# Cubic and quartic points on modular curves using generalised symmetric
Chabauty

Josha Box1, Stevan Gajović2 and Pip Goodman3 1University of Warwick,
2University of Groningen and 3University of Bristol  Correspondence to be sent
to<EMAIL_ADDRESS>
###### Abstract.

Answering a question of Zureick-Brown, we determine the cubic points on the
modular curves $X_{0}(N)$ for $N\in\\{53,57,61,65,67,73\\}$ as well as the
quartic points on $X_{0}(65)$. To do so, we develop a “partially relative”
symmetric Chabauty method. Our results generalise current symmetric Chabauty
theorems, and improve upon them by lowering the involved prime bound. For our
curves a number of novelties occur. We prove a “higher order” Chabauty theorem
to deal with these cases. Finally, to study the isolated quartic points on
$X_{0}(65)$, we rigorously compute the full rational Mordell–Weil group of its
Jacobian.

##  1\. Introduction

This work started at the 2020 Arizona Winter School with a question from David
Zureick–Brown: Is it possible to determine the finitely many cubic points on
$X_{0}(65)$ despite the infinitude of quadratic points? In this article we
answer this question affirmatively by developing a “partially relative”
symmetric Chabauty method (Theorem 2.6). This theorem generalises the work of
Siksek [Sik09] on symmetric Chabauty, and moreover can be used with a larger
set of small primes. In certain cases this “first order” method still fails.
To overcome this, we develop a Chabauty method (Theorem 2.24) that takes into
account higher order terms of the relevant expansions of differential forms.

The case of $X_{0}(65)$ does not stand alone. Often for higher degrees $d>2$,
it happens that a curve admits infinitely many points of lower degree $e<d$,
while the set of degree $d$ points is finite. This is the first Chabauty
method which has the potential to compute all degree $d$ points in some of
such cases, provided the rank of the Mordell–Weil group of the curve’s
Jacobian is not too large. This opens the way for the provable determination
of all points of fixed degree $d$ (notably when $d>2$) on various classes of
interesting curves. We illustrate this with the following result.

######  Theorem 1.1.

The set of cubic points on each of the curves

$X_{0}(53),\;\;X_{0}(57),\;\;X_{0}(61),\;\;X_{0}(65),\;\;X_{0}(67)\text{ and
}X_{0}(73)$

is finite and listed in Section 5. The quartic points on $X_{0}(65)$ form an
infinite set. This infinite set consists of inverse images of quadratic points
on the quotient curve $X^{+}_{0}(65)$ and a finite number of isolated points.
The isolated points are listed in §LABEL:subsection_quarticpoints.

For a curve $X$ over a number field $K$, denote by $X^{(d)}$ its $d$th
symmetric power. The $K$-rational points on $X^{(d)}$ are exactly the
$K$-rational effective degree $d$ divisors on $X$, and the
$\mathrm{Gal}(\overline{K}/K)$-orbit of each point $P\in X$ defined over a
degree $d$ extension of $K$ gives rise to such a divisor. By studying the
$K$-rational points on $X^{(d)}$ we can thus study the $L$-rational points on
$X$ for all degree $d$ extensions $L/K$ simultaneously.

An important tool to study $X^{(d)}$ is the Abel–Jacobi map. Given a
$K$-rational degree $d$ divisor $D_{0}$ on $X$, we define it as

$\iota\colon\;X^{(d)}\to J(X),\;D\mapsto[D-D_{0}].$

So when does $X/K$ have infinitely many points of degree $d$ over $K$?
Certainly when there is a degree $d$ map $\rho\colon X\to\mathbb{P}^{1}$, as
the inverse image of $\mathbb{P}^{1}(K)$ provides such an infinite set. More
generally, when there is a map $\rho\colon X\to C$ of degree $e$ to a curve
$C$ such that $C^{(f)}(K)$ is infinite, $e\cdot f\leq d$ and there exists
$P\in X^{(d-ef)}(K)$: then such an infinite set is

$P+\rho^{*}C^{(f)}(K)\subset X^{(d)}(K),$

and $\iota$ maps this set into a translate of the abelian subvariety
$\rho^{*}J(C)$ of $J(X)$.

Define $P\in X^{(d)}(K)$ to be an _isolated point_ when it is neither in the
inverse image of $\mathbb{P}^{1}(K)$ under a degree $d$ map
$X\to\mathbb{P}^{1}$, nor does $\iota(P)$ lie in a translate of a positive
rank abelian subvariety of $J(X)$ contained in $\iota(X^{(d)})$. Recently,
Bourdon, Ejder, Liu, Odumodu and Viray [BEL+19, Theorem 4.2] have shown that

  * (i)

$X$ has infinitely many points of degree $d$ over $K$ if and only if
$X^{(d)}(K)$ contains a non-isolated point, and

  * (ii)

there are finitely many isolated points in $X^{(d)}(K)$.

This result provides a road map for studying $X^{(d)}(K)$: first describe each
infinite set, then determine the finite set of isolated points. This is what
we have done for the modular curves listed above, using a generalisation of
Chabauty’s method.

Chabauty’s method is, classically, a method for effectively computing the
rational points on a curve in the special case when the rank $r$ of its
Mordell–Weil group is strictly smaller than its genus $g$. The effectiveness
is due to Coleman [Col85a], who realised that Chabauty’s finiteness proof
[Cha41] could be made effective using the machinery of locally analytic
$p$-adic functions.

While nowadays many focus on weakening the $r<g$ condition in Chabauty’s
method via non-abelian generalisations (see e.g. [Kim05] and [BBB+]), the
reach of Chabauty’s method is still being increased. After partial results of
Klassen [Kla93], Siksek [Sik09] extended Coleman’s ideas to obtain an
effective method for computing the $K$-rational points (when finite) on the
$d$th symmetric powers of curves over any number field $K$, provided the
stronger condition

$r<g-(d-1)$

is satisfied. A similar result was obtained by Derickx, Kamienny, Stein and
Stoll [DKSS] using the theory of formal immersions, which they applied to
points of degree up to 7 on several modular curves.

Their method has led to several important results: it was used by Derickx,
Najman and Siksek to prove that all elliptic curves over totally real cubic
fields are modular [DNS20], and helped solve the puzzle of classifying the
finite groups that appear as the torsion subgroup of the Mordell–Weil group of
an elliptic curve over a cubic field [DEvH+21]. In the latter article, the
Chabauty-like method for symmetric powers was used to determine
$X_{1}(22)^{(3)}(\mathbb{Q})$, $X_{1}(25)^{(3)}(\mathbb{Q})$ and the image of
$X_{1}(65)^{(3)}(\mathbb{Q})$ in $X_{0}(65)^{(3)}(\mathbb{Q})$. Using our
method, we can now describe the complete set $X_{0}(65)^{(3)}(\mathbb{Q})$ and
list the finitely many cubic points on $X_{0}(65)$.

Furthermore, Siksek [Sik09] developed a relative version of his symmetric
Chabauty method, which can be used to determine the isolated points on
$X^{(d)}$ if the infinite set consists entirely of pullbacks. To be precise,
given a map $\rho\colon X\to C$ of curves over $K$ of degree $d$, Siksek’s
relative Chabauty method can determine the remainder
$X^{(d)}(K)\setminus\rho^{*}C(K)$ if that consists entirely of isolated
points. This method was employed by the first named author [Box21] to describe
the quadratic points on $X_{0}(N)$ for $N\in\\{43,53,57,61,65,67,73\\}$.

The curve $X_{0}(65)$ admits a degree 2 map $\rho\colon X_{0}(65)\to
X_{0}(65)/\langle w_{65}\rangle=\colon X^{+}_{0}(65)$, where $w_{65}$ is the
Atkin–Lehner involution. This quotient $X_{0}^{+}(65)$ is an elliptic curve of
rank 1, and $\rho^{*}(X_{0}^{+}(65)(\mathbb{Q}))$ is an infinite set of degree
2 points. In [Box21], Siksek’s relative Chabauty method was used to show that
the only isolated degree 2 points are sums of two cusps. For degree 3,
however, we obtain for each cusp $c\in X_{0}(65)(\mathbb{Q})$ such an infinite
set

$c+\rho^{*}(X_{0}^{+}(65)(\mathbb{Q}))\subset X_{0}(65)^{(3)}(\mathbb{Q}),$

and Siksek’s method cannot be applied anymore to study the isolated points of
degree 3, as the infinite sets are not pullbacks.

It is exactly this problem that we solved, by generalising Siksek’s ideas to
obtain a “partially relative symmetric Chabauty method” that has the potential
to determine the isolated degree $d$ points on a curve $X$ if the infinite
sets are of the form

$P+\rho_{1}^{*}C_{1}^{(\ell_{1})}(K)+\cdots+\rho_{n}^{*}C_{n}^{(\ell_{n})}(K),$

where $P\in X^{(e)}(K)$ and $\rho_{i}\;\colon X\to C_{i}$ are maps of degree
$d_{i}$ such that $e+\ell_{1}d_{1}+\cdots+\ell_{n}d_{n}=d$. The result that
makes this possible is Theorem 2.6. This theorem has already contributed in
[Box] to the proof that all elliptic curves over quartic fields not containing
$\sqrt{5}$ are modular.

Our work on the modular curves $X_{0}(N)$ for $N\in\\{53,57,61,65,67,73\\}$
extends a string of papers studying the modular curves $X_{0}(N)$ of genus
$g\in\\{2,\ldots,5\\}$. Bruin and Najman [BN15] determined the isolated
quadratic points on the hyperelliptic $X_{0}(N)$ of these genera with finite
Mordell–Weil group. Subsequently, Özman and Siksek [OS19] determined the
finitely many quadratic points on the non-hyperelliptic $X_{0}(N)$ of genus
$g\in\\{2,\ldots,5\\}$ with finite Mordell–Weil group, and the first named
author [Box21] described the quadratic points on those with infinite
Mordell–Weil group.

The curves for which we determine the cubic points are exactly those
$X_{0}(N)$ of genus $g\in\\{2,\ldots,5\\}$ that have infinite Mordell–Weil
group and for which the Chabauty condition on the rank holds true. Only two of
these curves also satisfy the rank condition for quartic points: $X_{0}(65)$
and $X_{0}(57)$. However, a genus 5 curve admits infinitely many degree four
maps to $\mathbb{P}^{1}$ (see §5.1). This prevents us from determining the
quartic points on $X_{0}(57)$. For $X_{0}(65)$ on the other hand, we are in
luck. All degree four maps to $\mathbb{P}^{1}$ defined over $\mathbb{Q}$
factor through the elliptic curve $X_{0}^{+}(65)$. Applying our partial
relative Chabauty method with respect to this quotient, we determine all of
the isolated quartic points on $X_{0}(65)$. The Magma [BCP97] code to verify
all computations made in this paper can be found at

$\texttt{\href https://github.com/joshabox/cubicpoints/}\;.$

##  2\. A general symmetric Chabauty theorem

In this section we present a common generalisation of Theorems 3.2 and 4.3 in
[Sik09], which are Chabauty-type theorems for computing $K$-rational points on
symmetric powers of curves.

Before this, we give an overview of Siksek’s Chabauty method.

###  2.1. Uniformisers and differentials

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ with ring of integers $R$
and residue field $k$. Consider a curve $X/K$ together with a minimal proper
regular model $\mathcal{X}/R$ for $X$. We write $\widetilde{X}$ for the
special fibre of $\mathcal{X}$, and similarly denote reductions of objects
associated to $\mathcal{X}$ with a tilde. Denote by $\mathcal{O}_{S}$ the
sheaf of functions on a scheme $S$. For a sheaf $\mathcal{F}$ on $S$, we
denote by $\mathcal{F}_{s}$ its stalk at $s$. When $A$ is a discrete valuation
ring (DVR), we denote by $\widehat{A}$ its completion. Finally, when $s\in S$
is such that $\mathcal{O}_{S,s}$ is a DVR, $U\subset S$ is an open subset
containing $s$ and $\mathcal{F}$ is an $\mathcal{O}_{S}$-module, we denote by
$\mathrm{loc}_{s}$ the map $\mathcal{F}(U)\to\mathcal{F}_{s}$. For $x\in
X^{(d)}(K)$ and $y\in\widetilde{X}^{(d)}(k)$, we denote by $D(x)$ and $D(y)$
the points in $X^{(d)}(K)$ reducing to $\widetilde{x}$ and $y$ respectively.

When $x\in X$ is non-singular, a uniformiser $t\in\widehat{\mathcal{O}}_{X,x}$
is called a _local coordinate_ at $x\in X$. When $t$ also reduces to a
uniformiser in the reduction
$\widehat{\mathcal{O}}_{\widetilde{X},\widetilde{x}}$, we say that it is a
_well-behaved local coordinate_ or _well-behaved uniformiser_. This just means
that the maximal ideal of $\widehat{\mathcal{O}}_{\mathcal{X},\widetilde{x}}$
is generated by $t$ and $p$ as an $R$-module. The following facts can be found
for example in [LT02]. When $t$ is a well-behaved local coordinate at $x$, we
can evaluate $t$ at points in the residue disc $D(x)$ of $x$, yielding a
bijection between $D(x)$ and the maximal ideal of $R$.

Denote by $\Omega_{X/K}$ and $\Omega_{\mathcal{X}/R}$ (sometimes abbreviated
to $\Omega$) the sheaves of regular differentials on $X$ and $\mathcal{X}$
respectively. A choice of uniformiser $t\in\widehat{\mathcal{O}}_{X,x}$ gives
rise to the identifications $\widehat{\mathcal{O}}_{X,x}=K\llbracket
t\rrbracket$ and $\widehat{\Omega}_{X,x}=K\llbracket t\rrbracket\mathrm{d}t$.
If $t$ is moreover well-behaved, we have
$\widehat{\mathcal{O}}_{\mathcal{X},\widetilde{x}}=R\llbracket t\rrbracket$
and $\widehat{\Omega}_{\mathcal{X},\widetilde{x}}=R\llbracket
t\rrbracket\mathrm{d}t$.

Now consider $\omega\in H^{0}(X,\Omega_{X/K})$. Since
$H^{0}(\mathcal{X},\Omega_{\mathcal{X}/R})$ is a lattice in
$H^{0}(X,\Omega_{X/K})$, after multiplication by a constant in $R$, we may
assume that $\omega\in H^{0}(\mathcal{X},\Omega_{\mathcal{X}/R})$. In sum,
given a well-behaved uniformiser $s$ at a point $Q\in X(K)$, we can write

(1)
$\displaystyle\mathrm{loc}_{Q}(\omega)=\sum_{n=0}^{\infty}a_{n}s^{n}\mathrm{d}s\text{
with }a_{n}\in R\text{ for all }n.$

Consider any point $P_{0}\in X(\overline{K})$.

######  Lemma 2.1.

The map $\iota\colon X\to J(X),\;P\mapsto[P-P_{0}]$ induces an isomorphism

$\iota^{*}\colon\;H^{0}(J,\Omega_{J/K})\simeq H^{0}(X,\Omega_{X/K})$

of global differential forms independent of the choice of $P_{0}$.

###### Proof.

See e.g. [Sik09, Proposition 2.1]. ∎

We shall thus use $\iota^{*}$ to pass between these two spaces.

Now suppose that we have a map $\rho\colon X\to C$ between two curves over
$K$. We obtain a pushforward map $\rho_{*}\colon\;J(X)\to J(C)$ and pullback
map $\rho^{*}\colon\;J(C)\to J(X)$, leading to a decomposition up to isogeny

$J(X)\sim J(C)\times A,$

where $A\subset J(X)$ is an abelian subvariety. Denote by $\pi_{A}$ the map
$J(X)\to A$. We also obtain a push-forward, or “trace map” on meromorphic
differentials

$\mathrm{Tr}\colon\;\Omega_{K(X)/K}\to\Omega_{K(C)/K}.$

We consider $H^{0}(X,\Omega_{X/K})$ as a subspace of $\Omega_{K(X)/K}$, and
similarly for $C$.

######  Lemma 2.2.

We have

  * (i)

$\Omega_{K(X)/K}=\rho^{*}\Omega_{K(C)/K}\oplus\mathrm{Ker}(\mathrm{Tr})$, and

  * (ii)

$\iota^{*}\pi_{A}^{*}H^{0}(A,\Omega_{A/K})=\mathrm{Ker}(\mathrm{Tr})\cap
H^{0}(X,\Omega_{X/K})$.

###### Proof.

Part (i) follows from surjectivity of the trace map. For part (ii) it suffices
to compare dimensions and to check that
$\iota^{*}\pi_{A}^{*}H^{0}(A,\Omega_{A/K})\subset\mathrm{Ker}(\mathrm{Tr})$. ∎

###  2.2. Coleman integration

We consider the notation from the previous subsection. Again let $K$ be a
finite extension of $\mathbb{Q}_{p}$. Also let $B/K$ be any abelian variety
with good reduction at all primes in $K$ above $p$. In [Col85b, Section II],
Coleman defines a pairing, now called _Coleman integration_ :

(2) $\displaystyle H^{0}(B,\Omega_{B/K})\times B(K)\to
K,\;\;(\omega,P)\mapsto\int_{P}\omega.$

We note that Coleman defines this integration in a much more general setting
than this, but we shall only be concerned with the above case of abelian
varieties and differentials of the first kind.

