# Brauer relations, isogenies and parities of ranks

Alexandros Konstantinou  University College London, London WC1H 0AY, UK
<EMAIL_ADDRESS>
###### Abstract.

The present paper illustrates the utility of Brauer relations, Galois covers
of curves and the theory of regulator constants in the context of studying
isogenies between Jacobians and their relevance to the parity conjecture. This
framework presents a unified approach, enabling the reconstruction of a
diverse array of classical isogenies and the derivation of local expressions
for Selmer rank parities, drawing from an extensive body of existing
literature. These include the local expressions found in the works of
Mazur–Rubin (dihedral extensions), Coates–Fukaya–Kato–Sujatha ($p^{g}$
isogenies), Kramer–Tunnell (quadratic twists of elliptic curves),
Dokchitser–Maistret (Richelot isogenies), and Docking (prym construction).

###### Contents

  1. 1 Introduction
  2. 2 Background
  3. 3 Hyperelliptic Jacobians over $K(\sqrt{d})$
  4. 4 Richelot Isogeny
  5. 5 Prym varieties of trigonal curves
  6. 6 Elliptic curves with cyclic isogeny
  7. 7 Genus $2$ curves with elliptic subcovers

##  1\. Introduction

###  1.1. Uniform results for isogenies and rank parity expressions

The main motivation for this paper stems from [10] in which Brauer relations
are used to establish the parity conjecture for a certain class of twists of
elliptic curves. Building upon this groundwork, [13] introduces an expanded
framework, enabling a broader scope by allowing $K$-automorphism subgroups on
curves and pseudo Brauer relations. A Brauer relation is defined as a
$\mathbb{Z}$-linear combination of subgroups of a finite group $G$, say
$\sum_{i}H_{i}-\sum_{j}H_{j}^{\prime}$, arising from a $G$-representation
isomorphism

$\bigoplus_{i}\textup{Ind}_{H_{i}}^{G}\mathbf{1}\cong\bigoplus_{j}\textup{Ind}_{H_{j}^{\prime}}^{G}\mathbf{1}\
\ (H_{i},H_{j}^{\prime}\leq G).$

The notion of a pseudo Brauer relation provides an extension of these and is
recalled in Definition 2.6 below. One of the main results in [13] asserts
that, for a finite group $G$ acting on a curve $X$, a pseudo Brauer relation
represented by $\sum_{i}H_{i}-\sum_{j}H_{j}^{\prime}$ induces an isogeny
between Jacobians

(1.1)
$\prod_{j}\textup{Jac}_{X/H_{j}^{\prime}}\to\prod_{i}\textup{Jac}_{X/H_{i}}.$

Let us call such an isogeny pseudo Brauer verifiable, see Definition 1.6 for
details. One of the central results in this paper establishes pseudo Brauer
verifiability for a range of classical isogenies between Jacobians.

######  Theorem 1.2 (Theorems 3.2, 4.9, 5.2, 6.6, 7.8).

In each of the following cases, there exists an isogeny $A\to A^{\prime}$, and
this isogeny is pseudo Brauer verifiable.

  1. (1)

$A$ is the Weil restriction
$\operatorname{Res}_{K(\sqrt{d})/K}\operatorname{Jac}_{Y}$ and $A^{\prime}$ is
$\operatorname{Jac}_{Y}\times\operatorname{Jac}_{Y^{d}},$ where $Y$ is a
hyperelliptic curve and $Y^{d}$ its quadratic twist by $d\in K^{\times}$,

  2. (2)

$A$ and $A^{\prime}$ are Jacobians of C2D4 genus $2$ curves admitting a
polarized $(2,2)$ isogeny (also known as a Richelot isogeny),

  3. (3)

$A$ is $\operatorname{Jac}_{\tilde{Z}}$ and $A^{\prime}$ is
$\operatorname{Jac}_{Z}\times\textup{Prym}(\tilde{Z}/Z)$ where $\tilde{Z}$ is
a (potentially trivial) twist of a curve admitting an unramified double cover
to a trigonal curve $Z$,

  4. (4)

$A$ and $A^{\prime}$ are elliptic curves admitting an isogeny of prime degree
$p$,

  5. (5)

$A$ is $\textup{Jac}_{Y}$, where $Y$ is a curve of genus $2$ with a generic
cover to an elliptic curve $Y\to E$ of prime degree $p$, $A^{\prime}$ is
$E\times E^{\prime}$ for some complimentary elliptic curve $E^{\prime}$.

