambda}_{w}(2P)=4\hat{\lambda}_{w}(P)+\frac{1}{4}w\left(\frac{\bigl{(}2y(P)+a_{1}x(P)+a_{3}\bigr{)}^{4}}{\Delta}\right).$

  * (b)

With local heights normalized as in (a), we have

${\hat{h}}(P)=\frac{1}{[{\mathbb{K}}:{\mathbb{F}}]}\sum_{w\in
M_{\mathbb{K}}}e(w)\hat{\lambda}_{w}(P).\quad\text{for all
$P\in{E({\mathbb{K}})}\smallsetminus 0$.}$ (5)

  * (c)

Let ${\mathbb{L}}/{\mathbb{K}}$ be a finite separable extension, let
$w\in{M_{\mathbb{K}}}$, and let $u\in{M_{\mathbb{L}}}$ with $u\mid{w}$. Then

$\hat{\lambda}_{u}(P)=\hat{\lambda}_{w}(P)\quad\text{for all $P\in
E({\mathbb{K}}_{w})\subseteq E({\mathbb{L}}_{u})$.}$

###### Proof.

(a) This is essentially [12, Theorem VI.1.1]; see also [12, Exercise 6.3]. We
note that the quantity $x^{6}/\Delta$ is a well-defined function on $E$ that
does not depend on the choice of Weierstrass equation, since any other
Weierstrass equation for $E$ is related via $x^{\prime}=u^{2}x$ and
$\Delta^{\prime}=u^{12}\Delta$ for some non-zero $u$; see [13, III §1]. So the
formulas in (ii) and (iii) are independent of the choice of Weierstrass
equation.

(b) See [12, Theorem VI.2.1].

(c) We have normalized the valuations so that the restriction of $u$ to
${\mathbb{K}}_{w}$ is equal to $w$. Hence the restriction of
$\hat{\lambda}_{u}$ to $E({\mathbb{K}}_{w})$ satisfies (i)–(iii), so by the
uniqueness of $\hat{\lambda}_{w}$, we have
$\hat{\lambda}_{u}\big{|}_{E({\mathbb{K}}_{w})}=\hat{\lambda}_{w}$. ∎

We next give two definitions and then recall/prove some explicit formulas and
estimates for local canonical heights on elliptic curves.

######  Definition 7.

For $w\in{M_{\mathbb{K}}}$ and $v\in{M_{\mathbb{F}}}$ with $w\mid{v}$, we
define222We remark that in [6], our quantity $J_{w}$ was denoted by $\nu_{w}$.
We have switched notation, because $\nu$ and $v$ look too much alike.

$\displaystyle J_{w}$
$\displaystyle:=\max\bigl{\\{}0,\operatorname{ord}_{w}(j_{E}^{-1})\bigr{\\}}$
$\displaystyle=\max\bigl{\\{}0,e(w)\operatorname{ord}_{v}(j_{E}^{-1})\bigr{\\}}\quad\text{since
$j_{E}\in{\mathbb{F}}$,}$
$\displaystyle=e(w)\max\bigl{\\{}0,\operatorname{ord}_{v}(j_{E}^{-1})\bigr{\\}}$
$\displaystyle=e(w)\cdot\max\bigl{\\{}0,v(j_{E}^{-1})\bigr{\\}}\quad\text{since
$v=\operatorname{ord}_{v}$ for $v\in M_{\mathbb{F}}$.}$

######  Definition 8.

The _periodic second Bernoulli polynomial_ is the function

$\operatorname{\textsf{B}}_{2}:{\mathbb{R}}\longrightarrow{\mathbb{R}}/{\mathbb{Z}}\longrightarrow{\mathbb{R}},$

defined by

$\operatorname{\textsf{B}}_{2}(t)=t^{2}-t+\frac{1}{6}\quad\text{for $0\leq
t\leq 1$,}$

and extended periodically by requiring that

$\operatorname{\textsf{B}}_{2}(t+n)=\operatorname{\textsf{B}}_{2}(t)\quad\text{for
all $t\in{\mathbb{R}}$ and all $n\in{\mathbb{Z}}$.}$

######  Lemma 9.

With notation as in (1), let $v\in{M_{\mathbb{F}}}$, let
${\mathbb{K}}/{\mathbb{F}}$ be a finite separable extension, let
$w\in{M_{\mathbb{K}}}$ with $w\mid{v}$, and let

$w({\mathcal{D}}_{E/{\mathbb{K}}_{w}})=\text{valuation of the minimal
discriminant of $E$ at~{}$w$.}$

  * (a)

If $v(j_{E})\geq 0$, or equivalently if $E$ has potential good reduction at
$v$, then

$\hat{\lambda}_{w}(P)\geq 0\quad\text{for all $P\in
E({\mathbb{K}}_{w})\smallsetminus 0$.}$

  * (b)

If

$w({\mathcal{D}}_{E/{\mathbb{K}}_{w}})=v(j_{E}^{-1})>0,$

or equivalently if $E$ has multiplicative reduction at $w$, then

$\hat{\lambda}_{w}(P)\geq\frac{1}{2}\operatorname{\textsf{B}}_{2}\bigl{(}\varphi_{w}(P)\bigr{)}v(j_{E}^{-1}),$

where $\varphi_{w}$ is a homomorphism

$\varphi_{w}:E({\mathbb{K}}_{w})\longrightarrow\tfrac{1}{J_{w}}{\mathbb{Z}}/{\mathbb{Z}}\subset{\mathbb{Q}}/{\mathbb{Z}}.$

  * (c)

If

$w({\mathcal{D}}_{E/{\mathbb{K}}_{w}})-6=v(j_{E}^{-1})>0,$

or equivalently if $E$ has additive, potential multiplicative reduction, at
$w$, then

$\hat{\lambda}_{w}(P)\geq\frac{1}{2}\operatorname{\textsf{B}}_{2}\bigl{(}\varphi_{w}(P)\bigr{)}v(j_{E}^{-1}),$

where $\varphi_{w}$ is a homomorphism

$\varphi_{w}:E({\mathbb{K}}_{w})\longrightarrow\tfrac{1}{2}{\mathbb{Z}}/{\mathbb{Z}}\subset{\mathbb{Q}}/{\mathbb{Z}}.$

###### Proof.

(a) The assumption that $v(j_{E})\geq 0$ implies that $E$ has potential good
reduction at $v$ [13, Proposition VII.5.5], so we can find a finite separable
extension ${\mathbb{L}}/{\mathbb{K}}$ and a valuation
$w^{\prime}\in{M_{\mathbb{L}}}$ satisfying $w^{\prime}\mid{w}\mid{v}$ such
that $E$ has good reduction at $w^{\prime}$.333The separability follows from
the fact that ${\mathbb{L}}$ can be generated over ${\mathbb{K}}$ by adjoining
square and cube roots, and we have assumed that
$\operatorname{char}({\mathbb{F}})\neq 2,3$. It follows from [12, Theorem
VI.4.1] (see also [12, Remark VI.4.1.1]) that if we take a minimal Weierstrass
equation for $E/{\mathbb{L}}_{u}$, then

$\hat{\lambda}_{w^{\prime}}(P)=\frac{1}{2}\max\Bigl{\\{}w^{\prime}\bigl{(}x(P)^{-1}\bigr{)},\,0\Bigr{\\}}.\quad\text{for
all $P\in E({\mathbb{L}}_{w^{\prime}})\smallsetminus 0$.}$

In particular, we see that

$\hat{\lambda}_{w^{\prime}}(P)\geq 0\quad\text{for all $P\in
E({\mathbb{L}}_{u})\smallsetminus 0$.}$

We also know that the local height is invariant under field extension (Theorem
6(c)), i.e.,
$\hat{\lambda}_{w^{\prime}}|_{E({\mathbb{K}}_{w})}=\hat{\lambda}_{w}$, which
allows us to conclude that

$\hat{\lambda}_{w}(P)\geq 0\quad\text{for all $P\in
E({\mathbb{K}}_{w})\smallsetminus 0$.}$

(b) This follows from Tate’s explicit formula for local height in the case of
multiplicative reduction; see [12, Theorem VI.4.2].

(c) Our assumptions say that $E$ has reduction type $I_{M}^{*}$ at $w$ for
some integer $M\geq 1$, where444We note that $M\neq 0$ since $I_{0}^{*}$ is a
potential good reduction and we’ve assumed potential multiplicative reduction.
And the equalities in (6) are valid since
$\operatorname{char}({\mathbb{K}})\neq 2$.

$M=w({\mathcal{D}}_{E/{\mathbb{K}}_{w}})-6=w(j_{E}^{-1})=v(j_{E}^{-1}).$ (6)

The special fiber of the Néron model for $I_{M}^{*}$ reduction consists of
four multiplity-$1$ components; see Figure 1 in Appendix B, where we have
labeled the components ${\boldsymbol{0}}$, ${\boldsymbol{\alpha}}$,
${\boldsymbol{\beta}}$, and ${\boldsymbol{\beta}}^{\prime}$. We remark that
$E_{0}({\mathbb{K}}_{w})$ is the set of points that intersect the
${\boldsymbol{0}}$-component. Formulas for $\hat{\lambda}_{w}(P)$ may be found
in various sources, but for completeness we give two proofs, one in Appendix B
using explicit Weierstrass equations, and one in Appendix C using intersection
theory. These formula imply that

$\displaystyle\hat{\lambda}_{w}(P)$ $\displaystyle\geq 0$ if $P$ hits the
${\boldsymbol{0}}$-component, $\displaystyle\hat{\lambda}_{w}(P)$
$\displaystyle=\frac{1}{12}v(j_{E}^{-1})$ if $P$ hits the
${\boldsymbol{\alpha}}$-component, $\displaystyle\hat{\lambda}_{w}(P)$
$\displaystyle=-\frac{1}{24}v(j_{E}^{-1})$ if $P$ hits the
${\boldsymbol{\beta}}$ or ${\boldsymbol{\beta}}^{\prime}$-component.