######  Proposition 2.3.

The integration pairing (2) is

  * (i)

locally analytic in $P\in B(K)$,

  * (ii)

$\mathbb{Z}$-linear on the right,

  * (iii)

$K$-linear on the left,

  * (iv)

its left-hand kernel is zero,

  * (v)

its right-hand kernel is $B(K)_{\mathrm{tors}}$, and

  * (vi)

if $g\colon B\to B^{\prime}$ is a morphism of abelian varieties over $K$, then

$\int_{P}g^{*}\omega=\int_{g(P)}\omega$

for all $\omega\in H^{0}(B^{\prime},\Omega_{B^{\prime}/K})$ and $P\in B(K)$.

###### Proof.

Parts (i) and (ii) are [Col85b, Theorem 2.8]. Part (iii) follows from part (i)
and [Col85b, Proposition 2.4 (i), (iii)]. Part (iv) follows from [Col85b,
Theorem 2.8 (ii)], and Part (v) is [Col85b, Theorem 2.11]. Finally, part (vi)
is [Col85b, Theorem 2.7]. ∎

######  Lemma 2.4.

Suppose that $G\subset B(K)$ is a subgroup of rank $r$. Then there is a vector
space $\mathcal{V}\subset H^{0}(B,\Omega_{B/K})$ of dimension at least
$\mathrm{dim}(B)-r$ such that for all $\omega\in\mathcal{V}$, we have

$\int_{P}\omega=0\text{ for all }P\in\overline{G},$

where the closure is inside the $p$-adic topology on $B(K)$.

###### Proof.

Suppose that $D_{1},\ldots,D_{r}$ generate $G$ up to torsion. Then
$\overline{G}\otimes K=KD_{1}+\cdots+KD_{r}$, of dimension $\leq r$. Now we
find $\mathcal{V}$ because (2) extends to an exact pairing between
$H^{0}(B,\Omega_{B/K})$ and $B(K)\otimes K$ by Proposition 2.3. ∎

When $B=J(X)/K$ and $G=J(X)(K)$, we call this space $\mathcal{V}$ the _space
of annihilating differentials_. Moreover, when we have a map $\rho\colon X\to
C$ as in the previous section, we call
$\mathcal{V}\cap\mathrm{Ker}(\mathrm{Tr})$ the _space of annihilating
differentials with trace zero_. This is the pullback along $\pi_{A}$ of the
space of annihilating differentials on $A$, where $A$ is such that $J(X)\sim
J(C)\times A$.

More notation: by pulling back along $\iota$, we obtain integrals

$H^{0}(X,\Omega_{X/K})\times X(K)^{2}\to
K,\;\;(\omega,Q,P)\mapsto\int_{Q}^{P}\omega:=\int_{[P-Q]}(\iota^{*})^{-1}(\omega).$

######  Proposition 2.5.

  * (i)

For $P,Q\in X(K)$ such that $P\in D(Q)$, and a well-behaved uniformiser $s$ at
$Q$, we have for each $\omega\in H^{0}(X,\Omega_{X/K})$

$\int_{Q}^{P}\omega=\sum_{n=0}^{\infty}\frac{a_{n}}{n+1}s(P)^{n+1},$

where $\mathrm{loc}_{Q}(\omega)=\sum_{n}a_{n}s^{n}$ as in (1). We call such an
integral between points in the same residue class a _tiny integral_.

  * (ii)

If $\rho\colon X\to C$ is a non-constant morphism of curves over $K$ with good
reduction, then

$\int_{\rho^{*}D}\omega=\int_{D}\mathrm{Tr}(\omega)$

for all $\omega\in H^{0}(X,\Omega_{X/K})$ and every degree 0 divisor $D$ on
$C$ defined over $K$.

###### Proof.

Part (i) follows from Proposition 2.3 (i) together with the Fundamental
Theorem of Calculus proved in [Col85b, Proposition 2.4 (ii)]. Part (ii) is
[Sik09, Lemma 2.2]. ∎

###  2.3. An overview of the symmetric Chabauty–Coleman method

We refer the reader to [Wet97] and [MP12] for a clear overview of the
Chabauty–Coleman method in the classical case. Here we give an overview of
Chabauty and Coleman’s original ideas in the symmetric power setting, after
which we explain what needs to be changed in the relative case. For simplicity
we work over $\mathbb{Q}$, but everything generalises to number fields.

We consider an integer $d$, a prime $p$ and a curve $X/\mathbb{Q}$ of genus
$g_{X}\geq 2$ and minimal proper regular model $\mathcal{X}/\mathbb{Z}_{p}$
for $X_{\mathbb{Q}_{p}}$. To determine $X^{(d)}(\mathbb{Q})$, it suffices to
determine the rational points contained in each of the residue discs
separately. So consider $\widetilde{\mathcal{Q}}\in X^{(d)}(\mathbb{F}_{p})$
and its inverse image under the reduction map, the residue disc
$D(\widetilde{\mathcal{Q}})\subset X^{(d)}(\mathbb{Q}_{p})$. Assume there
exists $D_{0}\in X^{(d)}(\mathbb{Q})$. We use it to define the Abel–Jacobi
map:

$\iota\colon X^{(d)}\to J(X),\;\;D\mapsto[D-D_{0}].$

Chabauty’s idea was to consider the diagram

(3) ${D(\widetilde{\mathcal{Q}})\cap
X^{(d)}(\mathbb{Q})}$${J(X)(\mathbb{Q})}$${D(\widetilde{\mathcal{Q}})}$${J(X)(\mathbb{Q}_{p})}$$\scriptstyle{\iota}$$\scriptstyle{\iota}$

and instead determine

(4)
$\displaystyle\iota(D(\widetilde{\mathcal{Q}}))\cap\overline{J(X)(\mathbb{Q})},$

where the closure is inside the $p$-adic topology on $J(X)(\mathbb{Q}_{p})$.
This set contains $\iota(D(\widetilde{\mathcal{Q}})\cap X^{(d)}(\mathbb{Q}))$.
Let $r$ be the rank of $J(X)(\mathbb{Q})$. By Lemma 2.4, we obtain a space
$\mathcal{V}\subset H^{0}(X,\Omega_{X/K})$ of dimension
$\mathrm{dim}(\mathcal{V})\geq g_{X}-r$ such that for all
$\omega\in\mathcal{V}$, we have

$\int_{D}\omega=0\text{ for all }D\in\overline{J(X)(\mathbb{Q})}.$

In particular, when the _Chabauty condition_

(5) $\displaystyle r<g_{X}-(d-1)$

is satisfied, there are $d$ such linearly independent differentials, and
dimensions suggest that

$\mathcal{Z}_{p}(\widetilde{\mathcal{Q}})\colon=\left\\{\mathcal{P}\in
D(\widetilde{\mathcal{Q}})\mid\int_{\iota(\mathcal{P})}\omega=0\text{ for all
}\omega\in\mathcal{V}\right\\}$

is finite. In the case $d=1$, Chabauty [Cha41] proved that
$\overline{J(\mathbb{Q})}\cap\iota(D(\widetilde{\mathcal{Q}}))$ is indeed
finite when (5) is satisfied. For $d>1$, $X^{(d)}(\mathbb{Q})$ can still be
infinite even when (5) holds, e.g. due to the existence of a map $\rho\colon
X\to C$ of degree $\leq d$. For now, however, let us assume that $X$ satisfies
(5), and that we have no reason to believe that $X^{(d)}(\mathbb{Q})\cap
D(\widetilde{\mathcal{Q}})$ is infinite regardless.

It was Coleman’s idea [Col85a] to introduce integration of differentials and
compute $\mathcal{Z}_{p}(\widetilde{\mathcal{Q}})$ instead. Such zero sets can
be computed for $d=1$ by evaluating Coleman integrals between
$\mathbb{Q}_{p}$-rational points. In Sage on hyperelliptic curves, the
computation of Coleman integrals over $\mathbb{Q}_{p}$ has been implemented by
Balakrishnan, Bradshaw, and Kedlaya [BBK10], [Bal15], while Balakrishnan and
Tuitman [BT20] wrote a Magma implementation for plane curves. For $d>1$,
however, one would need to evaluate Coleman integrals between points $P,Q\in
X(K)$ for extensions $K/\mathbb{Q}_{p}$ of degree $>1$; this has only been
done for superelliptic curves over unramified extensions of $\mathbb{Q}_{p}$,
by recent work of Best [Bes].

Instead, therefore, we shall restrict our attention to tiny (i.e. directly
computable) integrals only: this suffices when combining the information from
multiple primes $p$ using the Mordell–Weil sieve (see Section 3). The drawback
here is that information on $J(X)(\mathbb{Q})$ is needed. Thanks to the sieve,
we need to consider only residue discs $D(\widetilde{\mathcal{Q}})$ containing
a known point $\mathcal{Q}\in X^{(d)}(\mathbb{Q})$. For each such point
$\mathcal{Q}$, (assuming that $X^{(d)}(\mathbb{Q})$ is finite) there is a
prime $p$ such that $X^{(d)}(\mathbb{Q})\cap
D(\widetilde{\mathcal{Q}})=\\{\mathcal{Q}\\}$. To compute
$X^{(d)}(\mathbb{Q})$, it then suffices to have a criterion to decide whether
$\mathcal{Z}_{p}(\widetilde{\mathcal{Q}})=\\{\mathcal{Q}\\}$.

Given the known point $\mathcal{Q}$, we can choose $D_{0}=\mathcal{Q}$ to
define $\iota$. Then for $\mathcal{P}\in D(\widetilde{\mathcal{Q}})$, the
integral $\int_{\iota(\mathcal{P})}\omega$ is a sum of $d$ tiny integrals. By
studying the power series obtained from these tiny integrals via Proposition
2.5 (i), Siksek [Sik09, Theorem 3.2] found a criterion for deciding whether
$Z_{p}(\widetilde{\mathcal{Q}})=\\{\mathcal{Q}\\}$.

###  2.4. An overview of our relative Chabauty–Coleman method

We continue the notation from the previous subsection. We now assume that we
_do_ have reason to believe that $X^{(d)}(\mathbb{Q})\cap
D(\widetilde{\mathcal{Q}})$ is infinite, due to the existence of a curve $C$
with minimal proper regular model $\mathcal{C}/\mathbb{Z}_{p}$, a map
$\rho\colon\mathcal{X}\to\mathcal{C}$ such that $\rho\colon X\to C$ has degree
$f$, a positive integer $e$ such that $e\cdot f\leq d$, and known points $P\in
X^{(d-ef)}(\mathbb{Q})$ and $Q\in C^{(e)}(\mathbb{Q})$ such that
$\mathcal{Q}:=P+\rho^{*}Q\in D(\widetilde{\mathcal{Q}})$. Indeed, we then have
a family

$P+\rho^{*}C^{(e)}(\mathbb{Q})\subset X^{(d)}(\mathbb{Q})$

intersecting $D(\widetilde{\mathcal{Q}})$. Even when $X$ satisfies the
Chabauty condition (5), the approach from the previous subsection fails when
either $C^{(e)}(\mathbb{Q})$ is infinite or when $C^{(e)}$ does not satisfy
the Chabauty condition $r_{C}<g_{C}-(e-1)$ itself, where
$r_{C}=\mathrm{rk}J(C)(\mathbb{Q})$ and $g_{C}$ is the genus of $C$.

As in Section 2.1, there is an abelian variety $A\subset J(X)$ such that
$J(X)\sim J(C)\times A$, and we define $\pi_{A}\colon J(X)\to A$. As before,
define the Abel–Jacobi map $\iota$ using $\mathcal{Q}$. We now replace $J(X)$
in Diagram (3) by $A$, and note that

$\pi_{A}\circ\iota\left(P+\rho^{*}C^{(e)}(\mathbb{Q})\right)=\\{0\\},$

so the entire family has been collapsed to a single point on $A$. It may be
easier to determine
$\overline{A(\mathbb{Q})}\cap\pi_{A}(\iota(D(\widetilde{\mathcal{Q}})))$ than
it is to determine
$\overline{J(X)(\mathbb{Q})}\cap\iota(D(\widetilde{\mathcal{Q}}))$. Again by
Lemma 2.2, we find a space $\mathcal{V}\subset H^{0}(A,\Omega_{A/K})$ of
dimension
$\mathrm{dim}(\mathcal{V})\geq\mathrm{dim}(A)-\mathrm{rank}(A(\mathbb{Q}))$
such that for all $\omega\in\mathcal{V}$

$\int_{D}\omega=0\text{ for each }D\in\overline{A(\mathbb{Q})}.$

Let $r_{X}$ be the rank of $J(X)(\mathbb{Q})$, $r_{C}$ the rank of
$J(C)(\mathbb{Q})$, and $g_{X}$ and $g_{C}$ be the genera of $X$ and $C$
respectively. If the Chabauty condition

(6) $r_{X}-r_{C}<g_{X}-g_{C}-(d-1)$

is satisfied, then $\mathrm{dim}(\mathcal{V})\geq d$ and the dimensions
suggest that the common zero set of $\int\omega$ for $\omega\in\mathcal{V}$
has finite intersection with $\pi_{A}\circ\iota(D(\widetilde{\mathcal{Q}}))$.
Define

$\mathcal{Z}_{p,A}(\widetilde{\mathcal{Q}}):=\left\\{\mathcal{P}\in
D(\widetilde{\mathcal{Q}})\mid\int_{\mathcal{Q}}^{\mathcal{P}}\omega=0\text{
for all }\omega\in\mathcal{V}\cap\mathrm{Ker}(\mathrm{Tr})\right\\},$

where $\mathrm{Tr}\colon\;\Omega_{K(X)/K}\to\Omega_{K(C)/K}$ is the trace map.
By Lemma 2.2 (ii) and Proposition 2.3 (vi), this is the inverse image to
$D(\widetilde{\mathcal{Q}})$ of the common zero set of the integrals
$\int\omega$ for $\omega\in\mathcal{V}$.

Analogous to the case of the previous section, we now desire a criterion to
decide if

$\mathcal{Z}_{p,A}(\widetilde{\mathcal{Q}})=(P+\rho^{*}C^{(e)}(\mathbb{Q}))\cap
D(\widetilde{\mathcal{Q}}).$

This is the purpose of Theorem 2.6.

###  2.5. The main theorem

Consider a point $Q$ on a curve $X$ over a field $K$, and a regular 1-form
$\omega\in H^{0}(X,\Omega_{X/K})$. We expand $\omega$ around $Q$ in terms of a
uniformiser $t_{Q}$ at $Q$, giving $\mathrm{loc}_{Q}(\omega)=\sum_{j\geq
0}a_{j}t_{Q}^{j}\mathrm{d}t_{Q}$. We define

$\displaystyle v(\omega,t_{Q},k)$
$\displaystyle:=\left(-a_{0},a_{1},\ldots,(-1)^{k}a_{k-1}\right).$

When $\mathcal{Q}$ is an effective $K$-rational divisor on $X$, we denote by
$K(\mathcal{Q})$ the (Galois) extension obtained by adjoining to $K$ all
points in the support of $\mathcal{Q}$.

######  Theorem 2.6.

Let $\rho_{j}\colon X\rightarrow C_{j}$ for $j\in\\{1,\ldots,h\\}$ be degree
$d_{j}$ maps of curves over a number field $K$, and consider given an
effective divisor

$\mathcal{Q}=\mathcal{Q}_{0}+\mathcal{Q}_{1}+\cdots+\mathcal{Q}_{h},\text{
where }\mathcal{Q}_{0}\in X^{(e)}(K)\text{ and
}\mathcal{Q}_{j}\in\rho_{j}^{*}C_{j}^{(\ell_{j})}(K)\text{ for }j\geq 1.$

Let $n=e+d_{1}\ell_{1}+\cdots+d_{h}\ell_{h}$. Suppose that $\mathfrak{r}$ is a
prime in $\mathcal{O}_{K}$ of good reduction for $X$ and each $C_{j}$. Let $p$
be the rational prime contained in $\mathfrak{r}$.

  * (1)

Assume that the supports of $\mathcal{Q}_{1},\ldots,\mathcal{Q}_{h}$ are
pairwise disjoint, and no point in the support of any $\mathcal{Q}_{i}$ for
$i\geq 1$ has ramification degree under $\rho_{i}$ divisible by $p$.

  * (2)

Let $N$ be the maximum of the ramification indices of $p$ in $K(Q_{i},Q_{j})$
for $i,j\in\\{1,\ldots,k\\}$. Assume that $p\geq N+2$.

Write $\lambda=\ell_{1}+\cdots+\ell_{h}$. Let $\mathcal{V}_{0}$ be the space
of annihilating differentials on $X$ with trace zero with respect to each
$\rho_{j}$, and consider a basis
$\widetilde{\omega}_{1},\ldots,\widetilde{\omega}_{q}$ for the image of
$\mathcal{V}_{0}\cap H^{0}(\mathcal{X},\Omega)$ under the mod $\mathfrak{r}$
reduction map on differentials. Let $\mathfrak{p}$ be a prime in
$K(\mathcal{Q})$ above $\mathfrak{r}$, and denote reductions of points with
respect to $\mathfrak{p}$ with a tilde. Write
$\mathcal{Q}=n_{1}Q_{1}+\cdots+n_{k}Q_{k}$ with $Q_{1},\ldots,Q_{k}\in X$
distinct points and each $n_{i}\geq 1$, and let $t_{\widetilde{Q}_{i}}$ be a
uniformiser at $\widetilde{Q}_{i}$ for each $i$.