The isogenies listed in Theorem 1.2 are classically well-known. The key
insight is that one can deduce their existence in a uniform way using a
suitable pseudo Brauer relation.

######  Remark 1.3.

Writing $\phi:A\to A^{\prime}$ to denote an isogeny in each of the cases
listed in Theorem 1.2, the following table records the pseudo Brauer relation
$\Theta_{\phi}$ and the corresponding automorphism group $G_{\phi}$. For
details regarding the curve $X_{\phi}$ being acted upon by $G_{\phi}$, see the
relevant theorem.

$\begin{array}[]{|
c||c|c|c|}\hline\cr\textup{Case}&G_{\phi}&\Theta_{\phi}&\textup{Relevant
theorem}\\\ \hline\cr(1)&C_{2}\times
C_{2}&\\{e\\}+2(C_{2}\mskip-3.0mu\mskip-3.0mu\times\mskip-3.0mu\mskip-3.0muC_{2})-C_{2}^{a}-C_{2}^{b}-C_{2}^{c}&\ref{pseudo_brauer_KT}\\\
\hline\cr(2)&D_{8}&C_{2}^{a}-(C_{2}^{2})^{a}+C_{2}^{b}-(C_{2}^{2})^{b}&\ref{pseudo_brauer_richelot}\\\
\hline\cr(3)&S_{4}&C_{2}^{2}-D_{8}-S_{3}+S_{4}&\ref{pseudo_brauer_verifiability_prym}\\\
\hline\cr(4)&C_{p}\rtimes C_{m}&C_{m}-(C_{p}\rtimes
C_{m})&\ref{pseudo_brauer_verifiability_EC}\\\ \hline\cr(5)&S_{p}\times
C_{2}&(S_{p-1}\times\\{e\\})-(S_{p}\times\\{e\\})-(S_{p-2}\times
C_{2})&\ref{complementary_elliptic_curve}\\\ \hline\cr\end{array}$

######  Remark 1.4.

We note that a Brauer relation for a group $G$ is automatically a pseudo
Brauer relation for an automorphism group $G$ acting on a curve $X/K$. We note
that the pseudo Brauer relations $\Theta_{\phi}$ in the cases listed (1)-(3)
in Theorem 1.2 are Brauer relations for the underlying group $G_{\phi}$.

Adding on, the theory of regulator constants utilises isogenies induced from
pseudo Brauer relations to provide local expressions for the parity of certain
Selmer ranks. To be precise, in view of [13, Corollary 7.3], recalled in
Theorem 1.14 below, to each pseudo Brauer relation $\Theta$ for $G$ and $X$,
we can associate an expression of the form

$\sum_{\rho\in S_{\Theta,p}}m_{\rho}\equiv(\textup{explicit local
invariants})\quad\textup{mod }2.$

In this expression, $S_{\Theta,p}$ denotes a special set consisting of
$\mathbb{Q}_{p}[G]$-representations (Definition 2.7), while $m_{\rho}$ denotes
the multiplicities of these in $\mathcal{X}_{p}(\textup{Jac}_{X})$, the dual
$p^{\infty}$-Selmer group of $\textup{Jac}_{X}$. This specifies a mechanism
for deriving local expressions. One of our main results is deduced from the
application of this in various settings.

######  Theorem 1.5 (Theorems 3.3, 4.11, 5.5, 6.10, 7.11).

Let $A$ be one of the abelian varieties listed in Theorem 1.2. Then, by
applying Theorem 1.14 with $p=2$ in cases (1)-(3), and with any rational prime
$p$ in cases (4)-(5), we acquire

$\textup{rk}_{p}(A)=\sum_{v\ \textup{place of}\ K}\textup{ord}_{p}\
\Lambda_{\Theta_{\phi}}(X_{\phi}/K_{v})\quad\textup{mod }2,\ $

where $\Theta_{\phi}$ is the pseudo Brauer relation given in Remark 1.3, while
$\Lambda_{\Theta}(X_{\phi}/K_{v})$ is a local invariant as in Definition 2.10.