Finally, we note that the group of components
$E({\mathbb{K}}_{w})/E_{0}({\mathbb{K}}_{w})$ is either
$({\mathbb{Z}}/2{\mathbb{Z}})^{2}$ of $({\mathbb{Z}}/4{\mathbb{Z}})$ depending
on whether $M$ is even or odd, with the ${\boldsymbol{\beta}}$ and
${\boldsymbol{\beta}}^{\prime}$-components being the elements of order $4$ in
the latter case. These formulas give the desired result. ∎

######  Remark 10.

Lemma 9(a) says that $\hat{\lambda}_{w}\geq 0$ at places of potential good
reduction. More precisely, if we write $E_{\nu}({\mathbb{K}}_{w})$ with
$\nu\geq 0$ for the formal group filtration of $E({\mathbb{K}}_{w})$, then one
can show that

$\hat{\lambda}_{w}(P)=\begin{cases}\nu+\frac{1}{12}w({\mathcal{D}}_{E/{\mathbb{K}}_{w}})&\text{for
$P\in E_{\nu}({\mathbb{K}}_{w})\smallsetminus
E_{\nu+1}({\mathbb{K}}_{w})$,}\\\ 0&\text{for $P\notin
E_{0}({\mathbb{K}}_{w})$.}\end{cases}$

##  4\. Reprise of Material from Previous Papers

######  Lemma 11.

([6, Proposition 1.2]) With notation as in (1), let $v\in{M_{\mathbb{F}}}$
with $v(j_{E}^{-1})>0$, let ${\mathbb{K}}/{\mathbb{F}}$ be a finite separable
extension, let $w\in{M_{\mathbb{K}}}$ with $w\mid{v}$, and let

$P_{0},\ldots,P_{N}\in{E}({\mathbb{K}}_{w})$

be distinct points. Then

$\sum_{\begin{subarray}{c}j,k=0\\\ j\neq k\\\
\end{subarray}}^{N}\hat{\lambda}_{w}(P_{j}-P_{k})\geq\frac{1}{12}\left(\frac{N+1}{J_{w}}\right)^{2}v(j_{E}^{-1})-\frac{N+1}{12}v(j_{E}^{-1}),$

where $J_{w}=e(w)v(j_{E}^{-1})$ is as in Definition 7.

###### Proof.

This is proved in [6, Proposition 1.2] in the case that $E$ has multiplicative
reduction at $w$, using the lower bound for $\hat{\lambda}_{w}$ in Lemma 9(b)
in terms of $\operatorname{\textsf{B}}_{2}$ and the homomorphism

$\varphi_{w}:E({\mathbb{K}}_{w})\longrightarrow
J_{w}^{-1}{\mathbb{Z}}/{\mathbb{Z}}\subset{\mathbb{Q}}/{\mathbb{Z}}.$

In the case that $E$ has additive, potential multiplicative, reduction, we can
use the analogous lower bound for $\hat{\lambda}_{w}$ in Lemma 9(c) and repeat
the proof in [6, Proposition 1.2] to obtain the stronger lower bound

$\displaystyle\sum_{\begin{subarray}{c}j,k=0\\\ j\neq k\\\
\end{subarray}}^{N}\hat{\lambda}_{w}(P_{j}-P_{k})$
$\displaystyle\geq\frac{1}{12}\left(\frac{N+1}{2}\right)^{2}v(j_{E}^{-1})-\frac{N+1}{12}v(j_{E}^{-1})$
$\displaystyle=\frac{(N+1)(N-3)}{48}v(j_{E}^{-1}).\qed$

######  Lemma 12.

([6, Proposition 1.3]) With notation as in (1), let $v\in{M_{\mathbb{F}}}$,
and let $w\in{M_{\mathbb{K}}}$ with $w\mid{v}$. Let $A\geq 1$ be an integer,
and let

$Q_{0},\ldots,Q_{6AN}\in{E({\mathbb{K}}_{w})}$

be distinct points. Then there is a subset

$\\{P_{0},\ldots,P_{N}\\}\subseteq\\{Q_{0},\ldots,Q_{6AN}\\}$

such that

$\hat{\lambda}_{w}(P_{j}-P_{k})\geq\frac{1-A^{-1}}{12}\cdot\max\bigl{\\{}v(j_{E}^{-1}),0\bigr{\\}}\quad\text{for
all $j\neq k$.}$ (7)

###### Proof.

If $E$ has multiplicative reduction at $w$, then Lemma 12 is proven in [6,
Proposition 1.3]. So we may suppose that $E$ has good or additive reduction at
$w$.

If $E$ has potential good reduction, i.e., if $v(j_{E})\geq 0$, then (7) is
the assertion that $\hat{\lambda}_{w}(P_{j}-P_{k})\geq 0$. This is an
immediate consequence of Lemma 9(a), and any subset of
$\\{Q_{0},\ldots,Q_{6AN}\\}$ will suffice.

If $E$ has additive, potential multiplicative, reduction, then Lemma 9(c)
implies that there is a subgroup $\Gamma\subset{E({\mathbb{K}}_{w})}$ of index
$2$ with the property that555We could alternatively use the fact that the
group of components on the Néron model has order $4$, but that would lose a
factor of $2$.

$\hat{\lambda}_{w}(P)\geq\frac{1}{2}\operatorname{\textsf{B}}(0)v(j_{E}^{-1})=\frac{1}{12}v(j_{E}^{-1})\quad\text{for
all $P\in\Gamma$.}$

We map

$\\{Q_{0},\ldots,Q_{2N}\\}\longrightarrow E({\mathbb{K}}_{w})/\Gamma$

and use the pigeon-hole principle to find a subset

$\\{P_{0},\ldots,P_{N}\\}\subseteq\\{Q_{0},\ldots,Q_{2N}\\}\quad\text{such
that}\quad P_{j}-P_{k}\in\Gamma\quad\text{for all $j\neq k$.}$

This yields

$\hat{\lambda}_{w}(P_{j}-P_{k})\geq\frac{1}{12}v(j_{E}^{-1})\quad\text{for all
$j\neq k$,}$

which is stronger than the stated result. ∎

######  Lemma 13.

Let $\alpha,\beta,e_{0},e_{1},\ldots,e_{r}$ be positive real numbers
satisfying

$e_{0}=\max\\{e_{0},\ldots,e_{r}\\}.$

Then

$\alpha
e_{0}+\beta\sum_{i=1}^{r}\frac{1}{e_{i}}\geq\biggl{(}\frac{27}{4}\alpha^{2}\beta\sum_{i=0}^{r}e_{i}\biggr{)}^{1/3}.$
(8)

###### Proof.

This was proven in [6] with unspecified constants and in [8, Lemma 9.4]
without the $27/4$ factor. We use a different approach in Appendix A to prove
(8), as well as to eliminate an annoying extra assumption required in the
proof of [8, Lemma 9.4]. We also show that any lower bound of the form
$C(\alpha,\beta)(\sum{e_{i}})^{1/3+\epsilon}$ with a constant that is
independent of $r$ and $e_{0},\ldots,e_{r}$ necessarily has $\epsilon\leq 0$,
and that for $\epsilon=0$, we necessarily have
$C(\alpha,\beta)\leq{C}(\alpha^{2}\beta)^{1/3}$ for some absolute constant
$C$. ∎

##  5\. Proofs of the Main Theorems

###### Proof of Theorem 3.

We set the notation

$\Sigma(B)=\Sigma(B,E/{\mathbb{F}},{\mathbb{K}}):=\bigl{\\{}P\in
E({\mathbb{K}}):{\hat{h}}_{E/{\mathbb{F}}}(P)\leq B\bigr{\\}}.$ (9)

We choose a place $v^{\prime}\in{M_{\mathbb{F}}}$ satisfying

$v^{\prime}(j_{E}^{-1})=\max_{v\in M_{\mathbb{F}}}v(j_{E}^{-1})\geq 1,$

where the lower bound follows from the assumption that $j_{E}\notin{k}$. Let

$w^{\prime}_{0},w^{\prime}_{1},\ldots,w^{\prime}_{r}\in{M_{\mathbb{K}}}$

be the places of $K$ that lie over $v^{\prime}$, so in particular

$J_{w^{\prime}_{i}}=e(w^{\prime}_{i})\cdot v^{\prime}(j_{E}^{-1})>0.$ (10)

For convenience, we relabel the $w^{\prime}_{i}$ so that

$J_{w^{\prime}_{0}}=\max_{0\leq i\leq r}J_{w^{\prime}_{i}}.$

We let

$B\quad\text{and}\quad N\geq M$

be parameters that we will specify later. Suppose that we have distinct points

$Q_{0},Q_{1},\ldots,Q_{12N}\subset\Sigma(B).$

We apply Lemma 12 with $v^{\prime}$ and $w^{\prime}_{0}$ and $A=2$ to find
distinct points

$P_{0},\ldots,P_{N}\subseteq\\{Q_{0},Q_{1},\ldots,Q_{12N}\\}$

satisfying

$\hat{\lambda}_{w_{0}^{\prime}}(P_{j}-P_{k})\geq\frac{1}{24}\cdot
v^{\prime}(j_{E}^{-1})>0\quad\text{for all $j\neq k$.}$ (11)

