$P_{1}\cdot P_{2}=1$? The answer is given in Theorem 4.7.8, which was only a
conjecture at the time of our investigation on conic bundles and could only be
proven after a more careful study of intersection numbers of sections. This
motivating problem lead to a broader broader investigation of the possible
values of intersection numbers of sections, whose results are gathered in this
chapter.

The main tool we need is the Mordell-Weil lattice (Section 2.7), which
consists of a lattice structure on $E(K)/E(K)_{\text{tor}}$ having a close
connection with the Néron-Severi lattice. We take advantage of the fact that
all possibilities for the Mordell-Weil lattices of $X$ have been classified in
[OS91], so that the well-know information of the height-pairing gives us
information about the intersection numbers of sections of $X$.

By adopting this strategy, we often come across a classical problem in number
theory, which is the representation of integers by positive-definite quadratic
forms. Indeed, if the free part of $E(K)$ is generated by $r$ terms, then the
height $h(P):=\langle P,P\rangle$ induces a positive-definite quadratic form
on $r$ variables with coefficients in $\mathbb{Q}$. If $O\in E(K)$ is the
neutral section and $R$ is the set of reducible fibers of $\pi$, then by the
height formula (2.3)

$h(P)=2+2(P\cdot O)-\sum_{v\in R}\text{contr}_{v}(P),$

where the sum over $v$ is a rational number which can be estimated (Table
2.4). By clearing denominators, we see that the possible values of $P\cdot O$
depend on a certain range of integers represented by a positive-definite
quadratic form with coefficients in $\mathbb{Z}$.

We organize this chapter as follows. In Section 4.1 we define some terminology
and dedicate Section 4.2 to explaining the role of torsion sections in our
investigation. The technical core of this chapter is in Sections 4.3, 4.4 and
4.5, where we find necessary and sufficient conditions for
$k\in\mathbb{Z}_{\geq 0}$ to be the intersection number of two sections. In
Section 4.6 we list the total of sufficient conditions obtained. The main
results are in Section 4.7, namely: the description of gap numbers when $E(K)$
is torsion-free with $r=1$ (Subsection 4.7.3), the absence of gap numbers for
$r\geq 5$ (Subsection 4.7.1), density of gap numbers when $r\leq 2$
(Subsection 4.7.2) and the classification of surfaces with a $1$-gap
(Subsection 4.7.4). We use the appendix in Chapter 6 to obtain the information
we need about Mordell-Weil lattices.

###  4.1 Gap numbers

We introduce some convenient terminology to express the possibility of finding
a pair of sections with a given intersection number.

######  Definition 4.1.1.

If there are no sections $P_{1},P_{2}\in E(K)$ such that $P_{1}\cdot P_{2}=k$,
we say that $X$ has a $k$-gap or that $k$ is a gap number of $X$.

######  Definition 4.1.2.

We say that $X$ is gap-free if for every $k\in\mathbb{Z}_{\geq 0}$ there are
sections $P_{1},P_{2}\in E(K)$ such that $P_{1}\cdot P_{2}=k$.

######  Remark 4.1.3.

In case the Mordell-Weil rank is $r=0$, we have $E(K)=E(K)_{\text{tor}}$. In
particular, any two distinct sections are disjoint [SS19, Cor. 8.30], hence
every $k\geq 1$ is a gap number of $X$. For positive rank, the description of
gap numbers is less trivial, hence our focus on $r\geq 1$.

###  4.2 Intersection with a torsion section

Before dealing with more technical details in Sections 4.3 and 4.4, we explain
how torsion sections can be of help in our investigation.

We first note some general properties of torsion sections. As the height
pairing is positive-definite on $E(K)/E(K)_{\text{tor}}$, torsion sections are
inert in the sense that for each $Q\in E(K)_{\text{tor}}$ we have $\langle
Q,P\rangle=0$ for all $P\in E(K)$. Moreover, in the case of rational elliptic
surfaces, torsion sections also happen to be mutually disjoint by Theorem
2.2.7.

By considering the properties above, we use torsion sections to help us find
$P_{1},P_{2}\in E(K)$ such that $P_{1}\cdot P_{2}=k$ for a given
$k\in\mathbb{Z}_{\geq 0}$. This is particularly useful when $\Delta\geq 2$, in
which case $E(K)_{\text{tor}}$ is not trivial by Lemma 2.9.1.

The idea works as follows. Given $k\in\mathbb{Z}_{\geq 0}$, suppose we can
find $P\in E(K)^{0}$ with height $h(P)=2k$. By the height formula (2.3),
$P\cdot O=k-1<k$, which is not yet what we need. In the next lemma we show
that replacing $O$ with a torsion section $Q\neq O$ gives $P\cdot Q=k$, as
desired.

######  Lemma 4.2.1.

Let $P\in E(K)^{0}$ such that $h(P)=2k$. Then $P\cdot Q=k$ for all $Q\in
E(K)_{\text{tor}}\setminus\\{O\\}$.

###### Proof.

Assume there is some $Q\in E(K)_{\text{tor}}\setminus\\{O\\}$. By Theorem
2.2.7, $Q\cdot O=0$ and by the height formula (2.3), $2k=2+2(P\cdot O)-0$,
hence $P\cdot O=k-1$. We use the height formula (2.2) for $\langle P,Q\rangle$
in order to conclude that $P\cdot Q=k$. Since $P\in E(K)^{0}$, it intersects
the neutral component $\Theta_{v,0}$ of every reducible fiber $\pi^{-1}(v)$,
so $\text{contr}_{v}(P,Q)=0$ for all $v\in R$. Hence

$\displaystyle 0$ $\displaystyle=\langle P,Q\rangle$ $\displaystyle=1+P\cdot
O+Q\cdot O-P\cdot Q-\sum_{v\in R}\text{contr}_{v}(P,Q)$
$\displaystyle=1+(k-1)+0-P\cdot Q-0$ $\displaystyle=k-P\cdot Q.$

∎

###  4.3 Necessary conditions

If $k\in\mathbb{Z}_{\geq 0}$, we state necessary conditions for having
$P_{1}\cdot P_{2}=k$ for some sections $P_{1},P_{2}\in E(K)$. We note that the
value of $\Delta$ is not relevant in this section, but plays a decisive role
for sufficient conditions in Section 4.4.

######  Lemma 4.3.1.

Let $k\in\mathbb{Z}_{\geq 0}$. If $P_{1}\cdot P_{2}=k$ for some
$P_{1},P_{2}\in E(K)$, then one of the following holds:

  1. i)

$h(P)=2+2k$ for some $P\in E(K)^{0}$.

  2. ii)

$h(P)\in[2+2k-c_{\text{max}},\,2+2k-c_{\text{min}}]$ for some $P\notin
E(K)^{0}$.

###### Proof.

Without loss of generality we may assume that $P_{2}$ is the neutral section,
so that $P_{1}\cdot O=k$. By the height formula (2.3), $h(P_{1})=2+2k-c$,
where $c:=\sum_{v}\text{contr}_{v}(P_{1})$. If $P_{1}\in E(K)^{0}$, then $c=0$
and $h(P_{1})=2+2k$, hence i) holds. If $P_{1}\notin E(K)^{0}$, then
$c_{\text{min}}\leq c\leq c_{\text{max}}$ by Lemma 2.8.4. But
$h(P_{1})=2+2k-c$, therefore $2+2k-c_{\text{max}}\leq h(P_{1})\leq
2+2k-c_{\text{min}}$, i.e. ii) holds. ∎

######  Corollary 4.3.2.

Let $k\in\mathbb{Z}_{\geq 0}$. If $P_{1}\cdot P_{2}=k$ for some
$P_{1},P_{2}\in E(K)$, then $Q_{X}$ represents some integer in
$[d\cdot(2+2k-c_{\text{max}}),d\cdot(2+2k)]$, where $d:=\det E(K)^{0}$.

###### Proof.

We apply Lemma 4.3.1 and rephrase it in terms of $Q_{X}$. If i) holds, then
$Q_{X}$ represents $d\cdot(2+2k)$ by Lemma 2.10.3. If ii) holds, then
$h(P)\in[2+2k-c_{\text{max}},2+2k-c_{\text{min}}]$ and by Lemma 2.10.3,
$Q_{X}$ represents $d\cdot
h(P)\in[d\cdot(2+2k-c_{\text{max}}),d\cdot(2+2k-c_{\text{min}})]$. In both i)
and ii), $Q_{X}$ represents some integer in
$[d\cdot(2+2k-c_{\text{max}}),d\cdot(2+2k)]$. ∎

###  4.4 Sufficient conditions when $\Delta\leq 2$

In this section we state sufficient conditions for having $P_{1}\cdot P_{2}=k$
for some $P_{1},P_{2}\in E(K)$ under the assumption that $\Delta\leq 2$. By
Lemma 2.9.1, this covers almost all cases (more precisely, all but No. 41, 42,
59, 60 in Table LABEL:table:MWL_data). We treat $\Delta<2$ and $\Delta=2$
separately, as the latter needs more attention.

####  4.4.1 The case $\Delta<2$

We first prove Lemma 4.4.1, which gives sufficient conditions assuming
$\Delta<2$, then Corollary 4.4.3, which states sufficient conditions in terms
of integers represented by $Q_{X}$. This is followed by Corollary 4.4.4, which
is a simplified version of Corollary 4.4.3.

######  Lemma 4.4.1.

Assume $\Delta<2$ and let $k\in\mathbb{Z}_{\geq 0}$. If
$h(P)\in[2+2k-c_{\text{max}},2+2k-c_{\text{min}}]$ for some $P\notin
E(K)^{0}$, then $P_{1}\cdot P_{2}=k$ for some $P_{1},P_{2}\in E(K)$.

###### Proof.

Let $O\in E(K)$ be the neutral section. By the height formula (2.3),
$h(P)=2+2(P\cdot O)-c$, where $c:=\sum_{v}\text{contr}_{v}(P)$. Since
$h(P)\in[2+2k-c_{\text{max}},2+2k-c_{\text{min}}]$, then

$\displaystyle 2+2k-c_{\text{max}}$ $\displaystyle\leq 2+2(P\cdot O)-c\leq
2+2k-c_{\text{min}}$ $\displaystyle\Rightarrow\frac{c-c_{\text{max}}}{2}$
$\displaystyle\leq P\cdot O-k\leq\frac{c-c_{\text{min}}}{2}.$

Therefore $P\cdot O-k$ is an integer in
$I:=\left[\frac{c-c_{\text{max}}}{2},\frac{c-c_{\text{min}}}{2}\right]$. We
prove that $0$ is the only integer in $I$, so that $P\cdot O-k=0$, i.e.
$P\cdot O=k$. First notice that $c\neq 0$, as $P\notin E(K)^{0}$. By Lemma
2.8.4 iii), $c_{\text{min}}\leq c\leq c_{\text{max}}$, consequently
$\frac{c-c_{\text{max}}}{2}\leq 0\leq\frac{c-c_{\text{min}}}{2}$, i.e. $0\in
I$. Moreover $\Delta<2$ implies that $I$ has length
$\frac{c_{\text{max}}-c_{\text{min}}}{2}=\frac{\Delta}{2}<1$, so $I$ contains
no integer except $0$ as desired. ∎

######  Remark 4.4.2.

Lemma 4.4.1 also applies when $c_{\text{max}}=c_{\text{min}}$, in which case
the closed interval degenerates into a point.

The following corollary of Lemma 4.4.1 states a sufficient condition in terms
of integers represented by the quadratic form $Q_{X}$ (Subsection 2.10).

######  Corollary 4.4.3.

Assume $\Delta<2$ and let $d:=\det E(K)^{0}$. If $Q_{X}$ represents an integer
not divisible by $d$ in the interval
$[d\cdot(2+2k-c_{\text{max}}),d\cdot(2+2k-c_{\text{min}})]$, then $P_{1}\cdot
P_{2}=k$ for some $P_{1},P_{2}\in E(K)$.

###### Proof.

Let $a_{1},...,a_{r}\in\mathbb{Z}$ such that
$Q_{X}(a_{1},...,a_{r})\in[d\cdot(2+2k-c_{\text{max}}),d\cdot(2+2k-c_{\text{min}})]$
with $d\nmid Q_{X}(a_{1},...,a_{r})$. If $P_{1},...,P_{r}$ are generators of
the free part of $E(K)$, let $P:=a_{1}P_{1}+...+a_{r}P_{r}$. Then $d\nmid
Q_{X}(a_{1},...,a_{r})=d\cdot h(P)$, which implies that
$h(P)\notin\mathbb{Z}$. In particular $P\notin E(K)^{0}$ since $E(K)^{0}$ is
an integer lattice. Moreover
$h(P)=\frac{1}{d}Q_{X}(a_{1},...,a_{r})\in[2+2k-c_{\text{max}},2+2k-c_{\text{min}}]$
and we are done by Lemma 4.4.1. ∎

The next corollary, although weaker than Corollary 4.4.3, is more practical
for concrete examples and is frequently used in Subsection 4.7.4. It does not
involve finding integers represented by $Q_{X}$, but only finding perfect
squares in an interval depending on the minimal norm $\mu$ (Definition
2.7.10).

######  Corollary 4.4.4.

Assume $\Delta<2$. If there is a perfect square
$n^{2}\in\left[\frac{2+2k-c_{\text{max}}}{\mu},\frac{2+2k-c_{\text{min}}}{\mu}\right]$
such that $n^{2}\mu\notin\mathbb{Z}$, then $P_{1}\cdot P_{2}=k$ for some
$P_{1},P_{2}\in E(K)$.

###### Proof.

Take $P\in E(K)$ such that $h(P)=\mu$. Since $h(nP)=n^{2}\mu\notin\mathbb{Z}$,
we must have $nP\notin E(K)^{0}$ as $E(K)^{0}$ is an integer lattice. Moreover
$h(nP)=n^{2}\mu\in[2+2k-c_{\text{max}},2+2k-c_{\text{min}}]$ and we are done
by Lemma 4.4.1. ∎

####  4.4.2 The case $\Delta=2$

The statement of sufficient conditions for $\Delta=2$ is almost identical to
the one for $\Delta<2$, the only difference being that the closed interval
Lemma 4.4.1 is substituted by a right half-open interval in Lemma 4.4.6. This
small change, however, is associated with a technical difficulty in the case
of a section with minimal contribution term, hence the separate treatment for
$\Delta=2$.