We note that Theorem 1.5 recovers local expressions found in the existing
literature. In particular, the corresponding local expressions can be found in
the works of Kramer [21] and Morgan [27] for (1), V. Dokchitser–Maistret [14]
for (2), Docking [8] for (3), Cassels (found in [9, Appendix]) for (4),
Green–Maistret [18] for $p=2$ and Coates–Fukaya–Kato–Sujatha [6] for odd $p$
for (5). To our best knowledge, these are all known instances of local
expressions in the case of Jacobians.

###  1.2. Brauer relations and isogenies

The work of Kani and Rosen [19] concerns the study of idempotent relations in
the algebra of rational endomorphisms of arbitrary abelian varieties. Their
central result in the setting of automorphisms on Jacobians proves that Brauer
relations induce isogenies between Jacobians as in (1.1). The usefulness of
this theorem rests ultimately on one’s ability to deduce the existence of an
isogeny by presenting an explicit Brauer relation. As a result, applications
of this work are widespread across the existing literature, see for example
[1], [5], [28] and [32].

One noteworthy application includes Edixhoven’s work [16] which presents an
explicit Brauer relation for $\textup{GL}_{2}(\mathbb{F}_{p})$ in order to
verify the existence of an isogeny between Jacobians of certain modular
curves. The existence of this isogeny was originally deduced in Chen’s work by
utilising the Selberg trace formula, see [4, Theorem 1]. This provides the
main stimulus for the contents of Theorem 1.2 mentioned above which asserts
that a range of classical isogenies are pseudo Brauer verifiable in the
following sense.

######  Definition 1.6.

Let $Y_{1},\dots,Y_{m}$ and $Y_{1}^{\prime},\dots,Y_{r}^{\prime}$ be curves
defined over a number field $K$. In addition, suppose that there exists a
$K$-rational isogeny
$\prod_{i=1}^{m}\textup{Jac}_{Y_{i}}\to\prod_{j=1}^{r}\textup{Jac}_{Y_{j}^{\prime}}$.
We say that a pseudo Brauer relation $\Theta$ for $G$ and a curve $X$ verifies
an isogeny
$\prod_{i=1}^{m}\textup{Jac}_{Y_{i}}\to\prod_{j=1}^{r}\textup{Jac}_{Y_{j}^{\prime}}$
if we can find subgroups $H_{1},\dots,H_{m}$ and
$H_{1}^{\prime},\dots,H_{r}^{\prime}$ of $G$ such that

  1. (1)

the quotient curves satisfy $X/H_{i}\cong Y_{i}$ and $X/H_{j}^{\prime}\cong
Y_{j}^{\prime}$ for all $i$ and $j$,

  2. (2)

and $\Theta=\sum_{i=1}^{m}H_{i}-\sum_{j=1}^{r}H_{j}^{\prime}$ is the
underlying pseudo Brauer relation.

In addition, we say that an isogeny
$\prod_{i=1}^{m}\textup{Jac}_{Y_{i}}\to\prod_{j=1}^{r}\textup{Jac}_{Y_{j}^{\prime}}$
is pseudo Brauer verifiable if we can find a group $G$ acting on a curve $X$
and a pseudo Brauer relation $\Theta$ for these such that conditions (1)-(2)
are satisfied.

######  Remark 1.7.

We note that the concept of pseudo Brauer verifiability is primarily focused
on verifying the existence of some isogeny between
$\prod_{i}\textup{Jac}_{X/H_{i}}$ and
$\prod_{j}\textup{Jac}_{X/H_{j}^{\prime}}$. We deliberately refrain from
making any specific references to an explicit morphism
$\prod_{i}\textup{Jac}_{X/H_{i}}\to\prod_{j}\textup{Jac}_{X/H_{j}^{\prime}}$
in this definition.

It would be interesting to know whether any isogeny between Jacobians is
pseudo Brauer verifiable. A slightly more ambitious problem would be deriving
similar results in the case of general abelian varieties. A result in this
direction can be found in the work of Yu [32, Corollary 10], where the author
deduces an isogeny between $\textup{Res}_{F/K}A$ for $F/K$ an abelian
extension and $\prod_{\chi\in\textup{Gal}(F/K)}A^{\chi}$, a product of its
twists. This is done by using twisted idempotent relations, his generalisation
of the Kani-Rosen relations. The underlying isogeny in this case is
classically well-known and can be found in [20].

