#  On the Mordell–Weil lattice of $y^{2}=x^{3}+bx+t^{3^{n}+1}$ in
characteristic $3$

Gauthier Leterrier

Abstract – We study the elliptic curves given by $y^{2}=x^{3}+bx+t^{3^{n}+1}$
over global function fields of characteristic $3$ ; especially we perform an
explicit computation of the $L$-function, by relating it to the zeta function
of a certain superelliptic curve $u^{3}+bu=v^{3^{n}+1}$.

More specifically, using the Néron-Tate height on the Mordell–Weil group, we
get lattices in dimension $2\cdot 3^{n}$ for every $n\geq 1$, which for $n=1$
have the same packing density as the lattice $E_{6}$ in dimension $6$, and as
the best currently known sphere packing in dimension $54$ (case $n=3$). It
improves on the currently best known sphere packing densities in dimensions
162 (case $n=4$) and 486 (case $n=5$).

††Date : July 2021. Keywords : elliptic curves, function fields,
$L$-functions, Mordell–Weil group, sphere packings. MSC 2010 (Math. Subject
Classification) : 11G05, 11M38, 11T24, 11H31.

## 1  $\cdot$ Introduction and main results

Following ideas of N. Elkies [ElkiesMW1, ElkiesMW2, ElkiesMW3] and T. Shioda
[Shioda_MW_and_sphere_packings] (from the 1990’s), one can use elliptic curves
over global function fields to get interesting lattice sphere packings, which
can have arbitrarily large rank. This is an opportunity to study their
arithmetic, and in particular their $L$-function. Interestingly, for our
family of elliptic curves, we can compute the $L$-function very explicitly in
an elementary way, and deduce the main arithmetic invariants of our curves.

For a given positive integer $n\geq 1$ and for an element
$b\in\mathbb{F}_{3^{n}}^{\times}$, we consider the elliptic curves given by
the (affine) Weierstrass equation :

$\displaystyle E_{n,b}\;:\;y^{2}=x^{3}+bx+t^{3^{n}+1}$ (1.1)

over $\mathbb{F}_{3^{n}}(t)$. One of our main results here is the explicit
computation of the $L$-function of $E_{n,b}$ over $\mathbb{F}_{3^{2n}}(t)$ for
some choice of the parameter $b$. In particular, we can determine the exact
value of the (analytic) rank of those elliptic curves.

######  Theorem A.

Let $n\geq 1$ be an integer and set $q=3^{n}$. Let
$b\in\mathbb{F}_{q}^{\times}$ be any element such that
$b^{\tfrac{q-1}{2}}=(-1)^{n+1}$. In other words, if $n$ is odd, $b$ is a
square in $\mathbb{F}_{q}^{\times}$ (for instance $b=1$), and if $n$ is even,
$b\in\mathbb{F}_{q}^{\times}$ is not a square.

Then the $L$-function (as defined in equation 2.2) of the elliptic curve
$E_{n,b}$ over $\mathbb{F}_{q^{2}}(t)$, given by 1.1, is equal to

$L\left(E_{n,b}/\mathbb{F}_{q^{2}}(t),\;T\right)=(1-q^{2}T)^{2\cdot 3^{n}}.$
(1.2)

In particular, the analytic rank of $E_{n,b}$ over $\mathbb{F}_{q^{2}}(t)$ is
equal to $2\cdot 3^{n}$.

Let us explain how to construct a lattice from those curves. In general, if
$E$ is an elliptic curve over a global function field $K=k(X)$, where $X$ is a
smooth projective algebraic curve over a finite field $k$ (we will mostly
focus on the case $X=\mathbb{P}^{1}$), then $E(K)$ is a finitely generated
abelian group, by Mordell–Weil theorem (generalized by Lang and Néron ; see
[SilvermanAAEC, theorem III.6.1]).

Given a Weierstrass equation for $E$ over $k(X)$, we have a degree-2 cover
$x:E\to\mathbb{P}^{1}$ given by the $x$-coordinate. For every $P\in
E(k(X))\smallsetminus\\{O_{E}\\}$, we can see $x(P)\in k(X)$ as a rational map
$X\dashrightarrow\mathbb{P}^{1}$. We can therefore define the naive height as

$\begin{array}[]{lrcl}h:&E(K)&\longrightarrow&\mathbb{Z}_{\geq 0}\\\
&P&\longmapsto&\begin{cases}\mathsf{\deg}(x(P))&\text{if }P\neq O_{E}\\\
0&\text{else}\end{cases}\end{array}$

(if $x(P)\in k$ is constant, its degree is set to be $0$). We define the
(canonical) Néron–Tate height as

$\hat{h}(P):=\lim_{n\to+\infty}4^{-n}h(2^{n}P)\in\mathbb{R}$ (1.3)

for every $P\in E(K)$. It is a quadratic form, which is positive-definite on
$E(K)/E(K)_{\operatorname{tors}}$ ([SilvermanAAEC, theorem III.4.3]), where
$E(K)_{\operatorname{tors}}$ denotes the torsion subgroup of $E(K)$.
Therefore, we obtain a lattice, called the Mordell–Weil lattice of $E$ over
$K$. We introduce a convenient sublattice, namely :

######  Definition 1.1.

The narrow Mordell–Weil lattice of $E$ over $K$ consists of all the points
$P\in E(K)$ such that for every place $v$ of $K$, the reduction $\overline{P}$
is a non-singular point on the reduction $\overline{E_{v}}$ of a minimal
integral Weierstrass model $E_{v}$ of $E$ at $v$. It is denoted by
$E(K)^{0}\subset E(K)$.

Now, given a lattice $L\hookrightarrow\mathbb{R}^{d}$, let

$\lambda_{1}(L):=\min\left\\{\|v\|\;:\;v\in L\smallsetminus\\{0\\}\right\\}$
(1.4)

be the length of one of its shortest non-zero vectors. Then the translates
$B+L$ of the euclidean ball
$B=B\left(0,\frac{\lambda_{1}(L)}{2}\right)\subset\mathbb{R}^{d}$ by points of
$L$ defines a lattice packing of balls. Its density is defined as the
proportion of the space covered by $B+L$, i.e.

$D(L):=\limsup\limits_{r\to+\infty}\dfrac{\operatorname{vol}((B+L)\cap
B(0,r))}{\operatorname{vol}(B(0,r))}\in[0,1].$ (1.5)

In the case of such a lattice packing, we can simplify the expression
into$D(L)=\dfrac{(\lambda_{1}(L)/2)^{d}\cdot\operatorname{vol}(B(0,1))}{\operatorname{vol}(\mathbb{R}^{d}/L)}$.
This motivates us to consider the following normalization :

######  Definition 1.2.

The center density of a packing of balls given by a lattice
$L\hookrightarrow\mathbb{R}^{d}$ as

$\delta(L):=\dfrac{(\lambda_{1}(L)/2)^{d}}{\operatorname{vol}(\mathbb{R}^{n}/L)}.$

As a corollary of A and of results of Shioda, we get a lower bound on the
sphere packing density of the narrow Mordell–Weil lattice of the elliptic
curves $E_{n,b}$.

######  Corollary A.

Let $n\geq 1$ be an integer, fix $b\in\mathbb{F}_{3^{n}}^{\times}$ as in A,
and set $q=3^{n}$.

Let $L_{n}:=E_{n,b}\left(\mathbb{F}_{q^{2}}(t)\right)^{0}$ be the narrow
Mordell–Weil lattice of the elliptic curve $E_{n,b}$ over
$\mathbb{F}_{q^{2}}(t)$, as defined in definition 1.1.

Then the rank of $L_{n}$ is $2\cdot 3^{n}$ and its center density satisfies
the lower bound

$\displaystyle\delta(L_{n})\geq\dfrac{\left((3^{n-1}+1)/4\right)^{3^{n}}}{3^{1/2}\cdot
3^{n\cdot(3^{n-1}-1)/2}}$ (1.6)

In particular, for $n\in\\{1,...,7\\}$, we get the following values, gathered
in the table below.

$n$ |  rank of $L_{n}$ |  $\log_{2}\left(\delta(L_{n})\right)\geq$ |  Best _lattice_ packing density known so far
---|---|---|---
$1$ |  $6$ |  $\log_{2}\left(\sqrt{3}/24\right)\simeq-3.79248$ |  $\delta(E_{6})=\frac{1}{8\sqrt{3}}$ ([Conway_Sloane_book], p. xix)
$2$ |  $18$ |  $\log_{2}\left(\tfrac{\sqrt{3}}{27}\right)\simeq-3.962406$ |  $-3.79248$ [Conway_Sloane_book], p. xix
$3$ |  $54$ |  $\log_{2}\left(\tfrac{\sqrt{3}\cdot 5^{27}}{2^{27}\cdot 3^{13}}\right)\simeq 15.88002$ |  15.88 (Elkies [Conway_Sloane_book], p. xviii)
$4$ |  $162$ |  144.1852 |  130.679 [Refinement_Craig]
$5$ |  $486$ |  741.1001 |  703.05 [Ball_lower_bound_on_packing_density]
$6$ |  $1458$ |  3172.032 |  3236.6 [Ball_lower_bound_on_packing_density]

We see that in dimensions $6$ and $54$, we get the same density as the
previous densest known lattice packings of balls. Moreover, in dimensions 162
and 486, we improve the current records. But in dimension 18, another
construction achieves a higher packing density, and in dimensions above 1458,
non-constructive lower bounds are the best known so far.