  * (3)

Assume that the matrix

$\widetilde{\mathcal{A}}:=\begin{pmatrix}v(\widetilde{\omega}_{1},t_{\widetilde{Q}_{1}},n_{1})&v(\widetilde{\omega}_{1},t_{\widetilde{Q}_{2}},n_{2})&\cdots&v(\widetilde{\omega}_{1},t_{\widetilde{Q}_{k}},n_{k})\\\
v(\widetilde{\omega}_{2},t_{\widetilde{Q}_{1}},n_{1})&v(\widetilde{\omega}_{2},t_{\widetilde{Q}_{2}},n_{2})&\cdots&v(\widetilde{\omega}_{2},t_{\widetilde{Q}_{k}},n_{k})\\\
\vdots&\ddots&\ddots&\vdots\\\
v(\widetilde{\omega}_{q},t_{\widetilde{Q}_{1}},n_{1})&v(\widetilde{\omega}_{q},t_{\widetilde{Q}_{2}},n_{2})&\cdots&v(\widetilde{\omega}_{q},t_{\widetilde{Q}_{k}},n_{k})\end{pmatrix}$

has rank $n-\lambda$.

Then every $\mathcal{P}\in X^{(n)}(K)$ in the mod $\mathfrak{r}$ residue disc
of $\mathcal{Q}$ is in fact contained in

$\mathcal{Q}_{0}+\rho_{1}^{*}C_{1}^{(\ell_{1})}(\overline{K})+\cdots+\rho_{h}^{*}C_{h}^{(\ell_{h})}(\overline{K}).$

We postpone the proof of this theorem to Section 2.8.

######  Remark 2.7.

By Proposition 2.17 in the next section, $n-\lambda$ is in fact the maximal
possible rank of this matrix. Moreover, note that condition (3) is not
satisfied when two distinct points $Q_{i}\neq Q_{j}$ have the same reduction
mod $\mathfrak{p}$. If they do, at least two columns in
$\widetilde{\mathcal{A}}$ would agree, reducing its rank to less than
$n-\lambda$ due to the nature of the linear equations between the columns
obtained from Proposition 2.17.

######  Remark 2.8.

Our lower bound $p\geq N+2$ leads to a general lower bound $p\geq n(n-1)+2$.
In particular, for $n=2$, $p\geq 5$ suffices; for $n=3$, $p\geq 11$ suffices;
and for $n=4$, $p\geq 17$ suffices.

######  Remark 2.9.

This theorem generalises [Sik09, Theorems 3.2 and 4.3]. More precisely, the
case $\mathcal{Q}=\mathcal{Q}_{0}$ corresponds to [Sik09, Theorem 3.2] and the
case $\mathcal{Q}=\mathcal{Q}_{1}$ corresponds to [Sik09, Theorem 4.3]. In
both cases, we significantly improve the lower bound on the prime $p$ compared
to [Sik09]. Part of the improvement is due to the introduction in the proof of
elementary symmetric polynomials replacing power sums. This removes
denominators in the power series expansion and hence in the matrix
$\widetilde{\mathcal{A}}$; this is an idea also used in [DKSS]. Another
improvement comes from a Galois theory argument, exploiting the fact that
$\mathcal{P}$ is $K$-rational.

######  Remark 2.10.

When $p$ is small, it sometimes happens that $X^{(n)}(\mathbb{Q})$ surjects
onto $X^{(n)}(\mathbb{F}_{p})$, in which case this theorem may be used to
determine $X^{(n)}(\mathbb{Q})$ directly (i.e. without sieving). This has the
advantage of requiring no information on the generators of the Mordell–Weil
group of $J(X)$. When $\mathcal{Q}=\mathcal{Q}_{0}$ and the annihilating
differentials come from a rank zero quotient of $J(X)$, it is best to use the
formal immersion criterion of Derickx, Kamienny, Stein and Stoll [Der16,
Chapter 3 Proposition 3.7] instead, which gives the same statement but works
for all primes $p\geq 2$ of good reduction. See for example [DEvH+21, §6.1]
where this is done for $X_{1}(22)^{(3)}(\mathbb{Q})$ and
$X_{1}(25)^{(3)}(\mathbb{Q})$ with $p=3$. Our theorem covers all relative and
positive rank cases, and can often be used with $p=3$.

######  Remark 2.11.

While we have stated the theorem for multiple maps $\rho_{1},\ldots,\rho_{h}$,
we only need the case $h=1$ in our examples, in which case
$\mathcal{P}\in\mathcal{Q}_{0}+\rho_{1}^{*}C_{1}^{(\ell_{1})}(K)$ (the pull-
back part is $K$-rational since $\mathcal{P}$ and $\mathcal{Q}_{0}$ are). Also
note that condition (1) is quite limiting when $h>1$. If, for example, there
are two degree 2 maps $\rho_{1}\colon X\to C_{1}$ and $\rho_{2}\colon X\to
C_{2}$ and $Q\in X(\mathbb{Q})$ then
$\rho_{1}^{*}\rho_{1}(Q)+\rho_{2}^{*}\rho_{2}(Q)\in X^{(4)}(\mathbb{Q})$ does
not satisfy (1).

###  2.6. Trace maps and ramification

We note that all statements in this section are essentially well-known results
in algebraic number theory. We nonetheless give proofs because we could not
find the exact statements in the literature.

Any map $\rho\colon\;X\to C$ of curves as in the statement of Theorem 2.6 may
be ramified at certain points of degree at most $d$.

######  Example 2.12.

When $\rho$ is the quotient map $\rho\colon\;X_{0}(67)\to X_{0}^{+}(67)$,
there is a non-cuspidal rational point $Q\in X_{0}(67)(\mathbb{Q})$ that
ramifies. Ramifying here means that $w_{67}(Q)=Q$. In order to deal with
degree 3 effective divisors such as $3Q=Q+\rho^{*}\rho(Q)\in
X_{0}(67)^{(3)}(\mathbb{Q})$, we study in more detail how differentials
transform under (ramified) maps of curves.

First, we briefly recall some facts about maps between Krull domains. We
consider a integral extension $A\to B$ of Krull domains. Krull domains can be
viewed as higher-dimensional generalisations of Dedekind domains; see e.g.
[Mat89] for their definition and theory. Consider a minimal prime ideal
$\mathfrak{p}\subset A$. Then the localisation $A_{\mathfrak{p}}$ is a DVR. We
denote by $k(\mathfrak{p})$ the residue field of $A_{\mathfrak{p}}$, and for
$f\in A$ by $f(\mathfrak{p})$ the image of $f$ in $k(\mathfrak{p})$. Now
$\mathrm{Frac}(B)/\mathrm{Frac}(A)$ is a finite extension. This field
extension comes with a trace map

$\mathrm{Tr}_{B/A}\colon\mathrm{Frac}(B)\to\mathrm{Frac}(A),$

which by integrality of $B/A$ satisfies $\mathrm{Tr}_{B/A}(B)\subset A$. Let
$\mathfrak{q}_{1},\ldots,\mathfrak{q}_{r}$ be the minimal prime ideals of $B$
above $\mathfrak{p}$, and denote by $e_{i}\geq 1$ the valuation of
$\mathfrak{p}$ in the DVR $B_{\mathfrak{q}_{i}}$. We similarly obtain local
trace maps
$\mathrm{Tr}_{\widehat{B}_{\mathfrak{q}_{i}}/\widehat{A}_{\mathfrak{p}}}$ and
$\mathrm{Tr}_{k(\mathfrak{q}_{i})/k(\mathfrak{p})}$. Denote by
$\mathrm{loc}_{\mathfrak{r}}\colon\mathrm{Frac}(R)\to\mathrm{Frac}(\widehat{R}_{\mathfrak{r}})$
the localisation map at the minimal prime ideal $\mathfrak{r}$ of a Krull
domain $R$.

######  Lemma 2.13.

We have

$\mathrm{loc}_{\mathfrak{p}}\circ\mathrm{Tr}_{B/A}=\sum_{i}\mathrm{Tr}_{\widehat{B}_{\mathfrak{q}_{i}}/\widehat{A}_{\mathfrak{p}}}\circ\mathrm{loc}_{\mathfrak{q}_{i}}.$

Moreover, for each $f\in B$ we have

$\mathrm{Tr}_{B/A}(f)(\mathfrak{p})=\sum_{i}e_{i}\mathrm{Tr}_{k(\mathfrak{q}_{i})/k(\mathfrak{p})}(f(\mathfrak{q}_{i})).$

###### Proof.

Upon localising $A$ and $B$ at the multiplicative subset
$A\setminus\mathfrak{p}$, we may assume that $A$ is a DVR with maximal ideal
$\mathfrak{p}$, and $B$ is a Dedekind domain with maximal ideals
$\mathfrak{q}_{1},\ldots,\mathfrak{q}_{r}$.

Now consider the completion of $B$ with respect to $\mathfrak{p}B$ and apply
the Chinese Remainder Theorem:

(7)
$\displaystyle\widehat{B}\colonequals\lim_{\begin{subarray}{c}\longleftarrow\\\
n\end{subarray}}B/\mathfrak{p}^{n}B=\prod_{i}\widehat{B}_{\mathfrak{q}_{i}}.$

Consider $b\in\widehat{B}\otimes_{\widehat{A}}\mathrm{Frac}(\widehat{A})$. We
define the trace $\mathrm{Tr}_{\widehat{B}/\widehat{A}}(b)$ by taking the
trace of the $\mathrm{Frac}(\widehat{A})$-linear map given by multiplication-
by-$b$ on $\widehat{B}\otimes_{\widehat{A}}\mathrm{Frac}(\widehat{A})$. Here
$B$ is finite and free as an $A$-module, and a basis for $B/A$ also determines
a basis for $\widehat{B}/\widehat{A}$. In particular, for
$b\in\mathrm{Frac}(B)$, we have
$\mathrm{Tr}_{B/A}(b)=\mathrm{Tr}_{\widehat{B}/\widehat{A}}(b)$, which is the
trace of a block matrix on
$\prod_{i}\mathrm{Frac}(\widehat{B}_{\mathfrak{q}_{i}})$. This gives the first
equality.

For the second equation, we begin by reducing the first modulo $\mathfrak{p}$.
It then suffices to show the equality
$e_{i}\mathrm{Tr}_{k(\mathfrak{q}_{i})/k(\mathfrak{p})}(f(\mathfrak{q}_{i}))=\mathrm{Tr}_{\widehat{B}_{\mathfrak{q}_{i}}/\widehat{A}_{\mathfrak{p}}}(f)(\mathfrak{p})$
for each $i$. To compute
$\mathrm{Tr}_{\widehat{B}_{\mathfrak{q}_{i}}/\widehat{A}_{\mathfrak{p}}}(f)(\mathfrak{p})$,
we may first determine the image of $f$ in the $k(\mathfrak{p})$-vector space
$\widehat{B}_{\mathfrak{q}_{i}}/\mathfrak{p}$, and then take the trace down to
$k(\mathfrak{p})$.

We note that

$\widehat{B}_{\mathfrak{q}_{i}}/\mathfrak{p}=\widehat{B}_{\mathfrak{q}_{i}}/\mathfrak{q}_{i}^{e_{i}}\simeq\widehat{B}_{\mathfrak{q}_{i}}/\mathfrak{q}_{i}\times\mathfrak{q}_{i}/\mathfrak{q}_{i}^{2}\times\cdots\times\mathfrak{q}_{i}^{e_{i}-1}/\mathfrak{q}_{i}^{e_{i}}$

as $k(\mathfrak{p})$-vector spaces. The right-hand side is a
$k(\mathfrak{q}_{i})$-vector space, and to compute the trace to
$k(\mathfrak{p})$ we may first compute the trace to $k(\mathfrak{q}_{i})$.
Finally, we may consider a uniformiser $t\in\widehat{B}_{\mathfrak{q}_{i}}$.
If $f=\sum_{n=0}^{\infty}a_{n}t^{n}\in\widehat{B}_{\mathfrak{q}_{i}}$ then the
multiplication-by-$f$ map on $\widehat{B}_{\mathfrak{q}_{i}}/\mathfrak{p}$
has, by the above, $e_{i}$ diagonal entries equal to $a_{0}$, so the trace
becomes $e_{i}a_{0}=e_{i}f(\mathfrak{q}_{i})$. ∎

When $M$ is an $R$-module and $\mathfrak{r}$ is a prime ideal in $R$, denote
by $M_{\mathfrak{r}}$ the localisation of $M$ at $\mathfrak{r}$, and by
$\widehat{M}_{\mathfrak{r}}$ its completion. Now suppose that $A$ and $B$ are
Dedekind domains and that $K$ is a field such that the $A$ and $B$ are
$K$-algebras and $A\to B$ is an embedding of $K$-algebras.

We obtain an embedding
$\Omega_{\mathrm{Frac}(A)/K}\to\Omega_{\mathrm{Frac}(B)/K}$ of the spaces of
Kähler differentials. Now given $s\in A$ (but $s\notin K)$, $\mathrm{d}s$ is a
$\mathrm{Frac}(B)$-basis for $\Omega_{\mathrm{Frac}(B)/K}$, and we obtain a
trace map

$\mathrm{Tr}_{B/A}\colon\;\Omega_{\mathrm{Frac}(B)/K}\to\Omega_{\mathrm{Frac}(A)/K},\;\;f\mathrm{d}s\mapsto\mathrm{Tr}_{B/A}(f)\mathrm{d}s$

which is independent of the choice of $s$. We also denote by
$\mathrm{loc}_{\mathfrak{r}}$ the localisation map on Kähler differentials
with respect to a prime ideal $\mathfrak{r}$.

######  Corollary 2.14.

The equation

$\mathrm{loc}_{\mathfrak{p}}\circ\mathrm{Tr}_{B/A}=\sum_{i}\mathrm{Tr}_{\widehat{B}_{\mathfrak{q}_{i}}/\widehat{A}_{\mathfrak{p}}}\circ\mathrm{loc}_{\mathfrak{q}_{i}}$

also holds true on $\Omega_{\mathrm{Frac}(B)/K}$.

We note that traces of $n$-th roots are easily computed.

######  Lemma 2.15.

Suppose that $F(\alpha)/F$ is a field extension of degree $n$ defined by the
minimal polynomial $\alpha^{n}=a$ for some $a\in F$. Then

$\mathrm{Tr}_{F(\alpha)/F}(\alpha^{i})=0\text{ unless }n\mid i.$

###### Proof.

This follows directly from the shape of the minimal polynomial. ∎

We now consider a map $\rho\colon X\to C$ of curves over a field $K$ and a
$K$-rational point $R\in C(K)$. As divisors, we write
$\rho^{*}R=\sum_{Q\mapsto R}e_{Q/R}Q$ as a sum of points $Q$ on $X$. Assume
that we have chosen $K$ such that $Q\in X(K)$ for each $Q$ mapping to $R$.
Denote by $v_{Q}$ the valuation of the discrete valuation ring
$\widehat{\mathcal{O}}_{X,Q}$.

From now on, suppose that $K$ is a finite extension of $\mathbb{Q}_{p}$ for a
prime $p$ of good reduction for $X$ and $C$, with ring of integers
$\mathcal{O}_{K}$ and prime ideal $\mathfrak{p}$. Then $\rho$ extends to an
$\mathcal{O}_{K}$-morphism $\mathcal{X}\to\mathcal{C}$ between the minimal
proper regular models of $X$ and $C$.

Suppose no two distinct points mapping to $R$ have equal reduction mod
$\mathfrak{p}$ and $p\nmid e_{Q/R}$ for all points $Q\mapsto R$.

######  Lemma 2.16.

Let $t\in\widehat{\mathcal{O}}_{C,R}$ be a well-behaved local coordinate, and
consider $Q\in X$ mapping to $R$. Then, after base changing to an unramified
extension of $K$ of degree at most $e_{Q/R}$, there exists a well-behaved
local coordinate $s_{Q}\in\widehat{\mathcal{O}}_{X,Q}$ such that
$s_{Q}^{e_{Q/R}}=t$. Moreover,
$\rho^{*}\colon\widehat{\mathcal{O}}_{C,R}\to\widehat{\mathcal{O}}_{X,Q}$
induces an embedding of fraction fields of degree $e_{Q/R}$ defined by the
equation $x^{e_{Q/R}}-t=0$.