The results are presented in the following order. First we prove a statement
about sections with minimal contribution term (Lemma 4.4.5). Next we provide
sufficient conditions when $\Delta=2$ in Lemma 4.4.6, then prove Corollaries
4.4.7 and 4.4.8.

######  Lemma 4.4.5.

Assume $\Delta=2$. If there is $P\in E(K)$ such that $\sum_{v\in
R}\text{contr}_{v}(P)=c_{\text{min}}$, then $P\cdot Q=P\cdot O+1$ for every
$Q\in E(K)_{\text{tor}}\setminus\\{O\\}$.

###### Proof.

If $Q\in E(K)_{\text{tor}}\setminus\\{O\\}$, then $Q\cdot O=0$ by Theorem
2.2.7. By the height formula (2.2),

$0=\langle P,Q\rangle=1+P\cdot O+0-P\cdot
Q-\sum_{v}\text{contr}_{v}(P,Q).\,\,(*)$

Hence it suffices to show that $\text{contr}_{v}(P,Q)=0$ $\forall v\in R$. By
Lemma 2.8.4 iv), $\text{contr}_{v^{\prime}}(P)=c_{\text{min}}$ for some
$v^{\prime}$ and $\text{contr}_{v}(P)=0$ for all $v\neq v^{\prime}$. In
particular $P$ meets $\Theta_{v,0}$, hence $\text{contr}_{v}(P,Q)=0$ for all
$v\neq v^{\prime}$. Thus from $(*)$ we see that
$\text{contr}_{v^{\prime}}(P,Q)$ is an integer, which we prove is $0$.

We claim that $T_{v^{\prime}}=A_{1}$, so that
$\text{contr}_{v^{\prime}}(P,Q)=0$ or $\frac{1}{2}$ by Table 2.4. In this
case, as $\text{contr}_{v^{\prime}}(P,Q)$ is an integer, it must be $0$, and
we are done. To see that $T_{v^{\prime}}=A_{1}$ we analyse
$\text{contr}_{v^{\prime}}(P)$. Since $\Delta=2$, then
$c_{\text{min}}=\frac{1}{2}$ by Table 2.6 and
$\text{contr}_{v^{\prime}}(P)=c_{\text{min}}=\frac{1}{2}$. By Table 2.4, this
only happens if $T_{v^{\prime}}=A_{n-1}$ and $\frac{1}{2}=\frac{i(n-i)}{n}$
for some $0\leq i<n$. The only possibility is $i=1$, $n=2$,
$T_{v^{\prime}}=A_{1}$. ∎

With the aid of Lemma 4.4.5 we are able to state sufficient conditions for
having $P_{1}\cdot P_{2}=k$ for some $P_{1},P_{2}\in E(K)$ when $\Delta=2$.

######  Lemma 4.4.6.

Assume that $\Delta=2$ and let $k\in\mathbb{Z}_{\geq 0}$. If
$h(P)\in[2+2k-c_{\text{max}},2+2k-c_{\text{min}})$ for some $P\notin
E(K)^{0}$, then $P_{1}\cdot P_{2}=k$ for some $P_{1},P_{2}\in E(K)$.

###### Proof.

By the height formula (2.3), $h(P)=2+2(P\cdot O)-c$, where
$c:=\sum_{v}\text{contr}_{v}(P)$. We repeat the arguments from Lemma 4.4.1, in
this case with the right half-open interval, so that the hypothesis that
$h(P)\in[2+2k-c_{\text{max}},2+2k-c_{\text{min}})$, implies that $P\cdot O-k$
is an integer in
$I^{\prime}:=\left[\frac{c-c_{\text{max}}}{2},\frac{c-c_{\text{min}}}{2}\right)$.

Since $I^{\prime}$ is half-open with length
$\frac{c_{\text{max}}-c_{\text{min}}}{2}=\frac{\Delta}{2}=1$, then
$I^{\prime}$ contains exactly one integer. If $0\in I^{\prime}$, then $P\cdot
O-k=0$, i.e. $P\cdot O=k$ and we are done. Hence we assume $0\notin
I^{\prime}$.

We claim that $P\cdot O=k-1$. First, notice that if $c>c_{\text{min}}$, then
the inequalities $c_{\text{min}}<c\leq c_{\text{max}}$ give
$\frac{c-c_{\text{max}}}{2}\leq 0<\frac{c-c_{\text{min}}}{2}$, i.e. $0\in
I^{\prime}$, which is a contradiction. Hence $c=c_{\text{min}}$. Since
$\Delta=2$, then $I^{\prime}=[-1,0)$. Thus $P\cdot O-k=-1$, i.e. $P\cdot
O=k-1$, as claimed.

Finally, let $Q\in E(K)_{\text{tor}}\setminus\\{O\\}$, so that $P\cdot
Q=P\cdot O+1=k$ by Lemma 4.4.5 and we are done. We remark that
$E(K)_{\text{tor}}$ is not trivial by Table 2.6, therefore such $Q$ exists. ∎

The following corollaries are analogues to Corollary 4.4.3 and Corollary 4.4.4
adapted to $\Delta=2$. Similarly to the case $\Delta<2$, Corollary 4.4.7 is
stronger than Corollary 4.4.8, although the latter is more practical for
concrete examples. We remind the reader that $\mu$ denotes the minimal norm
(Definition 2.7.10).

######  Corollary 4.4.7.

Assume $\Delta=2$ and let $d:=\det E(K)^{0}$. If $Q_{X}$ represents an integer
not divisible by $d$ in the interval
$[d\cdot(2+2k-c_{\text{max}}),d\cdot(2+2k-c_{\text{min}}))$, then $P_{1}\cdot
P_{2}=k$ for some $P_{1},P_{2}\in E(K)$.

###### Proof.

We repeat the arguments in Corollary 4.4.3, in this case with the half-open
interval. ∎

######  Corollary 4.4.8.

Assume $\Delta=2$. If there is a perfect square
$n^{2}\in\left[\frac{2+2k-c_{\text{max}}}{\mu},\frac{2+2k-c_{\text{min}}}{\mu}\right)$
such that $n^{2}\mu\notin\mathbb{Z}$, then $P_{1}\cdot P_{2}=k$ for some
$P_{1},P_{2}\in E(K)$.

###### Proof.

We repeat the arguments in Corollary 4.4.4, in this case with the half-open
interval. ∎

###  4.5 Necessary and sufficient conditions when $\Delta\leq 2$

For completeness, we present a unified statement of necessary and sufficient
conditions for having $P_{1}\cdot P_{2}=k$ for some $P_{1},P_{2}\in E(K)$
assuming $\Delta\leq 2$, which follows naturally from results in Sections 4.3
and 4.4.

######  Lemma 4.5.1.

Assume $\Delta\leq 2$ and let $k\in\mathbb{Z}_{\geq 0}$. Then $P_{1}\cdot
P_{2}=k$ for some $P_{1},P_{2}\in E(K)$ if and only if one of the following
holds:

  1. i)

$h(P)=2+2k$ for some $P\in E(K)^{0}$.

  2. ii)

$h(P)\in[2+2k-c_{\text{max}},2+2k-c_{\text{min}})$ for some $P\notin
E(K)^{0}$.

  3. iii)

$h(P)=2+2k-c_{\text{min}}$ and $\sum_{v\in
R}\text{contr}_{v}(P)=c_{\text{min}}$ for some $P\in E(K)$.

###### Proof.

If i) or iii) holds, then $P\cdot O=k$ directly by the height formula (2.3).
But if ii) holds, it suffices to to apply Lemma 4.4.1 when $\Delta<2$ and by
Lemma 4.4.6 when $\Delta=2$.

Conversely, let $P_{1}\cdot P_{2}=k$. Without loss of generality, we may
assume that $P_{2}=O$, so that $P_{1}\cdot O=k$. By the height formula (2.3),
$h(P_{1})=2+2k-c$, where $c:=\sum_{v}\text{contr}_{v}(P_{1})$.

If $c=0$, then $P_{1}\in E(K)^{0}$ and $h(P_{1})=2+2k$, so i) holds. Hence we
let $c\neq 0$, i.e. $P_{1}\notin E(K)^{0}$, so that $c_{\text{min}}\leq c\leq
c_{\text{max}}$ by Lemma 2.8.4. In case $c=c_{\text{min}}$, then
$h(P_{1})=2+2k-c_{\text{min}}$ and iii) holds. Otherwise $c_{\text{min}}<c\leq
c_{\text{max}}$, which implies $2+2k-c_{\text{max}}\leq
h(P_{1})<2+2k-c_{\text{min}}$, so ii) holds. ∎

###  4.6 Summary of sufficient conditions

For the sake of clarity, we summarize in a single proposition all sufficient
conditions for having $P_{1}\cdot P_{2}=k$ for some $P_{1},P_{2}\in E(K)$
proven in this chapter.

######  Proposition 4.6.1.

Let $k\in\mathbb{Z}_{\geq 0}$. If one of the following holds, then $P_{1}\cdot
P_{2}=k$ for some $P_{1},P_{2}\in E(K)$.

  1. 1)

$h(P)=2+2k$ for some $P\in E(K)^{0}$.

  2. 2)

$h(P)=2k$ for some $P\in E(K)^{0}$ and $E(K)_{\text{tor}}$ is not trivial.

  3. 3)

$\Delta<2$ and there is a perfect square
$n^{2}\in\left[\frac{2+2k-c_{\text{max}}}{\mu},\frac{2+2k-c_{\text{min}}}{\mu}\right]$
with $n^{2}\mu\notin\mathbb{Z}$, where $\mu$ is the minimal norm (Definition
2.7.10). In case $\Delta=2$, consider the right half-open interval.

  4. 4)

$\Delta<2$ and the quadratic form $Q_{X}$ represents an integer not divisible
by $d:=\det E(K)^{0}$ in the interval
$[d\cdot(2+2k-c_{\text{max}}),d\cdot(2+2k-c_{\text{min}})]$. In case
$\Delta=2$, consider the right half-open interval.

###### Proof.

In 1) a height calculation gives $2+2k=h(P)=2+2(P\cdot O)-0$, so $P\cdot O=k$.
For 2), we apply Lemma 4.4.5 to conclude that $P\cdot Q=k$ for any $Q\in
E(K)_{\text{tor}}\setminus\\{O\\}$. In 3) we use Corollary 4.4.4 when
$\Delta<2$ and Corollary 4.4.8 when $\Delta=2$. In 4), we apply Corollary
4.4.3 if $\Delta<2$ and Corollary 4.4.7 if $\Delta=2$. ∎

###  4.7 Applications

We prove the four main theorems of this chapter, which are independent
applications of the results from Sections 4.3 and 4.4. The first two are
general attempts to describe when and how gap numbers occur: Theorem 4.7.2
tells us that large Mordell-Weil groups prevent the existence of gap numbers,
more precisely for Mordell-Weil rank $r\geq 5$; in Theorem 4.7.4 we show that
for a smaller Mordell-Weil group, more precisely when $r\leq 2$, gap numbers
occur with probability $1$. The last two theorems, on the other hand, deal
with explicit values of gap numbers: Theorem 4.7.7 provides a complete
description of gap numbers in certain cases, whereas Theorem 4.7.8 is a
classification of all cases with a $1$-gap.

####  4.7.1 No gap numbers in $r\geq 5$

We show that if $E(K)$ has rank $r\geq 5$, then $X$ is gap-free. Our strategy
is to prove that for every $k\in\mathbb{Z}_{\geq 0}$ there is some $P\in
E(K)^{0}$ such that $h(P)=2+2k$, and by Proposition 4.6.1 1) we are done. We
accomplish this in two steps. First we show that this holds when there is an
embedding of $A_{1}^{\oplus}$ or of $A_{4}$ in $E(K)^{0}$ (Lemma 4.7.1).
Second, we show that if $r\geq 5$, then such embedding exists, hence $X$ is
gap-free (Theorem 4.7.2).

######  Lemma 4.7.1.

Assume $E(K)^{0}$ has a sublattice isomorphic to $A_{1}^{\oplus 4}$ or
$A_{4}$. Then for every $\ell\in\mathbb{Z}_{\geq 0}$ there is $P\in E(K)^{0}$
such that $h(P)=2\ell$.

###### Proof.

First assume $A_{1}^{\oplus 4}\subset E(K)^{0}$ and let
$P_{1},P_{2},P_{3},P_{4}$ be generators for each factor $A_{1}$ in
$A_{1}^{\oplus 4}$. Then $h(P_{i})=2$ and $\langle P_{i},P_{j}\rangle=0$ for
distinct $i,j=1,2,3,4$. By Lagrange’s four-square theorem [HW79, §20.5] there
are integers $a_{1},a_{2},a_{3},a_{4}$ such that
$a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}=\ell$. Defining
$P:=a_{1}P_{1}+a_{2}P_{2}+a_{3}P_{3}+a_{4}P_{4}\in A_{1}^{\oplus 4}\subset
E(K)^{0}$, we have

$h(P)=2a_{1}^{2}+2a_{2}^{2}+2a_{3}^{2}+2a_{4}^{2}=2\ell.$

Now let $A_{4}\subset E(K)^{0}$ with generators $P_{1},P_{2},P_{3},P_{4}$.
Then $h(P_{i})=2$ for $i=1,2,3,4$ and $\langle P_{i},P_{i+1}\rangle=-1$ for
$i=1,2,3$. We need to find integers $x_{1},...,x_{4}$ such that $h(P)=2\ell$,
where $P:=x_{1}P_{1}+...+x_{4}P_{4}\in A_{4}\subset E(K)^{0}$. Equivalently,
we need that

$\ell=\frac{1}{2}\langle
P,P\rangle=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}-x_{1}x_{2}-x_{2}x_{3}-x_{3}x_{4}.$

Therefore $\ell$ must be represented by
$q(x_{1},...,x_{4}):=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}-x_{1}x_{2}-x_{2}x_{3}-x_{3}x_{4}$.
We prove that $q$ represents all positive integers. Notice that $q$ is
positive-definite, since it is induced by $\langle\cdot,\cdot\rangle$. By
Bhargava-Hanke’s 290-theorem [BH][Thm. 1], $q$ represents all positive
integers if and only if it represents the following integers:

$2,3,5,6,7,10,13,14,15,17,19,21,22,23,26,$
$29,30,31,34,35,37,42,58,93,110,145,203,290.$

The representation for each of the above is found in Table 4.1. ∎

We now prove the main theorem of this section.