Recalling that $D=[{\mathbb{K}}:{\mathbb{F}}]$, we compute

$\displaystyle\sum_{\begin{subarray}{c}j,k=0\\\ j\neq k\\\
\end{subarray}}^{N}$ $\displaystyle{\hat{h}}_{E/{\mathbb{F}}}(P_{j}-P_{k})$
(12) $\displaystyle=\sum_{\begin{subarray}{c}j,k=0\\\ j\neq k\\\
\end{subarray}}^{N}\frac{1}{D}\cdot\sum_{w\in
M_{\mathbb{K}}}e(w)\hat{\lambda}_{w}(P_{j}-P_{k})\quad\text{from
\eqref{eqn:hhatPsumlhatwP},}$
$\displaystyle\geq\sum_{\begin{subarray}{c}j,k=0\\\ j\neq k\\\
\end{subarray}}^{N}\frac{1}{D}\cdot\sum_{\begin{subarray}{c}w\in
M_{\mathbb{K}}\\\ w(j_{E})<0\\\
\end{subarray}}e(w)\hat{\lambda}_{w}(P_{j}-P_{k})\quad\begin{tabular}[t]{@{}l@{}}from
Lemma~{}\ref{lemma:potentialgoodreduction}(a), which\\\ says that
$\hat{\lambda}_{w}\geq 0$ if $w(j_{E})\geq 0$,\\\ \end{tabular}$ (15)
$\displaystyle=\frac{1}{D}\sum_{\begin{subarray}{c}w\in M_{\mathbb{K}}\\\
w(j_{E})<0\\\ \end{subarray}}e(w)\biggl{(}\sum_{\begin{subarray}{c}j,k=0\\\
j\neq k\\\ \end{subarray}}^{N}\hat{\lambda}_{w}(P_{j}-P_{k})\biggr{)}$
$\displaystyle\geq\frac{1}{D}\sum_{\begin{subarray}{c}v\in M_{\mathbb{F}}\\\
v(j_{E})<0\\\ \end{subarray}}\sum_{\begin{subarray}{c}w\in M_{\mathbb{K}}\\\
w\mid v\\\ \end{subarray}}e(w)\biggl{(}\sum_{\begin{subarray}{c}j,k=0\\\ j\neq
k\\\ \end{subarray}}^{N}\hat{\lambda}_{w}(P_{j}-P_{k})\biggr{)}$
$\displaystyle=\frac{1}{D}e(w^{\prime}_{0})\biggl{(}\sum_{\begin{subarray}{c}j,k=0\\\
j\neq k\\\
\end{subarray}}^{N}\hat{\lambda}_{w^{\prime}_{0}}(P_{j}-P_{k})\biggr{)}$ (16)
$\displaystyle\qquad{}+\frac{1}{D}\sum_{i=1}^{r}e(w^{\prime}_{i})\biggl{(}\sum_{\begin{subarray}{c}j,k=0\\\
j\neq k\\\
\end{subarray}}^{N}\hat{\lambda}_{w^{\prime}_{i}}(P_{j}-P_{k})\biggr{)}$ (17)
$\displaystyle\qquad{}+\frac{1}{D}\sum_{\begin{subarray}{c}v\in
M_{\mathbb{F}}\\\ v(j_{E})<0\\\ v\neq v^{\prime}\\\
\end{subarray}}\sum_{\begin{subarray}{c}w\in M_{\mathbb{K}}\\\ w\mid v\\\
\end{subarray}}e(w)\biggl{(}\sum_{\begin{subarray}{c}j,k=0\\\ j\neq k\\\
\end{subarray}}^{N}\hat{\lambda}_{w}(P_{j}-P_{k})\biggr{)}.$ (18)

We use (11) to estimate (16), we use Proposition 11 to estimate (17), and we
use Proposition 11 without the positive term on the right-hand side to
estimate (18). This yields

$\displaystyle\sum_{\begin{subarray}{c}j,k=0\\\ j\neq k\\\
\end{subarray}}^{N}$ $\displaystyle{\hat{h}}_{E/{\mathbb{F}}}(P_{j}-P_{k})$
$\displaystyle\geq\frac{1}{D}e(w^{\prime}_{0})\sum_{\begin{subarray}{c}j,k=0\\\
j\neq k\\\ \end{subarray}}^{N}\frac{1}{24}\cdot v^{\prime}(j_{E}^{-1})$
$\displaystyle\qquad{}+\frac{1}{D}\sum_{i=1}^{r}e(w^{\prime}_{i})\left(\frac{1}{12}\left(\frac{N+1}{J_{w^{\prime}_{i}}}\right)^{2}v^{\prime}(j_{E}^{-1})-\frac{N+1}{12}v^{\prime}(j_{E}^{-1})\right)$
$\displaystyle\qquad{}+\frac{1}{D}\sum_{\begin{subarray}{c}v\in
M_{\mathbb{F}}\\\ v(j_{E})<0\\\ v\neq v^{\prime}\\\
\end{subarray}}\sum_{\begin{subarray}{c}w\in M_{\mathbb{K}}\\\ w\mid v\\\
\end{subarray}}e(w)\cdot\left(-\frac{N+1}{12}v(j_{E}^{-1})\right)$
$\displaystyle\geq\left(\frac{N^{2}+N}{24D}\cdot
v^{\prime}(j_{E}^{-1})\right)\cdot e(w^{\prime}_{0})$ (19)
$\displaystyle\qquad{}+\frac{(N+1)^{2}}{12Dv^{\prime}(j_{E}^{-1})}\cdot\sum_{i=1}^{r}\frac{1}{e(w^{\prime}_{i})}$
(20) $\displaystyle\qquad{}-\frac{N+1}{12D}\sum_{\begin{subarray}{c}v\in
M_{\mathbb{F}}\\\ v(j_{E})<0\\\ \end{subarray}}\sum_{\begin{subarray}{c}w\in
M_{\mathbb{K}}\\\ w\mid v\\\ \end{subarray}}e(w)\cdot v(j_{E}^{-1})$ (21)
using $J_{w^{\prime}_{i}}=e(w^{\prime}_{i})\cdot{v^{\prime}(j_{E}^{-1})}$ from
(10).

We evaluate (21) (with minus sign omitted) as

$\displaystyle\frac{N+1}{12D}$ $\displaystyle\sum_{\begin{subarray}{c}v\in
M_{\mathbb{F}}\\\ v(j_{E})<0\\\ \end{subarray}}\sum_{\begin{subarray}{c}w\in
M_{\mathbb{K}}\\\ w\mid v\\\ \end{subarray}}e(w)\cdot v(j_{E}^{-1})$
$\displaystyle=\frac{N+1}{12D}\sum_{\begin{subarray}{c}v\in M_{\mathbb{F}}\\\
v(j_{E})<0\\\ \end{subarray}}D\cdot
v(j_{E}^{-1})\quad\text{from~{}\eqref{eqn:sumeifi},}$
$\displaystyle=\frac{N+1}{12}h_{\mathbb{F}}(j_{E})\quad\begin{tabular}[t]{@{}l@{}}from
the definition~{}\eqref{eqn:defweilht} of height,\\\ where we note that
$j_{E}^{-1}\in{F}$\\\ and that
$h_{\mathbb{F}}(j_{E}^{-1})=h_{\mathbb{F}}(j_{E})$.\\\ \end{tabular}$

To ease notation, we let

$\alpha=\frac{N^{2}+N}{24D}\cdot
v^{\prime}(j_{E}^{-1})\quad\text{and}\quad\beta=\frac{(N+1)^{2}}{12Dv^{\prime}(j_{E}^{-1})}.$
(22)

Then our inquality becomes

$\sum_{\begin{subarray}{c}j,k=0\\\ j\neq k\\\
\end{subarray}}^{N}{\hat{h}}_{E/{\mathbb{F}}}(P_{j}-P_{k})\geq\alpha
e(w_{0}^{\prime})+\beta\sum_{i=1}^{r}\frac{1}{e(w^{\prime}_{i})}-\frac{N+1}{12}h_{\mathbb{F}}(j_{E}).$
(23)

We are going to use Lemma 13 to estimate the first two terms, where in the
lemma we take

$e_{i}=e_{w^{\prime}_{i}},\quad\text{so we have}\quad e_{0}+\cdots+e_{r}=D.$

This yields

$\displaystyle\sum_{\begin{subarray}{c}j,k=0\\\ j\neq k\\\
\end{subarray}}^{N}$ $\displaystyle{\hat{h}}_{E/{\mathbb{F}}}(P_{j}-P_{k})$
$\displaystyle\geq\left(\frac{27}{4}\alpha^{2}\beta
D\right)^{1/3}-\frac{N+1}{12}h_{\mathbb{F}}(j_{E})\quad\text{from
Lemma~{}\ref{lemma:LS-Lemma9.4} and~{}\eqref{eqn:sumjkgestuff1},}$
$\displaystyle=\left(\frac{27}{4}\frac{(N^{2}+N)^{2}}{(24D)^{2}}\cdot
v^{\prime}(j_{E}^{-1})^{2}\cdot\frac{(N+1)^{2}}{12Dv^{\prime}(j_{E}^{-1})}\cdot
D\right)^{1/3}-\frac{N+1}{12}h_{\mathbb{F}}(j_{E})$ using the values (22) of
$\alpha$ and $\beta$,
$\displaystyle\geq(N+1)^{2}\left(\left(\frac{N^{2}}{2^{10}\cdot(N+1)^{2}\cdot
D^{2}}\right)^{1/3}-\frac{h_{\mathbb{F}}(j_{E})}{12(N+1)}\right)$ simplifying
and using $v^{\prime}(j_{E}^{-1})\geq 1$,
$\displaystyle\geq(N+1)^{2}\left(\left(\frac{1}{2^{5}\cdot(1+M^{-1})\cdot
D}\right)^{2/3}-\frac{h_{\mathbb{F}}(j_{E})}{12(N+1)}\right)$ simplifying and
using $N\geq{M}$.5 (24)

Hence

$\displaystyle B$ $\displaystyle\geq\max_{0\leq i\leq
12N}{\hat{h}}_{E/{\mathbb{F}}}(Q_{i})\quad\text{since $Q_{i}\in\Sigma(B)$,}$
$\displaystyle\geq\max_{0\leq i\leq
N}{\hat{h}}_{E/{\mathbb{F}}}(P_{i})\quad\text{since $\\{P_{i}\\}_{0\leq i\leq
N}\subset\\{Q_{i}\\}_{0\leq i\leq 12N}$}$
$\displaystyle\geq\frac{1}{2(N+1)^{2}}\sum_{\begin{subarray}{c}j,k=0\\\ j\neq
k\\\ \end{subarray}}^{N}{\hat{h}}_{E/{\mathbb{F}}}(P_{j}-P_{k})\quad\text{from
Lemma~{}\ref{lemma:hhatineq},}$
$\displaystyle\geq\frac{1}{2}\left(\left(\frac{1}{2^{5}\cdot(1+M^{-1})\cdot
D}\right)^{2/3}-\frac{h_{\mathbb{F}}(j_{E})}{12(N+1)}\right)\quad\text{from
\eqref{eqn:hPjPkN2N12}.}$ (25)

We let

$B=B_{\delta,M}:=\frac{1-\delta}{2}\left(\frac{1}{2^{5}\cdot(1+M^{-1})\cdot
D}\right)^{2/3}.$ (26)

Then a bit of algebra with (5) yields

$N+1\leq\frac{h_{\mathbb{F}}(j_{E})}{12\delta}\left(\frac{2^{5}\cdot(1+M^{-1})\cdot
D}{1}\right)^{2/3}.$ (27)

To recapitulate, we have shown that for $B_{\delta,M}$ as given in (26), if

$\\#\Sigma(B_{\delta})\geq 12N,$

for some $N\geq{M}$, then $N$ is bounded above by (27). Using the specific
values in (26) and (27) yields

$\\#\Sigma\left(\frac{1-\delta}{2}\left(\frac{1}{2^{5}\cdot(1+M^{-1})\cdot
D}\right)^{2/3}\right)\\\
\leq\max\left\\{12M,\,\frac{h_{\mathbb{F}}(j_{E})}{\delta}\left(2^{5}\cdot(1+M^{-1})\cdot
D\right)^{2/3}\right\\}.$

Setting $\epsilon=M^{-1}$ completes the proof of Theorem 3. ∎

###### Proof of Theorem 2.

Continuing with the notation (9) for $\Sigma(B)$ as in the proof of Theorem 3,
we write

$\displaystyle\Cl{thm2-1}$
$\displaystyle=\Cr{thm2-1}(\delta,\epsilon):=\frac{1-\delta}{2}\left(\frac{1}{2^{5}\cdot(1+\epsilon)}\right)^{2/3},$
(28) $\displaystyle\Cl{thm2-2}$
$\displaystyle=\Cr{thm2-2}(\delta,\epsilon):=\frac{1}{\delta}\left(2^{5}\cdot(1+\epsilon)\right)^{2/3},$
(29)

for the constants in Theorem 3 such that

$\\#\Sigma\left(\Cr{thm2-1}\cdot
D^{-2/3}\right)\leq\max\Bigl{\\{}12\epsilon^{-1},\,\Cr{thm2-2}\cdot
h_{\mathbb{F}}(j_{E})\cdot D^{2/3}\Bigr{\\}}.$ (30)

(We will choose $\delta$ and $\epsilon$ later.)