### 1.1  $\cdot$ Outline of the proofs

A is proved in section 2 by performing an explicit computation of the
$L$-function. This requires counting the number of points on the reduction of
$E_{n,b}$ modulo all the places of $\mathbb{F}_{q^{2}}(t)$, which involves
sums of Legendre symbols, introduced in subsection 2.2.

Those sums can be determined thanks to an auxiliary superelliptic curve over
$\mathbb{F}_{q}$ (see subsection 2.3), and using the fact that $x\mapsto
x^{3}+bx$ is an additive map in characteristic $3$ (see lemmas 2.4 and 2.6).
Finally, the number of points over $\mathbb{F}_{q^{2}}$ of this auxiliary
superelliptic curve can be computed essentially because its jacobian is
isogenous to a power of a supersingular elliptic curve.

The idea behind this approach was inspired by the work of N. Elkies
[ElkiesMW1], where a counting argument about hyperelliptic curves has been
used. In our case, this will get replaced by a _superelliptic_ curve (see
subsection 2.3).

In both works, the elliptic curves (over function fields of characteristic 2
and 3 respectively) are isotrivial – we also say equivalently "potentially
constant". But in our case $E_{n,b}$ is a cubic twist of a constant curve
(i.e. defined over $\mathbb{F}_{q}$ ; see 3.5), while the elliptic curves
studied by Elkies were quadratic twists, which made the computation of the
rank (and then of the $L$-function) especially easy.

Finally, A, proved in section 3, follows from the use of Birch–Swinnerton-Dyer
formula (known in this case, because $E_{n,b}$ is isotrivial, see theorem 3.4
and 3.5), as well as a result of Shioda on the lower bound of the height of
points in the narrow Mordell-Weil lattice (theorem 3.6).

For the convenience of the reader, some frequently used notations are gathered
in a LABEL:SectNomenclature at the end of this document.

######  Remark 1.3.

Before continuing, we mention that Shioda’s results in
[Shioda_algorithm_Picard_nb] (especially remark 10 therein) tell us that if
$m$ is an even integer and $3^{e}\equiv-1\pmod{2m}$ for some integer $e\geq
1$, taken to be minimal, then the rank of
$\mathrm{E}_{m}(\overline{\mathbb{F}_{3}}(t))$ is $f(\mathrm{E}_{m})-4$, where
$f(\mathrm{E}_{m})$ denotes the conductor of the elliptic curve
$\mathrm{E}_{m}:y^{2}=x^{3}+x+t^{m}$ over $\mathbb{F}_{3}(t)$.

We are going to investigate the situation where $m=3^{n}+1$ for some integer
$n\geq 1$, which is _not_ directly covered by the above result.

Even if it did apply (e.g. via theorem 1, _ibid._), we would need to know over
what finite field of constants the rank is achieved, so anyway we need to use
another technique to determine the rank. ∎

## 2  $\cdot$ Proof of A

### 2.1  $\cdot$ Definition of the _L_ -function

In this paragraph, we consider a finite field $k=\mathbb{F}_{|k|}$, and we set
$K=k(t)$. Recall that the set of places $v$ of $K$ (i.e. an equivalence class
of absolute values on $K$, which are necessarily trivial on $k$ and are non-
archimedean) is in bijection with the set of closed points of
$\mathbb{P}^{1}_{k}$, which is itself in bijection with the set of Galois
orbits of $\bar{k}$-rational points in $\mathbb{P}^{1}(\overline{k})$.

We denote by $\infty$ the place of $K$ associated to the point
$[1:0]\in\mathbb{P}^{1}_{k}$ (it is given by $-\deg$).

Let $E$ be an elliptic curve over $K$. For any place $v$ of $K$, we let
$E_{v}$ be a minimal integral Weierstrass model at $v$. We let
$\overline{E_{v}}$ be its reduction modulo an uniformizer $\pi_{v}$ of the
ring of integers $\mathcal{O}_{v}\subset K_{v}$ of the completion $K_{v}$ of
$K$ at $v$, that is:
$\overline{E_{v}}:=E_{v}\times_{\mathcal{O}_{v}}\mathcal{O}_{v}/(\pi_{v})$.
This is a projective curve (possibly singular) over the finite field
$\mathbb{F}_{v}:=\mathcal{O}_{v}/(\pi_{v})$, and its isomorphism class does
not depend on the choice of a minimal integral Weierstrass model $E_{v}$ (this
follows from proposition VII.1.3 (b) in [SilvermanAEC]).

We now define the integers

$\begin{split}A_{E}(v,j)&:=|k|^{j}+1-\left|\overline{E_{v}}(\mathbb{F}_{|k|^{j}})\right|,\\\
a_{v}(E)&:=A_{E}(v,\deg(v))=|\mathbb{F}_{v}|+1-\left|\overline{E_{v}}(\mathbb{F}_{v})\right|,\end{split}$
(2.1)

where $j\geq 1$ is any integer multiple of $\deg(v):=[\mathbb{F}_{v}:k]$ (so
in particular $\mathbb{F}_{|k|^{j}}$ is an extension of $\mathbb{F}_{v}$).
Notice that $a_{v}(E)$ is equal $0$ if $E$ has additive reduction at $v$, and
$\pm 1$ if $E$ has multiplicative reduction at $v$ (this follows from
proposition III.2.5 in [SilvermanAEC], see also section 2.10 in
[Washington_EllCurves]).

We define the local factor at $v$ as

$L_{v}(E/K,T):=\begin{cases}1-a_{v}(E)T^{\deg(v)}+|k|^{\deg(v)}T^{2\deg(v)}&\text{if
$E$ has good reduction at $v$}\\\
1-a_{v}(E)T^{\deg(v)}&\text{else.}\end{cases}$

The L-function is defined as

$L(E/K,T):=\prod_{\begin{subarray}{c}v\text{ place}\\\ \text{of
}K\end{subarray}}L_{v}(E/K,T)^{-1}\in\mathbb{Z}[[T]].$ (2.2)

One can re-write the $L$-function as follows (this is Lemme 1.3.15 in
[Griffon_PhD]), by an elementary computation, where $[w]$ is the place
corresponding to $w$ :

$\displaystyle\log L(E/K,T)$ $\displaystyle=\sum_{j\geq
1}\Bigg{(}\sum_{w\in\mathbb{P}^{1}(\mathbb{F}_{|k|^{j}})}A_{E}([w],j)\Bigg{)}\dfrac{T^{j}}{j}.$
(2.3)

### 2.2  $\cdot$ Definition of the relevant Legendre sums

We first analyze the reduction types of the elliptic curve $E_{n,b}$ over
$\mathbb{F}_{q^{2}}(t)$, which we state as a proposition for later use. To
this end, we recall some standard notations.

######  Definition 2.1.

Let $k$ be a finite field, and let $X$ be a smooth projective geometrically
irreducible algebraic curve over $k$. Denote by $g_{X}$ its genus. Set
$K=k(X)$ and let $E$ be an elliptic curve over $K$.

  1. 1.

We denote by $\Delta_{\min}(E/K)$ the minimal discriminant of $E/K$ (as in
[SilvermanAAEC, exercise 3.35]). It is a divisor on the curve $X$.

  2. 2.

We denote by $f(E/K)$ the degree of the conductor divisor of $E/K$ (see
[SilvermanAAEC, exercise 3.36]).

  3. 3.

For each place $v$ of $K$, we denote by $c_{v}(E/K)$ the local Tamagawa factor
of $E/K$ at $v$, i.e. the number of irreducible components of the special
fiber of the Néron model of $E$ at $v$ that have multiplicity 1 and are
defined over the residue field $\mathbb{F}_{v}$ at $v$. We also set
$c(E/K):=\prod\limits_{v\in|X|}c_{v}(E/K)$.

######  Proposition 2.2.

Let $E_{n,b}$ be the elliptic curve $y^{2}=x^{3}+bx+t^{3^{n}+1}$ over
$K_{n}:=\mathbb{F}_{3^{2n}}(t)$ (where $b\in\mathbb{F}_{3^{n}}^{\times}$ and
$n\geq 1$ are fixed).

Then $E_{n,b}$ has good reduction at all places $v\neq\infty$ and has bad
additive reduction of type IV at $v=\infty$, with the following invariants :

$\displaystyle\deg\left(\Delta_{\min}(E_{n,b}/K_{n})\right)$
$\displaystyle=12\lceil(3^{n}+1)/6\rceil=2\cdot(3^{n}+3),$ $\displaystyle
f(E_{n,b}/K_{n})$
$\displaystyle=\deg\left(\Delta_{\min}(E_{n,b}/K_{n})\right)-2,$
$\displaystyle c(E_{n,b}/K_{n})$ $\displaystyle=c_{\infty}(E_{n,b}/K_{n})=3.$

[ Proof —]Its Weierstrass discriminant is
$-b^{3}\in\mathbb{F}_{3^{n}}^{\times}$ according to proposition A.1.1.(b) in
[SilvermanAEC]. In particular, $E_{n,b}$ had good reduction at all places
$v\neq\infty$ and $y^{2}=\linebreak x^{3}+bx+t^{3^{n}+1}$ is a minimal
integral Weierstrass model at all $v\neq\infty$.

We follow Tate’s algorithm as written down in [SilvermanAAEC], IV.9, p. 366.
Let $m:=3^{n}+1$ and $\mu:=\lceil\frac{m}{6}\rceil\geq 1$. It is easy to see
that $m\equiv 4\pmod{6}$ (e.g. by induction on $n$), so $6\mu-m=2$.