###### Proof.

Consider $Q\in\rho^{-1}(\\{R\\})$. As $Q$ is a $K$-rational point, its residue
field is $k(Q)=K$. Let $\pi\in\widehat{\mathcal{O}}_{X,Q}$ and
$t\in\widehat{\mathcal{O}}_{C,R}$ be two well-behaved uniformisers, and write
$e=e_{Q/R}$. We can write $\rho^{*}(t)=b_{0}\pi^{e}+b_{1}\pi^{e+1}+\cdots$,
with each $b_{i}\in\mathcal{O}_{K}$. Define
$u=b_{0}+b_{1}\pi+b_{2}\pi^{2}+\cdots\in\widehat{\mathcal{O}}_{X,Q}^{\times}$,
so that $\rho^{*}(t)=u\pi^{e}$. Then the mod $\mathfrak{p}$ reduction of $u$
is $\widetilde{u}=\sum_{n=0}^{\infty}\widetilde{b}_{n}\widetilde{\pi}^{n}$
since $\pi$ is well-behaved. Since also $t$ is well-behaved and
$\widetilde{\rho}$ still has ramification degree $e$ at
$\widetilde{Q}/\widetilde{R}$ by assumption, we must have
$\widetilde{b}_{0}\neq 0$ (c.f Section 2.1). Let $k^{\prime}/k$ be an
extension of the residue field of $K$ containing a root of
$X^{e}-\widetilde{b}_{0}$. Then $k^{\prime}$ corresponds to an unramified
extension $K^{\prime}/K$ and by Hensel’s Lemma (applicable because
$b_{0}\in\mathcal{O}_{K}^{\times}$ and $p\nmid e$), $K^{\prime}$ contains a
root of $X^{e}-b_{0}$. Then also $\widehat{\mathcal{O}}_{X_{K^{\prime}},Q}$
contains an $e$th root of $u$ by Hensel’s lemma, and we simply define
$s_{Q}=u^{1/e}\pi$. The map between fraction fields now has degree $e$ because
$\rho^{*}$ is an isomorphism on residue fields. ∎

Using the lemma, we extend $K$ and define $t$ and $s_{Q}$ (for each $Q\mapsto
R$) to be well-behaved uniformisers satisfying $s_{Q}^{e_{Q/R}}=t$. We
consider the Dedekind domain

$\mathcal{O}_{X,\rho^{*}R}\colonequals\\{f\in K(X)\mid v_{Q}(f)\geq 0\text{
for all }Q\mapsto R\\},$

which is an integral extension of the DVR $\mathcal{O}_{C,R}$. This Dedekind
domain has localisations $\mathcal{O}_{X,Q}$ at all places $Q$ mapping to $R$.
Denote by $\mathrm{Tr}_{Q/R}$ the trace map from
$\mathrm{Frac}(\widehat{\mathcal{O}}_{X,Q})$ to
$\mathrm{Frac}(\widehat{\mathcal{O}}_{C,R})$. Now suppose that $\omega$ is a
global meromorphic differential on $X/K$. We can interpret $\omega$ as a
Kähler differential in $\Omega_{K(X)/K}$. From the integral extension
$\rho^{*}\colon\mathcal{O}_{C,R}\to\mathcal{O}_{X,\rho^{*}R}$ of Dedekind
domains, we thus obtain a trace map

$\mathrm{Tr}\colon\;\Omega_{K(X)/K}\to\Omega_{K(C)/K},$

which for $\omega\in H^{0}(X,\Omega)$ equals the trace map defined in Section
2.1.

######  Proposition 2.17.

Write
$\mathrm{loc}_{Q}(\omega)=\sum_{i=0}^{\infty}a_{i}(Q)s_{Q}^{i}\mathrm{d}s_{Q}$
for the expansion of $\omega$ in
$\Omega_{\mathrm{Frac}(\widehat{\mathcal{O}}_{X,Q})/K}$. Then
$\mathrm{loc}_{R}(\mathrm{Tr}(\omega))=\sum_{Q\mapsto R}\sum_{j\geq
1}a_{je_{Q/R}-1}(Q)t^{j-1}\mathrm{d}t$.

In particular, if $\mathrm{Tr}(\omega)=0$, then for each $j\geq 1$ we have

$\sum_{Q\mapsto R}a_{j\cdot e_{Q/R}-1}(Q)=0.$

######  Remark 2.18.

Note that this equality determines a linear relation between the columns of
the matrix $\widetilde{\mathcal{A}}$ defined in the statement of Theorem 2.6.
For example, when $\rho$ has degree 2 and $Q\in X$ ramifies, the equation says
$a_{1}(Q)=0$, and $\widetilde{\mathcal{A}}$ has a vanishing column, c.f.
Example 4.3.

###### Proof.

We note that $K(X)=\mathrm{Frac}(\mathcal{O}_{X,\rho^{*}R})$ and
$K(C)=\mathrm{Frac}(\mathcal{O}_{C,R})$, and recall that the extension
$\mathcal{O}_{X,\rho^{*}R}/\mathcal{O}_{C,R}$ is integral. We also obtain
local trace maps
$\mathrm{Tr}_{Q/R}\colon\;\Omega_{\mathrm{Frac}(\widehat{\mathcal{O}}_{X,Q})/K}\to\Omega_{\mathrm{Frac}(\widehat{\mathcal{O}}_{C,R})/K}$.
Applying Corollary 2.14 to $\mathcal{O}_{X,\rho^{*}R}/\mathcal{O}_{C,R}$, we
obtain

$\mathrm{loc}_{R}(\mathrm{Tr}(\omega))=\sum_{Q\mapsto
R}\mathrm{Tr}_{Q/R}(\mathrm{loc}_{Q}(\omega))=\sum_{Q\mapsto
R}\mathrm{Tr}_{Q/R}\left(\sum_{i=1}^{\infty}a_{i-1}(Q)s_{Q}^{i-1}\mathrm{d}s_{Q}\right).$

Next, we recall that by Lemma 2.15 we have $\mathrm{Tr}_{Q/R}(s_{Q}^{i})=0$,
unless $i$ is a multiple of $e_{Q/R}$, in which case
$\mathrm{Tr}_{Q/R}(s_{Q}^{je_{Q/R}})=e_{Q/R}t^{j}$. Now
$i\mathrm{Tr}_{Q/R}(s_{Q}^{i-1}\mathrm{d}s_{Q})=\mathrm{d}\mathrm{Tr}_{Q/R}(s_{Q}^{i}),$
from which we deduce that
$\mathrm{loc}_{R}(\mathrm{Tr}(\omega))=\sum_{Q\mapsto
R}\sum_{j=1}^{\infty}a_{je_{Q/R}-1}(Q)t^{j-1}\mathrm{d}t$. Finally, we recall
from Section 2.1 that $\Omega_{\widehat{\mathcal{O}}_{C,R}/K}=K\llbracket
t\rrbracket\mathrm{d}t$, so the equality of power series yields a coefficient-
wise equality. ∎

Finally, we show that traces of uniformisers can be computed as expected.

######  Lemma 2.19.

Consider given $Q\in X(K)$ with $\rho(Q)=R$, again such that no other point
mapping to $R$ has the same reduction as $Q$. Suppose $P\in D(R)$ and write,
as divisors, $\rho^{*}P\cap D(Q)=\sum_{i=1}^{e}P_{i}$ (with possible
repetition), where $e$ is the common ramification index of $Q/R$ and
$\widetilde{Q}/\widetilde{R}$. We base change $X$ and $C$ from $K$ to
$K(P_{1},\ldots,P_{e})$, the extension of $K$ containing all coordinates of
$P_{1},\ldots,P_{e}$. Then for a well-behaved uniformiser $s$ at $Q$ we have

$\mathrm{Tr}_{Q/R}(s)(P)=\sum_{i=1}^{e}s(P_{i}).$

Consequently, if $f(X)=1-b_{1}X+b_{2}X^{2}-\cdots+(-1)^{e}b_{e}X^{e}$ is the
reverse minimal polynomial of $s\in\widehat{\mathcal{O}}_{X,Q}$ over
$\widehat{\mathcal{O}}_{C,R}$, then $b_{i}(P)$ is equal to the $i$th symmetric
polynomial in $s(P_{1}),\ldots,s(P_{e})$.

###### Proof.

We denote by $\widehat{\mathcal{O}}_{\mathcal{X},\widetilde{Q}}$ the completed
local ring (and Krull domain) at the closed point $\widetilde{Q}$ and by
$\widehat{\mathcal{O}}_{\mathcal{X},Q}$ the completed local ring (and DVR) at
the non-closed point (or subscheme of codimension 1) $Q$. If
$\mathfrak{m}_{Q}$ is the (minimal) prime ideal of
$\widehat{\mathcal{O}}_{\mathcal{X},\widetilde{Q}}$ corresponding to $Q$, then
$\widehat{\mathcal{O}}_{\mathcal{X},Q}$ is the localisation of
$\widehat{\mathcal{O}}_{\mathcal{X},\widetilde{Q}}$ at $\mathfrak{m}_{Q}$.
Well-behaved means that
$s\in\widehat{\mathcal{O}}_{\mathcal{X},\widetilde{Q}}$. We note that the map
$\rho^{*}\colon\widehat{\mathcal{O}}_{\mathcal{C},\widetilde{R}}\to\widehat{\mathcal{O}}_{\mathcal{X},\widetilde{Q}}$
determines an integral extension of Krull domains. We define
$A=\widehat{\mathcal{O}}_{\mathcal{C},\widetilde{R}}$ and
$B=\widehat{\mathcal{O}}_{\mathcal{X},\widetilde{Q}}$ and consider the minimal
prime ideal $\mathfrak{m}_{R}\subset A$. To compute $\mathrm{Tr}_{B/A}(s)$, we
apply the first part of Lemma 2.13 to $A\to B$ with
$\mathfrak{p}=\mathfrak{m}_{R}$. By assumption, $\mathfrak{m}_{Q}$ is the only
minimal prime of $B$ above $\mathfrak{m}_{R}$, so we find
$\mathrm{Tr}_{B/A}(s)=\mathrm{Tr}_{Q/R}(s)$. Similarly, the minimal primes
above $\mathfrak{m}_{P}$ are
$\mathfrak{m}_{P_{1}},\ldots,\mathfrak{m}_{P_{e}}$ (with possible repetition)
and we apply the second part of Lemma 2.13 to $A\to B$ and the ideal
$\mathfrak{p}=\mathfrak{m}_{P}$. This yields
$\mathrm{Tr}_{B/A}(s)(P)=\sum_{i=1}^{e}s(P_{i})$, as desired. ∎

###  2.7. Lemmas to control the prime bound

The following lemmas are needed to control the lower bound on the prime $p$ in
Theorems 2.6 and 2.24.

######  Lemma 2.20.

Let $N$, $T$ and $\ell>T$ be positive integers, and $p>N+T$ a prime number.
Then we have

(8) $\ell-T\geq N\mathrm{ord}_{p}(\ell)+1.$

###### Proof.

Write $\ell=p^{a}b$, where $a=\mathrm{ord}_{p}(\ell)$. When $a=0$, this is
true because $\ell>T$. Otherwise, we note that $\ell\geq p^{a}$, and

$p^{a}-T\geq(N+T+1)^{a}-T\geq 1+a(N+T)-T\geq 1+aN,$

as desired. ∎

Next, we consider the inclusions of polynomial rings
$\mathbb{Q}[s_{1},\ldots,s_{n}]=\mathbb{Q}[e_{1},\ldots,e_{n}]\subset\mathbb{Q}[z_{1},\ldots,z_{n}]$,
where $s_{k}=\sum_{\ell=1}^{n}z_{\ell}^{k}$ for each $k\geq 1$ (we also make
this definition for $k>n$), and $e_{k}$ is the $k$th elementary symmetric
polynomial in $z_{1},\ldots,z_{n}$. We adopt the conventions $e_{k}=0$ for
$k>n$ and $e_{0}=1$. Let $\mathfrak{m}\subset\mathbb{Q}[e_{1},\ldots,e_{n}]$
be the ideal generated by $e_{1},\ldots,e_{n}$. Then in the ring of formal
power series $\mathbb{Q}[z_{1},\ldots,z_{n}]\llbracket X\rrbracket$, Newton’s
identities can be given the following compact form:

(9)
$\displaystyle\sum_{k=0}^{\infty}(-1)^{k}e_{k}X^{k}=\prod_{i=1}^{n}(1-z_{i}X)=\mathrm{exp}\left(-\sum_{k=1}^{\infty}\frac{s_{k}}{k}X^{k}\right).$

######  Lemma 2.21.

  * (1)

For each $k\geq 1$, the difference $s_{k}/k-(-1)^{k-1}e_{k}$ is in
$\mathfrak{m}^{2}$, and equals a sum of terms $b\cdot e_{i_{1}}\cdots
e_{i_{\ell}}$ (for $\ell\geq 2$ and $i_{1},\ldots,i_{\ell}\geq 1$) with
coefficient $b\in\mathbb{Q}$ of denominator dividing $\ell$.

  * (2)

If $k>n$ we have

$\frac{s_{k}}{k}-\left(\frac{(-1)^{k}}{2}\sum_{i=k-n}^{n}e_{i}e_{k-i}\right)\in\mathfrak{m}^{3},$

and it equals again a sum of terms $b\cdot e_{i_{1}}\cdots e_{i_{\ell}}$ (for
$\ell\geq 3$) with coefficient $b$ of denominator dividing $\ell$. In
particular, if $k>2n$ then $s_{k}/k\in\mathfrak{m}^{3}$.

  * (3)

Suppose that $S$ is a ring with an ideal $I$, and we have
$a_{1},\ldots,a_{2n}\in I$ satisfying for each $i\leq n$ that $a_{i}\equiv
a_{n+i}\textrm{ mod }I^{w}$, where $w\in\mathbb{Z}_{\geq 1}$. Then
$a_{1}\cdots a_{n}\equiv a_{n+1}\cdots a_{2n}\textrm{ mod }I^{w+n-1}$.

###### Proof.

For part (3), it suffices to note the equality

$a_{1}\cdots a_{n}-a_{n+1}\cdots
a_{2n}=\sum_{i=1}^{n}(a_{i}-a_{i+n})a_{i+1}a_{i+2}\cdots a_{i+n-1}.$

Parts (1) and (2) follow directly from (9) after applying $\mathrm{Log}$ to
both sides of the equation and comparing coefficients for $X^{k}$. ∎

###  2.8. Proof of Theorem 2.6

For simplicity of exposition (to avoid using triple indices), we assume that
$h=1$. Then there is one map $\rho\colon X\to C$, and we assume moreover that

$\mathcal{Q}_{1}=\rho^{*}(m\cdot R)\text{ for }R\in C(K)\text{ and
}m\in\mathbb{Z}_{>0}.$

For the general case, one can simply sum all arguments that follow over the
points in the curves $C_{j}$, because $\mathcal{Q}_{1},\ldots,\mathcal{Q}_{h}$
are disjoint. We write $\rho^{*}R=\sum_{i=1}^{k}m_{i}Q_{i}$, where $m_{i}\geq
0$ for each $i$, and

$\mathcal{Q}=\sum_{i=1}^{k}n_{i}Q_{i},$

where $n_{i}\geq m\cdot m_{i}$ for each $i$. We choose the ordering so that
$m_{i}=0$ for all $i>k^{\prime}$, where $k^{\prime}\leq k$. Note that
$\mathcal{Q}_{0}=\sum_{i=1}^{k}(n_{i}-mm_{i})Q_{i}$ and
$\mathcal{Q}_{1}=\sum_{i=1}^{k^{\prime}}mm_{i}Q_{i}$.

Let $L=K(\mathcal{Q})$ be the field obtained by adjoining the coordinates of
$Q_{1},\ldots,Q_{k}$ and consider all points, maps and curves as base-changed
to $L$. Let $t_{R}$ be a well-behaved uniformiser at $R$. Using Lemma 2.16, at
each of the distinct points $Q_{i}$, consider a well-behaved uniformiser
$t_{Q_{i}}\in\widehat{\mathcal{O}}_{X,Q_{i}}$ satisfying
$t_{Q_{i}}^{m_{i}}=t_{R}$ if $i\leq k^{\prime}$. For this we need to extend
$L$ by an unramified extension, which we still denote by $L$. Recall that
$\mathcal{P}$ is the residue disc of $\mathcal{Q}$. Consider
$F:=L(\mathcal{P})$ and let $\mathfrak{q}$ be a prime in $F$ above
$\mathfrak{p}$. We write

$\mathcal{P}=\sum_{i=1}^{k}(P_{i,1}+\cdots+P_{i,n_{i}}),\text{ where
}P_{i,j}\equiv Q_{i}\textrm{ mod }\mathfrak{q}\text{ for each
}j\in\\{1,\ldots,n_{i}\\}.$

We write $z_{i,j}:=t_{Q_{i}}(P_{i,j})$ and
$s_{i,\ell}=\sum_{j=1}^{n_{i}}z_{i,j}^{\ell}$, and let $e_{i,\ell}$ be the
$\ell$th elementary symmetric polynomial in $z_{i,1},\ldots,z_{i,n_{i}}$. Note
that each $z_{i,j}\in\mathfrak{q}\mathcal{O}_{F_{\mathfrak{q}}}$ because
$t_{Q_{i}}$ is well-behaved. Fix one annihilating differential $\omega$ of
trace zero for all maps and write $\mathrm{loc}_{Q_{i}}(\omega)=\sum_{j\geq
0}a_{j}(Q_{i},\omega)t_{Q_{i}}^{j}\mathrm{d}t_{Q_{i}}$. After multiplying
$\omega$ with a scalar, we may assume $\omega\in H^{0}(\mathcal{X},\Omega)$
and each $a_{j}(Q_{i},\omega)\in\mathcal{O}_{L_{\mathfrak{p}}}$.

For $i>k^{\prime}$, we know that $e_{i,1}=\cdots=e_{i,n_{i}}=0$ implies that
$z_{i,j}=0$ for all $j\in\\{1,\ldots,n_{i}\\}$, and hence $P_{i,j}=Q_{i}$ for
all $j\in\\{1,\ldots,n_{i}\\}$ by injectivity of $t_{Q_{i}}$ on residue discs.
For $i\leq k^{\prime}$, we would like to be able to read off from the
$e_{i,\ell}$ whether $\mathcal{P}$ is (partially) a pullback.