We end this subsection with various examples of pseudo Brauer verifiable
isogenies.

######  Example 1.8 (Modular Curves).

Let $X(p)/\mathbb{Q}$ be the modular curve classifying elliptic curves with
full level $p$ structure equipped with its usual action by
$\textup{GL}_{2}(\mathbb{F}_{p})$. As already discussed, the main result
presented in [16] proves that the isogeny
$\textup{Jac}_{X_{\textup{sp}}^{+}(p)}\to\textup{Jac}_{X_{0}(p)}\times\textup{Jac}_{X_{\textup{nsp}}^{+}(p)}$
between certain modular Jacobians can be deduced from a Brauer relation for
$\textup{PGL}_{2}(\mathbb{F}_{p})$. As a result, the underlying isogeny is
pseudo Brauer verifiable.

######  Example 1.9 (Fermat Curves, cf. [19] §5).

Let $X_{m}$ be the Fermat curve with affine model $x^{m}+y^{m}=1$. Let $K$ be
any field containing $\zeta_{p}$ where $p|m$. Then,
$C_{p}\mskip-3.0mu\times\mskip-3.0muC_{p}$ acts on $X_{m}$ via
$\sigma(x,y)=(\zeta_{p}x,y)$, while $\tau(x,y)=(x,\zeta_{p}y)$. Let
$\\{C_{p}^{i}\\}_{i=0}^{p}$ denote all proper subgroups of
$C_{p}\mskip-3.0mu\times\mskip-3.0muC_{p}$, and write $X_{i}{{}^{\prime}}$ to
denote the (normalisation of) the curve with affine model
$y_{i}^{m}=x_{i}^{im/p}(1-x_{i}^{m/p}).$ Then, the Brauer relation
$\\{e\\}+p(C_{p}\mskip-3.0mu\times\mskip-3.0muC_{p})-\sum_{i=0}^{p}C_{p}^{i}$
for $C_{p}\mskip-3.0mu\times\mskip-3.0muC_{p}$ and $X_{m}$ verifies an isogeny
$\textup{Jac}_{X_{m}}\times\textup{Jac}_{X_{m/p}}^{p}\to\prod_{i=0}^{p}\textup{Jac}_{X_{i}^{\prime}}.$

######  Example 1.10 (Hyperelliptic curve with extra involution).

Let $X/K$ be a hyperelliptic curve which admits an affine model
$y^{2}=f(x^{2})$ for $f$ a square-free polynomial. The morphisms

$\displaystyle\phi_{1}:X$ $\displaystyle\to Y_{1}=\\{y_{1}^{2}=f(x_{1})\\},\ \
\ \ \phi_{2}:X$ $\displaystyle\to Y_{2}=\\{y_{2}^{2}=x_{2}f(x_{2})\\}$
$\displaystyle(x,y)$ $\displaystyle\mapsto(x^{2},y),\ \ \ \hskip
56.9055pt(x,y)$ $\displaystyle\mapsto(x^{2},xy)$

induce an isogeny
$(\phi_{1})^{*}+(\phi_{2})^{*}:\textup{Jac}_{Y_{1}}\times\textup{Jac}_{Y_{2}}\to\textup{Jac}_{X}.$
By considering the $C_{2}\times C_{2}$-action on $X/K$ given by separately
negating $x$ and $y$, we acquire:

$K(x^{2})$$K(x)$$K(Y_{1})=K(x^{2},y)$$K(X)=K(x,y)$$K(Y_{2})=K(x^{2},xy)$

Since $\Xi=\\{e\\}+2(C_{2}\times C_{2})-C_{2}^{a}-C_{2}^{b}-C_{2}^{c}$ is a
Brauer relation for $C_{2}\times C_{2}$, we deduce that the underlying isogeny
is pseudo Brauer verifiable.