The affine equation $E_{\infty}:y^{2}=x^{3}+bxt^{-4\mu}+t^{m-6\mu}$ is a
minimal integral Weierstrass model at $v=\infty$, where we take
$\pi_{v}:=t^{-1}$ as uniformizer.

We have the following coefficients, as defined in [SilvermanAAEC], IV.9, p.
364 :

$b_{2}=0,\qquad b_{4}=2bt^{-4\mu},\qquad b_{6}=4t^{m-6\mu},\qquad
b_{8}=-\frac{1}{4}\cdot(2bt^{-4\mu})^{2}.$

The singular point on the reduction of $E_{\infty}$ modulo $\pi$ is
$(\overline{0},\overline{0})$, which means that the condition in _Step 2_ of
Tate’s algorithm (as in [SilvermanAAEC], IV.9, p. 366) is satisfied.

Since $6\mu-m=2$, the constant coefficient $a_{6}:=t^{m-6\mu}$ is equal to
(hence divisible by) $\pi^{2}$. Moreover, $b_{8}$ is divisible by $\pi^{3}$,
but $b_{6}$ is not divisible by $\pi^{3}$. Therefore, Tate’s algorithm stops
at _Step 5_ , which states that $E$ has bad additive reduction of
Kodaira–Néron type IV.

From there, we know that the Tamagawa number at $v=\infty$ is
$c_{\infty}(E)=3$, and that the local conductor is $f_{v}=v(\Delta)-2$ and
$v_{\pi}(\Delta)=12\mu=12\lceil m/6\rceil$, since the Weierstrass discriminant
is $\Delta=-8b_{4}^{3}-27b_{6}^{2}=-8\cdot 8b^{3}\pi^{12\mu}$.

When $k$ is a finite field of odd cardinality, let
$\lambda_{k}:k^{\times}\to\\{\pm 1\\}\hookrightarrow\mathbb{C}^{\times}$ be
the Legendre symbol (i.e. the unique character of order $2$, given explicitly
by $x\mapsto x^{\frac{|k|-1}{2}}$).

######  Remark 2.3.

It is important to be careful about the subscript $k$ in $\lambda_{k}$,
because when $q^{\prime}$ is a power of some odd prime power $q$, the
restriction of $\lambda_{\mathbb{F}_{q^{\prime}}}$ to the subfield
$\mathbb{F}_{q}$ is _not_ equal to $\lambda_{\mathbb{F}_{q}}$ (e.g.
$\lambda_{\mathbb{F}_{q^{2}}}(x)=1$ for every $x\in\mathbb{F}_{q}$). ∎

According to equation 2.3, computing the $L$-function of $E_{n,b}$ amounts to
determining the sums

$S_{b}(n,j):=\sum_{w\in\mathbb{P}^{1}(\mathbb{F}_{(q^{2})^{j}})}A_{E_{n,b}}(w,j),$
(2.4)

where $q=3^{n}$, as we have $\displaystyle\log
L(E_{n,b}/\mathbb{F}_{q^{2}}(t),T)=\sum\limits_{j\geq
1}S_{b}(n,j)\frac{T^{j}}{j}$. From the definition 2.1 of $A_{E}(w,j)$ and of
$E_{n,b}$, we see that the above sum is equal to

$S_{b}(n,j)=-\sum_{w,x\in\mathbb{F}_{(q^{2})^{j}}}\lambda_{\mathbb{F}_{(q^{2})^{j}}}\left(x^{3}+bx+w^{3^{n}+1}\right).$
(2.5)

Notice that we can discard the terms with $w=[1:0]$ since we have
$A_{E_{n,b}}(\infty,j)=0$ for every $j\geq 1$ by 2.2.

The strategy to evalute those sums $S_{b}(n,j)$ consists of two steps :

  1. 1

First, we will compute the number of points on a certain superelliptic curve
$C_{n,b}$, given by $v^{3^{n}+1}=u^{3}+bu$ over $\mathbb{F}_{3^{2nj}}$ for
every $j\geq 1$, where $b\in\mathbb{F}_{3^{n}}^{\times}$ is chosen as in A.

  2. 2

Secondly, we study the sums

$\sigma_{b}(j,t):=\sum_{x\in\mathbb{F}_{3^{2nj}}}\lambda_{\mathbb{F}_{3^{2nj}}}(x^{3}+bx+t),$
(2.6)

where $t\in\mathbb{F}_{3^{2nj}}$ and $j\geq 1$ is any integer.

### 2.3  $\cdot$ Number of points on the superelliptic curve $C_{n,b}$

For any $n\geq 1$, let $C_{n,b}^{\rm aff}$ be the affine curve
$v^{3^{n}+1}=u^{3}+bu$, defined over $\mathbb{F}_{3^{n}}$, where
$b\in\mathbb{F}_{3^{n}}^{\times}$ satisfies $b^{(3^{n}-1)/2}=(-1)^{n+1}$ as in
the statement of A.

There is a smooth projective irreducible curve $C_{n,b}$ over
$\mathbb{F}_{3^{n}}$ (unique up to isomorphism) such that its function field
is the same as the one of $C_{n,b}^{\rm aff}$. We say that $C_{n,b}$ is a
_superelliptic curve_.

It turns out that $C_{n,b}$ has a unique point at infinity, defined over
$\mathbb{F}_{3^{n}}$ (see proposition 2 in
[Galbraith_Paulus_Smart_Superelliptic]), so that
$|C_{n,b}(k)|=\left|C_{n,b}^{\rm aff}(k)\right|+1$ for every finite extension
$k$ of $\mathbb{F}_{3^{n}}$.

The key point is that we will be able to deduce the number of points
$|C_{n,b}(\mathbb{F}_{3^{2nj}})|$, for all $j\geq 1$, just from the
computation of $|C_{n,b}(\mathbb{F}_{3^{2n}})|$. Now, we can compute
$|C_{n,b}(\mathbb{F}_{3^{2n}})|$ because the norm map

$\mathbb{F}_{3^{2n}}^{\times}\longrightarrow\mathbb{F}_{3^{n}}^{\times},\qquad
v\longmapsto v^{3^{n}}\cdot v=v^{3^{n}+1}=:w$

is a surjective morphism, with kernel of size
$\dfrac{3^{2n}-1}{3^{n}-1}=3^{n}+1$.

Therefore, we get

$|C_{n,b}(\mathbb{F}_{3^{2n}})|=1+3+(3^{n}+1)\sum_{w\in\mathbb{F}_{3^{n}}^{\times}}\\#\\{u\in\mathbb{F}_{3^{2n}}\;:\;u^{3}+bu=w\\}$
(2.7)

We determine each term in the latter sum in the following lemma (applied to
the case where $p:=3$).

######  Lemma 2.4.

Let $p$ be an odd prime, $n\geq 1$ be an integer, set $q=p^{n}$ and let
$b\in\mathbb{F}_{p^{n}}^{\times}$ be any element such that

$\mathrm{Nr}_{\mathbb{F}_{q}/\mathbb{F}_{p}}(b)\;=\;b^{\tfrac{p^{n}-1}{p-1}}=(-1)^{n+1}.$
(2.8)

Then we have

$\\#\\{x\in\mathbb{F}_{q^{2}}\;:\;x^{p}+bx\in\mathbb{F}_{q}\\}=p^{n+1}=p\cdot
q.$

[ Proof —]Consider the maps $f,g_{b}:\mathbb{F}_{q^{2}}\to\mathbb{F}_{q^{2}}$
defined by $f:x\mapsto x^{q}-x$ and $g_{b}:x\mapsto x^{p}+bx$. The key point
is that these maps are endomorphisms of the additive group
$(\mathbb{F}_{q^{2}},+)$ seen as vector space over $\mathbb{F}_{p}$, and we
can describe the set
$\\{x\in\mathbb{F}_{q^{2}}\;:\;x^{p}+bx\in\mathbb{F}_{q}\\}$ as the kernel of
$f\circ g_{b}$. Thereby, the proof essentially boils down to a basic argument
of linear algebra. A direct computation shows that $f\circ g_{b}=g_{b}\circ f$
(using the fact that $b\in\mathbb{F}_{q}^{\times}$).

The rank-nullity theorem yields

$\dim(\ker(g_{b}\circ
f))=\dim(\ker(f))+\dim(\ker(g_{b})\cap\operatorname{Im}(f)),$ (2.9)

where the dimensions are taken over $\mathbb{F}_{p}$.

It is clear that $\dim(\ker(f))=n$, since $q=p^{n}$, and that $\ker(g_{b})$
has dimension $1$ since it consists of roots in $\overline{\mathbb{F}_{p}}$ of
the separable polynomial $X^{p}+bX$ which has degree $p$, and all those roots
actually lie in $\mathbb{F}_{q^{2}}$. Indeed, if $x^{p}=-bx$ then

$\displaystyle\begin{split}x^{p^{n}}&=\hskip
3.01385pt(-b)^{1+p+\cdots+p^{n-1}}\cdot x=(-b)^{\tfrac{p^{n}-1}{p-1}}\cdot
x\overset{\hyperref@@ii[eq_hypothesis_on_b]{\textup{\tagform@{\ref*{eq_hypothesis_on_b}}}}}{=}(-1)^{\tfrac{p^{n}-1}{p-1}}\cdot(-1)^{n+1}x\\\
&\overset{p\text{ odd}}{=}(-1)^{n}\cdot(-1)^{n+1}x\;=\;-x,\end{split}$ (2.10)

which implies that $x^{q^{2}}=(x^{q})^{q}=(-x)^{q}=x$, i.e.
$x\in\mathbb{F}_{q^{2}}$ as claimed.