Let $P\in{E({\mathbb{K}})}$ be a non-torsion point, and let
$\Cl{thm1-1}=\Cr{thm1-1}(\delta,\epsilon)$ denote a constant whose value we
will specify later. We suppose that

${\hat{h}}_{E/{\mathbb{F}}}(P)\leq\frac{\Cr{thm1-1}}{h_{\mathbb{F}}(j_{E})^{2}\cdot
D^{2}}.$ (31)

Then for all $m$ satisfying

$|m|\leq(\Cr{thm2-1}/\Cr{thm1-1})^{1/2}\cdot h_{\mathbb{F}}(j_{E})\cdot
D^{2/3},$

we have

${\hat{h}}_{E/{\mathbb{F}}}(mP)=m^{2}{\hat{h}}_{E/{\mathbb{F}}}(P)\leq\Cr{thm2-1}\cdot
D^{-2/3}.$

Thus (31) implies that

$\bigl{\\{}mP:|m|\leq(\Cr{thm2-1}/\Cr{thm1-1})^{1/2}\cdot
h_{\mathbb{F}}(j_{E})\cdot D^{2/3}\bigr{\\}}\subseteq\Sigma(\Cr{thm2-1}\cdot
D^{-2/3}).$ (32)

We compute

$\displaystyle 2(\Cr{thm2-1}/\Cr{thm1-1})^{1/2}$ $\displaystyle\cdot
h_{\mathbb{F}}(j_{E})\cdot D^{2/3}-1$
$\displaystyle\leq\\#\Bigl{\\{}m\in{\mathbb{Z}}:|m|\leq(\Cr{thm2-1}/\Cr{thm1-1})^{1/2}\cdot
h_{\mathbb{F}}(j_{E})\cdot D^{2/3}\Bigr{\\}}$ using $2t-1\leq
2\lfloor{t}\rfloor+1$,
$\displaystyle\leq\\#\Bigl{\\{}mP:|m|\leq(\Cr{thm2-1}/\Cr{thm1-1})^{1/2}\cdot
h_{\mathbb{F}}(j_{E})\cdot D^{2/3}\Bigr{\\}}$ since $P$ is non-torsion, so the
$mP$ are distinct, $\displaystyle\leq\\#\Sigma(\Cr{thm2-1}\cdot
D^{-2/3})\quad\text{from \eqref{eqn:setmPinSigma},}$
$\displaystyle\leq\max\Bigl{\\{}12\epsilon^{-1},\,\Cr{thm2-2}\cdot
h_{\mathbb{F}}(j_{E})\cdot
D^{2/3}\Bigr{\\}}.\quad\text{from~{}\eqref{eqn:thm2C1C2}.}$

We thus obtain a contradiction to (31) if we choose $\Cr{thm1-1}$ to satisfy

$\Cr{thm1-1}<\max\left\\{\frac{4\Cr{thm2-1}h_{\mathbb{F}}(j_{E})^{2}D^{4/3}}{(12\epsilon^{-1}+1)^{2}},\;\frac{4\Cr{thm2-1}}{\bigl{(}\Cr{thm2-2}+h_{\mathbb{F}}(j_{E})^{-1}\cdot
D^{-2/3}\bigr{)}^{2}}\right\\}.$

Since we know that $h_{\mathbb{F}}(j_{E})\geq 1$ and $D\geq 2$, we have a
contradiction if we choose $\Cr{thm1-1}$ to satisfy

$\Cr{thm1-1}<\max\left\\{\frac{4\Cr{thm2-1}}{(12\epsilon^{-1}+1)^{2}\vphantom{\bigl{(}^{2}}},\;\frac{4\Cr{thm2-1}}{\bigl{(}\Cr{thm2-2}+1\bigr{)}^{2}}\right\\}.$

Using the values of $\Cr{thm2-1}$ and $\Cr{thm2-2}$ and the fact that we may
choose any $\delta$ and $\epsilon$ between $0$ and $1$, we deduce that

${\hat{h}}_{E/{\mathbb{F}}}(P)\geq\max_{\begin{subarray}{c}0<\delta<1\\\
0<\epsilon<1\\\
\end{subarray}}\min\left\\{\frac{1-\delta}{2^{4/3}\cdot(1+\epsilon)^{2/3}}\cdot\frac{\epsilon^{2}}{(12+\epsilon)^{2}\vphantom{\bigl{(}^{2}}},\right.\\\
\left.\frac{1-\delta}{2^{4/3}\cdot(1+\epsilon)^{2/3}\vphantom{\bigl{(}^{1})}}\cdot\frac{\delta^{2}}{\bigl{(}2^{10/3}\cdot(1+\epsilon)^{2/3}+\delta\bigr{)}^{2}}\right\\}\cdot\frac{1}{h_{\mathbb{F}}(j_{E})^{2}\cdot
D^{2}}.$

We are free to choose any $\delta$ and $\epsilon$ between $0$ and $1$, and a
short computation suggests taking the values

$(\delta,\epsilon)=(0.580153,0.5220101).$

This yields

${\hat{h}}_{E/{\mathbb{F}}}(P)\geq
0.00021883436\cdot\cdot\frac{1}{h_{\mathbb{F}}(j_{E})^{2}\cdot
D^{2}}\geq\frac{1}{4570\cdot h_{\mathbb{F}}(j_{E})^{2}\cdot D^{2}}.\qed$ (33)

######  Remark 14.

If we let $D\to\infty$ and $\epsilon\to 0$, then taking $\delta=\frac{2}{3}$
allows us to replace the $4570$ appearing in (33) with $1152$. We mention this
to indicate the limit of how good a constant this method can yield without
some further idea.

##  6\. Some Final Remarks

The various explicit formulas for the local height can be used to show that

$\hat{\lambda}_{w}(P)\in\frac{1}{144e(w)}{\mathbb{Z}}\quad\text{for all $P\in
E({\mathbb{K}})$ and all $w\in M_{\mathbb{K}}$.}$

(The $144$ can probably be improved.) Summing the local heights yields a
fraction whose denominator divides

$e({\mathbb{K}}/{\mathbb{F}})=\operatorname{LCM}\bigl{\\{}e(w):w\in
M_{\mathbb{K}}\bigr{\\}}.$

In particular, since ${\hat{h}}{E/{\mathbb{F}}}(P)>0$ for non-torsion points,
we obtain the lower bound

${\hat{h}}_{E/{\mathbb{F}}}(P)\geq\frac{1}{144e({\mathbb{K}}/{\mathbb{F}})}\quad\text{for
all non-torsion $P\in E({\mathbb{K}})$.}$

In general, all that we know about the ramification indices is that they
satisfy

$1\leq e(w)\leq D:=[{\mathbb{K}}:{\mathbb{F}}],$

which gives

$e({\mathbb{K}}/{\mathbb{F}})\leq\operatorname{LCM}[1,2,\ldots,D]=e^{D+o(D)}.$

(The last estimate follows from the prime number theorem.) So we obtain an
exponentially decaying Lehmer estimate of the form

${\hat{h}}_{E/{\mathbb{F}}}(P)\geq\frac{\Cl{pnt}(\epsilon)}{e^{D+\epsilon}},$

which is notable only in that the constant $\Cr{pnt}(\epsilon)$ is independent
of $E/{\mathbb{F}}$.

For Galois extensions, we can do better. Thus

${\mathbb{K}}/{\mathbb{F}}$ is Galois
$\displaystyle\quad\Longrightarrow\quad\text{$e(w)\mid D$ for all $w\in
M_{\mathbb{K}}$,}$ $\displaystyle\quad\Longrightarrow\quad
e({\mathbb{K}}/{\mathbb{F}}):=\operatorname{LCM}\bigl{\\{}e(w):w\in
M_{\mathbb{K}}\bigr{\\}}\leq D$
$\displaystyle\quad\Longrightarrow\quad{\hat{h}}_{E/{\mathbb{F}}}(P)\geq\frac{1}{144D}\quad\text{for
all non-torsion $P\in E({\mathbb{K}})$.}$

This shows that Lehmer’s conjecture (Conjecture 1) is true (and quite
elementary) in the function field setting for Galois extensions
${\mathbb{K}}/{\mathbb{F}}$, again with a constant that is independent of
$E/{\mathbb{F}}$.

Over Galois extensions of number fields, there are similarly strong, but much
less elementary, results. For example, if ${\mathbb{Q}}(\alpha)/{\mathbb{Q}}$
is a Galois extension and $h(\alpha)>0$, then the classical Lehmer’s
conjecture $h(\alpha)\geq\C/D$ is true [1], and indeed the stronger bound
$h(\alpha)\geq\C(\epsilon)/D^{\epsilon}$ holds for all $\epsilon>0$ [2]. For
elliptic curves, if $P$ is a non-torsion point generating a Galois extension,
then the elliptic Lehmer conjecture
${\hat{h}}_{E/{\mathbb{F}}}(P)\geq\C(E/{\mathbb{F}})/D$ is true [5].