######  Theorem 4.7.2.

If $r\geq 5$, then $X$ is gap-free.

###### Proof.

We show that for every $k\geq 0$ there is $P\in E(K)^{0}$ such that
$h(P)=2+2k$, so that by Proposition 4.6.1 1) we are done. By Lemma 4.7.1 it
suffices to prove that $E(K)^{0}$ has a sublattice isomorphic to
$A_{1}^{\oplus 4}$ or $A_{4}$.

The cases with $r\geq 5$ are No. 1-7 (Table LABEL:table:MWL_data). In No. 1-6,
$E(K)^{0}=E_{8},E_{7},E_{6},D_{6},D_{5},A_{5}$ respectively. Each of these
admit an $A_{4}$ sublattice [Nis96, Lemmas 4.2, 4.3]. In No. 7 we claim that
$E(K)^{0}=D_{4}\oplus A_{1}$ has an $A_{1}^{\oplus 4}$ sublattice. This is the
case because $D_{4}$ admits an $A_{1}^{\oplus 4}$ sublattice [Nis96, Lemma 4.5
(iii)].

∎

$n$ |  $x_{1},x_{2},x_{3},x_{4}$ with $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}-x_{1}x_{2}-x_{2}x_{3}-x_{3}x_{4}=n$
---|---
$1$ | $1,0,0,0$
$2$ | $1,0,1,0$
$3$ | $1,1,2,0$
$5$ | $1,0,2,0$
$6$ | $1,1,-2,-1$
$7$ | $1,1,-2,0$
$10$ | $1,0,3,0$
$13$ | $2,0,3,0$
$14$ | $1,2,5,1$
$15$ | $1,5,5,2$
$17$ | $1,0,4,0$
$19$ | $1,5,3,-1$
$21$ | $1,5,0,0$
$22$ | $1,5,0,-1$
$23$ | $1,6,6,2$
$26$ | $1,0,5,0$
$29$ | $2,0,5,0$
$30$ | $1,5,0,-3$
$31$ | $1,3,-4,-2$
$34$ | $3,0,5,0$
$35$ | $1,2,-2,4$
$37$ | $1,0,6,0$
$42$ | $1,1,-4,3$
$58$ | $3,0,7,0$
$93$ | $1,1,-10,0$
$110$ | $1,-2,3,-8$
$145$ | $1,0,12,0$
$203$ | $1,-5,-9,8$
$290$ | $1,0,17,0$
Table 4.1: Representation of the critical integers in Bhargava-Hanke’s
290-theorem.

####  4.7.2 Gaps with probability $1$ in $r\leq 2$

Fix a rational elliptic surface $\pi:X\to\mathbb{P}^{1}$ with Mordell-Weil
rank $r\leq 2$. We prove that if $k$ is a uniformly random natural number,
then $k$ is a gap number with probability $1$. More precisely, if
$G:=\\{k\in\mathbb{N}\mid k\text{ is a gap number of }X\\}$ is the set of gap
numbers, then $G\subset\mathbb{N}$ has density $1$, i.e.

$d(G):=\lim_{n\to\infty}\frac{\\#G\cap\\{1,...,n\\}}{n}=1.$

We adopt the following strategy. If $k\in\mathbb{N}\setminus G$, then
$P_{1}\cdot P_{2}=k$ for some $P_{1},P_{2}\in E(K)$ and by Corollary 4.3.2 the
quadratic form $Q_{X}$ represents some integer $t$ depending on $k$. This
defines a function $f:\mathbb{N}\setminus G\to T$, where $T$ is the set of
integers represented by $Q_{X}$. Since $Q_{X}$ is a quadratic form on $r\leq
2$ variables, $T$ has density $0$ in $\mathbb{N}$ by Lemma 4.7.3. By analyzing
the pre-images of $f$, in Theorem 4.7.4 we conclude that
$d(\mathbb{N}\setminus G)=d(T)=0$, hence $d(G)=1$ as desired.

######  Lemma 4.7.3.

Let $Q$ be a positive-definite quadratic form on $r=1,2$ variables with
integer coefficients. Then the set of integers represented by $Q$ has density
$0$ in $\mathbb{N}$.

###### Proof.

Let $S$ be the set of integers represented by $Q$. If $d$ is the greatest
common divisor of the coefficients of $Q$, let $S^{\prime}$ be the set of
integers representable by the primitive form $Q^{\prime}:=\frac{1}{d}\cdot Q$.
By construction $S^{\prime}$ is a rescaling of $S$, so $d(S)=0$ if and only if
$d(S^{\prime})=0$.

If $r=1$, then $Q^{\prime}(x_{1})=x_{1}^{2}$ and $S^{\prime}$ is the set of
perfect squares, so clearly $d(S^{\prime})=0$. If $r=2$, then $Q^{\prime}$ is
a binary quadratic form and the number of elements in $S^{\prime}$ bounded
from above by $x>0$ is given by $C\cdot\frac{x}{\sqrt{\log x}}+o(x)$ with
$C>0$ a constant and $\lim_{x\to\infty}\frac{o(x)}{x}=0$ [Ber12, p. 91]. Thus

$d(S^{\prime})=\lim_{x\to\infty}\frac{C}{\sqrt{\log x}}+\frac{o(x)}{x}=0.$

∎

We now prove the main result of this section.

######  Theorem 4.7.4.

Let $\pi:X\to\mathbb{P}^{1}$ be a rational elliptic surface with Mordell-Weil
rank $r\leq 2$. Then the set $G:=\\{k\in\mathbb{N}\mid k\text{ is a gap number
of }X\\}$ of gap numbers of $X$ has density $1$ in $\mathbb{N}$.

###### Proof.

If $r=0$, then all sections are torsion sections and the claim is obvious by
Theorem 2.2.7. Assume $r=1,2$. We prove that $S:=\mathbb{N}\setminus G$ has
density $0$. If $S$ is finite, there is nothing to prove. Otherwise, let
$k_{1}<k_{2}<...$ be the increasing sequence of all elements of $S$. By
Corollary 4.3.2, for each $n$ there is some $t_{n}\in
J_{k_{n}}:=[d\cdot(2+2k_{n}-c_{\text{max}}),d\cdot(2+2k_{n})]$ represented by
the quadratic form $Q_{X}$. Let $T$ be the set of integers represented by
$Q_{X}$ and define the function $f\colon S\to T$ by $k_{n}\mapsto t_{n}$.
Since $Q_{X}$ has $r=1,2$ variables, $T$ has density $0$ by Lemma 4.7.3.

For $N>0$, let $S_{N}:=S\cap\\{1,...,N\\}$ and $T_{N}:=T\cap\\{1,...,N\\}$.
Since $T$ has density zero, $\\#T_{N}=o(N)$, i.e. $\frac{\\#T_{N}}{N}\to 0$
when $N\to\infty$ and we need to prove that $\\#S_{N}=o(N)$. We analyze the
function $f$ restricted to $S_{N}$. Notice that as $t_{n}\in J_{k_{n}}$, then
$k_{n}\leq N$ implies $t_{n}\leq d\cdot(2+2k_{n})\leq d\cdot(2+2N)$. Hence the
restriction $g:=f|_{S_{N}}$ can be regarded as a function $g:S_{N}\to
T_{d\cdot(2+2k)}$.

We claim that $\\#g^{-1}(t)\leq 2$ for all $t\in T_{d\cdot(2+2N)}$, in which
case $\\#S_{N}\leq 2\cdot\\#T_{d\cdot(2+2N)}=o(N)$ and we are done. Assume by
contradiction that $g^{-1}(t)$ contains three distinct elements, say
$k_{\ell_{1}}<k_{\ell_{2}}<k_{\ell_{3}}$ with
$t=t_{\ell_{1}}=t_{\ell_{2}}=t_{\ell_{3}}$. Since $t_{\ell_{i}}\in
J_{k_{\ell_{i}}}$ for each $i=1,2,3$, then $t\in J_{k_{\ell_{1}}}\cap
J_{k_{\ell_{2}}}\cap J_{k_{\ell_{3}}}$. We prove that $J_{k_{\ell_{1}}}$ and
$J_{k_{\ell_{3}}}$ are disjoint, which yields a contradiction. Indeed, since
$k_{\ell_{1}}<k_{\ell_{2}}<k_{\ell_{3}}$, in particular
$k_{\ell_{3}}-k_{\ell_{1}}\geq 2$, therefore $d\cdot(2+2k_{\ell_{1}})\leq
d\cdot(2+2k_{\ell_{3}}-4)$. But $c_{\text{max}}<4$ by Lemma 2.8.4, so
$d\cdot(2+2k_{\ell_{1}})<d\cdot(2+2k_{\ell_{3}}-c_{\text{max}})$, i.e. $\max
J_{k_{\ell_{1}}}<\min J_{k_{\ell_{3}}}$. Thus $J_{k_{\ell_{1}}}\cap
J_{k_{\ell_{3}}}=\emptyset$, as desired. ∎

####  4.7.3 Identification of gaps when $E(K)$ is torsion-free with rank $r=1$

The results in Subsections 4.7.1 and 4.7.2 concern the existence and the
distribution of gap numbers. In the following subsections we turn our
attention to finding gap numbers explicitly. In this subsection we give a
complete description of gap numbers assuming $E(K)$ is torsion-free with rank
$r=1$. Such descriptions are difficult in the general case, but our assumption
guarantees that each $E(K),E(K)^{0}$ is generated by a single element and that
$\Delta<2$ by Lemma 2.9.1, which makes the problem more accessible.

We organize this subsection as follows. First we point out some trivial facts
about generators of $E(K),E(K)^{0}$ when $r=1$ in Lemma 4.7.5. Next we state
necessary and sufficient conditions for having $P_{1}\cdot P_{2}=k$ when
$E(K)$ is torsion-free with $r=1$ in Lemma 4.7.6. As an application of the
latter, we prove Theorem 4.7.7, which is the main result of the subsection.

######  Lemma 4.7.5.

Let $X$ be a rational elliptic surface with Mordell-Weil rank $r=1$. If $P$
generates the free part of $E(K)$, then

  1. a)

$h(P)=\mu$.

  2. b)

$1/\mu$ is an even integer.

  3. c)

$E(K)^{0}$ is generated by $P_{0}:=(1/\mu)P$ and $h(P_{0})=1/\mu$.

###### Proof.

Item a) is clear. Items b), c) follow from the fact that $E(K)^{0}$ is an even
lattice and that $E(K)\simeq L^{*}\oplus E(K)_{\text{tor}}$, where
$L:=E(K)^{0}$ [OS91, Main Thm.]. ∎

In what follows we use Lemma 4.7.5 and results from Section 4.4 to state
necessary and sufficient conditions for having $P_{1}\cdot P_{2}=k$ for some
$P_{1},P_{2}\in E(K)$ in case $E(K)$ is torsion-free with $r=1$.

######  Lemma 4.7.6.

Assume $E(K)$ is torsion-free with rank $r=1$. Then $P_{1}\cdot P_{2}=k$ for
some $P_{1},P_{2}\in E(K)$ if and only if one of the following holds:

  1. i)

$\mu\cdot(2+2k)$ is a perfect square.

  2. ii)

There is a perfect square
$n^{2}\in\left[\frac{2+2k-c_{\text{max}}}{\mu},\frac{2+2k-c_{\text{min}}}{\mu}\right]$
such that $\mu\cdot n\notin\mathbb{Z}$.

###### Proof.

By Lemma 4.7.5, $E(K)$ is generated by some $P$ with $h(P)=\mu$ and $E(K)^{0}$
is generated by $P_{0}:=n_{0}P$, where $n_{0}:=\frac{1}{\mu}\in 2\mathbb{Z}$.

First assume that $P_{1}\cdot P_{2}=k$ for some $P_{1},P_{2}$. Without loss of
generality we may assume $P_{2}=O$. Let $P_{1}=nP$ for some $n\in\mathbb{Z}$.
We show that $P_{1}\in E(K)^{0}$ implies i) while $P_{1}\notin E(K)^{0}$
implies ii).

If $P_{1}\in E(K)^{0}$, then $n_{0}\mid n$, hence $P_{1}=nP=mP_{0}$, where
$m:=\frac{n}{n_{0}}$. By the height formula (2.3),
$2+2k=h(P_{1})=h(mP_{0})=m^{2}\cdot\frac{1}{\mu}$. Hence
$\mu\cdot(2+2k)=m^{2}$, i.e. i) holds.

If $P_{1}\notin E(K)^{0}$, then $n_{0}\nmid n$, hence $\mu\cdot
n=\frac{n}{n_{0}}\notin\mathbb{Z}$. Moreover, $h(P_{1})=n^{2}h(P)=n^{2}\mu$
and by the height formula (2.3), $n^{2}\mu=h(P)=2+2k-c$, where
$c:=\sum_{v}\text{contr}_{v}(P_{1})\neq 0$. The inequalities
$c_{\text{min}}\leq c\leq c_{\text{max}}$ then give
$\frac{2+2k-c_{\text{max}}}{\mu}\leq
n^{2}\leq\frac{2+2k-c_{\text{min}}}{\mu}$. Hence ii) holds.

Conversely, assume i) or ii) holds. Since $E(K)$ is torsion-free, $\Delta<2$
by Lemma 2.9.1, so we may apply Lemma 4.4.1. If i) holds, then
$\mu\cdot(2+2k)=m^{2}$ for some $m\in\mathbb{Z}$. Since $mP_{0}\in E(K)^{0}$
and $h(mP_{0})=\frac{m^{2}}{\mu}=2+2k$, we are done by Lemma 4.4.1 i). If ii)
holds, the condition $\mu\cdot n\notin\mathbb{Z}$ is equivalent to $n_{0}\nmid
n$, hence $nP\notin E(K)^{0}$. Moreover
$n^{2}\in\left[\frac{2+2k-c_{\text{max}}}{\mu},\frac{2+2k-c_{\text{min}}}{\mu}\right]$,
implies $h(nP)=n^{2}\mu\in[2+2k-c_{\text{max}},2+2k-c_{\text{min}}]$. By Lemma
4.4.1 ii), we are done. ∎

An application of Lemma 4.7.6 to all possible cases where $E(K)$ is torsion-
free with rank $r=1$ yields the main result of this subsection.