The above computation 2.10 also shows that any element $x\in\ker(g_{b})$
satisfies $x^{p^{n}}=-x$, so that $f(x)=-2x$, which shows that
$x\in\operatorname{Im}(f)$ (recall that $p$ is odd, so
$-2\in\mathbb{F}_{p}^{\times}$ is invertible). In other words, we have
$\ker(g_{b})\cap\operatorname{Im}(f)=\ker(g_{b})$. Finally we get
$\dim(\ker(f\circ g_{b}))=\dim(\ker(g_{b}\circ f))=n+1$ from equation 2.9,
which yields

$\\#\\{x\in\mathbb{F}_{q^{2}}\;:\;x^{p}+bx\in\mathbb{F}_{q}\\}=|\ker(f\circ
g_{b})|=p^{n+1},$

which is what we wanted to prove.

Therefore, from equation 2.7 and the above lemma 2.4 (applied to $p=3$), we
get

$\displaystyle|C_{n,b}(\mathbb{F}_{3^{2n}})|$
$\displaystyle=1+3+(3^{n}+1)\left(\\#\\{u\in\mathbb{F}_{3^{2n}}\;:\;u^{3}+bu\in\mathbb{F}_{3^{n}}\\}-3\right)$
$\displaystyle=1+3^{n}\cdot 3^{n+1}$

We now consider $C_{n,b}$ as a curve over $\mathbb{F}_{3^{2n}}$ (instead of a
curve over $\mathbb{F}_{3^{n}}$). Let us write $\omega_{k}$ for the
eigenvalues of the Frobenius endomorphism of $x\mapsto x^{3^{2n}}$ acting on
$H^{1}_{\operatorname{\text{\'{e}t}}}(C_{n,b}\times\overline{\mathbb{F}_{3}},\mathbb{Q}_{\ell})$,
where $1\leq k\leq 2\cdot g(C_{n,b})$ and $g(C_{n,b})$ denotes the genus of
$C_{n,b}$ and $\ell\neq 3$ is a prime. It is known from the Weil conjectures
that the $\omega_{k}$ are the reciprocal of the roots of the numerator (in
$\mathbb{Z}[T]$) of zeta function $Z(C_{n,b}/\mathbb{F}_{3^{2n}},T)$ ; in
particular, they can be seen as complex numbers and their modulus is known to
be equal to $|\omega_{k}|=\sqrt{3^{2n}}=3^{n}$. Thereby, Lefschetz trace
formula tells us that

$\displaystyle|C_{n,b}(\mathbb{F}_{3^{2n}})|=3^{2n}+1-\sum_{k=1}^{2g(C_{n,b})}\omega_{k}.$

The genus of $C_{n,b}$ is equal to $g(C_{n,b})=3^{n}$ (see proposition 2 in
[Galbraith_Paulus_Smart_Superelliptic]). Hence we get

$\displaystyle|C_{n,b}(\mathbb{F}_{3^{2n}})|=1+3^{2n+1}=3\cdot
3^{2n}+1=3^{2n}+1-\sum_{k=1}^{2\cdot 3^{n}}\omega_{k},$

which implies $-2\cdot 3^{2n}=\sum\limits_{k=1}^{2\cdot 3^{n}}\omega_{k}$.
Because the $\omega_{k}\in\mathbb{C}$ satisfy
$|\omega_{k}|=\sqrt{3^{2n}}=3^{n}$, this forces $\omega_{k}=-3^{n}$ for every
$k$ (e.g. by taking the real part of the latter sum). We conclude that for
every $n\geq 1$ and every $j\geq 1$ :

$\displaystyle|C_{n,b}(\mathbb{F}_{3^{2nj}})|=3^{2nj}+1-2\cdot
3^{n}\cdot(-3^{n})^{j}.$

This completes the step 1 announced above. We can sum up what we have obtained
above in terms of the zeta function of $C_{n,b}$ :

######  Proposition 2.5.

Let $n\geq 1$ be an integer and let $b\in\mathbb{F}_{3^{n}}^{\times}$ be as in
A. The zeta function of the superelliptic curve $C_{n,b}$ over
$\mathbb{F}_{3^{2n}}$ is given by

$Z\big{(}C_{n,b}/\mathbb{F}_{3^{2n}},\ T\big{)}\;=\;\dfrac{(1+3^{n}T)^{2\cdot
3^{n}}}{(1-T)(1-3T)}.$

In particular, for every $j\geq 1$, we have

$|C_{n,b}(\mathbb{F}_{3^{2nj}})|=3^{2nj}+1-2\cdot 3^{n}\cdot(-3^{n})^{j}.$

### 2.4  $\cdot$ Evaluating the sums $\sigma_{b}(j,t)$

This paragraph is devoted to the explicit computation of the sums
$\sigma_{b}(j,t)$ defined in equation 2.6. Then we will conclude the proof of
A.

######  Lemma 2.6.

Let $n\geq 1$ be an integer, set $q=3^{n}$ and fix $b\in\mathbb{F}_{3^{n}}$
such that $\lambda_{\mathbb{F}_{3^{n}}}(b)=(-1)^{n+1}$. Let $j\geq 1$ be any
integer. Consider the map $g_{b,j}:\mathbb{F}_{q^{2j}}\to\mathbb{F}_{q^{2j}}$
defined by $g_{b,j}:x\mapsto x^{3}+bx$.

Then for every $t\in\mathbb{F}_{q^{2j}}$ we have :

$\displaystyle\sigma_{b}(j,t)=\begin{cases}-2\cdot(-3^{n})^{j}&\text{ if
}t\in\operatorname{Im}(g_{b,j})\\\ (-3^{n})^{j}&\text{ otherwise.
}\end{cases}$

[ Proof —]

  1. Step 1 –

The first key point here is to use again the fact that the map $g_{b,j}$ is
additive, in order to deduce that $\sigma_{b}(j,t)$ takes only two values (for
fixed $j,b$ and variable $t$).

Indeed, if we pick any $x_{0}\in\mathbb{F}_{q^{2j}}$, then

$\begin{split}\sigma_{b}(j,t)&\overset{\hyperref@@ii[eq_definition_sums_sigmab]{\textup{\tagform@{\ref*{eq_definition_sums_sigmab}}}}}{=}\sum_{x\in\mathbb{F}_{q^{2j}}}\lambda_{\mathbb{F}_{q^{2j}}}\\!\left(g_{b,j}(x)+t\right)=\sum_{x^{\prime}\in\mathbb{F}_{q^{2j}}}\lambda_{\mathbb{F}_{q^{2j}}}\\!\left(g_{b,j}(x^{\prime}+x_{0})+t\right)\\\
&\;\;=\sum_{x^{\prime}\in\mathbb{F}_{q^{2j}}}\lambda_{\mathbb{F}_{q^{2j}}}\left(g_{b,j}(x^{\prime})\;+\;g_{b,j}(x_{0})+t\right)=\sigma_{b}(j,t+g_{b,j}(x_{0})).\end{split}$

In other words, $\sigma_{b}(j,t)$ only depends on the class of $t$ in the
quotient additive group
$\mathbb{F}_{q^{2j}}\big{/}\operatorname{Im}(g_{b,j})$. Moreover, notice that

$\begin{split}\sigma_{b}(j,t)&=\sum_{x^{\prime}\in\mathbb{F}_{q^{2j}}}\lambda_{\mathbb{F}_{q^{2j}}}\big{(}g_{b,j}(-x^{\prime})+t\big{)}=\sum_{x^{\prime}\in\mathbb{F}_{q^{2j}}}\lambda_{\mathbb{F}_{q^{2j}}}\big{(}\\!-\\!g_{b,j}(x^{\prime})+t\big{)}\\\
&=\lambda_{\mathbb{F}_{q^{2j}}}(-1)\cdot\sigma_{b}(j,-t)=\sigma_{b}(j,-t),\end{split}$

where the last equality holds because $-1$ is a square in $\mathbb{F}_{3^{2}}$
and hence in $\mathbb{F}_{3^{2nj}}$.

Since $[\mathbb{F}_{q^{2j}}:\operatorname{Im}(g_{b,j})]=|\ker(g_{b,j})|=3$
(because $-b\in\mathbb{F}_{q}$ is a square in
$\mathbb{F}_{q^{2}}\hookrightarrow\mathbb{F}_{q^{2j}}$), we deduce that
$\sigma_{b}(j,t)$ only takes two values (for fixed $j,b$ and variable $t$).
The first value occurs when $t\in\operatorname{Im}(g_{b,j})$ in which case
$\sigma_{b}(j,t)=\sigma_{b}(j,0)$. Let us denote by $\sigma^{*}$ the other
value of $\sigma_{b}(j,t)$, which occurs when
$t\not\in\operatorname{Im}(g_{b,j})$. Observe that the value of $\sigma^{*}$
can be deduced from the sum

$\sum_{\mathclap{t\in\mathbb{F}_{3^{2nj}}}}\sigma_{b}(j,t)=|\operatorname{Im}(g_{b,j})|\cdot\sigma_{b}(j,0)\;+\;\left(3^{2nj}-|\operatorname{Im}(g_{b,j})|\right)\cdot\sigma^{*}=3^{2nj}\left(\frac{1}{3}\sigma_{b}(j,0)\;+\;\frac{2}{3}\sigma^{*}\right)$

because the left-hand side sum vanishes :

$\sum_{\mathclap{t\in\mathbb{F}_{3^{2nj}}}}\sigma_{b}(j,t)\;=\sum_{x\in\mathbb{F}_{3^{2nj}}}\sum_{t\in\mathbb{F}_{3^{2nj}}}\lambda_{\mathbb{F}_{3^{2nj}}}(x^{3}+bx+t)=0,$

since all the inner sums are $0$ (they are sums of a non-trivial
multiplicative character over the whole group – recall also that
$\lambda_{\mathbb{F}_{3^{2nj}}}(0)=0$). Therefore
$\sigma^{*}=-\frac{1}{2}\sigma_{b}(j,0)$, so it is enough to determine the
value of $\sigma_{b}(j,0)$.