######  Example 1.11 (Elliptic curve with $K$-rational $p$-torsion).

Let $E$ be an elliptic curve with affine model $y^{2}=f(x)$ which has a
$K$-rational $p$-torsion point $P$. Then, there exists an isogeny $\phi:E\to
E^{\prime}=\\{v^{2}=g(u)\\}$ with $\textup{Ker}(\phi)=\langle P\rangle$. By
exploiting the $D_{2p}$-action on $E$ given by the translation-by-$P$
automorphism and the hyperelliptic involution, we acquire:

$K(u)$$K(E^{\prime})=K(u,v)$$K(x)$$K(E)=K(x,y)$

In view of the Brauer relation $\Psi=\\{e\\}+2D_{2p}-2C_{2}-C_{p}$ for
$D_{2p}$, we deduce that the $p$-isogeny $E\to E^{\prime}$ is pseudo Brauer
verifiable.

###  1.3. Local expressions, regulator constants and the parity conjecture

In addition, this paper addresses the systematic derivation of local rank
parity expressions for Jacobians. The existence of such expressions is
expected in view of the parity conjecture. This conjecture is based on
expectations made by the Birch and Swinnerton-Dyer conjecture combined with
the theoretical framework of $L$-functions.

######  Conjecture 1.12 (Parity conjecture).

For an abelian variety $A/K$ defined over a number field,

$\textup{rk}(A/K)\equiv\sum_{v\ \textup{place of}\
K}\mu(A/K_{v})\quad\textup{mod }2,$

where $(-1)^{\mu(A/K_{v})}=w(A/K_{v})$, and $w(A/K_{v})$ denotes the local
root number of $A$ at a place $v$.

Such global-to-local expressions have established themselves as highly
valuable. Their usefulness stems from the relative ease with which local
invariants can be computed, a stark contrast to the formidable challenge posed
by global invariants such as the rank of an abelian variety. As a result, they
prove to be a crucial step in many parity-related proofs, see for example [6],
[10], [14] and [22].

A notable instance of a global-to-local formula appears in the work of
Mazur–Rubin [25] in the context of base changing elliptic curves to
$D_{2p}$-extensions. The primary impetus behind the development of Theorem 1.5
can be traced back to a result in [11]. Within its pages, the authors employ
their theory of regulator constants to give a local expression for the same
rank parity as the formula appearing in Mazur–Rubin’s work. This theory was
subsequently extended in [13], where the new framework allows for a broader
scope by permitting the use of subgroups of $K$-automorphisms on curves and
pseudo Brauer relations. We now recall the main result of this advancement.

######  Notation 1.13.

We write $\langle\cdot,\cdot\rangle$ for the standard inner product of complex
characters of $G$, and we fix an arbitrary embedding
$\mathbb{Q}_{p}\subseteq\mathbb{C}$ for the purpose of comparing characters.
We write $\mathcal{X}_{p}(\textup{Jac}_{X})$ for the dual $p^{\infty}$-Selmer
group of $\textup{Jac}_{X}$ and $\textup{rk}_{p}(\textup{Jac}_{X})$ for its
$p^{\infty}$-Selmer rank.

######  Theorem 1.14 ([13] Corollary 7.3).

Let $\Theta$ be a pseudo Brauer relation for $G$ and $X/K$, and in addition
suppose that $\Omega^{1}(\textup{Jac}_{X})$ is self-dual as a
$G$-representation. Then, for each rational prime $p$,

$\sum_{\tau\in
S_{\Theta,p}}\langle\tau,\mathcal{X}_{p}(\textup{Jac}_{X})\rangle\equiv\sum_{v\
\textup{place of}\ K}\textup{ord}_{p}\
\Lambda_{\Theta}(X/K_{v})\quad\textup{mod }2,$

where $\Lambda_{\Theta}(X/K_{v})$ are local invariants as in Definition 2.10
and $S_{\Theta,p}$ as in Definition 2.7.

######  Remark 1.15.

The self-duality assumption on $\Omega^{1}$ is automatically satisfied when
all representations of $G$ are self-dual or $K$ has a real place (see [13,
Lemma 5.1]).

We conclude this section by demonstrating applications of this theorem in
examples considered in §1.2.

######  Example 1.16.

Looking at Example 1.10, we write $C_{2}^{a}$ and $C_{2}^{b}$ to denote the
subgroups generated by $(x,y)\mapsto(-x,y)$ and $(x,y)\mapsto(-x,-y)$
respectively. We let $\epsilon_{a}$ and $\epsilon_{b}$ be the non-identity,
irreducible characters of $C_{2}\times C_{2}$ on which $C_{2}^{a}$ and
$C_{2}^{b}$ act trivially. Then, the special subsets $S_{\Theta,p}$ in
Examples 1.10 and 1.11 are as follows.