######  Theorem 4.7.7.

If $E(K)$ is torsion-free with rank $r=1$, then all the gap numbers of $X$ are
described in Table 4.2.

No. | $T$ | $\begin{matrix}k\text{ is a gap number}\Leftrightarrow\text{none of}\\\ \text{the following are perfect squares}\end{matrix}$ | first gap numbers
---|---|---|---
43 | $E_{7}$ | $k+1$, $4k+1$ | $1,4$
45 | $A_{7}$ | $\frac{k+1}{4}$, $16k,...,16k+9$ | $8,11$
46 | $D_{7}$ | $\frac{k+1}{2}$, $8k+1,...,8k+4$ | $2,5$
47 | $A_{6}\oplus A_{1}$ | $\frac{k+1}{7}$, $28k-3,...,28k+21$ | $12,16$
49 | $E_{6}\oplus A_{1}$ | $\frac{k+1}{3}$, $12k+1,...,12k+9$ | $3,7$
50 | $D_{5}\oplus A_{2}$ | $\frac{k+1}{6}$, $24k+1,...,24k+16$ | $6,11$
55 | $A_{4}\oplus A_{3}$ | $\frac{k+1}{10}$, $40k-4,...,40k+25$ | $16,20$
56 | $A_{4}\oplus A_{2}\oplus A_{1}$ | $\frac{k+1}{15}$, $60k-11,...,60k+45$ | $22,27$
Table 4.2: Description of gap numbers when $E(K)$ is torsion-free with $r=1$.

###### Proof.

For the sake of brevity we restrict ourselves to No. 55. The other cases are
treated similarly. Here $c_{\text{max}}=\frac{2\cdot 3}{5}+\frac{2\cdot
2}{4}=\frac{11}{5}$,
$c_{\text{min}}=\min\left\\{\frac{4}{5},\frac{3}{4}\right\\}=\frac{3}{4}$ and
$\mu=1/20$.

By Lemma 4.7.6, $k$ is a gap number if and only if neither i) nor ii) occurs.
Condition i) is that $\frac{2+2k}{20}=\frac{k+1}{10}$ is a perfect square.
Condition ii) is that
$\left[\frac{2+2k-c_{\text{max}}}{\mu},\frac{2+2k-c_{\text{min}}}{\mu}\right]=[40k-4,40k+25]$
contains some $n^{2}$ with $20\nmid n$. We check that $20\nmid n$ for every
$n$ such that $n^{2}=40k+\ell$, with $\ell=-4,...,25$. Indeed, if $20\mid n$,
then $400\mid n^{2}$ and in particular $40\mid n^{2}$. Then
$40\mid(n^{2}-40k)=\ell$, which is absurd. ∎

####  4.7.4 Surfaces with a $1$-gap

In Subsection 4.7.3 we take each case in Table 4.2 and describe all its gap
numbers. In this subsection we do the opposite, namely, we fix a number and
describe all cases having it as a gap number. We remind the reader that our
motivating problem was to determine when there are sections $P_{1},P_{2}$ such
that $P_{1}\cdot P_{2}=1$, which induce a conic bundle having $P_{1}+P_{2}$ as
a reducible fiber. The answer for this question is the main theorem of this
subsection:

######  Theorem 4.7.8.

Let $X$ be a rational elliptic surface. Then $X$ has a $1$-gap if and only if
$r=0$ or $r=1$ and $\pi$ has a $\text{III}^{*}$ fiber.

Our strategy for the proof is the following. We already know that a $1$-gap
exists whenever $r=0$ (Theorem 2.2.7) or when $r=1$ and $\pi$ has a
$\text{III}^{*}$ fiber (Theorem 4.7.7, No. 43). Conversely, we need to find
$P_{1},P_{2}$ with $P_{1}\cdot P_{2}=1$ in all cases with $r\geq 1$ and $T\neq
E_{7}$.

First we introduce two lemmas, which solve most cases with little computation,
and leave the remaining ones for the proof of Theorem 4.7.8. In both Lemma
4.7.9 and Lemma 4.7.11 our goal is to analyze the narrow lattice $E(K)^{0}$
and apply Proposition 4.6.1 to detect cases without a $1$-gap.

######  Lemma 4.7.9.

If one of the following holds, then $h(P)=4$ for some $P\in E(K)^{0}$.

  1. a)

The Gram matrix of $E(K)^{0}$ has a $4$ in its main diagonal.

  2. b)

There is an embedding of $A_{n}\oplus A_{m}$ in $E(K)^{0}$ for some $n,m\geq
1$.

  3. c)

There is an embedding of $A_{n},D_{n}$ or $E_{n}$ in $E(K)^{0}$ for some
$n\geq 3$.

###### Proof.

Case a) is trivial. Assuming b), we take generators $P_{1},P_{2}$ from
$A_{n},A_{m}$ respectively with $h(P_{1})=h(P_{2})=2$. Since $A_{n},A_{m}$ are
in direct sum, $\langle P_{1},P_{2}\rangle=0$, hence $h(P_{1}+P_{2})=4$, as
desired. If c) holds, then the fact that $n\geq 3$ allows us to choose two
elements $P_{1},P_{2}$ among the generators of $L_{1}=A_{n},D_{n}$ or $E_{n}$
such that $h(P_{1})=h(P_{2})=2$ and $\langle P_{1},P_{2}\rangle=0$. Thus
$h(P_{1}+P_{2})=4$ as claimed. ∎

######  Corollary 4.7.10.

In the following cases, $X$ does not have a $1$-gap.

  * •

$r\geq 3:$ all cases except possibly No. 20.

  * •

$r=1,2:$ cases No. 25, 26, 30, 32-36, 38, 41, 42, 46, 52, 54, 60.

###### Proof.

We look at column $E(K)^{0}$ in Table LABEL:table:MWL_data to find which cases
satisfy one of the conditions a), b), c) from Lemma 4.7.9.

  1. a)

Applies to No. 12, 17, 19, 22, 23, 25, 30, 32, 33, 36, 38, 41, 46, 52, 54, 60.

  2. b)

Applies to No. 10, 11, 14, 15, 18, 24, 26, 34, 35, 42.

  3. c)

Applies to No. 1-10, 13, 16, 21.

In particular, this covers all cases with $r\geq 3$ (No. 1-24) except No. 20.
By Lemma 4.7.9 in each of these cases there is $P\in E(K)^{0}$ with $h(P)=4$
and we are done by Proposition 4.6.1 1). ∎

In the next lemma we also analyze $E(K)^{0}$ to detect surfaces without a
$1$-gap.

######  Lemma 4.7.11.

Assume $E(K)^{0}\simeq A_{n}$ for some $n\geq 1$ and that $E(K)$ has
nontrivial torsion part. Then $X$ does not have a $1$-gap. This applies to
cases No. 28, 39, 44, 48, 51, 57, 58 in Table LABEL:table:MWL_data.

###### Proof.

Take a generator $P$ of $E(K)^{0}$ with $h(P)=2$ and apply Proposition 4.6.1
2). ∎

We are ready to prove the main result of this subsection.

###### Proof of Theorem 4.7.8.

We need to show that in all cases where $r\geq 1$ and $T\neq E_{7}$ there are
$P_{1},P_{2}\in E(K)$ such that $P_{1}\cdot P_{2}=1$. This corresponds to
cases No. 1-61 except 43 in Table LABEL:table:MWL_data.

The cases where $r=1$ and $E(K)$ is torsion-free can be solved by Theorem
4.7.7, namely No. 45-47, 49, 50, 55, 56. Adding these cases to the ones
treated in Corollary 4.7.10 and Lemma 4.7.11, we have therefore solved the
following:

$\text{No.
}1\text{-}19,\,21\text{-}26,\,28,\,30,\,32\text{-}36,\,38,\,39,\,41\text{-}52,\,54\text{-}58,\,60.$

For the remaining cases, we apply Proposition 4.6.1 3), which involves finding
perfect squares in the interval
$\left[\frac{4-c_{\text{max}}}{\mu},\frac{4-c_{\text{min}}}{\mu}\right]$ (see
Table 4.3), considering the half-open interval in the cases with $\Delta=2$
(No. 53, 61).

No. | $T$ | $E(K)$ | $\mu$ | $I$ | $n^{2}\in I$
---|---|---|---|---|---
20 | $A_{2}^{\oplus 2}\oplus A_{1}$ | $A_{2}^{*}\oplus\langle 1/6\rangle$ | $\frac{1}{6}$ | $[13,23]$ | $4^{2}$
27 | $E_{6}$ | $A_{2}^{*}$ | $\frac{2}{3}$ | $[4,4]$ | $2^{2}$
29 | $A_{5}\oplus A_{1}$ | $A_{1}^{*}\oplus\langle 1/6\rangle$ | $\frac{1}{6}$ | $[12,21]$ | $4^{2}$
31 | $A_{4}\oplus A_{2}$ | $\frac{1}{15}\left(\begin{matrix}2&1\\\ 1&8\end{matrix}\right)$ | $\frac{2}{15}$ | $[16,21]$ | $4^{2}$
37 | $A_{3}\oplus A_{2}\oplus A_{1}$ | $A_{1}^{*}\oplus\langle 1/12\rangle$ | $\frac{1}{12}$ | $[22,28]$ | $5^{2}$
40 | $A_{2}^{\oplus 2}\oplus A_{1}^{\oplus 2}$ | $\langle 1/6\rangle^{\oplus 2}$ | $\frac{1}{6}$ | $\left[10,21\right]$ | $4^{2}$
53 | $A_{5}\oplus A_{1}^{\oplus 2}$ | $\langle 1/6\rangle\oplus\mathbb{Z}/2\mathbb{Z}$ | $\frac{1}{6}$ | $[9,12]$ | $3^{2}$
59 | $A_{3}\oplus A_{2}\oplus A_{1}^{\oplus 2}$ | $\langle 1/12\rangle\oplus\mathbb{Z}/2\mathbb{Z}$ | $\frac{1}{12}$ | $[16,42]$ | $4^{2},5^{2},6^{2}$
61 | $A_{2}^{\oplus 3}\oplus A_{1}$ | $\langle 1/6\rangle\oplus\mathbb{Z}/3\mathbb{Z}$ | $\frac{1}{6}$ | $[9,12]$ | $3^{2}$
Table 4.3: Perfect squares in the interval
$I:=\left[\frac{4-c_{\text{max}}}{\mu},\frac{4-c_{\text{min}}}{\mu}\right]$.

In No. 59 we have $\Delta>2$, so a particular treatment is needed. Let
$T=T_{v_{1}}\oplus T_{v_{2}}\oplus T_{v_{3}}\oplus T_{v_{4}}=A_{3}\oplus
A_{2}\oplus A_{1}\oplus A_{1}$. If $P$ generates the free part of $E(K)$ and
$Q$ generates its torsion part, then $h(P)=\frac{1}{12}$ and $4P+Q$ meets the
reducible fibers at
$\Theta_{v_{1},2},\Theta_{v_{2},1},\Theta_{v_{3},1},\Theta_{v_{4},1}$
[Kur14][Example 1.7]. By Table 2.4 and the height formula (2.3),

$\frac{4^{2}}{12}=h(4P+Q)=2+2(4P+Q)\cdot O-\frac{2\cdot 2}{4}-\frac{1\cdot
2}{3}-\frac{1}{2}-\frac{1}{2},$

hence $(4P+Q)\cdot O=1$, as desired. ∎

##  Chapter 5 Large rank jumps and the Hilbert property

Throughout this chapter $k$ denotes a number field and $X$ a geometrically
rational elliptic surface over $k$ with elliptic fibration
$\pi:X\to\mathbb{P}^{1}$ and Mordell-Weil rank $r$ over $k$. We study the
variation of Mordell-Weil rank $r_{t}$ (over $k$) of the fiber $\pi^{-1}(t)$
as $t$ runs through $t\in\mathbb{P}^{1}_{k}$ in comparison to the generic rank
$r$ and say that there is a rank jump if $r_{t}>r$. We cover some of the
approaches used in the literature, explain their limitations and how we adapt
them in order to prove the main theorem of this chapter, namely

######  Theorem 5.0.1.

Let $\pi:X\to\mathbb{P}^{1}$ be a geometrically rational elliptic surface over
a number field $k$ with generic rank $r$. Assume that $\pi$ admits a RNRF-
conic bundle (Definition 5.2.2). Then the subset of the base of the elliptic
fibration

$\\{t\in\mathbb{P}^{1}_{k}\mid r_{t}\geq r+3\\}\subset\mathbb{P}^{1}_{k}$

is not thin (Definition 2.4.1).

The starting point for our investigation is a specialization theorem by
Silverman [Sil83, Thm. C], which states that $r_{t}\geq r$ for all but
finitely many $t\in\mathbb{P}^{1}_{k}$. We remark that this result is a
development from a more general theorem by Néron [Né52, Thm. 6] over higher-
dimensional bases which says that the rank is at least the generic rank
outside of a thin subset (see Definition 2.4.1) of the base curve. In light of
this, our natural guiding question is: what is the nature of the set of fibers
where rank jumps occur?

Techniques to deal with this question include the study of the behavior of the
root numbers in families first carried out by Rohrlich [Roh93]; height theory
estimations as done by Billard [Bil98] and geometric techniques, more
precisely base change, introduced by Néron [Né52].

A considerable limitation of the root number approach is that it relies on the
Birch and Swinnerton-Dyer conjecture and, even with that assumption, one is
restricted to $\mathbb{Q}$ since the functional equation is only conjectural
over number fields in general. As for the use of height theory, oftentimes one
is also restricted to $\mathbb{Q}$, as many properties of the height machinery
are proven only over the rationals. The geometric techniques, on the other
hand, tend to allow more flexibility on the base field and have been used by
several autors, such as [Shi91, Sal12, Sal15, HS19, CT20]

We explain Néron’s geometric method. His goal in [Né56, §5] is to start with
$X$ with generic rank $8$ and build an elliptic surface $f:Y\to C$ such that
$Y$ is a degree $6$ cover of $X$ with generic rank at least $11=8+3$. More
precisely, the elliptic fibrations $\pi:X\to\mathbb{P}^{1}$ and $f:Y\to C$ are
related by the following commutative diagram.