  2. Step 2 –

Now we compute the sum
$\sigma_{b}(j,0)=\sum\limits_{x\in\mathbb{F}_{q^{2j}}}\lambda_{\mathbb{F}_{q^{2j}}}(x^{3}+bx)$.

The most conceptual (and easiest, or shortest) proof relies on the fact that
if ${\pi:Y\to X}$ is a surjective morphism between two smooth irreducible
projective algebraic curves (or even varieties) defined over a finite field,
then the numerator of the zeta function of $X$ divides the one of $Y$ in
$\mathbb{Z}[T]$. This can be argued using the Tate modules of the jacobians of
these curves, see for instance proposition 5 in
[Aubry_Perret_Zeta_divisibility].

In our case, we have the morphism

$\pi:C_{n,b}\to\mathcal{E}_{b}\qquad(u,v)\mapsto\Big{(}u,v^{\tfrac{3^{n}+1}{2}}\Big{)}$

where $\mathcal{E}_{b}$ is the elliptic curve given by $y^{2}=x^{3}+bx$ over
$\mathbb{F}_{3^{n}}$ (we defined the morphism on the affine charts, but it
extends uniquely to a morphism between the smooth projective curves
$C_{n,b}\to\mathcal{E}_{b}$). Being a non-constant morphism between
irreducible curves, $\pi$ must be surjective.

The numerator of $Z(C_{n,b}/\mathbb{F}_{3^{2n}},T)$ is $(1+3^{n}T)^{2\cdot
3^{n}}$ by 2.5. Therefore, the numerator of
$Z(\mathcal{E}_{b}/\mathbb{F}_{3^{2n}},T)$ is $(1+3^{n}T)^{2}$ (which implies
that $\mathcal{E}^{b}$ is supersingular). Thus we deduce from standard
arguments (see [SilvermanAEC], application V.1.3 and theorem V.2.3.1) that

$\displaystyle
1+3^{2nj}+\sigma_{b}(j,0)\;=\;|\mathcal{E}_{b}(\mathbb{F}_{3^{2nj}})|\;=\;1+3^{2nj}-2(-3^{n})^{j},$

which gives the claimed value for $\sigma_{b}(j,0)$. Therefore, from step 1 we
get the value $\sigma^{*}=(-3^{n})^{j}$ and this finishes the proof.

######  Remark 2.7.

It is possible to give more concrete and elementary (but computationally
longer) proofs of the identity $\sigma_{b}(j,0)=-2\cdot(-3^{n})^{j}$ from
lemma 2.6, via quartic Jacobi sums.

Moreover, when $n$ is odd, one can also give a direct proof of the step 2
above, because the change of variables $x\mapsto-x$ allows to determine the
number of points of the elliptic curve $\mathcal{E}_{b}:y^{2}=x^{3}+bx$ over
$\mathbb{F}_{3^{n}}$ (because $-1$ is not a square in $\mathbb{F}_{3^{n}}$)
and hence over any field extension thereof. ∎

We are now in position to prove our main result.

[ Proof of A —] By the identity just below equation 2.4, we recall
that$\displaystyle\log L(E_{n,b}/\mathbb{F}_{q^{2}}(t),T)=\sum\limits_{j\geq
1}S_{b}(n,j)\frac{T^{j}}{j}.$

From equations 2.5 and 2.6, one can write

$-S_{b}(n,j)=\sum_{w\in\mathbb{F}_{3^{2nj}}}\sigma_{b}(j,w^{3^{n}+1})$

(be careful of the minus sign). Define the set

$\Gamma_{b}(n,j):=\left\\{w\in\mathbb{F}_{3^{2nj}}\;:\;w^{3^{n}+1}\in\operatorname{Im}(g_{b,j})\right\\},$

where $g_{b,j}:\mathbb{F}_{3^{2nj}}\to\mathbb{F}_{3^{2nj}}$ denotes the map
$x\mapsto x^{3}+bx$ as in lemma 2.6.

Notice that all the fibers of the map

$C_{n,b}^{\rm
aff}(\mathbb{F}_{3^{2nj}})\longrightarrow\Gamma_{b}(n,j),\qquad(u,v)\longmapsto
v$

have size $3$ (they have the shape $\\{(u,v);(u\pm\beta,v)\\}$, where
$\beta\in\mathbb{F}_{3^{2n}}\hookrightarrow\mathbb{F}_{3^{2nj}}$ is an element
such that $\beta^{2}=-b$). Thereby, we deduce from 2.5 that

$|\Gamma_{b}(n,j)|=\dfrac{1}{3}(|C_{n,b}(\mathbb{F}_{3^{2nj}})|-1)=\dfrac{1}{3}(3^{2nj}-2\cdot
3^{n}\cdot(-3^{n})^{j})$ (2.11)

Therefore, using lemma 2.6 and the above expression of $S_{b}(n,j)$, we get

$\displaystyle-S_{b}(n,j)$
$\displaystyle=-2\cdot(-3^{n})^{j}\cdot|\Gamma_{b}(n,j)|\quad+\quad(-3^{n})^{j}\cdot\big{(}3^{2nj}-|\Gamma_{b}(n,j)|\big{)}$
$\displaystyle=(-3^{n})^{j}\cdot\big{(}3^{2nj}-3\cdot|\Gamma_{b}(n,j)|\big{)}$
$\displaystyle\overset{\hyperref@@ii[eq_Gamma_values]{\textup{\tagform@{\ref*{eq_Gamma_values}}}}}{=}(-3^{n})^{j}\cdot
2\cdot 3^{n}\cdot(-3^{n})^{j}$ $\displaystyle=2\cdot 3^{n(1+2j)}=2q^{1+2j},$

Finally, we conclude that

$\displaystyle\log L(E_{n,b}/\mathbb{F}_{q^{2}}(t),T)$
$\displaystyle=\sum\limits_{j\geq 1}S_{b}(n,j)\frac{T^{j}}{j}$
$\displaystyle=-2q\sum\limits_{j\geq 1}\frac{(q^{2}T)^{j}}{j}$
$\displaystyle=2q\cdot\log(1-q^{2}T),$

which precisely means that

$L\left(E_{n,b}/\mathbb{F}_{q^{2}}(t),\;T\right)=(1-q^{2}T)^{2\cdot 3^{n}},$

as desired. This finishes the proof.

######  Remark 2.8.

It seems that for primes $p\geq 5$, the curves $E:y^{2}=x^{3}+x+t^{p+1}$ over
$\mathbb{F}_{p^{2}}(t)$ do not have their $L$-function equal to
$(1-p^{2}T)^{r}$ for some $r\geq 0$. This can be shown for some primes by
computing (e.g. with SAGE) the values $S_{E}(j)/(p^{2})^{j}$ for
$j\in\\{1,2\\}$, where

$S_{E}(j):=\sum_{w\in\mathbb{P}^{1}(\mathbb{F}_{p^{2j}})}\left(p^{2j}+1-|\overline{E_{w}}(\mathbb{F}_{p^{2j}})|\right),$

(similarly to equation 2.4) and noticing that they are either not equal or not
even integers (they should be both equal to $-r\in\mathbb{Z}$ if the
$L$-function was $(1-p^{2}T)^{r}$). So the case of characteristic $3$ seems to
be very special.

On the other hand, the elliptic surface (of Delsarte type in Shioda’s
terminology from [Shioda_algorithm_Picard_nb]) associated to
$E:y^{2}=x^{3}+x+t^{m}$ is birationally equivalent to a quotient of the Fermat
surface $\mathcal{F}_{d}$ of degree $d$, where
$d:=\frac{4m}{\mathrm{gcd}(2,m)}$. So one could follow the approach taken in
[Ulmer_Large_Rank] (or [Griffon_Ulmer]) to express the $L$-function of $E$ in
terms of Jacobi sums, using corollary 7.7 _ibid_. However, it is likely that
proposition 8.1 _ibid._ would not apply to our case if $m=p+1$ (instead it
applies to $m=\frac{p+1}{2}$ ; see also our 1.3 above). ∎

## 3  $\cdot$ Proof of A

We now turn to the proof of the corollary concerning the narrow Mordell–Weil
lattice attached to the elliptic curves $E_{n,b}$ (see definition 1.1), and
the lower bound on its sphere packing density (see definition 1.2).

Estimating the sphere packing density of a lattice $L$ requires three steps :

  1. 1.