$\begin{array}[]{|c||c|c|c|c|}\hline\cr\textup{Example}&G&\textup{Brauer
relation}&p&S_{\Theta,p}\\\
\hline\cr\ref{extra_involution_example}&C_{2}\times
C_{2}&\Xi=\\{e\\}+2(C_{2}\mskip-3.0mu\mskip-3.0mu\times\mskip-3.0mu\mskip-3.0muC_{2})-C_{2}^{a}-C_{2}^{b}-C_{2}^{c}&2&\\{\epsilon_{a},\epsilon_{b}\\}\\\
\hline\cr\ref{p-isogeny}&D_{2p}&\Psi=2C_{2}+C_{p}-2D_{2p}-\\{e\\}&p&\\{\mathbf{1}\\}\\\
\hline\cr\end{array}$

By applying Theorem 1.14 to Example 1.10 with respect to the pseudo Brauer
relation $\Theta$ and $p=2$, we acquire the local formula

$\textup{rk}_{2}(\textup{Jac}_{X})=\sum_{v\ \textup{place of}\ K}\
\textup{ord}_{2}\ \Lambda_{\Xi}(X/K_{v})\quad\textup{mod }2.$

In addition, by applying Theorem 1.14 to Example 1.11 with respect to the
pseudo Brauer relation $\Psi$ and any prime $p$, we acquire the local formula

$\textup{rk}_{p}(E/K)=\sum_{v\ \textup{place of}\ K}\ \textup{ord}_{p}\
\Lambda_{\Psi}(E/K_{v})\quad\textup{mod }2.$

###  1.4. Outline

This paper is organised as follows. In $\S 2$, we provide relevant background
results. This includes a review of the definitions of pseudo Brauer relations,
regulator constants and some well-known facts concerning monodromy. The
remaining sections share a consistent structure. In each of these, we delve
into the study of the well–known isogeny specified by the section’s title.
Within each section, we establish pseudo Brauer verifiability for the isogeny
in consideration, and then we utilise the underlying pseudo Brauer relation to
derive a local expression using regulator constants.

###  1.5. General notation and conventions

Throughout this paper, we adhere to the following convention.

Convention. For each rational prime $p$, we fix an embedding
$\mathbb{Q}_{p}\hookrightarrow{}\mathbb{C}$ for the purposes of comparing
characters.