${Y}$${C}$${X}$${\mathbb{P}^{1}}$$\scriptstyle{6:1}$$\scriptstyle{f}$$\scriptstyle{\pi}$

By applying Néron’s specialization theorem, the fact that $f:Y\to C$ has rank
at least $11$ implies that there are infinitely many fibers of $\pi$ whose
ranks are at least $11=8+3$, hence a rank jump of $3$. The construction of $Y$
is performed by taking three rational bisections $L_{1},L_{2},L_{3}$ in $X$
(Definition 2.1.4) such that the base change of $\pi:X\to\mathbb{P}^{1}$ by
$C:=L_{1}\times_{\mathbb{P}^{1}}L_{2}\times_{\mathbb{P}^{1}}L_{3}$ gives the
new fibration $f:Y\to C$, which he shows admits $3$ new and independent
sections. A key feature of this construction is the fact that the morphisms of
the base $L_{i}\to\mathbb{P}^{1}$ share a common branch point, hence the curve
$L_{1}\times_{\mathbb{P}^{1}}L_{2}\times_{\mathbb{P}^{1}}L_{3}$ has genus $1$.

A generalization of this technique was made by Salgado in [Sal12] when $X$ is
any rational elliptic surface admitting a conic bundle over $k$. One key
difference in comparison to Néron’s construction is that the curves $L_{i}$
need not to be given explicitly. Instead, she shows a more general result for
elliptic surfaces considering base change with respect to curves on a linear
system, which produce a strictly greater rank:

######  Theorem 5.0.2.

[Sal12, Cor. 4.3] Let $f:S\to C$ be an elliptic surface over a number field
and $\mathscr{L}$ a pencil of curves on $S$ which does not contain a fiber of
$f$. Then for all but finitely many $D\in\mathscr{L}$, the base-changed
fibration $S\times_{C}D\to D$ has generic rank strictly greater than that of
$f:S\to C$.

A drawback in this construction, however, is that the morphisms to the base
$D\to\mathbb{P}^{1}$ which play role of $L_{i}\to\mathbb{P}^{1}$ in Néron’s
construction do not necessarily admit a common branch point. In this case
$D_{1}\times_{\mathbb{P}^{1}}D_{2}$ may already have genus $1$, which limits
the number of base changes to $2$. As a consequence, one is only allowed to
conclude about rank jumps of at least $2$, but not $3$, which is precisely the
case in [Sal12] and [LS22]. One way to overcome this difficulty is to consider
conic bundles with certain ramification properties, which we call RNRF-conic
bundles (for ramified at a nonreduced fiber, Definition 5.2.2), which allows
us to detect rank jumps of at least $3$.

We want, moreover, to study the structure of set of fibers with rank jump of
at least $3$. Given Néron’s specialization theorem, it is natural to ask
whether this set is thin or not. In the case of Néron’s construction of rank
jump $3$, the fact that the curves $L_{i}$ are rigid imply that the set is
thin. On the other hand, in Salgado’s construction [Sal12], the pencil of
curves implies a larger set of rank jumps which turns out to be not thin, as
proven later in collaboration with Loughran [LS22]. By combining the technique
from [Sal12] with the later developments from [LS22], we arrive at Theorem
5.0.1, which we prove in this chapter.

We organize the chapter as follows. In Section 5.1 we explore some properties
of rational elliptic surfaces which admit a nonreduced fiber. In Section 5.2
we introduce a key tool, which are the RNRF conic bundles and present their
constructions. Section 5.3 is a commentary on the need to consider nonreduced
fibers, which introduces the proof of the main theorem in Section 5.4. At
last, a series of examples is provided in Section 5.5.

###  5.1 Nonreduced fibers

The central hypothesis of Theorem 5.0.1 is the presence of an RNRF conic
bundle, i.e. a conic bundle ramified at a nonreduced fiber (Definition 5.2.2).
By nonreduced we mean a fiber with one or more components with multiplicity
$\geq 2$, which by Kodaira’s classification (Theorem 2.1.8) correspond to
starred types, i.e. $\text{I}^{*}_{n\geq
0},\text{II}^{*},\text{III}^{*},\text{IV}^{*}$. In this section we present
some results for the case when $\pi:X\to\mathbb{P}^{1}$ admits a nonreduced
fiber (Proposition 5.1.1, Corollaries 5.1.2 and 5.1.3) and prove Lemma 5.1.5,
which is helpful in the construction of conic bundles in Section 5.2.

We begin by considering a local Weierstrass form for our rational elliptic
surface, namely

$y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6},\,\text{ where }a_{i}\in
k(\mathbb{P}^{1})\text{ for all }i.$

Dokchitsers’ Table 2.1 allows us to readily see that rational elliptic
fibrations with a nonreduced fiber at $t=\infty$, except possibly
I${}_{0}^{*}$, admit a conic bundle over the $x$-line. More precisely, we have
the following result.

######  Proposition 5.1.1.

Let $\pi:X\rightarrow\mathbb{P}^{1}$ be a rational elliptic surface defined
over $k$ with a nonreduced fiber $F$ at $t=\infty$ which is not of type
$I_{0}^{*}$. Then $X$ admits a Weierstrass equation of the form

$y^{2}+a_{1}(t)xy+a_{3}(t)y=x^{3}+a_{2}(t)x^{2}+a_{4}(t)x+a_{6}(t),\,\text{
with }\deg a_{i}\leq 2\text{ for all }i.$

In particular, $X$ admits a conic bundle by projection over the $x$-line.

###### Proof.

To analyze the behavior of the fiber at infinity, i.e.
$t=(1:0)\in\mathbb{P}^{1}$, we write the (long) Weierstrass equation of $X$
with homogeneous coefficients:

$y^{2}+a_{1}(t,u)xy+a_{3}(t,u)y=x^{3}+a_{2}(t,u)x^{2}+a_{4}(t,u)x+a_{6}(t,u).$

Since $X$ is rational, $\deg a_{i}(t,u)=i$. Dokchitsers’ Table 2.1 tells us
that $v_{\infty}(\frac{a_{i}}{i})\geq\frac{2}{3}$, if $F$ is of type IV∗, III∗
or II∗. Hence $\deg_{u}a_{i}\geq\frac{2i}{3}$ or $a_{i}=0$, which immediately
bounds the degrees of $a_{1},a_{2},a_{3},a_{4}$ and $a_{6}$ in the variable
$t$ from above by 2, as claimed.

It remains to check that this also holds for fibers of type $\text{I}_{n}^{*}$
with $n\neq 0$. In this case, Dokschitsers’ table tells us that
$v_{\infty}(\frac{a_{i}}{i})\geq\frac{1}{2}$ and $v_{\infty}(d)\geq 6$. The
former implies that $\deg_{t}a_{i}(u,t)\leq 2$ or $a_{i}=0$, for $i=1,2,3$ and
$4$, while the latter gives us $\deg_{t}a_{6}(u,t)\leq 2$, or $a_{6}=0$. ∎

######  Corollary 5.1.2.

Let $\pi:X\to\mathbb{P}^{1}$ be a geometrically rational elliptic surface over
$k$. Suppose that $\pi$ admits a unique nonreduced fiber. Then $X$ is
$k$-unirational.

###### Proof.

We apply [KM17, Thm. 7] to the surface obtained by contracting the zero
section, which is a conic bundle of degree $1$. ∎

######  Corollary 5.1.3.

Let $\pi:X\rightarrow\mathbb{P}^{1}$ be a rational elliptic fibration defined
over $k$ with a nonreduced fiber that is not of type I${}_{0}^{*}$. Then the
subset

$\\{t\in\mathbb{P}^{1}_{k}\mid r_{t}\geq r+2\\}\subset\mathbb{P}^{1}_{k}$

is not thin.

###### Proof.

Proposition 5.1.1 puts us in a position to apply [LS22, Theorem 1.1]. ∎

######  Remark 5.1.4.

  1. i)

Rational elliptic surfaces with 2 fibers of type $\text{I}_{0}^{*}$ have been
treated separately in [LS22].

  2. ii)

Unfortunately one might not be able to use the conic bundles from Proposition
5.1.1 to show that there is a rank jump of 3, which is our main goal. Indeed,
the hypothesis that the restriction of $\pi$ to the conics has a common
ramification is crucial. By [LS22, Lemma 2.10], this can only happen over
nonreduced fibers. Example 5.5.0.2 presents a surface with a fiber of type
$\text{IV}^{*}$ whose conic bundle over the $x$-line is not ramified over the
nonreduced fiber.

The following lemma is relevant for the construction of conic bundles in
Section 5.2. It provides a supply of divisors that can be used to form the
support of a genus $0$ bisection over $k$.

######  Lemma 5.1.5.

Let $\pi:X\rightarrow\mathbb{P}^{1}$ be a rational elliptic surface defined
over $k$ with a nonreduced fiber $F$ with components $\Theta_{i}$’s as in
Table 2.2. Then one of the following holds:

  1. i)

If $F=\text{II}^{*}$ or $\text{III}^{*}$, then all its components are defined
over $k$.

  2. ii)

If $F=\text{IV}^{*}$, then $\Theta_{0},\Theta_{1},\Theta_{2}$ are defined over
$k$. The other components are defined over some extension of $k$ of degree at
most $2$.

  3. iii)

If $F=\text{I}_{n}^{*}$ ($n\geq 1$), then $\Theta_{0},\Theta_{1}$ and all the
nonreduced components are defined over $k$. The far components
$\Theta_{2},\Theta_{3}$ are defined over some extension of $k$ of degree at
most $2$.

  4. iv)

If $F=\text{I}_{0}^{*}$ and $F$ is the only reducible fiber of $\pi$, then
$\Theta_{0}$ and $\Theta_{4}$ are defined over $k$. The other components are
defined over an extension of $k$ of degree at most $3$.

###### Proof.

The fibration $\pi$ is defined over $k$, and, by assumption, so is its zero
section $O$. The Galois group $G:=\text{Gal}(\overline{k}/k)$ acts on the
Néron-Severi group preserving intersection multiplicities. Since $F$ is
assumed to be the unique fiber of its type, the elements of $G$ permute the
components of $F$.

In particular, the presence of conjugate components implies symmetries in the
intersection graph of $F$. For the sake of clarity and to avoid repetition in
what follows, we state as a fact one of the immediate consequences of the
intersection-preserving action of $G$ in the Néron-Severi group. We refer to
it as _I.F._ in what follows.

Intersection Fact (I.F.): Let $C,D,E\subset X$ be integral curves such that
$C$ meets $D$ but not $E$. If $C$ is stable under $G$, then $D,E$ cannot be
conjugate.

We also note that since $O$ is defined over $k$ (hence stable under $G$) and
meets $\Theta_{0}$, then $\Theta_{0}$ is necessarily stable under $G$. We now
analyze i), ii), iii), iv) with the notation from Table 5.1.

II${}^{*}\,(\widetilde{E}_{8})$ |  $\bullet$$2\Theta_{7}$$\bullet$$4\Theta_{6}$$\bullet$$6\Theta_{5}$$\bullet$$3\Theta_{8}$$\bullet$$5\Theta_{4}$$\bullet$$4\Theta_{3}$$\bullet$$3\Theta_{2}$$\bullet$$2\Theta_{1}$$\bullet$$\Theta_{0}$
---|---
III${}^{*}\,(\widetilde{E}_{7})$ |  $\bullet$$\Theta_{0}$$\bullet$$2\Theta_{1}$$\bullet$$3\Theta_{2}$$\bullet$$4\Theta_{3}$$\bullet$$2\Theta_{7}$$\bullet$$3\Theta_{4}$$\bullet$$2\Theta_{5}$$\bullet$$\Theta_{6}$
IV${}^{*}\,(\widetilde{E}_{6})$ |  $\bullet$$\Theta_{0}$$\bullet$$2\Theta_{1}$$\bullet$$3\Theta_{2}$$\bullet$$2\Theta_{5}$$\bullet$$\Theta_{6}$$\bullet$$2\Theta_{3}$$\bullet$$\Theta_{4}$
I${}_{n}^{*}\,(\widetilde{D}_{n+4})$ |  $\bullet$$\Theta_{0}$$\bullet$$\Theta_{1}$$\bullet$$2\Theta_{4}$$\bullet$$2\Theta_{5}$$\bullet$$2\Theta_{n+3}$$\bullet$$2\Theta_{n+4}$$\bullet$$\Theta_{2}$$\bullet$$\Theta_{3}$
Table 5.1: Nonreduced fibers of $\pi$

i) Let $F=\text{II}^{*}$. For lack of symmetry, each component of $F$ is
stable under $G$, as desired. Now let $F=\text{III}^{*}$. By symmetry
$\Theta_{3},\Theta_{7}$ are stable and the possible conjugate pairs are
$(\Theta_{0},\Theta_{6})$, $(\Theta_{1},\Theta_{5})$,
$(\Theta_{2},\Theta_{4})$. Since $\Theta_{0}$ is stable, then so is
$\Theta_{6}$. We use I.F. with $(C,D,E)=(\Theta_{0},\Theta_{1},\Theta_{5})$ so
that $\Theta_{1},\Theta_{5}$ are stable. Then for
$(C,D,E)=(\Theta_{1},\Theta_{2},\Theta_{4})$ we conclude
$\Theta_{2},\Theta_{4}$ are stable.

ii) Let $F=\text{IV}^{*}$. By symmetry, $\Theta_{2}$ is stable and the
possible $G$-orbits are $(\Theta_{0},\Theta_{4},\Theta_{6})$ and
$(\Theta_{1},\Theta_{3},\Theta_{5})$. Since $\Theta_{0}$ is stable, then
$\Theta_{4},\Theta_{6}$ are possibly conjugates, in which case both are
defined over some $L/k$ with $[L:k]\leq 2$. Applying I.F. with
$(C,D,E)=(\Theta_{0},\Theta_{1},\Theta_{3})$ and then with
$(C,D,E)=(\Theta_{0},\Theta_{1},\Theta_{5})$ we conclude that $\Theta_{1}$ is
stable and $(\Theta_{3},\Theta_{5})$ is possibly an orbit. In this case,
$\Theta_{3},\Theta_{5}$ are defined over some $L/k$ with $[L:k]\leq 2$.

iii) Let $F=\text{I}_{n\geq 1}^{*}$. By symmetry, the following pairs is a
possible $G$-orbits: $(\Theta_{i},\Theta_{n+j})$ where both
$\Theta_{i},\Theta_{n+j}$ are components with multiplicity $2$ in $F$ and
$i+j=8$. Since $\Theta_{0}$ is stable, we apply I.F. with
$(C,D,E)=(\Theta_{0},\Theta_{4},\Theta_{n+4})$ to conclude that
$\Theta_{4},\Theta_{n+4}$ are stable. We now use induction on $i$: assuming
$\Theta_{i}$ is stable and applying I.F. to
$(C,D,E)=(\Theta_{i},\Theta_{i+1},\Theta_{n+j+1})$ with $j+i=8$, we conclude
that $\Theta_{i+1},\Theta_{n+j+1}$ are stable, hence all components with
multiplicity $2$ are stable. We are left with $\Theta_{2},\Theta_{3}$. By
symmetry, $(\Theta_{0},\Theta_{1},\Theta_{2},\Theta_{3})$ is a possible orbit.
But $\Theta_{0},\Theta_{4}$ are stable, so $\Theta_{2},\Theta_{3}$ are
possibly conjugates, hence both are defined over some $L/k$ with $[L:k]\leq
2$.

iv) Let $F=\text{I}_{0}^{*}$. By symmetry, $\Theta_{4}$ is stable and
$(\Theta_{0},\Theta_{1},\Theta_{2},\Theta_{3})$ is a possible $G$-orbit. But
$\Theta_{0}$ is stable, so $\Theta_{1},\Theta_{2},\Theta_{3}$ are possibly
conjugates, all three defined over some $L/k$ with $[L:k]\leq 3$. ∎

###  5.2 Conic bundles ramified at a nonreduced fiber (RNRF)

We introduce conic bundles with the property that the restriction of the
elliptic fibration to all conics share a ramification point, which are the
main tool in the proof of Theorem 5.0.1.