Determine the rank of $L$. In the case of the Mordell–Weil lattice of
$E_{n,b}$, this is essentially done in A.

  2. 2.

Get an upper bound on the covolume of $L$. In our case, this is will achieved
by using the so-called Birch–Swinnerton-Dyer formula which we discuss below.

  3. 3.

Finally, get a lower bound on the minimal non-zero norm in $L$. In the context
of the narrow Mordell–Weil lattices, we use a result of Shioda (see theorem
3.6 below).

### 3.1  $\cdot$ Birch–Swinnerton-Dyer conjecture and formula

We briefly recall what the Birch–Swinnerton-Dyer (BSD) conjecture is, and what
is known about it. Originally, it was stated for elliptic curves over
$\mathbb{Q}$, but it was then generalized to abelian varieties over any global
field. However, for the sake of simplicity, we will stick to the case of
elliptic curves over function fields, as given in [Gross_lectures_BSD,
conjecture 2.10].

Theorem 2.6 _ibid._ states that the $L$-function $L(E/K,T)$ of any non-
constant elliptic curve over a global function field is a polynomial in $T$.
In particular, this allows us to speak of the order of vanishing of the
$L$-function at any given value of $T$ in the field of constants. In the case
of the curves $E_{n,b}$ defined above, A provides a proof of the fact
$L(E/K,T)\in\mathbb{Z}[T]$. Before stating the conjecture, we introduce some
(standard) notations :

######  Definition 3.1.

Let $k$ be a finite field, and let $X$ be a smooth projective geometrically
irreducible algebraic curve over $k$. Denote by $g_{X}$ its genus. Set
$K=k(X)$ and let $E$ be an elliptic curve over $K$.

  1. 1.

Given the Néron–Tate height $\hat{h}:E(K)\to\mathbb{R}_{\geq 0}$ as in
equation 1.3, we define the pairing

$\langle-,-\rangle:E(K)\to\mathbb{R},\quad(P,Q)\mapsto\frac{1}{2}\cdot\left(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)\right).$

Then the _regulator_ of $E/K$ is the discriminant of this pairing, and we
denote it by $\operatorname{Reg}(E/K):=\det\Big{(}(\langle
P_{i},P_{j}\rangle)_{\scriptsize 1\leq i,j\leq r}\Big{)}$, where
$\\{P_{1},...,P_{r}\\}$ is any $\mathbb{Z}$-basis of the free abelian group
$E(K)\big{/}E(K)_{\operatorname{tors}}$ (we set $\operatorname{Reg}(E/K)=1$ by
convention if the rank is $r=0$).

  2. 2.

We further set the _special value_ of the $L$-function of $E/K$ to be

$L^{*}(E/K)\;:=\;\dfrac{1}{\rho!}L^{(\rho)}(E/K,T)\Big{|}_{T=|k|^{-1}}$

where $\rho=\rho(E/K):=\operatorname*{ord}_{T=|k|^{-1}}L(E/K,T)$ denotes the
_analytic rank_.

  3. 3.

Finally, we define the _height_ of $E/K$ as
$H(E/K):=|k|^{\tfrac{\deg\left(\Delta_{\min}(E/K)\right)}{12}}.$

######  Remark 3.2.

Because the $L$-function is a rational function in $\mathbb{Q}(T)$, we also
have $L^{*}(E/K)=\left.\dfrac{L(E/K,T)}{(1-|k|T)^{\rho}}\right|_{T=|k|^{-1}}$
and this is a non-zero rational number.

There is another normalization of the Néron–Tate height, which is
$\hat{h^{\prime}}:=\log(|k|)\cdot\hat{h}$, as in [Gross_lectures_BSD] (lecture
3, §2). In that case, for the BSD formula to be true, one has to take the
special value of the _complex_ $L$-function, namely the value
$\mathcal{L}^{*}(E/K)$ such that

$\mathcal{L}(E/K,s):=L(E/K,|k|^{-s})\sim\mathcal{L}^{*}(E/K)\cdot(s-1)^{\rho},\qquad\text{as
$s\to 1$.}$

The two normalizations are consistent. Indeed, on the one hand, if one defines
$\operatorname{Reg}^{\prime}(E/K)$ as the discriminant with respect to the
pairing associated to $\hat{h^{\prime}}$ (as in definition 3.1), then one has
$\operatorname{Reg}^{\prime}(E/K)=\log(|k|)^{r}\,\operatorname{Reg}(E/K)$. On
the other hand, since $1-|k|^{1-s}\sim\log(|k|)(s-1)$ as $s\to 1$, we have
$\mathcal{L}^{*}(E/K)=\log(|k|)^{r}\,L^{*}(E/K)$.

We make the choice of using $\hat{h}$ and not $\hat{h^{\prime}}$ because then
the narrow Mordell–Weil group $E(K)^{0}$ (definition 1.1) becomes an
_integral_ lattice (see theorem 3.6). ∎

######  Conjecture 3.3 (Birch–Swinnerton-Dyer).

Let $k$ be a finite field, and let $X$ be a smooth projective geometrically
irreducible curve over $k$. Denote by $g_{X}$ its genus. Let $E$ be an
elliptic curve over the function field $K:=k(X)$.

Then the following statements hold (using notations from definitions 3.1 and
2.1):

  1. a)

The rank of the finitely generated111This result of finite generation of
$E(K)$ is known as Mordell–Weil theorem, further extended by Néron and Lang.
abelian group $E(K)$ is equal to the order of vanishing of the $L$-function of
$E/K$ at $T=|k|^{-1}$, i.e.

$\operatorname{rk}_{\mathbb{Z}}\\!\big{(}E(K)\big{)}=\operatorname*{ord}_{T=|k|^{-1}}L(E/K,T).$

  2. b)

The Tate–Shafarevitch group $\Sha(E/K)$ is finite.

  3. c)

(BSD formula) We have the identity

$L^{*}(E/K)\;=\;\dfrac{|\Sha(E/K)|\cdot\operatorname{Reg}(E/K)\cdot
c(E/K)}{|E(K)_{\operatorname{tors}}|^{2}\cdot|k|^{g_{X}-1}\cdot H(E/K)}.$
(3.1)

######  Theorem 3.4 (Artin, Tate, Milne).

Let $E$ be an elliptic curve the function field $K:=k(X)$, where $k$ is a
finite field, as in definition 3.1.

  1. 1.

The statements a, b and c in conjecture 3.3 are all equivalent.

  2. 2.

Assume that $E$ is a potentially constant (= isotrivial) elliptic curve, i.e.
there is a finite extension $K^{\prime}/K$ such that the base change
$E\times_{K}K^{\prime}$ is isomorphic to $E^{\prime}\times_{k}K^{\prime}$ for
some (constant) elliptic curve $E^{\prime}$ defined over $k$.

Then all the statements of conjecture 3.3 are true.

[ Proof —]The first part is stated as theorem 3.1 in [Gross_lectures_BSD]. The
second claim is stated in lecture 1, theorem 12.2 of
[Ulmer_elliptic_curves_over_FF], and is proved in lecture 3, theorem 8.1,
_ibid_.

######  Proposition 3.5.

The elliptic curve $E_{n,b}$ over $K=\mathbb{F}_{3^{2n}}(t)$ from A is
isotrivial. More precisely, it is a _cubic twist_ of the constant curve
$E^{\prime}:y^{\prime 2}=x^{\prime 3}+x^{\prime}$ over $\mathbb{F}_{3}$.

Moreover, the Mordell–Weil group $E_{n,b}(K)$ is torsion-free.

[ Proof —]The first statement is immediate from the change of variables
$y=y^{\prime},x=x^{\prime}+u$ where $u\in\overline{\mathbb{F}_{3}(t)}$
satisfies $u^{3}+bu=t^{3^{n}+1}$ (this exactly defines the superelliptic curve
from subsection 2.3). One can also see that the $j$-invariant of $E_{n,b}$ is
$0$, so it must be an isotrivial elliptic curve.

We now explain why $E_{n,b}(K)$ is torsion-free. From 2.2, we know that the
product of the Tamagawa numbers is equal to
$c(E_{n,b}/K)=\prod\limits_{v}c_{v}=3$. In particular, this is a square-free
integer. But proposition 6.31 in [MW_Lattices_Book] states that
$|E_{n,b}(K)_{\operatorname{tors}}|^{2}$ divides
$\prod\limits_{v}c_{v}(E_{n,b}/K)$, so we deduce that $E_{n,b}(K)$ is torsion-
free.

### 3.2  $\cdot$ Lower bound on the minimal norm and on the packing density

We start this subsection in a general framework : we let $E$ be an elliptic
curve over a global function field $K=k(X)$ as in definition 3.1, that is, $X$
is a smooth geometrically irreducible projective curve over a finite field
$k$.

One of the main features of the _narrow_ Mordell–Weil lattice $E(K)^{0}\subset
E(K)$ (definition 1.1) is that it is an _even integral_ lattice, and that we
have an explicit lower bound on the minimal height among non-zero vectors.

######  Theorem 3.6 (Shioda).

Let $E$ be an elliptic curve over a global function field $K=k(X)$. Then for
every $P\in E(K)^{0}\smallsetminus\\{0\\}$ we have

$\hat{h}(P)\geq\frac{1}{6}\deg\left(\Delta_{\min}(E/K)\right).$

In particular, $E(K)^{0}$ is a torsion-free. Moreover, $(E(K)^{0},\hat{h})$
forms an even integral lattice.