##  Appendix A Proof of an Inequality

In this section we prove Lemma 13, which we recall says that for all positive
real numbers $\alpha,\beta,e_{0},e_{1},\ldots,e_{r}$, we have

$e_{0}=\max\\{e_{0},\ldots,e_{r}\\}\quad\Longrightarrow\quad\alpha
e_{0}+\beta\sum_{i=1}^{r}\frac{1}{e_{i}}\geq\biggl{(}\frac{27}{4}\alpha^{2}\beta\sum_{i=0}^{r}e_{i}\biggr{)}^{1/3}.$

###### Proof of Lemma 13.

We denote the all-ones vector by

$\mathbbm{1}_{n}=(1,1,\ldots,1)\in{\mathbb{Z}}^{n}.$

For ${\boldsymbol{x}}=(x_{0},\ldots,x_{r})\in{\mathbb{R}}_{>0}^{r+1}$ and
$\alpha,\beta>0$ and $\epsilon\geq 0$, we define

$f_{\epsilon}(\alpha,\beta,{\boldsymbol{x}})=\frac{\alpha x_{0}+\beta
x_{1}^{-1}+\cdots+\beta
x_{r}^{-1}}{(x_{0}+x_{1}+\cdots+x_{r})^{1/3+\epsilon}}.$

The goal is to find a lower bound for
$f_{\epsilon}(\alpha,\beta,{\boldsymbol{x}})$ that depends only on $\alpha$,
$\beta$, and $\epsilon$, but is independent of both $r$ and
${\boldsymbol{x}}$.

We note that

$\displaystyle f_{\epsilon}(\alpha,\beta,\sqrt{\beta
r/\alpha}\cdot\mathbbm{1}_{r+1})$ $\displaystyle=\frac{\sqrt{\alpha\beta
r}+r\sqrt{\alpha\beta/r}}{\bigl{(}(r+1)\sqrt{\beta
r/\alpha}\bigr{)}^{1/3+\epsilon}}$ $\displaystyle=\frac{2\sqrt{\alpha\beta
r}}{r^{1/2+3\epsilon/2}\cdot(\beta/\alpha)^{1/6+\epsilon/2}\cdot(1+r^{-1})^{1/3+\epsilon}}$
$\displaystyle\xrightarrow{\;\;r\to\infty\;\;}\begin{cases}2(\alpha^{2}\beta)^{1/3}&\text{if
$\epsilon=0$.}\\\ 0&\text{if $\epsilon>0$.}\\\ \end{cases}$

This shows that we cannot hope for a lower bound in (8) that is independent of
$r$ using an exponent strictly larger that $1/3$, and that for exponent $1/3$,
the best lower bound will be a constant multiple of $(\alpha^{2}\beta)^{1/3}$.

We henceforth let

$f({\boldsymbol{x}}):=f_{0}(\alpha,\beta,{\boldsymbol{x}}):=\frac{\alpha
x_{0}+\beta x_{1}^{-1}+\cdots+\beta
x_{r}^{-1}}{(x_{0}+x_{1}+\cdots+x_{r})^{1/3}}.$

To ease notation, we let

$A=A({\boldsymbol{x}}):=\alpha
x_{0}+\beta\sum_{i=1}^{r}\frac{1}{x_{i}}\quad\text{and}\quad
D=D({\boldsymbol{x}}):=\sum_{i=0}^{r}x_{i},$

so

$f({\boldsymbol{x}})=\frac{A({\boldsymbol{x}})}{D({\boldsymbol{x}})^{1/3}}$

We want to minimize $f({\boldsymbol{x}})$ on the region

${\mathcal{R}}:=\Bigl{\\{}{\boldsymbol{x}}:\text{$0\leq x_{i}\leq x_{0}$ for
all $0\leq i\leq r$}\Bigr{\\}}.$

We note that as any coordinate $x_{j}\to 0$, we have
$f({\boldsymbol{x}})\to\infty$, so the minimum does not occur along any of the
coordinate planes. We first look for critical points in the interior of the
region ${\mathcal{R}}$. For any $1\leq{j}\leq{r}$ we have

$\frac{\partial f({\boldsymbol{x}})}{\partial
x_{j}}=\frac{D^{1/3}\beta(-x_{j}^{-2})-A\cdot\frac{1}{3}\cdot
D^{-2/3}}{D^{2/3}}=-\frac{3D\beta x_{j}^{-2}+A}{3D^{4/3}}.$ (34)

All of the quantities $A$, $D$, $\beta$, and $x_{j}^{2}$ are strictly
positive, so $\partial f({\boldsymbol{x}})/\partial x_{j}$ doesn’t vanish in
the interior of ${\mathcal{R}}$. Hence the minimum is attained on the boundary
of ${\mathcal{R}}$.

The boundary of ${\mathcal{R}}$ is the union of the $r$ regions, one region
for each $1\leq k\leq r$, having the form

$\Bigl{\\{}{\boldsymbol{x}}:\text{$0\leq x_{i}\leq x_{0}=x_{k}$ for all $0\leq
i\leq r$}\Bigr{\\}}.$

By symmetry, it suffices to take $k=1$. More generally, again by symmetry, it
suffices to look for the minimum of $f({\boldsymbol{x}})$ in the interior of
each of the regions

${\mathcal{R}}_{s}:=\Bigl{\\{}{\boldsymbol{x}}:\text{$0\leq x_{i}\leq
x_{0}=x_{1}=\cdots=x_{s}$ for all $0\leq i\leq r$}\Bigr{\\}}.$

To ease notation, we write points in ${\mathcal{R}}_{s}$ as

$({\boldsymbol{x}},{\boldsymbol{y}}):=(\underbrace{x,x,\ldots,x}_{\text{$s$
copies of $x$}},y_{1},\ldots,y_{t})\quad\text{with $t=r+1-s$.}$

For $t\geq 1$, if we look for critical points in the interior of
${\mathcal{R}}_{s}$, we find that666We note that if $t=r$, then this is the
same calculation (34) that we did earlier.

$\frac{\partial f({\boldsymbol{x}})}{\partial
y_{1}}=\frac{D^{1/3}\beta(-y_{1}^{-2})-A\cdot\frac{1}{3}\cdot
D^{-2/3}}{D^{2/3}}=-\frac{3D\beta y_{1}^{-2}+A}{3D^{4/3}}.$

Thus again there are no critical points in the boundary of
${\mathcal{R}}_{s}$. This proves that the minimum value of
$f({\boldsymbol{x}})$ is attained in the region ${\mathcal{R}}_{r+1}$ with
$t=0$. In other words, it is attained at a point of the form

$x\mathbbm{1}_{r+1}=(x,x,x,\ldots,x).$

We are thus reduced to studying

$f(x\mathbbm{1}_{r+1})=\frac{\alpha x+\beta r/x}{(r+1)^{1/3}x^{1/3}}.$

A short calculation yields

$\displaystyle\frac{df(x\mathbbm{1}_{r+1})}{dx}$ $\displaystyle=\frac{2(\alpha
x^{2}-2\beta r)}{(r+1)^{1/3}x^{7/3}}.$

There is thus a unique critical point at $x=\sqrt{2\beta{r}/\alpha}$, which is
necessarily a minimum since $f(x\mathbbm{1}_{r+1})\to\infty$ as $x\to 0^{+}$
and as $x\to\infty$. Hence

$\displaystyle\inf_{{\boldsymbol{x}}\in{\mathcal{R}}}f({\boldsymbol{x}})$
$\displaystyle=\inf_{x\in{\mathcal{R}}_{r+1}}f(x\mathbbm{1}_{r+1})$
$\displaystyle=f\bigl{(}\sqrt{2\beta r/\alpha}\,\mathbbm{1}_{r+1}\bigr{)}$
$\displaystyle=\frac{\alpha(2\beta r/\alpha)^{1/2}+\beta r(2\beta
r/\alpha)^{-1/2}}{(r+1)^{1/3}\cdot(2\beta r/\alpha)^{1/6}}$
$\displaystyle=\left(\sqrt{2}+\frac{1}{\sqrt{2}}\right)\cdot\left(\frac{\alpha^{2}\beta}{1+r^{-1}}\right)^{1/3}.$

We now let $r\to\infty$ to obtain a lower bound that holds for all $r$, and we
rewrite the constant using $2^{1/2}+2^{-1/2}=(27/4)^{1/3}$ to obtain the
desired result. ∎

##  Appendix B Local Height Formulas via Weierstrass Equations

In this section we derive formulas for the local height in the case of
$I_{M}^{*}$ reduction. We remark that these formulas appear already in [11,
Table 1]. They are also implicit in [3, Table 1.19] in the case that
${\mathbb{F}}$ is the function field of a curve over ${\mathbb{C}}$. But for
completeness, we give here a proof using explicit Weierstrass equations that
is valid is all residue characteristics other than $2$ and $3$, and in Section
C we give an alternative proof using intersection theory on the associated
elliptic surface.

Let $({\mathbb{F}}_{v},v)$ be a complete local field with valuation normalized
so that $v:{\mathbb{F}}_{v}^{*}\twoheadrightarrow{\mathbb{Z}}$, and let
$E/{\mathbb{F}}_{v}$ be an elliptic curve with additive, potential
multiplicative, reduction. Thus in the Kodaira–Néron classification ([14],
[13, Table 15.1]), the elliptic curve $E$ has reduction type $I_{M}^{*}$ for
some $M\geq 1$. The special fiber of a complete minimal model of $E$ at $v$ is
illustrated in Figure 1. The fiber consists of four components of multiplity
$1$ that we have labeled $\boldsymbol{0}$, $\boldsymbol{\alpha}$,
$\boldsymbol{\beta}$, and $\boldsymbol{\beta^{\prime}}$, together with $m+1$
components of multiplicty $2$. Points in $E_{0}({\mathbb{F}}_{v})$ are the
points that intersect the $\boldsymbol{0}$-component. More generally, we let

$E_{\nu}({\mathbb{F}}_{v}):=\bigl{\\{}P\in{E}({\mathbb{F}}_{v}):v\bigl{(}x(P)^{-1}\bigr{)}\geq
2\nu\bigr{\\}}$

denote the points in the $\nu$th level of the formal group filtration, where
$x$ is a coordinate on a minimal Weierstrass equation for $E$ at $v$. The
group of components $E({\mathbb{F}}_{v})/E_{0}({\mathbb{F}}_{v})$ is either
$({\mathbb{Z}}/2{\mathbb{Z}})^{2}$ of $({\mathbb{Z}}/4{\mathbb{Z}})$ depending
on whether $M$ is even or odd, with the $\boldsymbol{\beta}$ and
$\boldsymbol{\beta^{\prime}}$-components being the elements of order $4$ in
the latter case. We also note that the minimal discriminant of $E$ at $v$
satisfies