We begin by defining a multisection ramified at a fiber of
$\pi:X\to\mathbb{P}^{1}$, which leads to the central object of this chapter,
namely the RNRF-conic bundle. We remind the reader that the presence of a
conic bundle over $k$ is equivalent to the existence of a bisection of genus
$0$ over $k$ by Lemma 2.3.5.

######  Definition 5.2.1.

Let $C\subset X$ be a multisection $\pi:X\to\mathbb{P}^{1}$. We say that $C$
is ramified at the fiber $\pi^{-1}(t)$ if the finite morphism
$\pi|_{C}:C\to\mathbb{P}^{1}$ branches at $t$.

######  Definition 5.2.2.

Let $\varphi_{|D|}:X\to\mathbb{P}^{1}$ be a conic bundle induced by a
bisection $D\subset X$. If $\pi$ has a unique nonreduced fiber $F$, we say
that this is an RNRF-conic bundle if $D$ is ramified at $F$. Equivalently, if
each $C\in|D|$ intersects (transversally) a nonreduced component of $F$. The
acronym RNRF stands for Ramified at a Nonreduced Fiber.

RNRF-conic bundles arise naturally on rational elliptic surfaces with a
nonreduced fiber. The following result tells us that they exist over $k$
without any further assumption depending on the type of nonreduced fiber, and
gives conditions for it to happen in the remaining configurations.

######  Proposition 5.2.3.

Let $\pi:X\to\mathbb{P}^{1}$ be a rational elliptic surface over $k$. Assume
that one of the following holds:

  1. 1.

It admits a fiber of type II∗, III∗ or I${}_{n}^{*}$ for $n\in\\{2,3,4\\}$.

  2. 2.

It admits a fiber of type IV∗ or I${}_{m}^{*}$, for $m\in\\{0,1\\}$ and a
reducible, reduced fiber.

  3. 3.

It admits a fiber of type IV∗ and a non-trivial section defined over $k$.

  4. 4.

It admits a fiber of type I${}_{1}^{*}$ and a non-trivial section defined over
$k$ that intersects the near component.

  5. 5.

It admits a fiber of type I${}_{1}^{*}$ and two non-intersecting sections that
are conjugate under $\text{Gal}(\overline{k}/k)$.

  6. 6.

It admits 2 fibers of type I${}_{0}^{*}$ and a non-trivial 2-torsion section
defined over $k$.

Then $X$ admits a RNRF-conic bundle over $k$.

###### Proof.

For each possible configuration listed in the hypothesis of the theorem, we
provide an effective divisor $D$ on $X$ (see Table 5.2) such that $D^{2}=0$,
$D\cdot F=2$ and such that every $D^{\prime}\in|D|$ intersects a nonreduced
component of $F$. Moreover, by Lemma 5.1.5, the divisor $D$ is defined over
$k$, i.e. it remains invariant by the action of $\text{Gal}(\overline{k}/k)$.

To prove that $\pi|_{D}:D\to\mathbb{P}^{1}$ is ramified above $F$, we only
need to observe that each $D$ constructed meets $F$ at a nonreduced component.
∎

MW rank | nonreduced fiber | class of conics | extra information
---|---|---|---
0 | II∗ |  $O$$2$$\bullet$$\Theta_{0}$$2$$\bullet$$2$$\Theta_{1}$$\bullet$$2$$\Theta_{2}$$\bullet$$2$$\Theta_{3}$$\bullet$$2$$\Theta_{4}$$\bullet$$2$$\Theta_{5}$$\bullet$$\Theta_{8}$$\bullet$$\Theta_{6}$ |
0,1 | III∗ |  $O$$2$$\bullet$$\Theta_{0}$$2$$\bullet$$2$$\Theta_{1}$$\bullet$$2$$\Theta_{2}$$\bullet$$2$$\Theta_{3}$$\bullet$$\Theta_{7}$$\bullet$$\Theta_{4}$ |
0 | IV∗ |  $P_{1}$$\bullet$$\Theta_{0}$$\bullet$$\Theta_{1}$$\bullet$$\Theta_{2}$$\bullet$$\Theta_{3}$$\bullet$$\Theta_{4}$$P_{2}$ |
0 | I${}_{4}^{*}$ |  $O$$2$$\bullet$$\Theta_{0}$$2$$\bullet$$2$$\Theta_{1}$$\bullet$$\Theta_{2}$$\bullet$$\Theta_{3}$ |
0,1 | I${}_{3}^{*}$ |  $O$$2$$\bullet$$\Theta_{0}$$2$$\bullet$$2$$\Theta_{1}$$\bullet$$\Theta_{3}$$\bullet$$\Theta_{2}$ |
0,1,2 | I${}_{2}^{*}$ |  $2$$\bullet$$2$$\bullet$$2$$\bullet$$\bullet$ |
0 | I${}_{1}^{*}$ |  $\bullet$$\bullet$$\bullet$ |
1,2 | I${}_{1}^{*}$ |  $\bullet$$\Theta_{0}^{1}$$2$$O$$\bullet$$\Theta_{0}^{2}$ |
3 | I${}_{1}^{*}$ |  $P_{1}$$\bullet$$\bullet$$\bullet$$P_{2}$ |  $P_{1},P_{2}$ conjugate sections intersecting near components
1 | IV∗ |  $\bullet$$\Theta_{0}^{1}$$2$$O$$\bullet$$\Theta_{0}^{2}$ |
2 | IV∗ |  $P_{1}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$P_{2}$ |  $P_{1},P_{2}$ non-intersecting conjugate sections
Table 5.2: Class of conics for the proof of Proposition 5.2.3

######  Remark 5.2.4.

  1. i)

Conics in a RNRF-conic bundle are distinguished in the following sense.
Generally, a degree 2 base change of a rational elliptic surface produces a K3
surface. This is true if the base change is ramified in smooth fibers, and
remais true even if it ramifies at reduced singular fibers if one considers
the desingularization of the base changed surface. On the other hand, if a
degree 2 base change ramifies at a nonreduced fiber, then by an Euler number
calculation one can show that the base changed surface is still rational (see
Lemma 5.2.5).

  2. ii)

It is natural to wonder about the extra conditions on surfaces with fibers of
type $\text{IV}^{*}$, $\text{I}_{1}^{*}$, $\text{I}_{0}^{*}$. One can show
that surfaces with such fiber configuration always admit a conic bundle over
$k$. Nevertheless, such a conic bundle is not necessarily a RNRF-conic bundle.
It is worth noticing that, on the other hand, the isotrivial rational surfaces
with $2$I${}_{0}^{*}$ admit a RNRF-conic bundle. In this particular setting it
is called a Châtelet bundle (see [LS22, Lemma 3.3]) since they occur as conic
bundles on a Châtelet surface obtained after blowing down the sections of
$\pi$. Still, they are useless for our purposes as all fibers ramify above the
same $2$ nonreduced fibers, not leaving the degree of freedom needed to avoid
certain covers when verifying that the rank jump occurs on a not thin subset.

The following result justifies the study of RNRF-conic bundles on rational
elliptic surfaces. It tells us that the base change of a rational elliptic
fibration by a bisection in an RNRF-conic bundle is again a rational elliptic
fibration. This allows us to apply [LS22, Thm. 1.1] to the base-changed
rational elliptic surface and achieve a higher rank jump.

######  Lemma 5.2.5.

Let $\pi:X\to\mathbb{P}^{1}$ be a rational elliptic surface with only one
nonreduced fiber $F$. Let $D\subset X$ be a genus $0$ bisection ramified at
$F$, and let $X_{D}$ be the normalization of the base change surface
$X\times_{\mathbb{P}^{1}}D$. Then $X_{D}$ is a rational elliptic surface and
$|D|$ induces a conic bundle on $X_{D}$.

###### Proof.

Let $\varphi:=\pi|_{D}:D\to\mathbb{P}^{1}$ be the base change map. The curve
$D$ is rational, so by Hurwitz formula $\varphi$ has two branch points. By
hypothesis, these two points correspond to $F$ and some other reduced fiber.
By inspection of possible singular fibers in a rational elliptic surface
[Per90], $F$ is one of the following:

$\text{I}_{n}^{*}\,\text{with }0\leq n\leq
4,\,\text{II}^{*},\,\text{III}^{*}\text{ or IV}^{*}.$

By the ramification of $\varphi$, there is precisely one fiber $F^{\prime}$ in
$X_{D}$ above $F$. An inspection of Table 2.3, confirms a pattern for the
Euler number, namely, $e(F^{\prime})=2\,e(F)-12$ ($*$).

Moreover, if $G$ is any singular fiber of $\pi$ distinct from $F$, we claim
that $\sum_{G^{\prime}}e(G^{\prime})=2e(G)$ ($**$), where $G^{\prime}$ runs
through the fibers of $X_{D}$ above $G$ (since $\varphi$ has degree $2$, there
are at most $2$ possibilities for $G^{\prime}$). Indeed, if there is no
ramification associated to $G$, then there are two fibers
$G_{1}^{\prime},G_{2}^{\prime}$ isomorphic to $G$, hence
$e(G_{1}^{\prime})+e(G_{2}^{\prime})=2\,e(G)$. In case there is ramification,
there is only one possibility for $G^{\prime}$ and $e(G^{\prime})=2\,e(G)$ for
all cases by Table 2.3. This proves $(**)$.

We prove that $X_{D}$ is rational by checking that $e(X_{D})=12$ and applying
Lemma 2.2.4. Since $X$ is rational, then $e(X)=12$ and combining ($*$), ($**$)
we get

$\displaystyle e(X_{D})$ $\displaystyle=e(F^{\prime})+\sum_{G^{\prime}\neq
F^{\prime}}e(G^{\prime})$ $\displaystyle=e(F^{\prime})+\sum_{G\neq
F}\sum_{G^{\prime}}e(G^{\prime})$ $\displaystyle=2\,e(F)-12+2\sum_{G\neq
F}2\,e(G)$ $\displaystyle=2\,e(X)-12$ $\displaystyle=12,$

as desired. In order to prove that $|D|$ induces a conic bundle on $X_{D}$,
take an arbitrary $C\in|D|$ and let $\psi:=\pi|_{C}:C\to\mathbb{P}^{1}$. As in
$\varphi$, the map $\psi$ has $2$ branch points. Let $\nu:X_{D}\to
X\times_{\mathbb{P}^{1}}D$ be the normalization map and $E\subset X_{D}$ the
strict transform of $C\times_{\mathbb{P}^{1}}D$ under $\nu$. We prove that $E$
is smooth of genus $0$, so that $|E|$ induces a conic bundle on $X_{D}$. We
have the following diagram:

${E}$${C\times_{\mathbb{P}^{1}}D}$${C}$${D}$${\mathbb{P}^{1}}$$\scriptstyle{\nu}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{\psi^{\prime}}$$\scriptstyle{\psi}$$\scriptstyle{\varphi}$

The singularities of $C\times_{\mathbb{P}^{1}}D$ are the points $(c,d)\in
C\times D$ such that $\psi,\varphi$ ramify at $c,d$ respectively. These are
also singular points of $X\times_{\mathbb{P}^{1}}D$, which are eliminated by
$\nu$, hence $E$ is smooth. Since $\varphi$ has degree $2$, then $E\to C$ has
degree $2$ and, moreover, has $2$ branch points, namely the ones where $\psi$
ramifies. By Hurwitz formula, $g(E)=0$ as desired. ∎

###  5.3 RNRF and multiple base changes

In [Sal12] and [LS22], the authors use $2$ base changes by curves in a conic
bundle to show that the rank jumps by at least $2$. The final base for the
base changed fibration is an elliptic curve with positive Mordell-Weil rank.
One is tempted to consider a third base change to increase the rank jump
further. Unfortunately, this cannot be done in such generality. Indeed, the
genus of a new base after a third base change would be at least $2$. In
particular, the base curve would have a finite set of $k$-points thanks to
Faltings’ theorem [Fal83].

Rational elliptic surfaces with a nonreduced fiber on the other hand allow for
a sequence of 3 base changes with final base curve of genus 1. Further
hypotheses on the surface allow us to show that the genus 1 curve is an
elliptic curve with positive Mordell–Weil rank. Indeed, Lemma 5.2.5 assures
that the first base change by an RNRF-conic bundle produces a surface that is
again rational and moreover admits a conic bundle. In other words, admits a
bisection of arithmetic genus $0$ as in the hypothesis of [LS22, Theorem 2].
One can then take the latter as the starting point and apply [LS22, Theorem 2]
to conclude. This is explained in detail in the following section.

###  5.4 Rank Jump three times

In this section we make use of an RNRF-conic bundle on a rational elliptic
surface with a unique nonreduced fiber to show that the collection of fibers
for which the Mordell–Weil rank is at least the generic rank plus 3 is not
thin as a subset of the base of the fibration.