######  Lemma 2.22.

Consider the notation as defined so far, and define the effective divisor
$\mathcal{P}^{\prime}=\sum_{i=1}^{k^{\prime}}(P_{i,1}+\cdots+P_{i,n_{i}})$
(sum only up to $k^{\prime}$).

Then
$\mathcal{P^{\prime}}=\sum_{i=1}^{k^{\prime}}(n_{i}-mm_{i})Q_{i}+\rho^{*}\mathcal{S}$
for some $\mathcal{S}\in C^{(m)}(\overline{K})$ if (and only if) there exists
a polynomial $f\in\overline{K}_{\mathfrak{r}}[X]$ of degree $m$ with constant
coefficient 1 such that for all $i\in\\{1,\ldots,k^{\prime}\\}$ we have an
equality of polynomials

$1-e_{i,1}X+\cdots+(-1)^{n_{i}}e_{i,n_{i}}X^{n_{i}}=f(X^{m_{i}}).$

###### Proof.

Denote by $\mathcal{R}$ the set of inverses of roots of $f$. Choose
$i\in\\{1,\ldots,k^{\prime}\\}$. By assumption, we find that

$\prod_{j=1}^{n_{i}}(1-z_{i,j}X)=f(X^{m_{i}}).$

In particular, after reordering we find that

  * (a)

$z_{i,mm_{i}+1}=0,\ldots,z_{i,n_{i}}=0$, and moreover

  * (b)

$\\{z_{i,j}^{m_{i}}\mid j\in\\{1,\ldots,mm_{i}\\}\\}=\mathcal{R}$.

Because well-behaved uniformisers are injective on residue classes, (a)
implies that $Q_{i}=P_{i,mm_{i}+1}=\cdots=P_{i,n_{i}}$. Recall that
$z_{i,j}^{m_{i}}=t_{R}(\rho(P_{i,j}))$ for $j\leq mm_{i}$. From (b) and
because $t_{R}$ is injective on $D(R)$, we find that $\\{\rho(P_{ij})\mid
i\in\\{1,\ldots,k^{\prime}\\},j\in\\{1,\ldots,mm_{i}\\}\\}$ has size $m$.
Denote these points by $S_{1},\ldots,S_{m}$. They satisfy $S_{\ell}\equiv
R\textrm{ mod }\mathfrak{q}$ for each $\ell$, and
$\\{t_{R}(S_{\ell})\mid\ell\in\\{1,\ldots,m\\}\\}=\mathcal{R}$.

Write $\rho^{*}(S_{\ell})=\sum_{i=1}^{k^{\prime}}\sum_{j=1}^{m_{i}}P_{\ell
ij}^{\prime}$, where again $P_{\ell ij}^{\prime}\equiv Q_{i}$ modulo a fixed
prime above $\mathfrak{q}$ in $F(\rho^{*}S_{\ell})$ for each $i$. Write
$z_{\ell ij}^{\prime}=t_{Q_{i}}(P^{\prime}_{\ell ij})$, and define $e_{\ell
ij}^{\prime}$ to be the $j$th elementary symmetric polynomial in
$z_{\ell,i,1}^{\prime},\ldots,z_{\ell,i,m_{i}}^{\prime}$. Recall that
$t_{Q_{i}}^{m_{i}}=\rho^{*}t_{R}$. From Lemma 2.19, we see that
$e_{\ell,i,j}^{\prime}=0$ for each $j\in\\{1,\ldots,m_{i}-1\\}$ and
$e^{\prime}_{\ell,i,m_{i}}=(-1)^{m_{i}+1}t_{R}(S_{\ell})$. We conclude that

$\left(\prod_{\ell=1}^{m}\prod_{j=1}^{m_{i}}(1-z_{\ell,i,j}^{\prime}X)\right)=\prod_{\ell=1}^{m}(1-t_{R}(S_{\ell})X^{m_{i}})=f(X^{m_{i}})$

for each $i$, from which it follows by injectivity of $t_{Q_{i}}$ that

$P_{i,1}+\cdots+P_{i,mm_{i}}=\sum_{\ell=1}^{m}(P_{\ell,i,1}^{\prime}+\cdots+P_{\ell,i,m_{i}}^{\prime}),$

and thus
$\mathcal{P}=\sum_{i=1}^{k}(n_{i}-mm_{i})Q_{i}+\rho^{*}(S_{1}+\cdots+S_{m})$.
∎

We continue the proof of Theorem 2.6. As $[\mathcal{P}-\mathcal{Q}]\in J(K)$
and $\omega$ is a annihilating differential, we find that
$\int_{\mathcal{Q}}^{\mathcal{P}}\omega=0.$ This is a sum of tiny integrals
equal by Proposition 2.5 (i) to

(10)
$\sum_{i=1}^{k}\sum_{\ell=1}^{\infty}a_{\ell-1}(Q_{i},\omega)\dfrac{s_{i,\ell}}{\ell}=0.$

For $i\in\\{1,\ldots,k^{\prime}\\}$, write
$p_{i,\ell}:=\frac{1}{m_{k^{\prime}}}\sum_{j=1}^{n_{k^{\prime}}}\mathrm{Tr}_{Q_{i}/R}(t_{Q_{i}}^{\ell})(\rho(P_{k^{\prime},j}))$.
Then $p_{i,\ell}=0$ when $m_{i}\nmid\ell$ and
$\frac{p_{i,jm_{i}}}{jm_{i}}=\frac{s_{k^{\prime},jm_{k^{\prime}}}}{jm_{k^{\prime}}}$
for $j\geq 1$. Now we consider $i=k^{\prime}$, and subtract

$0=\frac{1}{m_{k^{\prime}}}\int_{n_{k^{\prime}}Q_{k^{\prime}}}^{P_{k^{\prime},1}+\cdots+P_{k^{\prime},n_{k^{\prime}}}}\rho^{*}\mathrm{Tr}(\omega)=\sum_{i=1}^{k^{\prime}}\sum_{\ell=1}^{\infty}\frac{a_{\ell-1}(Q_{i},\omega)}{\ell}p_{i,\ell}$

from (10). Here we used Proposition 2.17 to evaluate the integral.

We thus obtain

(11)
$\sum_{i=1}^{k^{\prime}}\sum_{\ell=1}^{\infty}a_{\ell-1}(Q_{i},\omega)\left(\dfrac{s_{i,\ell}-p_{i,\ell}}{\ell}\right)+\sum_{i=k^{\prime}+1}^{k}\sum_{\ell\geq
1}a_{\ell-1}(Q_{i},\omega)\dfrac{s_{i,\ell}}{\ell}=0.$

For $i\leq k^{\prime}$, define $r_{i,\ell}$ for $\ell\in\mathbb{Z}_{\geq 1}$
by $r_{i,\ell}=0$ when $m_{i}\nmid\ell$ and
$r_{i,jm_{i}}=(-1)^{jm_{i}-jm_{k^{\prime}}}e_{k^{\prime},jm_{k^{\prime}}}$.
For $i>k^{\prime}$, define $r_{i,\ell}=0$ for all $\ell$. We now use Lemma
2.21 (1) to rewrite traces/power sums in terms of elementary symmetric
polynomials:

$p_{i,jm_{i}}/jm_{i}=s_{k^{\prime},jm_{k^{\prime}}}/jm_{k^{\prime}}=(-1)^{jm_{i}-1}r_{i,jm_{i}}+\text{higher-
order terms}$

and $s_{i,\ell}/\ell=(-1)^{\ell-1}e_{i,\ell}+\text{higher-order terms}$. This
yields

(12) $\displaystyle
0=\sum_{i=1}^{k}\sum_{\ell=1}^{n_{i}}(-1)^{\ell-1}a_{\ell-1}(Q_{i},\omega)(e_{i,\ell}-r_{i,\ell})+\text{higher-
order terms},$

where the higher order terms are of the form

(13) $\displaystyle\frac{\alpha}{s}\left(e_{i,i_{1}}\cdots
e_{i,i_{s}}-r_{i,i_{1}}\cdots r_{i,i_{s}}\right)\text{ with }s\geq 2\text{ and
}\alpha\in\mathcal{O}_{L_{\mathfrak{p}}}.$

Now define
$f(X)=1+(-1)^{m_{k^{\prime}}}e_{k^{\prime},m_{k^{\prime}}}X+\cdots+(-1)^{mm_{k^{\prime}}}e_{k^{\prime},mm_{k^{\prime}}}X^{m}$.
Then for each $i\leq k^{\prime}$, we find that
$f(X^{m_{i}})=1-r_{i,1}X+\cdots+(-1)^{n_{i}}r_{i,n_{i}}X^{n_{i}}$. In view of
Lemma 2.22, it is our aim to show that for each
$i\in\\{1,\ldots,k^{\prime}\\}$, we have

$f(X^{m_{i}})=1-e_{i,1}X+\cdots+(-1)^{n_{i}}e_{i,n_{i}}X^{n_{i}},$

and for $i>k^{\prime}$ we have $e_{i,\ell}=0$ for all
$\ell\in\\{1,\ldots,n_{i}\\}$. In other words, we aim to show
$e_{i,\ell}=r_{i,\ell}$ for all $i\in\\{1,\ldots,k\\}$ and $\ell\leq n_{i}$.

We first show that $e_{i,\ell},r_{i,\ell}\in L_{\mathfrak{p}}$. Note that
$F/L$ is Galois. Suppose that
$\sigma\in\mathrm{Gal}(F_{\mathfrak{q}}/L_{\mathfrak{p}})$. For each
$P_{i,j}\in\mathcal{P}$ (that is, in the support of $\mathcal{P}$) reducing
mod $\mathfrak{q}$ to $\widetilde{P_{i,j}}=\widetilde{Q_{i}}$, also
$P_{i,j}^{\sigma}$ must reduce to
$\widetilde{Q_{i}}=\widetilde{Q_{i}^{\sigma}}$. Also
$P_{i,j}^{\sigma}\in\mathcal{P}$ as $\mathcal{P}$ is $K$-rational, so
$P_{i,j}^{\sigma}=P_{i,j^{\prime}}$ for some
$j^{\prime}\in\\{1,\ldots,n_{i}\\}$. As moreover
$t_{Q_{i}}^{\sigma}=t_{Q_{i}}$, we find that $e_{i,\ell}^{\sigma}=e_{i,\ell}$.
By definition, then also $r_{i,\ell}^{\sigma}=r_{i,\ell}$, and all
$e_{i,\ell}$ and $r_{i,\ell}$ are in $L_{\mathfrak{p}}$. We define

$\nu:=\mathrm{min}_{\begin{subarray}{c}i\geq 1\\\ \ell\leq
n_{i}\end{subarray}}v_{\mathfrak{p}}(e_{i,\ell}-r_{i,\ell}),$

where $v_{\mathfrak{p}}$ is $\mathfrak{p}$-adic valuation. We argue by
contradiction and assume that $\nu<\infty$. Recall that $\nu\geq 1$ because
$\mathcal{P}$ and $\mathcal{Q}$ are in the same residue disc.

Note that in fact $e_{i,\ell}\in K(Q_{i})$ and $r_{i,\ell}\in
K(Q_{k^{\prime}})$. Let $\mathfrak{p}_{i}$ be the prime of
$K(Q_{i},Q_{k^{\prime}})$ below $\mathfrak{p}$ and let
$\nu_{i}:=\lceil\nu/e_{\mathfrak{p}/\mathfrak{p}_{i}}\rceil$, where
$e_{\mathfrak{p}/\mathfrak{p}_{i}}$ denotes the ramification index. Recall
that $N\geq e_{\mathfrak{p}_{i}/p}$. We apply Lemma 2.21 (3) with
$I=\mathfrak{p}_{i}$, which yields $e_{i,i_{1}}\cdots e_{i,i_{s}}\equiv
r_{i,i_{1}}\cdots r_{i,i_{s}}\textrm{ mod }\mathfrak{p}_{i}^{\nu_{i}+s-1}$. By
Lemma 2.20 with $T=1$, and because $p\geq e_{\mathfrak{p}_{i}/p}+2$, we find
that $s-1\geq e_{\mathfrak{p}_{i}/p}\mathrm{ord}_{p}(s)+1$. We conclude that
each of the higher order terms (13) vanishes mod
$\mathfrak{p}_{i}^{\nu_{i}+1}$, hence mod $\mathfrak{p}^{\nu+1}$.

Now let $\omega_{1},\ldots,\omega_{q}\in H^{0}(\mathcal{X},\Omega)$ be
linearly independent generators for the space of annihilating differentials
with trace zero with respect to each $\rho_{i}$. We obtain equation (12) for
each $\omega\in\\{\omega_{1},\ldots,\omega_{q}\\}$. Let $\mathcal{A}$ be the
matrix made up of the $v(\omega_{j},t_{Q_{i}},n_{i})$, so that its reduction
mod $\mathfrak{p}$ is the matrix $\widetilde{\mathcal{A}}$ from the statement
of the theorem. Denote by $\mathcal{B}$ the matrix obtained from $\mathcal{A}$
by removing for each $j\in\\{1,\ldots,m\\}$ the column with entries

$((-1)^{jm_{k^{\prime}}}a_{jm_{k^{\prime}}-1}(Q_{k^{\prime}},\omega_{1}),\ldots,(-1)^{jm_{k^{\prime}}}a_{jm_{k^{\prime}}-1}(Q_{k^{\prime}},\omega_{q})).$

By Proposition 2.17, the columns of $\mathcal{A}$ satisfy $m$ linear
equations, each of which has coefficient 1 for exactly one of the removed
columns. Since $\widetilde{\mathcal{A}}$ has rank $n-m$, we conclude that the
reduction $\widetilde{\mathcal{B}}$ of $\mathcal{B}$ has full rank $n-m$.
Denote by $\mathbf{v}$ the vector of length $n-m$ with as entries the list
$e_{1,1}-r_{1,1},\ldots,e_{1,n_{1}}-r_{1,n_{1}},\ldots,e_{k,n_{k}}-r_{k,n_{k}}$,
from which we remove the elements
$e_{k^{\prime},jm_{k^{\prime}}}-r_{k^{\prime},jm_{k^{\prime}}}$ for
$j\in\\{1,\ldots,m\\}$ (each of which is zero by definition of $r_{i,\ell}$).
Then $\mathbf{v}=0\textrm{ mod }\mathfrak{p}^{\nu}$ by definition of $\nu$,
and from (12) and the analysis of the higher order terms, we conclude that

$\mathcal{B}\cdot\mathbf{v}=0\textrm{ mod }\mathfrak{p}^{\nu+1}.$

Since $\widetilde{\mathcal{B}}$ has full rank, we find that
$\mathbf{v}=0\textrm{ mod }\mathfrak{p}^{\nu+1}$, contradicting the definition
of $\nu$, unless indeed $\nu=\infty$ and $\mathbf{v}=0$. We find that
$e_{i,\ell}=r_{i,\ell}$ for all values of $i$ and $\ell$, as desired.

For $i>k^{\prime}$, this means $z_{i,j}=0$ for all $j\in\\{1,\ldots,n_{i}\\}$,
so that $\mathcal{P}_{i,j}=Q_{i}$ for each such $i$ and $j$. We then apply
Lemma 2.22 with
$\mathcal{P}^{\prime}=\sum_{i=1}^{k^{\prime}}(P_{i,1}+\cdots+P_{i,n_{i}})$ and
polynomial $f$. We conclude that $\mathcal{P}$ is of the desired form.
$\blacksquare$

###  2.9. Chabauty using the higher order terms

In some situations, there is an obstruction that prevents the matrix in
Theorem 2.6 to have sufficiently high rank at any prime, but we do expect the
outcome of the theorem to hold true.