$X/K$ |  a curve over a field $K$ (which is smooth, proper, not necessarily connected, see §1.6)
---|---
$A/K$ |  an abelian variety over a field $K$
$\textup{Res}_{F/K}A$ |  the Weil restriction of $A$ from a field extension $F$ to $K$
$\operatorname{Jac}_{X}$ |  the Jacobian variety of $X$, the identity component of the relative Picard functor $X/K$
$\mathcal{X}_{p}(A)$ |  $\text{Hom}_{\mathbb{Z}_{p}}(\varinjlim\text{Sel}_{p^{n}}(A),\mathbb{Q}_{p}/\mathbb{Z}_{p})\otimes\mathbb{Q}_{p}$, the dual $p^{\infty}$-Selmer group of $A$
$\operatorname{rk}_{p}A$ |  the $p^{\infty}$-Selmer rank of $A$, equivalently $\textup{dim}(\mathcal{X}_{p}(A))$
$V_{\ell}(A)$ |  the $\mathbb{Q}_{\ell}$ vector space of the $\ell$-adic Tate module for $A$
$\Omega^{1}(A)$ |  the $K$-vector space of regular differentials on $A$
$|\cdot|_{K}$, $|\cdot|_{v}$ |  the unique extension of the the normalised absolute value on a local field $K$ (resp. $K_{v}$) to $\overline{K}$ (resp. $\overline{K_{v}}$)
$c(A)$ |  the local Tamagawa number of $A$ defined over a non-Archimedean local field $\mathcal{K}$. When $\mathcal{K}$ is the completion of $K$ at a finite place $v$, then we write $c_{v}(A)$ to signify this.
$C(A,w)$ |  $c(A)\cdot|\omega/\omega^{0}|_{\mathcal{K}}$ when $\mathcal{K}$ is non-archimedean and $\omega^{0}$ is a Néron exterior form on $A$; $\int_{A(\mathcal{K})}|\omega|$ when $\mathcal{K}=\mathbb{R}$; $2^{\text{dim}(A)}\int_{A(\mathcal{K})}|\omega\wedge\bar{\omega}|$ when $\mathcal{K}=\mathbb{C}$; in all cases $\omega$ is an exterior form for $A/\mathcal{K}$
$\mu(X)$ |  encodes whether $X$ is deficient, see [13, Definition 5.5]
$X/H$ |  quotient of the curve $X$ by a group $H\leq\operatorname{Aut}_{K}(X)$
$\langle\cdot,\cdot\rangle$ |  the inner product on characters of $G$-representations
$\rho^{*}$, $\rho^{H}$ |  the dual, resp. $H$-invariant ($H\leq G$) vectors, of a $G$-representation $\rho$
$\Theta$ |  a pseudo Brauer relation relative to a $G$-representation $\mathcal{V}$, see Definition 2.2
$\mathcal{C}_{\Theta}(\mathcal{V})$ |  regulator constant for $\mathcal{V}$, see Definition 2.4
$S_{\Theta,p}$ |  set of self-dual and irreducible $\mathbb{Q}_{p}[G]$-subrepresentations of $\mathcal{X}_{p}$ with $\mathcal{C}_{\Theta}(\mathcal{X}_{p})\equiv p\quad\textup{mod }\mathbb{Q}_{p}^{\times 2}$, see Definition 2.7
$\Lambda_{\Theta}(X/{\mathcal{K}})$ |  arithmetic local invariant corresponding to pseudo Brauer relation $\Theta$, see Definition 2.10 (cf. [13, Definition 6.16])
$\mathbb{P}^{1}_{x}/K$ |  $\mathbb{P}^{1}$ together with a choice of $x$ which generates its function field over $K$
$D_{2p}$ |  the dihedral group of order $2p$
$C_{m}$ |  cyclic group of order $m$

We will usually write $K$ to denote a number field.

###  1.6. Notation and convention for curves

Throughout this paper, a curve $X/K$ is defined to be a $K$-variety which is
pure of dimension $1$ (equivalently, we suppose that all its irreducible
components have dimension $1$). We assume that all curves are smooth and
proper but not necessarily connected (and neither are the connected components
assumed to be geometrically connected). We refer the reader to [13, Appendix
A] for further details concerning these curves and their Jacobians.

We will call a curve $X/K$ nice if it is connected, and in addition we say it
is very nice if its geometric components are defined over the base field $K$.
Given a number field extension $L/K$ and a very nice curve $X/L$, we write
${}_{K}(X)$ to denote $X$ viewed as a scheme over $K$, i.e. the scheme $X$
equipped with the structure morphism
$X\to\textup{Spec}(L)\to\textup{Spec}(K)$.

### Acknowledgements

I would like to express my special thanks to Vladimir Dokchitser, not only for
suggesting this project but also for his guidance and support which proved to
be essential for the completion of this work. Thank you, Vladimir. I would
also like to thank Adam Morgan for his work in laying the theoretical
foundations for much of what will be presented here. Thank you, Adam.

##  2\. Background

In this section, we review existing results on pseudo Brauer relations, their
regulator constants, and the relevant local invariant $\Lambda_{\Theta}$
already established in [13]. To conclude, we present some well-known results
on monodromy in the case of Galois covers of curves.

###  2.1. Regulator constants, pseudo Brauer relations and isogenies

######  Notation 2.1.

We write $L$ to denote any field of characteristic $0$ which admits an
embedding into $\mathbb{C}$, $G$ is a finite group, and $\mathcal{V}$ is a
self-dual $L[G]$-representation.