$M=v({\mathcal{D}}_{E/KK})-6=v(j_{E}^{-1}).$ (35)

######  Proposition 15.

With notation as above for an elliptic curve with Type $I_{M}^{*}$ reduction,
the canonical local height

$\hat{\lambda}_{v}:E({\mathbb{F}}_{v})\smallsetminus
0\longrightarrow[0,\infty)$

is given by

$\hat{\lambda}_{v}(P)=\begin{cases}\phantom{-}\dfrac{v(j_{E}^{-1})}{12}+\nu&\text{if
$P\in E_{\nu}({\mathbb{F}}_{v})\smallsetminus
E_{\nu+1}({\mathbb{F}}_{v})$,}\\\\[6.0pt]
\phantom{-}\dfrac{v(j_{E}^{-1})}{12}=\frac{1}{2}\operatorname{\textsf{B}}_{2}(0)v(j_{E}^{-1})&\text{if
$P$ hits the $\boldsymbol{\alpha}$-component,}\\\\[6.0pt]
-\dfrac{v(j_{E}^{-1})}{24}=\frac{1}{2}\operatorname{\textsf{B}}_{2}(\frac{1}{2})v(j_{E}^{-1})&\text{if
$P$ hits the $\boldsymbol{\beta}$ or
$\boldsymbol{\beta^{\prime}}$-component.}\\\ \end{cases}$

$\boldsymbol{0}$$\boldsymbol{\alpha}$$\boldsymbol{\beta}$$\boldsymbol{\beta^{\prime}}$${\boldsymbol{\gamma}}_{0}$${\boldsymbol{\gamma}}_{1}$${\boldsymbol{\gamma}}_{2}$${\boldsymbol{\gamma}}_{M-2}$${\boldsymbol{\gamma}}_{M-1}$${\boldsymbol{\gamma}}_{M}$
Figure 1. The special fiber for $I_{M}^{*}$ reduction. It consists of $4$
fibers of multiplicity $1$, labeled $\boldsymbol{0}$, $\boldsymbol{\alpha}$,
$\boldsymbol{\beta}$ and $\boldsymbol{\beta^{\prime}}$, and $M+1$ components
of multiplicity $2$ labeled
${\boldsymbol{\gamma}}_{0},\ldots,{\boldsymbol{\gamma}}_{M}$.

###### Proof.

By a minor abuse of notation, we let

$\displaystyle E^{({\boldsymbol{0}})}({\mathbb{F}}_{v})$
$\displaystyle:=\bigl{\\{}P\in E({\mathbb{F}}_{v}):\text{$P$ hits the
${\boldsymbol{0}}$-component}\bigr{\\}},$ $\displaystyle
E^{({\boldsymbol{\alpha}})}({\mathbb{F}}_{v})$ $\displaystyle:=\bigl{\\{}P\in
E({\mathbb{F}}_{v}):\text{$P$ hits the
${\boldsymbol{\alpha}}$-component}\bigr{\\}},$ $\displaystyle
E^{({\boldsymbol{\beta}})}({\mathbb{F}}_{v})$ $\displaystyle:=\bigl{\\{}P\in
E({\mathbb{F}}_{v}):\text{$P$ hits the ${\boldsymbol{\beta}}$ or
${\boldsymbol{\beta}}^{\prime}$-component}\bigr{\\}}.$

Let $\pi$ be a uniformizer for $v$. The assumption that ${\mathbb{F}}_{v}$
does not have characteristic $2$ or $3$ means that we can start with a
Weierstrass equation in short normal form. Applying Tate’s algorithm ([14],
[12, IV §9]) along the branch that leads to Type $I_{M}^{*}$ shows that

$E:y^{2}=x^{3}+a_{4}x+a_{6}$

has Type $I_{M}^{*}$ reduction if and only if

$v(a_{4})=2,\quad v(a_{6})=3,\quad
v(\Delta)=v\bigl{(}-2^{4}\cdot(4a_{4}^{3}+27a_{6}^{2})\bigr{)}=6+M.$

We let $a_{4}=-3\pi^{2}a$ and $a_{6}=2\pi^{3}b$, giving the equation

$E:y^{2}=x^{3}-3\pi^{2}ax+2\pi^{3}b,$

with

$v(a)=v(b)=0,\quad
v(\Delta)=v\bigl{(}12^{3}\cdot\pi^{6}\cdot(a^{3}-b^{2})\bigr{)}=6+M.$

Since ${\mathbb{F}}_{v}$ is complete with algebraically closed residue field,
Hensel’s lemma tells us that $\alpha\in{\mathbb{F}}_{v}^{*}$ is a square if
and only if $v(\alpha)$ is even. In particular, we can find an
$A\in{\mathbb{F}}_{v}^{*}$ satisfying777The equality $v(A^{3}-b)=M$ serves to
fix the sign of $A$, since we know that $v(A^{6}-b^{2})=M$, and only one of
$A^{3}\pm{b}$ has positive valuation.

$A^{2}=a\quad\text{and}\quad v(A^{3}-b)=M.$

We change variables

$x\to{x+\pi{A}}\quad\text{and for convenience we set}\quad D=A^{3}-b.$

This yields the minmial Weierstrass equation

$E:y^{2}=x^{3}-3\pi Ax^{2}-2\pi^{3}D$

with

$\displaystyle v(A)$ $\displaystyle=0,\quad v(D)=M=v(j_{E}^{-1}),$
$\displaystyle{\mathcal{D}}_{E}$ $\displaystyle=12^{3}\pi^{6}D(2A^{3}-D),$
$\displaystyle v({\mathcal{D}}_{E})$ $\displaystyle=6+v(D)=6+M.$

For a Weierstrass equation of this form, the duplication formula [13,
III.2.3(d)] takes the form

$x(2P)=\frac{x^{4}+16\pi^{3}Dx+24\pi^{4}DA}{4y^{2}}.$

Let $P_{0}=(x_{0},y_{0})\in{E({\mathbb{F}}_{v})}$. Tracing the points through
Tate’s algorithm, we find that:

$\displaystyle P_{0}\in E^{({\boldsymbol{0}})}({\mathbb{F}}_{v})$
$\displaystyle\;\Longleftrightarrow\;v\bigl{(}x(P_{0})\bigr{)}\leq 0,$
$\displaystyle P_{0}\in E^{({\boldsymbol{\alpha}})}({\mathbb{F}}_{v})$
$\displaystyle\;\Longleftrightarrow\;v\bigl{(}x(P_{0})\bigr{)}=1,$ $P_{0}\in
E^{({\boldsymbol{\beta}})}({\mathbb{F}}_{v})$ and $M$ odd
$\displaystyle\;\Longleftrightarrow\;\text{$v\bigl{(}x(P_{0})\bigr{)}\geq\frac{M+3}{2}$
and $v\bigl{(}y(P_{0})\bigr{)}=\frac{M+3}{2}$,}$ $P_{0}\in
E^{({\boldsymbol{\beta}})}({\mathbb{F}}_{v})$ and $M$ even
$\displaystyle\;\Longleftrightarrow\;\text{$v\bigl{(}x(P_{0})\bigr{)}=\frac{M+2}{2}$
and $v\bigl{(}y(P_{0})\bigr{)}\geq\frac{M+4}{2}$.}$

### Case ${\boldsymbol{\alpha}}$:
$P_{0}\in{E^{({\boldsymbol{\alpha}})}({\mathbb{F}}_{v})}$.

In this case we always have $2P_{0}\in{E^{({\boldsymbol{0}})}}$. The three
monomials in the numerator of $x(2P_{0})$ have valuations

$v(x_{0}^{4})=4,\quad v(16\pi^{3}Dx_{0})=4+M,\quad v(24\pi^{4}DA)=4+M,$

so the valuation of the numerator of $x(2P_{0})$ is $4$. Since
$2P_{0}\in{E^{({\boldsymbol{0}})}}({\mathbb{F}}_{v})$, its local height is
given by

$\displaystyle\hat{\lambda}_{v}(2P_{0})$
$\displaystyle=\frac{1}{2}v\bigl{(}x(2P_{0})^{-1}\bigr{)}+\frac{1}{12}v({\mathcal{D}}_{E})$
$\displaystyle=\frac{1}{2}v(y_{0}^{2})-\frac{1}{2}\cdot
4+\frac{1}{12}v({\mathcal{D}}_{E})$
$\displaystyle=v(y_{0})-2+\frac{1}{12}v({\mathcal{D}}_{E}).$ (36)

We use this to compute

$\displaystyle\hat{\lambda}_{v}(P_{0})$
$\displaystyle=\frac{1}{4}\Bigl{(}\hat{\lambda}_{v}(2P_{0})-v(2y_{0})+\frac{1}{4}v({\mathcal{D}}_{E})\Bigr{)}\quad\text{duplication
formula,}$
$\displaystyle=\frac{1}{4}\left(\left\\{v(y_{0})-2+\frac{1}{12}v({\mathcal{D}}_{E})\right\\}-v(2y_{0})+\frac{1}{4}v({\mathcal{D}}_{E})\right)\quad\text{from
\eqref{eqn:lv2P0IMstar},}$
$\displaystyle=\frac{1}{12}v({\mathcal{D}}_{E})-\frac{1}{2}\quad\text{note
that $v(2)=0$,}$ $\displaystyle=\frac{v(j_{E}^{-1})}{12}\quad\text{since
$v({\mathcal{D}}_{E})=M+6=v(j_{E}^{-1})+6$.}$

### Case ${\boldsymbol{\beta}}$-even:
$P_{0}\in{E^{({\boldsymbol{\beta}})}({\mathbb{F}}_{v})}$ and $M$ even.

This case is characterized by

$v(x_{0})=\frac{M+2}{2}\quad\text{and}\quad v(y_{0})\geq\frac{M+4}{2}.$

Hence the three monomials in the numerator of $x(2P_{0})$ have valuations

$v(x_{0}^{4})=2M+6,\quad v(16\pi^{3}Dx_{0})=\frac{3}{2}M+4,\quad
v(24\pi^{4}DA)=4+M.$