######  Example 2.23.

Let us see an example of this obstruction. Consider the degree 2 map
$\rho\colon\;X_{0}(73)\to X_{0}^{+}(73)$. Let $c_{0},c_{\infty}\in
X_{0}(73)(\mathbb{Q})$ be the two cusps, and consider $\mathcal{Q}=3c_{0}$.
Note that $w_{73}$ interchanges these cusps, i.e.
$\rho(c_{0})=\rho(c_{\infty})$. Let $t$ be the pullback under $\rho$ of a
well-behaved uniformiser at $\rho(c_{0})$. Then $t$ is a well-behaved
uniformiser at $c_{0}$ and at $c_{\infty}$. The curve $X_{0}(73)$ has genus 5,
and its quotient $X_{0}^{+}(73)$ has genus 2. We find a 3-dimensional space of
annihilating differentials with trace zero on $X_{0}(73)$ (c.f. [Box21]).
Denote by $\omega_{1},\omega_{2},\omega_{3}$ a basis. By Proposition 2.17, if
we write $\mathrm{loc}_{c_{0}}(\omega_{i})=\sum_{j\geq
0}a_{i,j}t^{j}\mathrm{d}t$ then
$\mathrm{loc}_{c_{\infty}}(\omega_{i})=\sum_{j\geq
0}(-a_{i,j})t^{j}\mathrm{d}t$. Consider the matrices

$\mathcal{A}_{0}=(a_{i,j})_{\begin{subarray}{c}1\leq i\leq 3\\\ 0\leq j\leq
2\end{subarray}}\text{ and }\mathcal{A}_{\infty}=-\mathcal{A}_{0}.$

Then $\mathcal{A}:=(\mathcal{A}_{0}\mid\mathcal{A}_{\infty})$ satisfies
$\mathrm{rk}(\mathcal{A})=\mathrm{rk}(\mathcal{A}_{0})$. Also, $\mathcal{A}$
is the matrix whose mod $p$ reduction corresponds to the matrix in condition
(3) of Theorem 2.6 for the point $\mathcal{Q}=3c_{0}+3c_{\infty}$. If its
reduction $\widetilde{\mathcal{A}}$ modulo any large prime $p$ had rank 3, the
entire mod $p$ residue class of $3c_{0}+3c_{\infty}$ would be contained in
$\rho^{*}\left((X_{0}^{+}(73))^{(3)}(\mathbb{Q})\right)$

This is not the case, however. We compute that the Riemann–Roch space
$L(3c_{0}+3c_{\infty})$ is 3-dimensional, which is unusually large. In fact,
we find a degree 6 function $f\in L(3c_{0}+3c_{\infty})$ such that
$w_{73}^{*}f\neq f$, i.e. the corresponding map $f:X\to\mathbb{P}^{1}$ does
not factor via $\rho$. Now note that $3c_{0}+3c_{\infty}=f^{*}(1:0)$. So for
each prime $p$ of good reduction, the points $R\in\mathbb{P}^{1}(\mathbb{Q})$
such that $R\equiv(1:0)\textrm{ mod }p$ satisfy also that $f^{*}R\equiv
3c_{0}+3c_{\infty}$. But, as $f$ does not factor via $X_{0}^{+}(73)$, $f^{*}R$
is in general not the pullback of a degree 3 divisor on $X_{0}^{+}(73)$. By
Theorem 2.24, $\widetilde{\mathcal{A}}$ therefore cannot have rank 3, and
neither can $\widetilde{\mathcal{A}}_{0}$. Indeed, we find that $\omega_{3}$
satisfies $a_{3,0}=a_{3,1}=a_{3,2}=0$, so that the entire bottom row of
$\mathcal{A}_{0}$ is zero. Therefore, condition (3) in Theorem 2.6 is never
satisfied for $\mathcal{Q}=3c_{0}$.

A similar situation occurs on $X_{0}(57)$.

In such cases, it can help to look further into the expansion of the 1-forms.
We begin with some notation. Consider a point $Q$ on a curve $X$ and a regular
1-form $\omega\in H^{0}(X,\Omega)$. We expand $\omega$ around $Q$ in terms of
a uniformiser $t_{Q}$ at $Q$, giving $\mathrm{loc}_{Q}(\omega)=\sum_{j\geq
0}a_{j}t_{Q}^{j}$, and define

$\displaystyle v(\omega,t_{Q},\ell,k)$
$\displaystyle:=\left((-1)^{\ell-1}a_{\ell},(-1)^{\ell}a_{\ell+1},\ldots,(-1)^{k-2}a_{k-1}\right).$

With this notation, we have $v(\omega,t_{Q},k)=v(\omega,t_{Q},0,k)$. For
integers $j$ and $i$ such that $j+1\leq i\leq 2j$ and any $x_{1},\ldots,x_{j}$
in some field, we define

$\psi_{i}(x_{1},\ldots,x_{j})=\sum_{\ell=i-j}^{j}x_{\ell}x_{i-\ell}.$

######  Theorem 2.24.

Consider a number field $K$, a curve $X/K$ and
$\mathcal{Q}=\sum_{i=1}^{k}n_{i}Q_{i}\in X^{(n)}(K)$, where
$n=\sum_{i=1}^{k}n_{i}$ and $Q_{1},\ldots,Q_{k}\in X$ are distinct points. Let
$\mathfrak{r}$ be a prime of $K$, of good reduction for $X$, containing the
rational prime $p\in\mathfrak{r}$. Denote by
$\mathcal{X}/\mathcal{O}_{K_{\mathfrak{r}}}$ a minimal proper regular model of
$X/K$. Let $N$ be the ramification index of $p$ in $K(\mathcal{Q})$.

  * (1)

Suppose that $p\geq N+3$.

  * (2)

Suppose that the space $\mathcal{V}$ of annihilating differentials has
dimension at least $n$, and consider linearly independent
$\omega_{1},\dots,\omega_{n}\in\mathcal{V}$ on $X$. For each $Q_{i}$, choose a
uniformiser $t_{Q_{i}}$, and suppose that the $n\times n$-matrix

$\mathcal{A}:=\begin{pmatrix}v(\omega_{1},t_{Q_{1}},n_{1})&\cdots&v(\omega_{1},t_{Q_{k}},n_{k})\\\
\vdots&\ddots&\vdots\\\
v(\omega_{n},t_{Q_{1}},n_{1})&\cdots&v(\omega_{n},t_{Q_{k}},n_{k})\end{pmatrix}$

has rank $r<n$.

We thus choose uniformisers $t_{\widetilde{Q}_{i}}$ at $\widetilde{Q}_{i}$,
and a basis $\widetilde{\omega}_{1},\ldots,\widetilde{\omega}_{n}$ for the
image of $\mathcal{V}\cap H^{0}(\mathcal{X},\Omega)$ under reduction mod
$\mathfrak{r}$, such that the vectors
$v(\widetilde{\omega}_{j},t_{\widetilde{Q}_{i}},n_{i})$ for all $1\leq i\leq
k$ and $r<j\leq n$ are zero vectors.

  * (3)

Assume that the $r\times n$ matrix $\widetilde{\mathcal{A}}_{1}$ defined below
has rank $r$.

  * (4)

Let $\mathbb{F}$ be the residue field of a prime in $K(\mathcal{Q})$ above
$\mathfrak{r}$. Suppose moreover that there are no non-zero vectors
$(\widetilde{\mathbf{x}}_{1},\dots,\widetilde{\mathbf{x}}_{k})\in\mathbb{F}^{n}$
solving the system of equations
$\widetilde{\mathcal{L}}\cdot\widetilde{\mathbf{w}}=\textbf{0}$, where

$\widetilde{\mathbf{x}}_{i}=(\widetilde{x}_{i,1},\ldots,\widetilde{x}_{i,n_{i}}),\quad\widetilde{\mathcal{L}}=\begin{pmatrix}\widetilde{\mathcal{A}}_{1}&0\\\
0&\widetilde{\mathcal{A}}_{2}\end{pmatrix},\quad\widetilde{\mathcal{A}}_{1}:=\begin{pmatrix}v(\widetilde{\omega}_{1},t_{\widetilde{Q}_{1}},n_{1})&\cdots&v(\widetilde{\omega}_{1},t_{\widetilde{Q}_{k}},n_{k})\\\
\vdots&\ddots&\vdots\\\
v(\widetilde{\omega}_{r},t_{\widetilde{Q}_{1}},n_{1})&\cdots&v(\widetilde{\omega}_{r},t_{\widetilde{Q}_{k}},n_{k})\end{pmatrix},$
$\widetilde{\mathcal{A}}_{2}=\begin{pmatrix}-v(\widetilde{\omega}_{r+1},t_{\widetilde{Q}_{1}},n_{1},2n_{1})&\cdots&-v(\widetilde{\omega}_{r+1},t_{\widetilde{Q}_{k}},n_{k},2n_{k})\\\
\vdots&\ddots&\vdots\\\
-v(\widetilde{\omega}_{n},t_{\widetilde{Q}_{1}},n_{1},2n_{1})&\cdots&-v(\widetilde{\omega}_{n},t_{\widetilde{Q}_{k}},n_{k},2n_{k})\end{pmatrix},$

and

$\widetilde{\mathbf{w}}=(\widetilde{\mathbf{x}}_{1},\ldots,\widetilde{\mathbf{x}}_{k},\psi_{n_{1}+1}(\widetilde{\mathbf{x}}_{1}),\ldots,\psi_{2n_{1}}(\widetilde{\mathbf{x}}_{1}),\psi_{n_{2}+1}(\widetilde{\mathbf{x}}_{2}),\ldots,\psi_{2n_{2}}(\widetilde{\mathbf{x}}_{2}),\ldots,\psi_{n_{k}+1}(\widetilde{\mathbf{x}}_{k}),\ldots,\psi_{2n_{k}}(\widetilde{\mathbf{x}}_{k}))^{T}.$

Then $\mathcal{Q}\in X^{(n)}(K)$ is alone in its mod $\mathfrak{r}$ residue
class.

######  Remark 2.25.

We emphasize that $\mathrm{rk}(\mathcal{A})<n$ means that modulo any prime
$\mathfrak{p}$ of $K(\mathcal{Q})$ the reduction of $\mathcal{A}$ mod
$\mathfrak{p}$ has rank smaller than $n$, and Theorem 2.6 therefore cannot be
applied.

###### Proof.

Let $\mathfrak{p}$ be a prime above $\mathfrak{r}$ in $L:=K(\mathcal{Q})$ with
uniformiser $\pi\in\mathfrak{p}$. Let
$\mathcal{P}=\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}P_{i,j}\in X^{(n)}(K)$ belong to
the residue disc of $\mathcal{Q}$, where the sum is arranged such that points
$P_{i,j}$ reduce to the same point as $Q_{i}$ modulo a prime $\mathfrak{q}$ of
$F:=L(\mathcal{P})$ above $\mathfrak{p}$. For each $i$, choose $t_{Q_{i}}$ to
be a well-behaved uniformiser at $Q_{i}$, and for each $i$ and $j$, define
$z_{i,j}=t_{Q_{i}}(P_{i,j})\in\mathcal{O}_{F_{\mathfrak{q}}}$. Define
$s_{i,\ell}=\sum_{j=1}^{n_{i}}z_{i,j}^{\ell}$, and let $e_{i,\ell}$ be the
$\ell$th elementary symmetric polynomial in $z_{i,1},\ldots,z_{i,n_{i}}$.

In the statement of the theorem, condition (3) implies that
$\mathrm{Span}\\{\widetilde{\omega}_{r+1},\ldots,\widetilde{\omega}_{n}\\}$ is
exactly the space of differentials $\widetilde{\omega}$ on $\widetilde{X}$
satisfying $v(\widetilde{\omega},t_{\widetilde{Q}_{i}},n_{i})=0$ for all $i$.
Note that condition (4) is independent of the choice of bases for
$\mathrm{Span}\\{\widetilde{\omega}_{1},\ldots,\widetilde{\omega}_{r}\\}$ and
$\mathrm{Span}\\{\widetilde{\omega}_{r+1},\ldots,\widetilde{\omega}_{n}\\}$.
Now let $\omega_{1},\ldots,\omega_{n}\in\mathcal{V}\cap
H^{0}(\mathcal{X},\Omega)$ be linearly independent such that
$v(\omega_{j},t_{Q_{i}},n_{i})=0$ for all $i$ and for each
$j\in\\{r+1,\ldots,n\\}$. Having chosen well-behaved uniformisers and integral
differential forms we conclude that condition (4) holds for the reductions
$\widetilde{\omega}_{1},\ldots,\widetilde{\omega}_{n}$ of
$\omega_{1},\ldots,\omega_{n}$ modulo $\mathfrak{r}$. Moreover, the matrices
$\widetilde{\mathcal{L}},\widetilde{\mathcal{A}}_{1}$ and
$\widetilde{\mathcal{A}}_{2}$ are the mod $\mathfrak{p}$ reductions of
corresponding matrices over $\mathcal{O}_{L_{\mathfrak{p}}}$, defined in terms
of $\omega_{1},\ldots,\omega_{n}$.

As in the proof of Theorem 2.6, we find that each
$e_{i,j}\in\mathcal{O}_{L_{\mathfrak{p}}}$ by Galois theory. Define

$\nu:=\mathrm{min}_{\begin{subarray}{c}i\in\\{1,\ldots,k\\}\\\
j\in\\{1,\ldots,n_{i}\\}\end{subarray}}v_{\mathfrak{p}}(e_{i,j})$

and assume that $\nu<\infty$. We aim to find a contradiction, giving
$\nu=\infty$. Recall that $\nu\geq 1$ because $\mathcal{P}$ and $\mathcal{Q}$
are in the same residue disc. Now define $x_{i,\ell}$ by
$e_{i,\ell}=\pi^{\nu}x_{i,\ell}$. Their mod $\mathfrak{p}$ reductions
$\widetilde{x}_{i,\ell}$ will correspond to a non-zero solution of
$\widetilde{\mathcal{L}}\cdot\widetilde{\mathbf{w}}=\mathbf{0}$, which we
shall show in two parts.

For $\omega\in\\{\omega_{1},\ldots,\omega_{n}\\}$, we have at each $Q_{i}$ the
expansion $\mathrm{loc}_{Q_{i}}(\omega)=\sum_{\ell\geq
0}a_{\ell}(Q_{i},\omega)t_{Q_{i}}^{\ell}\mathrm{d}t_{Q_{i}}$. As before, we
obtain from $\int_{\mathcal{Q}}^{\mathcal{P}}\omega=0$ the equation

(14)
$0=\sum_{i=1}^{k}\sum_{\ell=1}^{\infty}a_{\ell-1}(Q_{i},\omega)\frac{s_{i,\ell}}{\ell}=0.$

We again use Lemma 2.21 (1) to rewrite this in terms of $e_{i,\ell}$s, and
apply Lemma 2.21 (3) and Lemma 2.20 (with $T=1$) to the higher order terms to
obtain

(15)
$\sum_{i=1}^{k}\sum_{\ell=1}^{n_{i}}(-1)^{\ell-1}a_{\ell-1}(Q_{i},\omega)e_{i,\ell}\equiv
0\pmod{\mathfrak{p}^{\nu+1}}.$

Dividing by $\pi^{\nu}$ and ranging over
$\omega\in\\{\omega_{1},\ldots,\omega_{r}\\}$, we obtain
$\widetilde{\mathcal{A}}_{1}\cdot(\widetilde{\mathbf{x}}_{1},\ldots,\widetilde{\mathbf{x}}_{k})^{T}=\mathbf{0}$,
the “upper half” of
$\widetilde{\mathcal{L}}\cdot\widetilde{\mathbf{w}}=\mathbf{0}$.

Consider now any of the remaining annihilating differential forms
$\omega\in\\{\omega_{r+1},\ldots,\omega_{n}\\}$. Recall that all
$a_{\ell-1}(Q_{i},\omega)=0$ for $1\leq i\leq k$, $1\leq\ell\leq n_{i}$.
Hence, using Lemma 2.21 (2), equation (14) in this case becomes

(16)
$\sum_{i=1}^{k}\sum_{\ell=n_{i}+1}^{2n_{i}}a_{\ell}(Q_{i},\omega)\cdot\dfrac{(-1)^{\ell}}{2}\sum_{m=\ell-
n_{i}}^{n_{i}}e_{i,m}e_{i,\ell-m}+\text{higher-order terms}=0,$

where each of the higher order terms is of the form $\frac{b}{s}\cdot
e_{i,i_{1}}\cdots e_{i,i_{s}}$ with $s\geq 3$ and
$b\in\mathcal{O}_{L_{\mathfrak{p}}}$. Now $s\geq 3$ implies by Lemma 2.20
(with $T=2$) that $v_{\mathfrak{p}}(e_{i,i_{1}}\cdots e_{i,i_{s}})\geq
2\nu+(s-2)\geq 2\nu+1+N\mathrm{ord}_{p}(s)$. We conclude that all higher order
terms vanish mod $\mathfrak{p}^{2\nu+1}$. Multiplication by 2 (note that
$2\notin\mathfrak{p}$) and division by $\pi^{2\nu}$ of (16) now yields

$\widetilde{\mathcal{A}}_{2}\cdot(\psi_{n_{1}+1}(\widetilde{\mathbf{x}}_{1}),\ldots,\psi_{2n_{1}}(\widetilde{\mathbf{x}}_{1}),\psi_{n_{2}+1}(\widetilde{\mathbf{x}}_{2}),\ldots,\psi_{2n_{k}}(\widetilde{\mathbf{x}}_{k}))^{T}=0,$

as desired. Note that
$(\mathbf{x}_{1},\ldots,\mathbf{x}_{k})\neq\mathbf{0}\textrm{ mod
}\mathfrak{p}$ by maximality of $\nu$. This solution of
$\widetilde{\mathcal{L}}\cdot\widetilde{\mathbf{w}}=\mathbf{0}$ contradicts
our assumption, so we must have $\nu=\infty$ and $e_{i,\ell}=0$ for all $i$
and $\ell$, so that also $z_{i,j}=0$ for all $i$ and $j$ and
$\mathcal{P}=\mathcal{Q}$ by injectivity of well-behaved uniformisers on
residue classes. ∎

##  3\. The Mordell–Weil sieve

###  3.1. A formal description

As mentioned in Section 2.3, Theorem 2.6 can in some cases be used to compute
the set of rational points on symmetric powers of curves when combined with
the Mordell–Weil sieve. In this section we describe this sieve, which is
similar to the sieves in [Sik09], [Box21] and [Box].

We consider a curve $X$ with maps $\rho_{i}:X\to C_{i}$ of degree $d_{i}$, for
$i\in\\{1,\ldots,s\\}$, and an integer $e$. Next, suppose we are given the
following:

  * (i)

A finite list of points $\mathcal{L}^{\prime}\subset X^{(e)}(\mathbb{Q})$.

  * (ii)

A $\mathbb{Q}$-rational degree $e$ divisor $D_{0}$ on $X$.

  * (ii)

Explicit independent generators $D_{1},\ldots,D_{r}$ for a subgroup $G\subset
J(X)(\mathbb{Q})$ and $I\in\mathbb{Z}_{\geq 1}$ such that $I\cdot
J(X)(\mathbb{Q})\subset G$. Here $r$ is the rank of $J(X)(\mathbb{Q})$.