Throughout this section, $\pi:X\rightarrow\mathbb{P}^{1}$ is a geometrically
rational elliptic surface with a unique nonreduced fiber $F$ and $D$ is a
bisection of $\pi$ such that $|D|$ is a RNRF-conic bundle.

We let $\psi_{i}:Y_{i}\rightarrow\mathbb{P}^{1}$, for $i\in I$, be an
arbitrary finite collection of finite morphisms of degree $\geq 2$, as in the
discussion at the end of Section 2.4. Our goal is to show that there is a
curve $X\supset C{\overset{\varphi}{\rightarrow}}\mathbb{P}^{1}$ such that:

  1. 1.

$C(k)$ is infinite;

  2. 2.

$\textrm{rank}(X\times_{\mathbb{P}^{1}}C)(k(C))\geq r+3$;

  3. 3.

$\exists P\in\varphi(C(k))\subset\mathbb{P}^{1}(k)$ such that
$P\notin\bigcup_{i}\psi(Y_{i}(k))$.

######  Lemma 5.4.1.

Assume $\pi:X\to\mathbb{P}^{1}$ has a unique non-reduced fiber. Then for all
$D_{1}\in|D|$, there are infinitely many pairs $(D_{2},D_{3})\in|D|\times|D|$
such that $D_{1}\times_{\mathbb{P}^{1}}D_{2}\times_{\mathbb{P}^{1}}D_{3}$ is a
curve of genus $1$.

###### Proof.

For a given a $D_{1}$, there are only finitely many $D_{2},D_{3}\in|D|$ such
that $D_{2}$ or $D_{3}$ share 2 common ramification points with $D_{1}$. We
exclude these. The 3 quadratic extensions $k(D_{i})$ are non-isomorphic as
they have precisely one ramification point in common. Hence they are linearly
disjoint and the curve
$C=D_{1}\times_{\mathbb{P}^{1}}D_{2}\times_{\mathbb{P}^{1}}D_{3}$ is
geometrically integral. Our hypothesis on the common ramification implies
moreover that the curve $D_{1}\times_{\mathbb{P}^{1}}D_{2}$ has genus $0$ (see
[LS22, Lemma 5.2]). Let $t,t_{i}\in\mathbb{P}^{1}(k)$ be the two branch points
of $D_{i}\to\mathbb{P}^{1}$ where $t$ is the common branch point corresponding
to the nonreduced fiber, and $t_{i}\neq t_{j}$, for $i\neq j$. Consider the
degree $2$ morphism
$\phi:D_{1}\times_{\mathbb{P}^{1}}D_{2}\times_{\mathbb{P}^{1}}D_{3}\to
D_{1}\times_{\mathbb{P}^{1}}D_{2}$. Then $\phi$ is ramified at the four points
above $t_{3}\in\mathbb{P}^{1}(k)$. A direct application of the Hurwitz formula
gives that $C$ is a curve of genus $1$. ∎

######  Proposition 5.4.2.

Assume $\pi:X\to\mathbb{P}^{1}$ has a unique nonreduced fiber and let
$\mathcal{P}=\\{P_{1},\cdots,P_{m}\\}$ be a finite subset of
$\mathbb{P}^{1}_{k}$. Then there are infinitely many $D_{1},D_{2},D_{3}\in|D|$
such that:

  * a)

$C:=D_{1}\times_{\mathbb{P}^{1}}D_{2}\times_{\mathbb{P}^{1}}D_{3}$ is an
elliptic curve with positive Mordell-Weil rank over $k$;

  * b)

$k(D_{1})\otimes k(D_{2})\otimes k(D_{3})$ is linearly disjoint with every
$k(Y_{i})$;

  * c)

The rank of the generic fiber of $X_{C}\rightarrow C$ is at least $r+3$;

  * d)

$C$ ramifies above exactly one of $P_{j}$.

###### Proof.

This proof is analogous to the proof of [LS22, Prop. 4.1]. We may assume that
$\mathcal{P}$ contains all branch points of $Y_{i}$ and the singular locus of
$\pi$. In particular, it contains a ramification point of all conics in $|D|$.
We call this point $P_{1}$. After choosing $D_{1}$ and $D_{2}$ in an infinite
set such that the rank of the generic fiber of the elliptic fibration
$\pi_{12}:X_{D_{1}\times_{B}D_{2}}\rightarrow
D_{1}\times_{\mathbb{P}^{1}}D_{2}$ is at least $r+2$ and
$D_{1}\times_{\mathbb{P}^{1}}D_{2}(k)$ is infinite and
$X_{D_{1}\times_{\mathbb{P}^{1}}D_{2}}(k)$ is Zariski dense as in [LS22],
allowing $D_{3}$ to vary in $|D|$ gives by Lemma 5.4.1 an infinite family of
elliptic curves with positive Mordell-Weil rank that are bisections of
$\pi_{12}$. By Theorem 5.0.2, all but finitely many of such curves can be used
to base change and obtain an elliptic surface $X_{C}\rightarrow C$ with
$C=D_{1}\times_{\mathbb{P}^{1}}D_{2}\times_{\mathbb{P}^{1}}D_{3}$ and generic
fiber of rank at least $r+3$.

The fibration $\pi$ has a unique nonreduced fiber so after excluding finitely
many curves when picking $D_{1},D_{2}$ and $D_{3}$ we may assume that
$D_{i}$’s are not ramified over other singular fibers of $\pi$, nor do they
share ramification points with $Y_{i}$ other than possibly $P_{1}$. This
proves d). ∎

The following result is parallel to [LS22, Lemma 5.5]. Since we construct
conic bundles whose members are always ramified at a nonreduced singular
fiber, we need to reprove the result. Fortunately, that comes with no cost as
elliptic fibrations defined over global fields have at least two singular
fibers.

######  Lemma 5.4.3.

The elliptic surface $X\times_{\mathbb{P}^{1}}C\rightarrow C$ is nonconstant,
i.e. not isomorphic to $E\times C\to C$, where $E$ is an elliptic curve.

###### Proof.

The surface $X\to\mathbb{P}^{1}$ is a relatively minimal geometrically
rational elliptic surface defined over a global field. In particular, it has
at least $2$ singular fibers. Hence there is at least $1$ reduced singular
fiber $F$. Since we chose $C$ such that $C\to\mathbb{P}^{1}$ is not ramified
at $F$, the pull-back of $F$ to $X\times_{\mathbb{P}^{1}}C\rightarrow C$ is a
singular fiber. ∎

We have at hand all the tools needed to prove our main result.

###### Proof of Theorem 5.0.1.

We choose $C$ as in Proposition 5.4.2. By construction, the curves
$D_{1},D_{2},D_{3}$, and $Y_{i}$ are smooth and the respective maps to
$\mathbb{P}^{1}$ share at most one branch point. On the other hand, the map
$C\times_{\mathbb{P}^{1}}Y_{i}\to C$ is branched on the ramification points of
$Y_{i}\to\mathbb{P}^{1}$. A direct application of the Riemann-Hurwitz formula
gives $g(C\times_{\mathbb{P}^{1}}Y_{i})\geq 2$. In particular, by Faltings’
theorem [Fal83], $C\times_{\mathbb{P}^{1}}Y_{i}(k)$ is finite. To conclude, we
invoke part c) of Proposition 5.4.2 and apply Theorem 5.0.2 to the nonconstant
elliptic surface $X\times_{\mathbb{P}^{1}}C\to C$. ∎

###  5.5 Examples

######  Example 5.5.0.1.

Let $X$ be an elliptic surface with Weierstrass equation

$y^{2}=x^{3}+a(t)x+b(t),$

with $\deg a(t),b(t)\leq 1$ and $a(t)$ and $b(t)$ not simultaneously constant.
Then $X$ admits a nonreduced fiber at infinity. More precisely: if $\deg
a(t)=1$ then $X$ admits a fiber of type $\text{III}^{*}$; and if $\deg a(t)=0$
then $X$ admits a fiber of type $\text{II}^{*}$.

The surface $X$ admits a RNRF-conic bundle over the $x$-line. In case $ii)$
this is the unique conic bundle on the surface. A nice geometric description
for this case is as follows. Let $C$ be a plane cubic with an inflection point
$P$ defined over $k$. Let $L$ be the line tangent to $C$ at $P$. We consider
the following pencil of plane cubics

$uC+t(3L)=0,\,\,(t:u)\in\mathbb{P}^{1},$

whose base locus is the nonreduced scheme $\\{9P\\}$. We consider its 9-fold
blow up and obtain a rational elliptic surface with a fiber of type II∗ at
infinity. The unique conic bundle on it is given by the strict transforms of
the pencil of lines through $P$. By following the blow ups one sees readily
that all conics intersect the unique double component of the fiber of type II∗
(Figure 5.1).

Figure 5.1: RNRF-conic bundle (pink) on $X$ with a $II^{*}$ fiber.

######  Example 5.5.0.2.

Let $F=y^{2}z-x^{3}+x^{2}z+xz^{2}-z^{3}$ and $G=z^{2}(y-z)$ be two plane
cubics. Let $P_{1}=(0:1:0)$ and $P_{i}=(x_{i}:1:1)$ with $x_{i}$, for
$i=2,3,4$, the 3 roots of the polynomial $x^{3}-x^{2}-x$. Hence $F,G$ meet at
the nonreduced scheme $F\cap G=\\{6P_{1},P_{2},P_{3},P_{4}\\}$. Let $X$ be the
blow up of $\mathbb{P}^{2}$ in $F\cap G$. Then its (affine) Weierstrass
equation is

$y^{2}-ty=x^{3}-x^{2}-x+(t-1).$

In particular, $X$ has a fiber of type IV∗ at $t=\infty$ and, as expected by
Proposition 5.1.1, it admits a conic bundle over the $x$-line. Geometrically,
the fiber of type IV∗ is given by $G^{\prime}-E_{1}-E_{2}-E_{3}-E_{4}$, where
$G^{\prime}$ is the proper transform of $G$ and $E_{i}$ is the exceptional
divisor above $P_{i}$. If $D$ is a fiber of the conic bundle over the $x$-line
then $D=\ell_{1}-E_{1}$, where $\ell_{1}$ is the proper transform of a line
through $P_{1}$. In particular, for all lines through $P_{1}$, except the $3$
lines that pass through $P_{2},P_{3}$ or $P_{4}$, the curve $D$ intersects the
fiber IV∗ transversally in the simple component given by $m-E_{2}-E_{3}-E_{4}$
where $m$ is the proper transform of the line $y=z$, and in a simple component
above the blow up of $P_{1}$. Hence the restriction of the elliptic fibration
to all but $3$ conics is not ramified at IV∗ and, in particular, cannot all
share a common ramification. In other words, the conic bundle over the
$x$-line is not a RNRF-conic bundle.

Nevertheless, $X$ admits a RNRF-conic bundle, namely the one given by
$|\ell_{2}-E_{2}|$, where $\ell_{2}$ is the proper transform of a line through
$P_{2}$. Indeed, $\ell_{2}$ intersects the double component of IV∗ above the
strict transform of $z^{2}=0$. Since $P_{2}=(0:1:1)$, the conic bundle is
defined over $\mathbb{Q}$. Thus there are infinitely many $t\in\mathbb{Q}$
such that $r_{t}\geq 1+3=4$ (Figure 5.2).

Figure 5.2: RNRF-conic bundle (pink) and a non-RNRF conic bundle (orange) on
$X$ with a IV∗ fiber.

######  Example 5.5.0.3.

Let $X$ be the rational elliptic surface studied by Washington in [Was87] with
Weierstrass equation

$y^{2}=x^{3}+tx^{2}-(t+3)x+1.$

The generic Mordell-Weil rank of this surface over $\mathbb{Q}$ is 1. Its
singular fibers are of type $(\text{I}_{2}^{*},2\text{II})$. In particular, by
Proposition 5.2.3, it admits a RNRF-conic bundle defined over $\mathbb{Q}$. We
can apply Theorem 5.0.1 to conclude that the subset of fibers of rank at least
4 is not thin.

For this surface, we can expect an even higher rank jump on a non-thin set.
Indeed, Rizzo proved in [Riz03, Thm. 1] that the root number of each fiber is
$-1$. Hence, under the Parity conjecture [Dok13], the rank of all fibers is
odd. This together with Theorem 5.0.1 would imply that the set of fibers with
rank at least 5 is not thin. In other words, under the Parity conjecture,
there is a rank jump of at least 4 for a non-thin set of fibers.

##  Chapter 6 Appendix

We reproduce part of the table in [OS91, Main Th.] with data on Mordell-Weil
lattices of rational elliptic surfaces with Mordell-Weil rank $r\geq 1$. We
add columns $c_{\text{max}},c_{\text{min}},\Delta$.