Finally, the index $\left[E(K):E(K)^{0}\right]$ divides the product
$c(E/K):=\prod\limits_{v}c_{v}(E/K)$ of the Tamagawa numbers.

[ Proof —]For the lower bound on the minimal non-zero norm and the fact that
the lattice $E(K)^{0}$ is even and integral, see theorem 6.44 in
[MW_Lattices_Book], as well as theorem 5.47 and corollary 5.50, _ibid_.

We now prove the result on the index $\left[E(K):E(K)^{0}\right]$. Let
$R\subset|X|$ be the set of bad places of $E$, where $|X|$ denotes the set of
closed points of $X$. For each $v\in R$, let
$G_{v}:=\dfrac{\tilde{\mathcal{E}_{v}}(\mathbb{F}_{v})}{\tilde{\mathcal{E}_{v}}^{0}(\mathbb{F}_{v})}$
be the component group at $v$, where $\mathcal{E}_{v}$ denotes the Néron model
of $E$ at $v$ and $\mathbb{F}_{v}$ is the residue field.

By definition and by [SilvermanAAEC, corollary IV.9.2.(c)], $E(K)^{0}$ is the
kernel of the map

$\theta:E(K)\longrightarrow\prod_{v\in R}G_{v}$

defined as follows: for each $v\in R$, there is a unique irreducible component
$\Theta_{v,i(v,P)}$ of $\tilde{\mathcal{E}_{v}}$ that contains the image
$\widetilde{P_{v}}$ of $P$ in $\tilde{\mathcal{E}_{v}}$. Then
$P\longmapsto(\Theta_{v,i(v,P)})_{v\in R}$ induces the above map $\theta$.

The map $\theta$ is a group homomorphism (see lemma 6.4 in
[Elliptic_Surfaces_Shioda_Schutt], or just notice that
$E(K_{v})\cong\mathcal{E}_{v}(\mathcal{O}_{v})\to\tilde{\mathcal{E}_{v}}(\mathbb{F}_{v})$
is a morphism) and therefore, we have an injective morphism

$E(K)/E(K)^{0}\;\lhook\joinrel\xrightarrow{\quad}\;\prod_{v\in R}G_{v},$

which shows the desired divisibility.

We can now give a lower bound on the sphere packing density of the narrow
Mordell–Weil sublattice $E(K)^{0}\subset E(K)$ (see definition 1.2).

######  Proposition 3.7.

Let $E$ be an elliptic curve over a global function field $K=k(X)$, where $X$
is a smooth projective curve of genus $g_{X}$ over a finite field $k$.

Assume that the $L$-function of $E/K$ is of the form $L(E/K,T)=(1-|k|T)^{r}$
where $r$ is the rank of $E(K)/E(K)_{\operatorname{tors}}$.

Then the center (sphere packing) density of the narrow Mordell–Weil
lattice$E(K)^{0}\subset E(K)$ (see definitions 1.1 and 1.2) is lower bounded
by

$\delta\left(E(K)^{0}\right)\;\geq\;\dfrac{\left(\dfrac{\deg\left(\Delta_{\min}(E/K)\right)}{24}\right)^{r/2}}{c(E/K)^{1/2}\cdot|E(K)_{\operatorname{tors}}|\cdot|k|^{g_{X}/2-1/2}\cdot
H(E/K)^{1/2}},$

where we use the notations from definitions 3.1 and 3.3.

[ Proof —]First of all, the hypothesis $L(E/K,T)=(1-|k|T)^{r}$ implies that
BSD formula is true, by part 1 of theorem 3.4 — more precisely we used the
implication a $\implies$ c. This hypothesis also forces the special value of
the $L$-function to be $L^{*}(E/K)=1$.

Because the cardinality of the finite group $\Sha(E/K)$ is at least $1$, BSD
formula allows us to get an upper bound on the discriminant of $E(K)$ :

$\operatorname{Reg}(E/K)\leq|E(K)_{\operatorname{tors}}|^{2}\cdot|k|^{g_{X}-1}\cdot
H(E/K)\cdot c(E/K)^{-1}$ (3.2)

From theorem 3.6, we have

$\lambda_{1}\left(E(K)^{0}\right):=\min\Big{\\{}\hat{h}(P)^{1/2}\;:\;P\in
L_{n}\smallsetminus\\{0\\}\Big{\\}}\;\geq\;\left(\dfrac{\deg\left(\Delta_{\min}(E/K)\right)}{6}\right)^{1/2}.$

Now the covolume of $E(K)^{0}$ is given by

$\operatorname{covol}\left(E(K)^{0}\right)=[E(K):E(K)^{0}]\cdot\operatorname{covol}(E(K))=[E(K):E(K)^{0}]\cdot\operatorname{Reg}(E/K)^{1/2}.$

Using the last statement of theorem 3.6, together with equation 3.2, we deduce

$\operatorname{covol}\left(E(K)^{0}\right)\leq
c(E/K)^{1/2}\cdot|E(K)_{\operatorname{tors}}|\cdot|k|^{g_{X}/2-1/2}\cdot
H(E/K)^{1/2}$

Thereby, combining the above inequalities, we see that the center density of
the lattice $L_{n}=E(K)^{0}$ is lower bounded by

$\delta\left(E(K)^{0}\right)\;\geq\;\dfrac{\left(\dfrac{\deg\left(\Delta_{\min}(E/K)\right)}{24}\right)^{r/2}}{c(E/K)^{1/2}\cdot|E(K)_{\operatorname{tors}}|\cdot|k|^{g_{X}/2-1/2}\cdot
H(E/K)^{1/2}},$

where $r$ is the rank of lattice $E(K)^{0}$. Notice that the narrow
Mordell–Weil$E(K)^{0}\subset E(K)$ is a _full-rank_ sublattice (this follows
for instance from the last statement of theorem 3.6 : its index in $E(K)$ is
finite), so its rank is the same as the rank of $E(K)$.

We can now conclude with the proof of our main corollary. [ Proof of A —] For
ease of notation, in what follows, we write $K_{n}:=\mathbb{F}_{3^{2n}}(t)$.

First of all, we notice that the rank of the lattice
$L_{n}:=E_{n,b}\left(K_{n}\right)^{0}$ is equal to $r=2\cdot 3^{n}$. Indeed,
theorem 3.4 and 3.5 imply that the BSD conjecture (item a) is fulfilled. In
particular, the algebraic rank of $E_{n,b}$ over $K_{n}$ agrees with the
analytic rank, which equals $2\cdot 3^{n}$ by A.

This very theorem also allows us to apply the above 3.7. Thereby, the values
from 2.2 and the last statement of 3.5 (namely the fact
$|E_{n,b}(K_{n})_{\operatorname{tors}}|=1$) yield

$\delta(L_{n})\;\geq\;\dfrac{\Big{(}(3^{n-1}+1)/4\Big{)}^{3^{n}}}{3^{1/2}\cdot
3^{n/2\,\cdot\,(3^{n-1}-1)}},$

which is exactly the lower bound stated in A. This concludes the proof.

### 3.3  $\cdot$ Discussion of the sharpness of the lower bound on the packing
density

In this paragraph, we shorty study sufficient conditions under which the
inequality in A is actually an equality. In fact, this lower bound is sharp if
and only the following conditions are all satisfied :

  * •

The index $[E(K):E(K)^{0}]$ is equal to $c(E/K)$ (instead of just dividing it,
as in theorem 3.6).

  * •

The lower bound on the minimal norm form theorem 3.6 is achieved, that is
there is a point $P\in E(K)^{0}$ such that
$\hat{h}(P)=\dfrac{1}{6}\deg\left(\Delta_{\min}(E/K)\right)$, which is equal
to $3^{n-1}+1$ when $E=E_{n,b}$ according to 2.2.

  * •

The Tate–Shafarevitch group $\Sha(E/K)$ is trivial.

As for the index $[E_{n,b}(K):E_{n,b}(K)^{0}]$, where
$K:=\mathbb{F}_{3^{2n}}(t)$, we can prove easily that it is in fact equal to
$c(E_{n,b}/K)=3$. First, we know from the last statement of theorem 3.6 that
$[E_{n,b}(K):E_{n,b}(K)^{0}]$ must divide $c(E_{n,b}/K)=3$, so it is either 1
or 3. We prove that the index cannot be equal to 1 by noticing that the point

$Q_{n}:=\left(0,t^{(3^{n}+1)/2}\right)\in
E_{n,b}(\mathbb{F}_{3}(t))\hookrightarrow E_{n,b}(K)$

does not belong to $E_{n,b}(K)^{0}$.

Indeed, if we set $\mu=\lceil(3^{n}+1)/6\rceil$, then the point $Q_{n}$ gets
mapped to the point $(Q_{n})_{\infty}:=\big{(}0,t^{(3^{n}+1)/2-3\mu}\big{)}$
on the minimal integral Weierstrass model $(E_{n,b})_{v}:y^{2}=\linebreak
x^{3}+bxt^{-4\mu}+t^{3^{n}+1-6\mu}$ of $E_{n,b}$ at $v:=\infty$ (via the map
$(x,y)\mapsto(xt^{-2\mu},yt^{-3\mu})$), as in 2.2. Then $(Q_{n})_{\infty}$
modulo $t^{-1}$ is the singular point $(\overline{0},\overline{0})$ of
$\overline{(E_{n,b})_{v}}$. Therefore, $Q_{n}\not\in E_{n,b}(K)^{0}$, as
claimed.