  * (iii)

A list $p_{1},\ldots,p_{n}$ of primes of good reduction for $X$.

Extend $\mathcal{L}^{\prime}$ to a (possibly infinite) set $\mathcal{L}$ by
adding for each point of the form
$\mathcal{P}+\rho_{1}^{*}(\mathcal{R}_{1})+\cdots+\rho_{s}^{*}(\mathcal{R}_{s})\in\mathcal{L}^{\prime}$,
with $\mathcal{R}_{i}\in C_{i}^{(m_{i})}(\mathbb{Q})$ ($m_{i}\geq 0$) for each
$i$, the entire set
$\mathcal{P}+\rho_{1}^{*}C^{(m_{1})}(\mathbb{Q})+\cdots+\rho_{s}^{*}C^{(m_{s})}(\mathbb{Q})$
to $\mathcal{L}$. The purpose of the sieve is to show that
$X^{(e)}(\mathbb{Q})=\mathcal{L}$.

To this end, we first consider the map

$\iota:\;X^{(e)}(\mathbb{Q})\longrightarrow G,\;\;D\mapsto I\cdot[D-D_{0}].$

Next, define $A$ to be the abstract finitely generated abelian group
isomorphic to $G$, with basis $e_{1},\ldots,e_{r}\in A$, and set

$\phi:A\to J(X)(\mathbb{Q}),\;\;e_{i}\mapsto D_{i}.$

For each $p\in\\{p_{1},\ldots,p_{n}\\}$, we obtain a commutative diagram

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where $\iota_{p}:D\mapsto I[D-\widetilde{D}_{0}]$, $\widetilde{D}_{0}$ is the
reduction of $D_{0}$, and $\phi_{p}=\mathrm{red}_{p}\circ\phi$.

######  Definition 3.1.

Define $\mathcal{M}_{p}\subset J(\widetilde{X})(\mathbb{F}_{p})$ to be the
subset of elements of
$\mathrm{red}_{p}(G)\cap\iota_{p}(\widetilde{X}^{(e)}(\mathbb{F}_{p}))$ that
are either

  * (1)

not in the image of $\mathcal{L}^{\prime}$, or

  * (2)

the image of
$\mathcal{Q}=\mathcal{P}+\rho_{1}^{*}(\mathcal{R}_{1})+\cdots+\rho_{s}^{*}(\mathcal{R}_{s})\in\mathcal{L}^{\prime}$,
with $\mathcal{R}_{i}\in C_{i}^{(m_{i})}(\mathbb{Q})$ ($m_{i}\geq 0$) for each
$i$, such that $\mathcal{Q}$ does _not_ satisfy the conditions of Theorems 2.6
and, when applicable, 2.24.

By definition, any hypothetical point $\mathcal{Q}\in
X^{(e)}(\mathbb{Q})\setminus\mathcal{L}$ satisfies
$\mathrm{red}_{p}\circ\iota(\mathcal{Q})\in\mathcal{M}_{p}$. We conclude the
following.

######  Proposition 3.2 (Mordell–Weil sieve).

If

$\bigcap_{i=1}^{n}\phi_{p_{i}}^{-1}(\mathcal{M}_{p_{i}})=\emptyset$

then $X^{(e)}(\mathbb{Q})=\mathcal{L}$.

######  Remark 3.3.

Naturally, this will only work in practice if the finite set
$\mathcal{L}^{\prime}$ contains all isolated points, as well as sufficiently
many points (partially) composed of pullbacks.

###  3.2. Implementing the sieve efficiently

The groups $J(\widetilde{X})(\mathbb{F}_{p})$ can be computed using the class
group algorithm of Hess [Hes02]. When $e$ and $p$ are large, it is important
to implement the sieve in an efficient way. Rather than first computing
$\mathcal{M}_{p_{i}}$ for each $i$ and intersecting afterwards, we compute
intersections recursively. It also turns out that computing
$\widetilde{X}^{(e)}(\mathbb{F}_{p})$ and $\iota_{p}$ for large values of $e$
and $p$ is suboptimal, so we compute Riemann–Roch spaces instead.

For a divisor $D$ on a curve $Y$ over a field $K$, define the Riemann–Roch
space

$L(D):=\\{f\in K(Y)^{\times}\mid\mathrm{div}(f)+D\geq 0\\}\cup\\{0\\}.$

Also define
$H_{i-1}:=\mathrm{ker}(\phi_{p_{1}})\cap\cdots\cap\mathrm{ker}(\phi_{p_{i-1}})$.
Suppose that we have computed the finite set $W_{i-1}$ of $H_{i-1}$-coset
representatives for $\cap_{j=1}^{i-1}\phi_{p_{j}}^{-1}(\mathcal{M}_{p_{j}})$,
and we want to compute $W_{i}$. First, we determine the (larger) set
$W_{i}^{\prime}$ of $H_{i}$-coset representatives for the same intersection.
For each $w\in W_{i}^{\prime}$, we then compute the $\mathbb{F}_{p}$-vector
spaces

$L(z+\widetilde{D}_{0})\text{ for each }z\text{ such that }I\cdot
z=\phi_{p_{i}}(w).$

Often these spaces will simply be 0-dimensional and $w$ can be removed from
$W_{i}^{\prime}$. For each non-zero $f\in L(z+D_{0})$, we obtain
$\mathrm{div}(f)+z+\widetilde{D}_{0}\in\widetilde{X}^{(e)}(\mathbb{F}_{p})$,
and we verify whether it is in the image of $\mathcal{L}^{\prime}$ and
satisfies the conditions of Theorem 2.6 or Theorem 2.24. This way, we
determine $W_{i}\subset W_{i}^{\prime}$ without the need to compute
$\widetilde{X}^{(e)}(\mathbb{F}_{p})$.

We mention another improvement. When $p_{1}$ and $p_{2}$ are distinct primes
such that $\\#J(X)(\mathbb{F}_{p_{1}})$ and $\\#J(X)(\mathbb{F}_{p_{2}})$ are
coprime, we have $\phi_{p_{2}}(W_{1})=\phi_{p_{2}}(A)$ by the Chinese
Remainder theorem. One thus tends to choose primes $p$ such that the numbers
$\\#J(\widetilde{X})(\mathbb{F}_{p})$ have prime factors in common. Often
these factors are powers of small primes, whereas
$\\#J(\widetilde{X})(\mathbb{F}_{p})$ tends to also be divisible by a large
prime factor, let us call it $r$. It is unlikely that $r$ will also divide
$\\#J(\widetilde{X})(\mathbb{F}_{q})$ for any other prime $q$ we consider,
hence the “mod $r$” information is of little use. It does, however, slow the
sieve down considerably, because information is stored in $H_{i}$-cosets and
the factor $r$ increases the index of $H_{i}$. We remedy this by composing
$\iota_{p}$ and $\phi_{p}$ with the multiplication-by-$r$ map $m_{r}$, thereby
“forgetting” the mod $r$ information.

##  4\. Applying Chabauty to modular curves

For each $X_{0}(N)$ with $N\in\\{53,61,65,67,73\\}$, let $\rho\colon
X_{0}(N)\to X_{0}^{+}(N)$ be the quotient map by $w_{N}$. Denote by
$X_{0}^{*}(N)$ the quotient of $X_{0}(N)$ by the full Atkin-Lehner group. To
determine the cubic points on these curves, we use the Mordell–Weil sieve
described in Section 3 in combination with Theorems 2.6 and 2.24 to determine
the isolated points in $X_{0}(N)^{(3)}(\mathbb{Q})$, i.e. those not of the
form $P+\rho^{*}Q$ with $P\in X_{0}(N)(\mathbb{Q})$ and $Q\in
X_{0}^{+}(N)(\mathbb{Q})$. For $X_{0}(57)$, we determine the entire set
$X_{0}(N)^{(3)}(\mathbb{Q})$. On $X_{0}(65)$, we moreover determine the points
in $X_{0}(65)^{(4)}(\mathbb{Q})$ that are not of the form $\rho^{*}D$ for
$D\in X_{0}^{+}(65)^{(2)}(\mathbb{Q})$ or of the form $P+\rho^{*}Q$ with $P\in
X_{0}(65)^{(2)}(\mathbb{Q})$ and $Q\in X_{0}^{+}(65)(\mathbb{Q})$.

In this section, we determine the information necessary to apply Theorems 2.6
and 2.24 and run the sieve.

###  4.1. Models, points and differentials

All the modular curves we consider were studied in [Box21], so we use the
models, subgroups of Mordell–Weil groups and annihilating differentials as
computed there.

We recall that these (canonical) models for each $X_{0}(N)$ were computed in
[Box21] using either the Small Modular Curves package in Magma or the code
written by Özman and Siksek [OS19]. Both algorithms simply compute equations
satisfied by $q$-expansions of weight 2 cusp forms, following Galbraith
[Gal02].

To find cubic and quartic points on a modular curve
$X_{0}(N)\subset\mathbb{P}^{n}$ (with coordinates $x_{0},\ldots,x_{n}$), we
have used code written by Özman and Siksek [OS19] to decompose intersections
$H\cap X_{0}(N)$ of hyperplanes $H\subset\mathbb{P}^{n}$ given by

$H:\;a_{0}x_{0}+\cdots+a_{n}x_{n}=0,$

with each $a_{i}\in\mathbb{Z}$ satisfying $|a_{i}|\leq C$ for a (small) bound
$C$. Each irreducible component of $X_{0}(N)\cap H$ of degree $d$ yields a
point of degree $d$ on $X_{0}(N)$.

Decomposing such intersections can be time-consuming on genus 5 curves, and we
did not always find all points this way. In such cases, we ran the sieve
described in Section 3 with this too small set of known points
$\mathcal{L}^{\prime}$. Consider the notation introduced in Section 3. After a
number of primes $p_{1},\ldots,p_{n}$, we found a non-empty set $W_{n}$ of
representatives for the possible $H_{n}$-cosets where unknown rational points
can map into. As the index of $H_{n}$ in $A$ is large, it is a priori unlikely
that one of these cosets $w+H_{n}$, for $w\in W_{n}$, contains a “small
vector”, say with all coefficients for free generators of $A$ smaller than 100
in absolute value. So, heuristically, small vectors should only occur with
good reason, for example as images of rational points.

Using the LLL-algorithm [LLL82], we searched for small vectors in these cosets
$w+H_{n}$. For each small vector $v\in w+H_{n}$ found, we compute
$L(D_{0}+\phi(v))$ and find that it is indeed 1-dimensional, generated by,
say, $f$. Then in each case, indeed $\mathrm{div}(f)+D_{0}+\phi(v)$ was a new
isolated rational point on the symmetric power.

Finally, in order to apply Theorems 2.6 and 2.24 to $X_{0}(N)$, we need for
each prime $p$ (of good reduction for $X_{0}(N)$) a basis for the image
$\widetilde{\mathcal{V}}_{0}$ of $\mathcal{V}_{0}\cap
H^{0}(\mathcal{X}_{N},\Omega)$ under the reduction map, where
$\mathcal{X}_{N}/\mathbb{Z}_{p}$ is a proper minimal model of $X_{0}(N)$ and
$\mathcal{V}_{0}$ is the space of annihilating differentials, with trace zero
to $X_{0}^{+}(N)$ when $N\neq 57$ and $X_{0}^{*}(57)$ otherwise. In [Box21,
Section 3.4], it was shown for each $N\in\\{53,61,65,67,73\\}$ that
$\mathcal{V}_{0}=\mathrm{Ker}(1+w_{N}^{*})$ and
$\widetilde{\mathcal{V}}_{0}=\mathrm{Ker}(1+\widetilde{w}_{N}^{*})$, where
$w_{N}^{*}$ and $\widetilde{w}_{N}^{*}$ are the pull-back morphisms on
$\Omega_{X_{0}(N)/\mathbb{Q}}$ resp.
$\Omega_{\widetilde{X}_{0}(N)/\mathbb{F}_{p}}$. Similarly, for $N=57$, it was
shown that
$\mathcal{V}_{0}=\mathrm{Ker}(1+w_{19}^{*})\cap\mathrm{Ker}(1+w_{57}^{*})$ and
$\widetilde{\mathcal{V}}_{0}=\mathrm{Ker}(1+\widetilde{w}_{19}^{*})\cap\mathrm{Ker}(1+\widetilde{w}_{57}^{*})$.
This allows us to compute those annihilating differentials and verify the
conditions of Theorems 2.6 and 2.24 in practice.

###  4.2. Computing Mordell-Weil groups

For each $N\in\\{53,57,61,65,67,73\\}$, the first named author [Box21]
computes a finite index subgroup of $J_{0}(N)(\mathbb{Q})$. Apart from $N=57$,
the index of these subgroups is shown to divide $2$. Thanks to our Mordell-
Weil sieve §3, this almost completely suffices for our purposes. The only
cases where they do not are $N=57,65$. For $N=57$ we produce a subgroup with
index dividing $2$; this appears to be needed purely for computational
reasons. Whereas when $N=65$ the issue is somewhat deeper. In this case we
verify the subgroup given in [Box21] is the entire Mordell-Weil group.

The method used in [Box21] to compute subgroups of $J_{0}(N)(\mathbb{Q})$ with
bounded index can be broken down into two steps. First, one finds a subgroup
of bounded index in the free part of $J_{0}(N)(\mathbb{Q})$. Then one computes
the torsion subgroup of $J_{0}(N)(\mathbb{Q})$. This is done by combining
theorems of Manin–Drinfeld and Mazur with bounds given by
$J_{0}(N)(\mathbb{F}_{p})$ for $p\nmid N$, see [Box21, Lemma 3.2]. We mention
that bounds may also be obtained using Hecke operators as in [DEvH+21, Section
4.3], although this is not needed here.

Let $X/\mathbb{Q}$ be a (projective, non-singular) curve and
$\Gamma\leq\operatorname{{\rm Aut}}_{\mathbb{Q}}(X)$ a finite subgroup. The
quotient curve $C=X/\Gamma$ is also defined over $\mathbb{Q}$ and the natural
map $\rho:X\rightarrow C$ has degree $\\#\Gamma$. Denote the Jacobian of $X$
by $J(X)$ and that of $C$ by $J(C)$. Choosing compatible base points for the
maps $\iota_{X}:X\rightarrow J(X)$, $\iota_{C}:C\rightarrow J(C)$, we obtain a
commutative diagram:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{X}}$$\scriptstyle{\rho}$$\textstyle{J(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{*}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{C}}$$\textstyle{J(C).}$

If the ranks of $J(C)(\mathbb{Q})$ and $J(X)(\mathbb{Q})$ are equal then
$\rho^{*}J(C)(\mathbb{Q})$ gives a subgroup of index dividing $\\#\Gamma$ in
the free part of $J(X)(\mathbb{Q})$ [Box21, Prop. 3.1].

The problem of computing subgroups of bounded index in $J(X)(\mathbb{Q})$ is
thus reduced to computing the free part of $J(C)(\mathbb{Q})$. When $C$ is an
elliptic curve this may be accomplished via Cremona’s method [Cre97]. Else, if
$C$ has genus 2, this may be achieved using techniques due to Stoll [Sto02].

We shall always take $X=X_{0}(N)$ and $C=X_{0}^{+}(N)$ with
$N\in\\{53,57,61,65,67,73\\}$. For all these values of $N$, the curve
$X_{0}^{+}(N)$ has genus one or two.

######  Example 4.1.

Let us give a few more details in the case $X=X_{0}(57)$. Here,
$C=X^{+}_{0}(57)$ has a model defined by
$y^{2}=x^{6}-2x^{5}+3x^{4}+3x^{2}-2x+1$. Stoll’s method [Sto17] may be used to
show the free part of $J_{0}^{+}(57)(\mathbb{Q})$ is generated by
$Q=\infty^{+}-\infty^{-}$. Pulling $Q$ back to $J_{0}(57)$ gives
$\rho^{*}(Q)=(1:1:0:1:0)+(6:9:-1:7:2)-P-\overline{P}$, where

$P=\left(-\sqrt{-2}+4:-4\sqrt{-2}+7:\sqrt{-2}-1:-2\sqrt{-2}+2\right).$

####  4.2.1. The Mordell-Weil group of $J_{0}(65)$

Let $\rho\colon X_{0}(65)\rightarrow X_{0}^{+}(65)$ be the quotient map. The
latter is an elliptic curve and so we may determine its Mordell-Weil group.
The group

$G=\rho^{*}(J_{0}^{+}(65)(\mathbb{Q}))J_{0}(65)(\mathbb{Q})_{\mathrm{tors}}$

is isomorphic to
$\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/84\mathbb{Z}$ as shown
in [Box21, Section 4.5]. Denote by $D$ a generator of the free part of $G$. We
use the Abel–Jacobi map $\iota:X_{0}(65)(\mathbb{Q})^{(4)}\rightarrow
J_{0}(\mathbb{Q})$ given by $P\mapsto[P-2\rho^{*}(0:1:0)]$.