Since

$4+M<\frac{3}{2}M+4<2M+6\quad\text{for all $M\geq 1$,}$

we find that

$v\bigl{(}x(2P_{0})\bigr{)}=4+M-2v(y_{0})=v({\mathcal{D}}_{E})-2-2v(y_{0}).$

Further, since
$E({\mathbb{F}}_{v})/E_{0}({\mathbb{F}}_{v})=({\mathbb{Z}}/2{\mathbb{Z}})^{2}$,
we again have $2P_{0}\in{E^{({\boldsymbol{0}})}}({\mathbb{F}}_{v})$, so we can
compute its local height as

$\displaystyle\hat{\lambda}_{v}(2P_{0})$
$\displaystyle=\frac{1}{2}v\bigl{(}x(2P_{0})^{-1}\bigr{)}+\frac{1}{12}v({\mathcal{D}}_{E})$
$\displaystyle=\frac{1}{2}\Bigl{(}-v({\mathcal{D}}_{E})+2+2v(y_{0})\Bigr{)}+\frac{1}{12}v({\mathcal{D}}_{E})$
$\displaystyle=v(y_{0})+1-\frac{5}{12}v({\mathcal{D}}_{E}).$ (37)

We use this to compute

$\displaystyle\hat{\lambda}_{v}(P_{0})$
$\displaystyle=\frac{1}{4}\Bigl{(}\hat{\lambda}_{v}(2P_{0})-v(2y_{0})+\frac{1}{4}v({\mathcal{D}}_{E})\Bigr{)}\quad\text{duplication
formula,}$
$\displaystyle=\frac{1}{4}\left(\left\\{v(y_{0})+1-\frac{5}{12}v({\mathcal{D}}_{E})\right\\}-v(2y_{0})+\frac{1}{4}v({\mathcal{D}}_{E})\right)\quad\text{from
\eqref{eqn:lv2P0IMstarbetaeven},}$
$\displaystyle=-\frac{1}{24}v({\mathcal{D}}_{E})+\frac{1}{4}\quad\text{note
that $v(2)=0$,}$ $\displaystyle=-\frac{v(j_{E}^{-1})}{24}\quad\text{since
$v({\mathcal{D}}_{E})=M+6=v(j_{E}^{-1})+6$.}$

### Case ${\boldsymbol{\beta}}$-odd:
$P_{0}\in{E^{({\boldsymbol{\beta}})}({\mathbb{F}}_{v})}$ and $M$ odd.

This case is characterized by

$v(x_{0})\geq\frac{M+3}{2}\quad\text{and}\quad v(y_{0})=\frac{M+3}{2}.$

Further, we have
$E({\mathbb{F}}_{v})/E_{0}({\mathbb{F}}_{v})\cong{\mathbb{Z}}/4{\mathbb{Z}}$,
with the ${\boldsymbol{\beta}}$ components being the elements of order $4$,
and thus $2P_{0}\in{E^{({\boldsymbol{\alpha}})}}$. It follows from a case that
we already did that

$\hat{\lambda}(2P_{0})=\frac{v(j_{E}^{-1})}{12}.$

Hence

$\displaystyle\hat{\lambda}_{v}(P_{0})$
$\displaystyle=\frac{1}{4}\Bigl{(}\hat{\lambda}_{v}(2P_{0})-v(2y_{0})+\frac{1}{4}v({\mathcal{D}}_{E})\Bigr{)}\quad\text{duplication
formula,}$
$\displaystyle=\frac{1}{4}\left(\frac{v(j_{E}^{-1})}{12}-\frac{M+3}{2}+\frac{1}{4}v({\mathcal{D}}_{E})\right)$
$\displaystyle=\frac{1}{4}\left(\frac{v(j_{E}^{-1})}{12}-\frac{v(j_{E}^{-1})+3}{2}+\frac{1}{4}\Bigl{(}6+v(j_{E}^{-1})\Bigr{)}\right)$
$\displaystyle=-\frac{1}{24}v(j_{E}^{-1}).\qed$ (38)

##  Appendix C Local Height Formulas via Intersection Theory

An alternative approach to canonical heights in the function field setting is
via intersection theory, an idea appearing already in [9]. Writing a global
intersection as a sum of local intersections gives another way to derive local
height formulas, as is done in [3] for elliptic surfaces over ${\mathbb{C}}$.
We sketch the proof and rederive the local height formula for $I_{M}^{*}$
reduction, omitting some details.

Let $\pi:{\mathcal{E}}\to{C}$ be a smooth minimal elliptic fibration over a
smooth projective curve, everything defined over an algebraically closed field
$k$. For this section we let ${\mathbb{F}}=k(C)$ be the global function field,
and for $v\in{M_{\mathbb{F}}}$, i.e., for points of $C$, we denote the
completion of ${\mathbb{F}}$ at $v$ by ${\mathbb{F}}_{v}$. Let
${\mathcal{K}}_{\mathcal{E}}\in\operatorname{Pic}({\mathcal{E}})$ be the
canonical bundle, and let
${\mathcal{D}}_{\mathcal{E}}\in\operatorname{Div}(C)$ be the minimal
discriminant of ${\mathcal{E}}/C$. Fixing a Weierstrass equation $y^{2}=f(x)$
and using the section $(dx)^{6}/\Delta$ to $\Omega_{\mathcal{E}}^{\otimes
12}$, a local computation yields

${\mathcal{O}}_{\mathcal{E}}(\pi^{*}{\mathcal{D}}_{\mathcal{E}})\cong{\mathcal{K}}_{{\mathcal{E}}/C}^{\otimes
12},$ (39)

where
${\mathcal{K}}_{{\mathcal{E}}/C}:={\mathcal{K}}_{\mathcal{E}}\otimes\pi^{*}{\mathcal{K}}_{C}^{-1}$
is the relative canonical bundle for the fibration $\pi:{\mathcal{E}}\to{C}$.

Let $O:C\to{\mathcal{E}}$ be the zero-section, and let $P:C\to{\mathcal{E}}$
be an arbitrary section. We identify $P$ and $O$ with their images in
${\mathcal{E}}$, so in particular $(P)$ and $(O)$ are divisors on
${\mathcal{E}}$. The Hodge index theorem, i.e., the fact that the intersection
pairing is negative definite except for one positive direction, allows one to
find a fibral divisor $\Phi_{P}\in\operatorname{Div}({\mathcal{E}})$ so that
the divisor888The divisor $\Phi_{P}$ is uniquely determined up to addition of
a divisor of the form $\pi^{*}\Gamma$ for some
$\Gamma\in\operatorname{Div}(C)$. We can pin down a particular $\Phi_{P}$ by
requiring that $i_{v}\bigl{(}\Phi_{P},(O)\bigr{)}=0$ for all $v$, i.e., the
support $\Phi_{P}$ does not contain any fibral components that intersect the
zero section; but any choice of $\Phi_{P}$ will yield the same formulas for
${\hat{h}}_{E}$ and $\hat{\lambda}_{v}$.

$D_{P}:=(P)-(O)+\Phi_{P}$

satisfies

$D_{P}\cdot F=0\quad\text{for all fibral divisors
$F\in\operatorname{Div}({\mathcal{E}})$.}$

Then one can show that

${\hat{h}}_{\mathcal{E}}(P)=-\frac{1}{2}D_{P}^{2}.$ (40)

We observe that the translation-by-$P$ automorphism
$\tau_{P}:{\mathcal{E}}\to{\mathcal{E}}$ satisfies $\tau_{P}^{*}(P)=(O)$, so
$(P)^{2}=(O)^{2}$. We compute

$\displaystyle(O)^{2}$
$\displaystyle=-(O)\cdot{\mathcal{K}}_{\mathcal{E}}+2g(C)-2\quad\text{the
adjunction formula,}$
$\displaystyle=-(O)\cdot{\mathcal{K}}_{\mathcal{E}}+\deg{\mathcal{K}}_{C}$
$\displaystyle=-(O)\cdot{\mathcal{K}}_{\mathcal{E}}+(O)\cdot\pi^{*}{\mathcal{K}}_{C}$
$\displaystyle=-(O)\cdot\left({\mathcal{K}}_{\mathcal{E}}\otimes\pi^{*}{\mathcal{K}}_{C}^{-1}\right)$
$\displaystyle=-(O)\cdot{\mathcal{K}}_{{\mathcal{E}}/C}$
$\displaystyle=-\frac{1}{12}(O)\cdot\pi^{*}{\mathcal{D}}_{\mathcal{E}}\quad\text{from
\eqref{eqn:OEpiDEKEC12},}$
$\displaystyle=-\frac{1}{12}\deg{\mathcal{D}}_{\mathcal{E}}.$ (41)

Putting this together, we find that

$\displaystyle{\hat{h}}_{\mathcal{E}}(P)$
$\displaystyle=-\frac{1}{2}D_{P}^{2}\quad\text{from \eqref{eqn:hecalP},}$
$\displaystyle=-\frac{1}{2}D_{P}\cdot\bigl{(}(P)-(O)+\Phi_{P}\bigr{)}\quad\text{by
definition of $D_{P}$,}$
$\displaystyle=-\frac{1}{2}D_{P}\cdot\bigl{(}(P)-(O)\bigr{)}\quad\begin{tabular}[t]{@{}l@{}}since
$D_{P}\cdot F=0$ for fibral $F$,\\\ and $\Phi_{P}$ is fibral,\\\
\end{tabular}$
$\displaystyle=-\frac{1}{2}\bigl{(}(P)-(O)+\Phi_{P}\bigr{)}\cdot\bigl{(}(P)-(O)\bigr{)}\quad\text{by
definition of $D_{P}$,}$
$\displaystyle=(P)\cdot(O)+-\frac{1}{2}(P)^{2}-\frac{1}{2}(O)^{2}-\frac{1}{2}\Phi_{P}\cdot\bigl{(}(P)-(O)\bigr{)}$
$\displaystyle=(P)\cdot(O)+\frac{1}{12}\deg{\mathcal{D}}_{\mathcal{E}}-\frac{1}{2}\Phi_{P}\cdot\bigl{(}(P)-(O)\bigr{)}\quad\text{from
\eqref{eqn:Osqr2112}.}$

For $P\neq{O}$, this allows a local decomposition by setting

$\hat{\lambda}_{v}(P):=i_{v}\bigl{(}(P),(O)\bigr{)}+\frac{1}{12}v({\mathcal{D}}_{\mathcal{E}})-\frac{1}{2}i_{v}\bigl{(}\Phi_{P},(P)-(O)\bigr{)}.$
(42)

We note that if $P\in E_{0}({\mathbb{F}}_{v})$, i.e., if $P$ goes through the
identity component of the fiber over $v$, then the fact that $\Phi_{P}$ is
fibral implies that the their term vanishes. Further, working on a minimal
Weierstrass equation, the first term is

$i_{v}\bigl{(}(P),(O)\bigr{)}=\frac{1}{2}\max\bigl{\\{}v\bigl{(}x(P)^{-1}\bigr{)},\,0\bigr{\\}}$

is the location of $P$ in the formal group filtration. Hence

$P\in
E_{0}({\mathbb{F}}_{v})\;\Longrightarrow\;\hat{\lambda}_{v}(P)=\nu+\frac{1}{12}v({\mathcal{D}}_{\mathcal{E}})\quad\text{for
$P\in E_{\nu}({\mathbb{F}}_{v})\smallsetminus E_{\nu+1}({\mathbb{F}}_{v})$.}$

Finally, for $P\notin{E_{0}({\mathbb{F}}_{v})}$, the local height
$\hat{\lambda}_{v}(P)$ depends only on the fibral component hit by $P$ and can
be determined by computing the divisor $\Phi_{P}$. To do this, we start by
computing the intersection matrix $A$ for the components of the special fiber.
Figure 2 gives the matrix $A(I_{M}^{*})$ for the $I_{M}^{*}$ configuration
illustrated in Figure 1.