Table 6.1: Mordell-Weil lattices of rational elliptic surfaces with Mordell-Weil rank $r\geq 1$. No. | $r$ | $T$ | $E(K)^{0}$ | $E(K)$ | $c_{\text{max}}$ | $c_{\text{min}}$ | $\Delta$
---|---|---|---|---|---|---|---
1 | $8$ | $0$ | $E_{8}$ | $E_{8}$ | 0 | 0 | 0
2 | $7$ | $A_{1}$ | $E_{7}$ | $E_{8}^{*}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$
3 | $6$ | $A_{2}$ | $E_{6}$ | $E_{6}^{*}$ | $\frac{2}{3}$ | $\frac{2}{3}$ | $0$
4 |  | $A_{1}^{\oplus 2}$ | $D_{6}$ | $D_{6}^{*}$ | $\frac{3}{2}$ | $1$ | $\frac{1}{2}$
5 | $5$ | $A_{3}$ | $D_{5}$ | $D_{5}^{*}$ | $1$ | $\frac{3}{4}$ | $\frac{1}{4}$
6 |  | $A_{2}\oplus A_{1}$ | $A_{5}$ | $A_{5}^{*}$ | $\frac{7}{6}$ | $\frac{1}{2}$ | $\frac{2}{3}$
7 |  | $A_{1}^{\oplus 3}$ | $D_{4}\oplus A_{1}$ | $D_{4}^{*}\oplus A_{1}^{*}$ | $\frac{3}{2}$ | $\frac{1}{2}$ | $1$
8 | $4$ | $A_{4}$ | $A_{4}$ | $A_{4}^{*}$ | $\frac{6}{5}$ | $\frac{4}{5}$ | $\frac{2}{5}$
9 |  | $D_{4}$ | $D_{4}$ | $D_{4}^{*}$ | $1$ | $1$ | $0$
10 |  | $A_{3}\oplus A_{1}$ | $A_{3}\oplus A_{1}$ | $A_{3}^{*}\oplus A_{1}^{*}$ | $\frac{3}{2}$ | $\frac{1}{2}$ | $1$
11 |  | $A_{2}^{\oplus 2}$ | $A_{2}^{\oplus 2}$ | ${A_{2}^{*}}^{\oplus 2}$ | $\frac{4}{3}$ | $\frac{2}{3}$ | $\frac{2}{3}$
12 |  | $A_{2}\oplus A_{1}^{\oplus 2}$ | $\left(\begin{matrix}4&-1&0&1\\\ -1&2&-1&0\\\ 0&-1&2&-1\\\ 1&0&-1&2\end{matrix}\right)$ | $\frac{1}{6}\left(\begin{matrix}2&1&0&-1\\\ 1&5&3&1\\\ 0&3&6&3\\\ -1&1&3&5\end{matrix}\right)$ | $\frac{5}{3}$ | $\frac{1}{2}$ | $\frac{7}{6}$
13 |  | $A_{1}^{\oplus 4}$ | $D_{4}$ | $D_{4}^{*}\oplus\mathbb{Z}/2\mathbb{Z}$ | $2$ | $\frac{1}{2}$ | $\frac{3}{2}$
14 |  | $A_{1}^{\oplus 4}$ | $A_{1}^{\oplus 4}$ | ${A_{1}^{*}}^{\oplus 4}$ | $2$ | $\frac{1}{2}$ | $\frac{3}{2}$
15 | $3$ | $A_{5}$ | $A_{2}\oplus A_{1}$ | $A_{2}^{*}\oplus A_{1}^{*}$ | $\frac{3}{2}$ | $\frac{5}{6}$ | $\frac{2}{3}$
16 |  | $D_{5}$ | $A_{3}$ | $A_{3}^{*}$ | $\frac{5}{4}$ | $1$ | $\frac{1}{4}$
17 |  | $A_{4}\oplus A_{1}$ | $\left(\begin{matrix}4&-1&1\\\ -1&2&-1\\\ 1&-1&2\end{matrix}\right)$ | $\frac{1}{10}\left(\begin{matrix}3&1&-1\\\ 1&7&3\\\ -1&3&7\end{matrix}\right)$ | $\frac{17}{10}$ | $\frac{1}{2}$ | $\frac{6}{5}$
18 |  | $D_{4}\oplus A_{1}$ | $A_{1}^{\oplus 3}$ | ${A_{1}^{*}}^{\oplus 3}$ | $\frac{3}{2}$ | $\frac{1}{2}$ | $1$
19 |  | $A_{3}\oplus A_{2}$ | $\left(\begin{matrix}2&0&-1\\\ 0&2&-1\\\ -1&-1&4\end{matrix}\right)$ | $\frac{1}{12}\left(\begin{matrix}7&1&2\\\ 1&7&2\\\ 2&2&4\end{matrix}\right)$ | $\frac{5}{3}$ | $\frac{2}{3}$ | $1$
20 |  | $A_{2}^{\oplus 2}\oplus A_{1}$ | $A_{2}\oplus\langle 6\rangle$ | $A_{2}^{*}\oplus\langle 1/6\rangle$ | $\frac{11}{6}$ | $\frac{1}{2}$ | $\frac{4}{3}$
21 |  | $A_{3}\oplus A_{1}^{\oplus 2}$ | $A_{3}$ | $A_{3}^{*}\oplus\mathbb{Z}/2\mathbb{Z}$ | $2$ | $\frac{1}{2}$ | $\frac{3}{2}$
22 |  | $A_{3}\oplus A_{1}^{\oplus 2}$ | $A_{1}\oplus\langle 4\rangle$ | $A_{1}^{*}\oplus\langle 1/4\rangle$ | $2$ | $\frac{1}{2}$ | $\frac{3}{2}$
23 |  | $A_{2}\oplus A_{1}^{\oplus 3}$ | $A_{1}\oplus\left(\begin{matrix}4&-2\\\ -2&4\end{matrix}\right)$ | $A_{1}^{*}\oplus\frac{1}{6}\left(\begin{matrix}2&1\\\ 1&2\end{matrix}\right)$ | $\frac{13}{6}$ | $\frac{1}{2}$ | $\frac{5}{3}$
24 |  | $A_{1}^{\oplus 5}$ | $A_{1}^{\oplus 3}$ | ${A_{1}^{*}}^{\oplus 3}\oplus\mathbb{Z}/2\mathbb{Z}$ | $\frac{5}{2}$ | $\frac{1}{2}$ | $2$
25 | $2$ | $A_{6}$ | $\left(\begin{matrix}4&-1\\\ -1&2\end{matrix}\right)$ | $\frac{1}{7}\left(\begin{matrix}2&1\\\ 1&4\end{matrix}\right)$ | $\frac{12}{7}$ | $\frac{6}{7}$ | $\frac{6}{7}$
26 |  | $D_{6}$ | $A_{1}^{\oplus 2}$ | ${A_{1}^{*}}^{\oplus 2}$ | $\frac{3}{2}$ | $1$ | $\frac{1}{2}$
27 |  | $E_{6}$ | $A_{2}$ | $A_{2}^{*}$ | $\frac{4}{3}$ | $\frac{4}{3}$ | $0$
28 |  | $A_{5}\oplus A_{1}$ | $A_{2}$ | $A_{2}^{*}\oplus\mathbb{Z}/2\mathbb{Z}$ | $2$ | $\frac{1}{2}$ | $\frac{3}{2}$
29 |  | $A_{5}\oplus A_{1}$ | $A_{1}\oplus\langle 6\rangle$ | $A_{1}^{*}\oplus\langle 1/6\rangle$ | $2$ | $\frac{1}{2}$ | $\frac{3}{2}$
30 |  | $D_{5}\oplus A_{1}$ | $A_{1}\oplus\langle 4\rangle$ | $A_{1}^{*}\oplus\langle 1/4\rangle$ | $\frac{7}{4}$ | $\frac{1}{2}$ | $\frac{5}{4}$
31 |  | $A_{4}\oplus A_{2}$ | $\left(\begin{matrix}8&-1\\\ -1&2\end{matrix}\right)$ | $\frac{1}{15}\left(\begin{matrix}2&1\\\ 1&8\end{matrix}\right)$ | $\frac{28}{15}$ | $\frac{2}{3}$ | $\frac{6}{5}$
32 |  | $D_{4}\oplus A_{2}$ | $\left(\begin{matrix}4&-2\\\ -2&4\end{matrix}\right)$ | $\frac{1}{6}\left(\begin{matrix}2&1\\\ 1&2\end{matrix}\right)$ | $\frac{5}{3}$ | $\frac{2}{3}$ | $1$
33 |  | $A_{4}\oplus A_{1}^{\oplus 2}$ | $\left(\begin{matrix}6&-2\\\ -2&4\end{matrix}\right)$ | $\frac{1}{10}\left(\begin{matrix}2&1\\\ 1&3\end{matrix}\right)$ | $\frac{11}{5}$ | $\frac{1}{2}$ | $\frac{17}{10}$
34 |  | $D_{4}\oplus A_{1}^{\oplus 2}$ | $A_{1}^{\oplus 2}$ | ${A_{1}^{*}}^{\oplus 2}$ | $2$ | $\frac{1}{2}$ | $\frac{3}{2}$
35 |  | $A_{3}^{\oplus 2}$ | $A_{1}^{\oplus 2}$ | ${A_{1}^{*}}^{\oplus 2}\oplus\mathbb{Z}/2\mathbb{Z}$ | $2$ | $\frac{3}{4}$ | $\frac{5}{4}$
36 |  | $A_{3}^{\oplus 2}$ | $\langle 4\rangle^{\oplus 2}$ | $\langle 1/4\rangle^{\oplus 2}$ | $2$ | $\frac{3}{4}$ | $\frac{5}{4}$
37 |  | $A_{3}\oplus A_{2}\oplus A_{1}$ | $A_{1}\oplus\langle 12\rangle$ | $A_{1}^{*}\oplus\langle 1/12\rangle$ | $\frac{13}{6}$ | $\frac{1}{2}$ | $\frac{5}{3}$
38 |  | $A_{3}\oplus A_{1}^{\oplus 3}$ | $A_{1}\oplus\langle 4\rangle$ | $A_{1}^{*}\oplus\langle 1/4\rangle\oplus\mathbb{Z}/2\mathbb{Z}$ | $\frac{5}{2}$ | $\frac{1}{2}$ | $2$
39 |  | $A_{2}^{\oplus 3}$ | $A_{2}$ | $A_{2}^{*}\oplus\mathbb{Z}/3\mathbb{Z}$ | $2$ | $\frac{2}{3}$ | $\frac{4}{3}$
40 |  | $A_{2}^{\oplus 2}\oplus A_{1}^{\oplus 2}$ | $\langle 6\rangle^{\oplus 2}$ | $\langle 1/6\rangle^{\oplus 2}$ | $\frac{7}{3}$ | $\frac{1}{2}$ | $\frac{11}{6}$
41 |  | $A_{2}\oplus A_{1}^{\oplus 4}$ | $\left(\begin{matrix}4&-2\\\ -2&4\end{matrix}\right)$ | $\frac{1}{6}\left(\begin{matrix}2&1\\\ 1&2\end{matrix}\right)$ | $\frac{8}{3}$ | $\frac{1}{2}$ | $\frac{13}{6}$
42 |  | $A_{1}^{\oplus 6}$ | $A_{1}^{\oplus 2}$ | ${A_{1}^{*}}^{\oplus 2}\oplus(\mathbb{Z}/2\mathbb{Z})^{2}$ | $3$ | $\frac{1}{2}$ | $\frac{5}{2}$
43 | $1$ | $E_{7}$ | $A_{1}$ | $A_{1}^{*}$ | $\frac{3}{2}$ | $\frac{3}{2}$ | $0$
44 |  | $A_{7}$ | $A_{1}$ | $A_{1}^{*}\oplus\mathbb{Z}/2\mathbb{Z}$ | $2$ | $\frac{7}{8}$ | $\frac{11}{8}$
45 |  | $A_{7}$ | $\langle 8\rangle$ | $\langle 1/8\rangle$ | $2$ | $\frac{7}{8}$ | $\frac{11}{8}$
46 |  | $D_{7}$ | $\langle 4\rangle$ | $\langle 1/4\rangle$ | $\frac{7}{4}$ | $1$ | $\frac{3}{4}$
47 |  | $A_{6}\oplus A_{1}$ | $\langle 14\rangle$ | $\langle 1/14\rangle$ | $\frac{31}{14}$ | $\frac{1}{2}$ | $\frac{12}{7}$
48 |  | $D_{6}\oplus A_{1}$ | $A_{1}$ | $A_{1}^{*}$ | $2$ | $\frac{3}{2}$ | $\frac{1}{2}$
49 |  | $E_{6}\oplus A_{1}$ | $\langle 6\rangle$ | $\langle 1/6\rangle$ | $\frac{11}{6}$ | $\frac{1}{2}$ | $\frac{4}{3}$
50 |  | $D_{5}\oplus A_{2}$ | $\langle 12\rangle$ | $\langle 1/12\rangle$ | $\frac{23}{12}$ | $\frac{2}{3}$ | $\frac{5}{4}$
51 |  | $A_{5}\oplus A_{2}$ | $A_{1}$ | $A_{1}^{*}\oplus\mathbb{Z}/3\mathbb{Z}$ | $\frac{13}{6}$ | $\frac{2}{3}$ | $\frac{3}{2}$
52 |  | $D_{5}\oplus A_{1}^{\oplus 2}$ | $\langle 4\rangle$ | $\langle 1/4\rangle\oplus\mathbb{Z}/2\mathbb{Z}$ | $\frac{9}{4}$ | $\frac{1}{2}$ | $\frac{7}{4}$
53 |  | $A_{5}\oplus A_{1}^{\oplus 2}$ | $\langle 6\rangle$ | $\langle 1/6\rangle\oplus\mathbb{Z}/2\mathbb{Z}$ | $\frac{5}{2}$ | $\frac{1}{2}$ | $2$
54 |  | $D_{4}\oplus A_{3}$ | $\langle 4\rangle$ | $\langle 1/4\rangle\oplus\mathbb{Z}/2\mathbb{Z}$ | $2$ | $\frac{3}{4}$ | $\frac{5}{4}$
55 |  | $A_{4}\oplus A_{3}$ | $\langle 20\rangle$ | $\langle 1/20\rangle$ | $\frac{11}{5}$ | $\frac{3}{4}$ | $\frac{29}{20}$
56 |  | $A_{4}\oplus A_{2}\oplus A_{1}$ | $\langle 30\rangle$ | $\langle 1/30\rangle$ | $\frac{71}{30}$ | $\frac{1}{2}$ | $\frac{28}{15}$
57 |  | $D_{4}\oplus A_{1}^{\oplus 3}$ | $A_{1}$ | $A_{1}^{*}$ | $\frac{5}{2}$ | $\frac{1}{2}$ | $2$
58 |  | $A_{3}^{\oplus 2}\oplus A_{1}$ | $A_{1}$ | $A_{1}^{*}\oplus\mathbb{Z}/4\mathbb{Z}$ | $\frac{5}{2}$ | $\frac{1}{2}$ | $2$
59 |  | $A_{3}\oplus A_{2}\oplus A_{1}^{\oplus 2}$ | $\langle 12\rangle$ | $\langle 1/12\rangle\oplus\mathbb{Z}/2\mathbb{Z}$ | $\frac{8}{3}$ | $\frac{1}{2}$ | $\frac{13}{6}$
60 |  | $A_{3}\oplus A_{1}^{\oplus 4}$ | $\langle 4\rangle$ | $\langle 1/4\rangle\oplus\mathbb{Z}/2\mathbb{Z}$ | $3$ | $\frac{1}{2}$ | $\frac{5}{2}$
61 |  | $A_{2}^{\oplus 3}\oplus A_{1}$ | $\langle 6\rangle$ | $\langle 1/6\rangle\oplus\mathbb{Z}/3\mathbb{Z}$ | $\frac{5}{2}$ | $\frac{1}{2}$ | $2$

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