Let us say a few words on the lower bound
$\hat{h}(P)\geq\dfrac{1}{6}\deg\left(\Delta_{\min}(E_{n,b}/K)\right)=3^{n-1}+1$
for $P\in E_{n,b}(K)^{0}\smallsetminus\\{0\\}$. We do not know whether it is a
sharp bound in general, but for $n\in\\{1,2,3\\}$ we can exhibit points that
achieve this bound. We first list those points explicitly, and then briefly
explain how to compute their Néron–Tate height.

  * —

When $n=1$ and $b=1$, the point

$P_{1}=(t^{2},-t^{3}+t)\;\in\;E_{1,1}(\mathbb{F}_{3}(t))\;\lhook\joinrel\xrightarrow{\quad}\;{}E_{1,1}(K)$

has Néron–Tate height $2$, i.e. $\hat{h}(P_{1})=3^{0}+1$. Notice that $P_{1}$
lies in the narrow Mordell–Weil sublattice, because at $v=\infty$, the point
$P_{1}$ gets mapped to $(1,-1+t^{-2})$ on the minimal integral Weierstrass
model at $\infty$, so it reduces to the smooth point
$(\overline{1},\overline{-1})$ modulo $t^{-1}$.

  * —

If $n=2$, let us write $\mathbb{F}_{3^{2}}\cong\mathbb{F}_{3}[X]/(X^{2}-X-1)$
and let $z$ be the class of $X$ in $\mathbb{F}_{3^{2}}$. One can take $b:=z$
since $z^{(3^{n}-1)/2}=z^{4}=-1$. The point

$P_{2}:=\left(t^{4}+(z+1)t^{2}-1\;,\;-t^{6}+t^{4}-t^{2}-z+1\right)\in
E_{2,b}(\mathbb{F}_{3^{2}}(t))\hookrightarrow E_{2,b}(\mathbb{F}_{3^{2n}}(t))$

has height 4. Again, $P_{2}$ lies in the narrow Mordell–Weil sublattice : its
reduction modulo $t^{-1}$ is $(\overline{1},\overline{-1})$ as for the $n=1$
case.

  * —

If $n=3$ and $b=1$, then

$P_{3}=\left(t^{10}+t^{8}+t^{2}\;,\;-t^{15}+t^{13}-t^{11}-t^{7}-t^{5}+t\right)\in
E_{3,1}(K)$

has height 10. Moreover, as before, $P_{3}$ lies in the narrow Mordell–Weil
sublattice.

Using theorem 6.24 of [MW_Lattices_Book], one can show that
$\hat{h}(P_{n})=3^{n-1}+1$ for $n\leq 3$ by checking that the intersection
product $(P_{n})\cdot(O)$ vanishes, that is, the sections $(P_{n})$ and $(O)$
– from $\mathbb{P}^{1}$ to the elliptic surface associated to $E_{n,b}$ – do
not intersect (we use the notations from Proposition 5.4 and Notation 5.5 in
[MW_Lattices_Book]).

One can argue as in the proof of Proposition 5.1 of
[Shioda_MW_and_sphere_packings] (even though the exact statement from there
does not directly apply in characteristic 3): both coordinates of $P_{n}$ are
polynomials in $t$, so have no pole on $\mathbb{A}^{1}$, and hence $(P_{n})$
and $(O)$ do not intersect at any point of $\mathbb{A}^{1}$. At
$v=\infty\in\mathbb{P}^{1}$, we let $\mu=\lceil(3^{n}+1)/6\rceil$ and observe
that under the map $(x(t),y(t))\mapsto(x(t)t^{-2\mu},x(t)t^{-3\mu})$, the
points $P_{n}$ get mapped to points $(P_{n})_{\infty}$ on the minimal integral
Weierstrass model of $E_{n,b}$ at $v=\infty$ such that both coordinates have
non-zero constant term. Hence, we see that both coordinates have no pole at
$v=\infty$, and we conclude that $(P_{n})$ and $(O)$ never intersect.

Finally, for the order of the Tate–Shafarevitch group, we can just point out
that it is a 3-group, i.e. it is equal to its 3-primary part
$\Sha(E_{n,b}/K)=\Sha(E_{n,b}/K)[3^{\infty}]$, where
$K=\mathbb{F}_{3^{2n}}(t)$. This follows from BSD formula : because
$L^{*}(E_{n,b}/K)=1$, $|E_{n,b}(K)_{\operatorname{tors}}|=1$ and
$c(E_{n,b}/K)=3$, we have

$|\Sha(E_{n,b}/K)|\cdot\operatorname{Reg}(E_{n,b}/K)=\dfrac{1}{3}\cdot(3^{2n})^{-1+\tfrac{1}{12}\deg\left(\Delta_{\min}(E_{n,b}/K)\right)}.$

But we have seen above that $[E_{n,b}(K):E_{n,b}(K)^{0}]=3$, and we know from
theorem 3.6 that $E_{n,b}(K)^{0}$ is an integral lattice, so it follows that
$\operatorname{Reg}(E_{n,b}/K)\in\frac{1}{3^{2}}\mathbb{Z}$.

Computations on MAGMA seem to indicate that $\Sha(E_{n,b}/K)$ is trivial when
$n=1$, but in analogy with [Shioda_MW_and_sphere_packings, proposition 4.3],
it is possible that it is non-trivial for $n$ large enough.

######  Remark 3.8.

In fact, when $n=1$, i.e. when the rank is $r=2\cdot 3^{1}=6$, it is known
that the $E_{6}$ lattice provides the best _lattice_ sphere packing in 6
dimensions [Blichfeldt_optimality_E6_7_8], and since the lower bound on the
density of the lattice $E_{1,1}(\mathbb{F}_{3^{2}}(t))^{0}$ agrees with the
density of $E_{6}$, the lower bound from A must be sharp when $n=1$, in
particular $\Sha(E_{1,1}/K)$ is trivial. ∎

######  Remark 3.9.

  1. 1.

We mention here that when $n\to+\infty$, we have the asymptotic lower bound
$\log_{2}(\delta(L_{n}))\geq 3^{n}\cdot n\cdot\log_{2}(3)-\frac{n\cdot
3^{n-1}}{2}\log_{2}(3)+o(n\cdot 3^{n})$ from A. Because the rank of $L_{n}$ is
$r=2\cdot 3^{n}$, this reads

$\log_{2}(\delta(L_{n}))\geq\left(\frac{1}{2}-\frac{1}{12}\right)r\log_{2}(r)+o(r\log_{2}(r)),$

which implies

$D(L_{n})\geq
2^{-\tfrac{1}{12}r\log_{2}(r)\cdot(1+o(1))}=r^{-r/12\cdot(1+o(1))},$

where $D(L_{n})\in[0,1]$ is the packing density as defined in equation 1.5.
Although this is far from attaining Minkowski–Hlawka lower bound $\geq
2^{-r}$, we get the same asymptotic density as in [ElkiesMW1, theorem 1] and
[Shioda_MW_and_sphere_packings, equation (1.12)].

  2. 2.

The densities of the narrow and the full Mordell–Weil lattices compare as
follows. Let $Q_{n}:=\left(0,t^{(3^{n}+1)/2}\right)$ be as above. Using
theorem 6.24 and Table 6.1 (p. 127) of [MW_Lattices_Book] and the fact that
the reduction of $E_{n,b}$ at $v=\infty$ has type IV (2.2), one can show that
$\hat{h}(Q_{n})=3^{n-1}+1-\frac{2}{3}$, using an argument similar as the one
for the computations of $\hat{h}(P_{n})$ above. Then

$\delta\big{(}E_{n,b}(\mathbb{F}_{3^{2n}}(t)\big{)}\leq\dfrac{\big{(}\hat{h}(Q_{n})^{1/2}/2\big{)}^{2\cdot
3^{n}}\cdot[E_{n,b}(\mathbb{F}_{3^{2n}}(t)):L_{n}]}{\operatorname{covol}(L_{n})}$

Thus we get, because $[E_{n,b}(\mathbb{F}_{3^{2n}}(t)):L_{n}]=3$ as mentioned
previously,

$\displaystyle\dfrac{\delta\big{(}E_{n,b}(\mathbb{F}_{3^{2n}}(t)\big{)}}{\delta(L_{n})}\leq
3\cdot\left(\dfrac{\hat{h}(Q_{n})}{\lambda_{1}(L_{n})^{2}}\right)^{3^{n}}\leq
3\cdot\left(\dfrac{3^{n-1}+1-2/3}{3^{n-1}+1}\right)^{3^{n}}=3\cdot\left(1-\dfrac{2}{3^{n}+3}\right)^{3^{n}}$

Thus the narrow Mordell–Weil lattice $L_{n}$ is always denser than the full
Mordell–Weil lattice, and the ratio of the densities tends to $3e^{-2}\simeq
0.406$ as $n\to+\infty$. ∎

#### Acknowledgments

I would like to thank my advisor, Prof. Maryna Viazovska, for her support and
for having suggested to study this topic.

This work was funded by the Swiss National Science Foundation (SNSF), Project
funding (Div. I-III), "Optimal configurations in multidimensional spaces",
184927.

Gauthier Leterrier, École Polytechnique Fédérale de Lausanne (EPFL), MA B3
424, Station 8, 1015 Lausanne, Switzerland

E-mail address : gauthier.leterrier at epfl dot ch or gauthier.leterrier at
gmail dot com