$\bordermatrix{&0&{\boldsymbol{\alpha}}&{\boldsymbol{\gamma}}_{0}&{\boldsymbol{\gamma}}_{1}&\cdots&\cdots&{\boldsymbol{\gamma}}_{M-1}&{\boldsymbol{\gamma}}_{M}&{\boldsymbol{\beta}}&{\boldsymbol{\beta}}^{\prime}\cr
0&-2&0&1\cr{\boldsymbol{\alpha}}&0&-2&1\cr{\boldsymbol{\gamma}}_{0}&1&1&-2&1\cr{\boldsymbol{\gamma}}_{1}&&&1&-2&1\cr\vdots&&&&\ddots&\ddots&\ddots\cr\vdots&&&&&\ddots&\ddots&\ddots\cr{\boldsymbol{\gamma}}_{M-1}&&&&&&1&-2&1\cr{\boldsymbol{\gamma}}_{M}&&&&&&&1&-2&1&1\cr{\boldsymbol{\beta}}&&&&&&&&1&-2&1\cr{\boldsymbol{\beta}}^{\prime}&&&&&&&&1&0&-2\cr}$
Figure 2. The intersecton matrix $A(I_{M}^{*})$ for the $I_{M}^{*}$ components
illustrated in Figure 1.

The null space of $A(I_{M}^{*})$ is spanned by the total fiber (with
multiplicities)

${\boldsymbol{0}}+{\boldsymbol{\alpha}}+2{\boldsymbol{\gamma}}_{0}+2{\boldsymbol{\gamma}}_{1}+\cdots+2{\boldsymbol{\gamma}}_{M}+{\boldsymbol{\beta}}+{\boldsymbol{\beta}}^{\prime}.$

This can be computed directly, or from the fact that all total fibers are
algebraically (and hence numerically) equivalent. We define a reduced matrix

$A(I_{M}^{*})^{\textup{red}}=\text{the matrix $A(I_{M}^{*})$ with the first
row and column deleted.}$

The matrix $A(I_{M}^{*})^{\textup{red}}$ is invertible, and indeed one can
check that

$\det\bigl{(}A(I_{M}^{*})^{\textup{red}}\bigr{)}=(-1)^{M}\cdot 4.$

We define a vector whose entries are the non-${\boldsymbol{0}}$ fibral
components and a vector whose entries are the coefficients of $\Phi_{P}$:

$\displaystyle{\boldsymbol{F}}$
$\displaystyle=({\boldsymbol{\alpha}},{\boldsymbol{\gamma}}_{0},{\boldsymbol{\gamma}}_{1},\ldots,{\boldsymbol{\gamma}}_{M},{\boldsymbol{\beta}},{\boldsymbol{\beta}}^{\prime}),$
$\displaystyle\tilde{\Phi}_{P}$
$\displaystyle=(a,c_{0},c_{1},\ldots,c_{M},b,b^{\prime}),$

so

$\Phi_{P}={\boldsymbol{F}}\cdot{{}^{t}\tilde{\Phi}_{P}}\quad\text{and}\quad
A(I_{M}^{*})^{\textup{red}}={{}^{t}\\!{\boldsymbol{F}}}\cdot{\boldsymbol{F}}.$

The $v$-fibral part of the (normalized) divisor $\Phi_{P}$ is determined by

$P\cdot{{}^{t}\\!{\boldsymbol{F}}}=A(I_{M}^{*})^{\textup{red}}\cdot{{}^{t}\tilde{\Phi}_{P}},\quad\text{and
thus}\quad{{}^{t}\tilde{\Phi}_{P}}=\bigl{(}A(I_{M}^{*})^{\textup{red}}\bigr{)}^{-1}\cdot\bigl{(}P\cdot{{}^{t}\\!{\boldsymbol{F}}}\bigr{)}.$

Since we have normalized $\Phi_{P}$ so that
$i_{v}\bigl{(}\Phi_{P},(O)\bigr{)}=0$, the quantity appearing in (42) is

$i_{v}\bigl{(}\Phi_{P},(P)\bigr{)}=P\cdot\Phi_{P}=P\cdot{\boldsymbol{F}}\cdot{{}^{t}\tilde{\Phi}_{P}}=(P\cdot{\boldsymbol{F}})\cdot\bigl{(}A(I_{M}^{*})^{\textup{red}}\bigr{)}^{-1}\cdot\bigl{(}P\cdot{{}^{t}\\!{\boldsymbol{F}}}\bigr{)}.$

Figure 3 gives an explicit formula for
$\bigl{(}A(I_{M}^{*})^{\textup{red}}\bigr{)}^{-1}$. But we really only need a
few of its entries.

<EMAIL_ADDRESS>1&2&2&\cdots&2&\frac{2}{2}&\frac{2}{2}\\\\[3.0pt]
1&2&3&\cdots&3&\frac{3}{2}&\frac{3}{2}\\\\[3.0pt]
&\vdots&&\ddots&\vdots&\vdots&\vdots\\\
1&2&3&\cdots&M+2&\frac{M+2}{2}&\frac{M+2}{2}\\\\[3.0pt]
\hline\cr\frac{1}{2}&\frac{2}{2}&\frac{3}{2}&\cdots&\frac{M+2}{2}&\frac{M+4}{2}&\frac{M+2}{2}\vphantom{{\frac{1}{1}}^{1}}\\\\[3.0pt]
\frac{1}{2}&\frac{2}{2}&\frac{3}{2}&\cdots&\frac{M+2}{2}&\frac{M+2}{2}&\frac{M+4}{2}\\\
\end{array}\right)$ Figure 3. The inverse of the reduced intersecton matrix
for $I_{M}^{*}$ reduction.

Indeed, since $P$ must go through a multiplicity-1 component and since we are
assuming that $P\notin{\mathcal{E}}^{({\boldsymbol{0}})}({\mathbb{F}}_{v})$,
there are only three cases:

$P\cdot{{}^{t}\\!{\boldsymbol{F}}}=\begin{cases}{}^{t}(1,0,0,\ldots,0,0,0)&\text{if
$P\in{\mathcal{E}}^{({\boldsymbol{\alpha}})}({\mathbb{F}}_{v})$,}\\\
{}^{t}(0,0,0,\ldots,0,1,0)&\text{if
$P\in{\mathcal{E}}^{({\boldsymbol{\beta}})}({\mathbb{F}}_{v})$,}\\\
{}^{t}(0,0,0,\ldots,0,0,1)&\text{if
$P\in{\mathcal{E}}^{({\boldsymbol{\beta}}^{\prime})}({\mathbb{F}}_{v})$.}\\\
\end{cases}$

Writing $B_{i,j}$ for the $(i,j)$th entry of a matrix $B$, Figure 3 gives the
formulas

$i_{v}\bigl{(}\Phi_{P},(P)\bigr{)}=\begin{cases}\bigl{(}A(I_{M}^{*})^{\textup{red}}\bigr{)}^{-1}_{1,1}=-1&\text{if
$P\cdot{\boldsymbol{\alpha}}=1$,}\\\
\bigl{(}A(I_{M}^{*})^{\textup{red}}\bigr{)}^{-1}_{M+3,M+3}=-\frac{M+4}{4}&\text{if
$P\cdot{\boldsymbol{\beta}}=1$,}\\\
\bigl{(}A(I_{M}^{*})^{\textup{red}}\bigr{)}^{-1}_{M+4,M+4}=-\frac{M+4}{4}&\text{if
$P\cdot{\boldsymbol{\beta}}^{\prime}=1$.}\\\ \end{cases}$

Hence

$\displaystyle P\in{\mathcal{E}}^{({\boldsymbol{\alpha}})}({\mathbb{F}}_{v})$
$\displaystyle\quad\Longrightarrow\quad\begin{aligned}
{\hat{h}}_{\mathcal{E}}(P)&=\frac{1}{12}v({\mathcal{D}}_{\mathcal{E}})-\frac{1}{2}\\\
&=\frac{1}{12}v(j_{E}^{-1}),\\\ \end{aligned}$ (43) $\displaystyle
P\in{\mathcal{E}}^{({\boldsymbol{\beta}})}({\mathbb{F}}_{v})$
$\displaystyle\quad\Longrightarrow\quad\begin{aligned}
{\hat{h}}_{\mathcal{E}}(P)&=\frac{1}{12}v({\mathcal{D}}_{\mathcal{E}})-\frac{M+4}{8}\\\
&=-\frac{1}{24}v(j_{E}^{-1})+\frac{1}{8}\bigl{(}v(j_{E}^{-1})-M\bigr{)},\\\
\end{aligned}$ (44)

where we have used the fact that

$v({\mathcal{D}}_{\mathcal{E}})=v(j_{E}^{-1})+6\quad\text{for $I_{M}^{*}$
reduction.}$

These formulas hold in all characteristics. If we further assume that the
residue characteristic is not $2$, then $v(j_{E}^{-1})=M$, and the second
formula simplifies.

######  Acknowledgements.

The author would like to thank David Masser for suggesting revisiting this
problem in the function field setting, Pascal Autissier for sharpening the
inequality in Lemma 5, and Jordan Ellenberg for a helpful observation. Various
numerical and algebraic calculations were done using Pari-GP [15]. The
author’s research was partially supported by Simons Collaboration Grant
#712332.

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