A priori $G\subset J_{0}(65)(\mathbb{Q})$ has at most index two. Whilst the
sieve described in §3 normally allows one to get away with a subgroup of known
finite index, this appears not to be the case here. Indeed, for every prime
$p$ there seems to be a point on $X_{0}(65)(\mathbb{F}_{p})^{(4)}$ which under
the Abel–Jacobi map $\iota_{p}$ (compatible with $\iota$ and reduction modulo
$p$) doubles to the reduction of $\rho^{*}([(-1:0:1)-(0:1:0)])$ (there also
appear to be similar problems for four points coming from the quotient
$X_{0}^{+}(65)$). It seems likely that for each prime $p$ there is an element
in the set of preimages of $\rho^{*}([(-1:0:1)-(0:1:0)])$ in
$J_{0}(65)(\overline{\mathbb{Q}})$ under multiplication by two whose reduction
belongs to $\iota_{p}(X_{0}(65)(\mathbb{F}_{p})^{(4)})$. We have not been able
to compute the full set of preimages to verify this, but this does suggest the
following method to prove $G=J_{0}(65)(\mathbb{Q})$.

First find a divisor $D_{0}$ defined over an extension $K/\mathbb{Q}$ such
that $2D_{0}=D$. Computing the 2-torsion of $J_{0}(65)(K)$ then allows us to
determine all points in $J_{0}(65)(K)$ which double to $D$. Indeed, if
$2D_{0}^{\prime}=D$, then $2(D_{0}-D_{0}^{\prime})=0$. Given this set, it then
suffices to decide whether any element belongs to $J_{0}(65)(\mathbb{Q})$.

Owing in part to $J_{0}^{+}(65)$ being an elliptic curve, it was
straightforward for us to find such a $D_{0}$. There are algorithms and
explicit formulas which compute Mumford representations for preimages under
multiplication by $2$ on odd degree hyperelliptic jacobians [Sto17, Zar19]. In
principle, this allows one to adapt our method to other curves.

######  Theorem 4.2.

The Mordell-Weil group $J_{0}(65)(\mathbb{Q})$ of $X_{0}(65)$ is the group
generated by the two subgroups

$\rho^{*}(J_{0}^{+}(65)(\mathbb{Q}))\text{ and
}J_{0}(65)(\mathbb{Q})_{\mathrm{tors}}.$

###### Proof.

Let us abbreviate $J_{0}(65)$ to $J$. Set
$G=\langle\rho^{*}(J_{0}^{+}(65)(\mathbb{Q})),J(\mathbb{Q})_{\mathrm{tors}}\rangle$.
In [Box21], it is shown that
$G\cong\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/84\mathbb{Z}$.
Explicit generators are given (they are the same as in
§LABEL:subsubsection_65cubic), let us denote them by $D,T_{2},T_{84}$
respectively. In particular, $D=\rho^{*}\left((1:0:1)-(0:1:0)\right)$
generates the free part of $G$ (here we use the models as given in
§LABEL:subsubsection_65cubic). By Proposition [Box21, Prop. 3.1] and the above
it suffices to show no element in $D+\langle T_{2},T_{84}\rangle$ is the
double of a point in $J(\mathbb{Q})$, or equivalently, none of $D$, $D+T_{2}$,
$D+T_{84}$ and $D+T_{2}+T_{84}$ are doubles. The last three may easily be
checked by reducing modulo $7$. We now focus our attention on $D$.

There are four points which double to $(1:0:1)$ on the elliptic curve
$X^{+}_{0}(65)$. Two of these are defined over $\mathbb{Q}(\sqrt{5})$, the
other two over $\mathbb{Q}(\sqrt{13})$. Let $d_{0}$ be any of these points.
The pullback $D_{0}:=\rho^{*}(d_{0}-(0:1:0))$ then satisfies $2D_{0}=D$.

We claim $J(K)_{\mathrm{tors}}=J(\mathbb{Q})_{\mathrm{tors}}$ for
$K=\mathbb{Q}(\sqrt{5}),\mathbb{Q}(\sqrt{13})$. To verify this, we compute
$J(\mathbb{F}_{p})$ as an abstract group for primes $p\neq 5,13$ split in $K$.
For these primes we have embeddings of abstract groups
$J(K)_{\mathrm{tors}}\hookrightarrow J(\mathbb{F}_{p})$ which allow us to show
the lower bound given by
$J(\mathbb{Q})_{\mathrm{tors}}\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/84\mathbb{Z}$
is tight. The primes 11 and 19 split in $\mathbb{Q}(\sqrt{5})$ and the
corresponding groups give

$J(\mathbb{F}_{11})=\mathbb{Z}/2\mathbb{Z}\times\frac{\mathbb{Z}}{2^{2}\cdot
3\cdot 5\cdot 7^{2}\cdot 37\mathbb{Z}}\text{ and
}J(\mathbb{F}_{19})=\frac{\mathbb{Z}}{2\cdot 3\cdot
23\mathbb{Z}}\times\frac{\mathbb{Z}}{2^{2}\cdot 3\cdot 7\cdot 13\cdot 23},$

from which it is quickly deduced that
$J(\mathbb{Q}(\sqrt{5}))_{\mathrm{tors}}=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/84\mathbb{Z}$.
For $K=\mathbb{Q}(\sqrt{13})$ one may use 17 and 23.

We thus have the equality of sets

$S:=\\{x\in J(K)\mid 2x=D\\}=\\{D_{0}+T\mid T\in J(\mathbb{Q})[2]\\}.$

Hence we are left to show $D_{0}$ is not defined over $\mathbb{Q}$. Denote by
$d_{0}^{c}$ the Galois conjugate of $d_{0}$. Then both $d_{0}+d_{0}^{c}$ and
$2d_{0}$ are in $X^{+}_{0}(65)(\mathbb{Q})$, so also $Q:=d_{0}-d_{0}^{c}\in
X^{+}_{0}(65)(\mathbb{Q})$. Now $Q=Q^{c}=-Q$, so $Q$ is the unique non-trivial
point in $X^{+}_{0}(65)(\mathbb{Q})[2]$. We find that
$D_{0}-D_{0}^{c}=\rho^{*}Q$, and it suffices to show that $\rho^{*}Q\neq 0\in
J(\mathbb{Q})$. We can check this modulo 7.

Hence none of the elements in $S$ is in $J(\mathbb{Q})$, and so we must have
that $J(\mathbb{Q})=G,$ as desired. ∎

###  4.3. Two examples

We now give explicit examples of Theorems 2.6 and 2.24 in use.

######  Example 4.3.

Consider the non-cuspidal rational point $Q$ ramified under the map
$\rho\colon X_{0}(67)\rightarrow X_{0}^{+}(67)$. This gives rise to the point
$3Q$ in $X_{0}(67)(\mathbb{Q})^{(3)}$. We will apply Theorem 2.6 with
$\mathfrak{r}=19$ to show any other element in its residue disc also belongs
to the set $Q+\rho^{*}X_{0}^{+}(67)(\mathbb{Q})$. Writing
$3Q=Q+\rho^{*}\rho(Q)$, we see it is enough to show the matrix
$\widetilde{\mathcal{A}}$ has rank two. Computing $\widetilde{\mathcal{A}}$
with respect to a uniformiser $s$ at $Q$ such that $s^{2}=\rho^{*}t$, where
$t$ is a uniformiser at $\rho(Q)$, we find

$\widetilde{\mathcal{A}}=\begin{pmatrix}15&0&0\\\ 2&0&2\\\
6&0&11\end{pmatrix}$

which clearly has rank 2. Note the second column is identically zero, in
agreement with Proposition 2.17.

######  Example 4.4.

As explained in Example 2.23, Theorem 2.6 will always fail for the points
$3c_{0},3c_{\infty}\in X_{0}(73)(\mathbb{Q})^{(3)}$, where
$c_{0},c_{\infty}\in X_{0}(73)(\mathbb{Q})$ are the cusps. To fix this we need
expand further into the coefficients of the differentials. We verify the
conditions of Theorem 2.24 at $\mathfrak{r}=19$. Given a choice of uniformiser
and differentials, the corresponding matrices for $3c_{\infty}$ are

$\mathcal{A}=\begin{pmatrix}2&-13&47\\\ 0&-1&11\\\ 0&0&0\end{pmatrix}$

which clearly has rank 2, and the $\mathbb{F}_{19}$-matrices

$\widetilde{\mathcal{A}}_{1}=\begin{pmatrix}9&1&1\\\ 0&16&9\end{pmatrix}\text{
and }\widetilde{\mathcal{A}}_{2}=\begin{pmatrix}2&6&3\end{pmatrix}.$

One then checks
$\widetilde{\mathcal{L}}\cdot(x,y,z,y^{2}+2xz,2yz,z^{2})^{T}=\mathbf{0}$ has
no non-zero solutions for $x,y,z\in\mathbb{F}_{19}$.

Note Theorem 2.24 does not require one to form $\widetilde{\mathcal{A}}_{1}$
from the reductions of the uniformiser and differentials used for computing
$\mathcal{A}$. For this reason, $\widetilde{\mathcal{A}}_{1}$ above does not
coincide with the reduction of the first two rows of $\mathcal{A}$.

##  5\. Results

###  5.1. $\mathbb{Q}$-curves

An elliptic curve $E$ defined over a number field $K$ is said to be a
$\mathbb{Q}$-curve if it is $\overline{\mathbb{Q}}$ isogenous to all of its
$\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ conjugates. For example, all
elliptic curves with rational $j$-invariant are $\mathbb{Q}$-curves, as are CM
elliptic curves (for elliptic curves with CM by the maximal order this may be
deduced from Propositions 1.2 and 2.1 in §II of [Sil94]).

For $N\in\\{53,57,61,65,67,73\\}$ there are only finitely many cubic points on
$X_{0}(N)$. These points are listed §5.2. We find these points give rise to
$\mathbb{Q}$-curves exactly when the elliptic curves have CM.

######  Proposition 5.1.

Let $K/\mathbb{Q}$ be a non-trivial extension of odd degree. Then non-CM
points in $X_{0}(N)(K)$ for $N\in\\{53,57,61,65,67,73\\}$ do not give rise to
$\mathbb{Q}$-curves.

###### Proof.

Let $P\in X_{0}(N)(K)$ be a non-CM point. While this is a well-known fact, we
explain why we can take a representative $(E,C)$ of $P$ such that $E$ is
defined over $K$ and $C\subset E[N]$ is a
$\mathrm{Gal}(\overline{K}/K)$-stable cyclic subgroup of order $N$. Take any
representative $(E,C)$ of $P$. The $j$-map $j\colon X_{0}(N)\to X(1)$ is
defined over $\mathbb{Q}$. Since $j(E)=j(P)\in K$, we may take a different
representative $(E^{\prime},C^{\prime})$ where $E^{\prime}$ is defined over
$K$ (e.g. by considering the Legendre form). As $P$ is a non-CM point,
$E^{\prime}$ is a quadratic twist of $E$ (see e.g. [CN21, Lemma A.4], [Sil86,
§X, Prop. 5.4]). The mod $\ell$ representations ($\ell$ prime) of $E^{\prime}$
and $E$ are either equal or differ by a quadratic twist, so in particular
$E^{\prime}$ has a $\mathrm{Gal}(\overline{K}/K)$-stable subgroup of order
$\ell$ if and only if $E$ has one also. In other words, we may choose
$C^{\prime}$ to be $\mathrm{Gal}(\overline{K}/K)$-stable. For $N\neq 65$, the
statement now follows from applying [CN21, Theorem 1.1] to $E^{\prime}$.

Suppose $P\in X_{0}(65)(K)$ is a non-CM point which gives rise to a
$\mathbb{Q}$-curve. Take the representative $E^{\prime}/K$ of $P$ as above.
Then by Theorem 2.7 of [CN21], $E^{\prime}$ is isogenous to an elliptic curve
with rational $j$-invariant and thus has rational isogenies of degrees $5$ and
$13$, by Corollary 3.4 in [CN21]. But $X_{0}(65)(\mathbb{Q})$ consists of
cusps, giving a contradiction. ∎

Cremona and Najman describe an algorithm to determine whether an elliptic
curve is a $\mathbb{Q}$-curve or not in §5.5 of [CN21]. We use this algorithm
to show the isolated quartic points on $X_{0}(65)$ which give rise to
$\mathbb{Q}$-curves are exactly the CM points. This algorithm has been
implemented in Sage [CN21]. However, as part of this implementation the
conductor is computed, and this appears to be too costly for our elliptic
curves. Instead, we partially implemented the algorithm in Magma and avoid
computing the conductor. In particular, for each of the non-CM elliptic curves
$E/K$ under consideration there was a rational prime $p$ and primes
$\mathfrak{p},\mathfrak{q}$ of $K$ above $p$ such that either $E$ had bad
potentially multiplicative reduction at $\mathfrak{p}$ but not $\mathfrak{q}$
or $E$ had good reduction at all primes above $p$ and the endomorphism
algebras of the reductions at $\mathfrak{p}$ and $\mathfrak{q}$ were not
isomorphic. These contradict certain isogeny properties satisfied by
$\mathbb{Q}$-curves, see Propositions 5.1 and 5.2 of [CN21].

The curve $X_{0}(65)$ has infinitely many quartic points: the quotient
$X_{0}^{+}(65)$ is an elliptic curve. That is, we have a degree two map to an
elliptic curve, and elliptic curves have infinitely many degree two maps to
$\mathbb{P}^{1}$. This yields infinitely many degree four maps to
$\mathbb{P}^{1}$ and in particular infinitely many quartic points on
$X_{0}(65)$.

In fact, by the Existence Theorem in Brill–Noether Theory [ACGH85, Chapter V,
pg206], genus 5 curves have infinitely many degree four maps to
$\mathbb{P}^{1}$. The fact $X_{0}(65)$ admits a degree two map to an elliptic
curve is our saving grace in this instance. Indeed, each such map turns out to
factor through this elliptic curve $X_{0}^{+}(65)$, allowing us to perform our
relative Chabauty methods with respect to the quotient. On the flip side, no
such quotient exists for $X_{0}(57)$, leaving us with an abundance of quartic
points from infinitely many sources.

Accordingly, every quartic point on $X_{0}(65)$ either arises from a quadratic
point on $X_{0}^{+}(65)$, or is a point listed in
§LABEL:subsection_quarticpoints. For points not coming from the quotient, we
find the $\mathbb{Q}$-curves are exactly those with CM, just as the case was
for the cubic points. The points arising from the quotient may be
$\mathbb{Q}$-curves (for example if they come from the full Atkin-Lehner
quotient $X_{0}^{*}(65)$), but are not necessarily. However, each of these
points will give rise to a $K$-curve for some quadratic field $K$. Indeed,
$w_{65}$ determines the isogeny between such a quartic point and one of its
Galois conjugates.

###  5.2. Cubic Points

Let $N\in\\{53,57,61,65,67,73\\}$. In this section we list all cubic points on
$X_{0}(N)$, or rather a representative from each
$\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-conjugacy class. As explained
in §4.2, subgroups $G\subseteq J_{0}(N)(\mathbb{Q})$ of index at most two are
computed using $J_{0}^{+}(N)(\mathbb{Q})$. We provide explicit generators for
$G$ and $J_{0}^{+}(N)(\mathbb{Q})$ below.

For $N\neq 57$, we apply the Mordell-Weil sieve in conjunction with Theorems
2.6, 2.24 relative to $X_{0}^{+}(N)$ in order to provably show all points on
$X_{0}^{(3)}(N)(\mathbb{Q})$ either arise as the sum of a rational point and
the pullback of a rational point from $X_{0}^{+}(N)$, or belong to a finite
set. This finite set consists of
($\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-stable sums of) the cubic
points and sums of rational and isolated degree two points. Rational and
isolated degree two points are listed in [Box21, §4].

We note for $N=57$ there is no need to use ‘relative Chabauty’ as there are
only finitely many quadratic points on $X_{0}(57)$ and these can be determined
provably using Chabauty. In fact Siksek’s symmetric Chabauty Theorem [Sik09,
Theorem 3.2] almost suffices to determine the cubic points. But, as mentioned
in Example 2.23, there are two points on $X_{0}^{(3)}(57)$ which require
Theorem 2.24. The only other curve where we are also required to apply Theorem
2.24 is $X_{0}(73)$.

Using the same models as in [Box21, §4], we list a representative for every
Galois conjugacy class of cubic points on $X_{0}(N)$, along with the
corresponding $j$-invariant and indicate whether the class of elliptic curves
have CM. The only points which give rise to $\mathbb{Q}$-curves are the CM
points, see §5.1. For this reason we do not include an extra column to
indicate which points are $\mathbb{Q}$-curves.

We recall that the Magma [BCP97] code to verify all computations can be found
at

$\texttt{\href https://github.com/joshabox/cubicpoints/}\;.$

## Funding

This work was supported by Deutsche Forschungsgemeinschaft [DFG-Grant MU
4110/1-1 to S.G.]; and the Engineering and Physical Sciences Research Council
[EP/N509796/1 to J.B., EP/N509619/1 to P.G.].

## Acknowledgements

The authors would like to warmly thank David Zureick-Brown for initiating this
project, and both Jackson Morrow and David for their help during the project
sessions of the 2020 Arizona Winter School. We are also grateful to the
organisers of the school for providing such excellent conditions in Tucson,
and the first and last named authors thank the school’s donors for their
financial support.

We thank Samir Siksek for many valuable comments and suggestions, which have
improved this work greatly. Likewise, we thank Tim Dokchitser, Tom Fisher,
Steffen Müller, Filip Najman, Lazar Radičević and Damiano Testa for their
help.

We are grateful to anonymous referees for many helpful comments that notably
improved our article.

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