######  Definition 2.2.

Let $H_{1},...,H_{n}$ and $H^{\prime}_{1},...,H_{m}^{\prime}$ be (not
necessarily distinct) subgroups of $G$. We say that
$\Theta=\sum_{i}H_{i}-\sum_{j}H_{j}^{\prime}$ is a pseudo Brauer relation
relative to $\mathcal{V}$ if there are $\mathbb{C}[G]$-representations
$\rho_{1}$ and $\rho_{2}$, satisfying
$\langle\rho_{1},\mathcal{V}\rangle=\langle\rho_{2},\mathcal{V}\rangle=0$,
such that

$\rho_{1}\oplus\bigoplus_{i}\textup{Ind}_{H_{i}}^{G}\mathbf{1}\cong\rho_{2}\oplus\bigoplus_{j}\textup{Ind}_{H_{j}^{\prime}}^{G}\mathbf{1}.$

######  Notation 2.3.

Let $\Theta=\sum_{i}H_{i}-\sum_{j}H_{j}^{\prime}$ be a pseudo Brauer relation
relative to $\mathcal{V}$. Given a non-degenerate, $G$-invariant, $L$-bilinear
pairing $\langle\mskip-3.0mu\langle,\rangle\mskip-3.0mu\rangle$ on
$\mathcal{V}$, we denote by $\langle,\rangle_{1}$ the pairing

$\langle,\rangle_{1}=\bigoplus_{i}\frac{1}{|H_{i}|}\langle\mskip-3.0mu\langle,\rangle\mskip-3.0mu\rangle\quad\textup{on
the vector space}\quad\bigoplus_{i}\mathcal{V}^{H_{i}},$

and define the pairing $\left\langle,\right\rangle_{2}$ on
$\bigoplus_{j}\mathcal{V}^{H_{j}^{\prime}}$ in an analogous way. Given a basis
$\mathcal{B}=\\{v_{i}\\}_{i}$ for $\bigoplus_{i}\mathcal{V}^{H_{i}}$, we
denote by $\langle\mathcal{B},\mathcal{B}\rangle_{1}$ the matrix with
$(i,j)^{th}$ entry $\langle v_{i},v_{j}\rangle_{1}$, and define
$\langle\mathcal{B}^{\prime},\mathcal{B}^{\prime}\rangle_{2}$ for a basis
$\mathcal{B}^{\prime}$ of $\bigoplus_{j}\mathcal{V}^{H_{j}^{\prime}}$
similarly.

######  Definition 2.4.

Let $\Theta=\sum_{i}H_{i}-\sum_{j}H_{j}^{\prime}$ be a pseudo Brauer relation
relative to $\mathcal{V}$, and let
$\langle\mskip-3.0mu\langle,\rangle\mskip-3.0mu\rangle$ be a non-degenerate,
$G$-invariant, $L$-bilinear pairing on $\mathcal{V}$ taking values in some
field extension $L^{\prime}$ of $L$. Given bases $\mathcal{B}$ for
$\bigoplus_{i}\mathcal{V}^{H_{i}}$ and $\mathcal{B}^{\prime}$ for
$\bigoplus_{j}\mathcal{V}^{H_{j}^{\prime}}$, we define

$\mathcal{C}^{\mathcal{B},\mathcal{B}^{\prime}}_{\Theta}(\mathcal{V})=\frac{\text{det}\langle\mathcal{B},\mathcal{B}\rangle_{1}}{\text{det}\langle\mathcal{B}^{\prime},\mathcal{B}^{\prime}\rangle_{2}}\in
L^{\prime\times}.$

We then define the regulator constant of $\mathcal{V}$ relative to $\Theta$,
denoted $\mathcal{C}_{\Theta}(\mathcal{V})$, to be the class of
$\mathcal{C}^{\mathcal{B},\mathcal{B}^{\prime}}_{\Theta}(\mathcal{V})$ in
$L^{\prime\times}/L^{\times 2}$ for any choice of bases $\mathcal{B}$,
$\mathcal{B}^{\prime}$.

######  Remark 2.5.

As noted in [13, Theorem 2.7], the value of
$\mathcal{C}_{\Theta}(\mathcal{V})$ is independent of the pairing
$\langle\mskip-3.0mu\langle,\rangle\mskip-3.0mu\rangle$ chosen on
$\mathcal{V}$, is well-defined as an element in $L^{\times}/L^{\times 2}$ and
is multiplicative in both $\Theta$ and $\mathcal{V}$.

The present paper will only utilise pseudo Brauer relations relative to the
$\ell$-adic Tate module $V_{\ell}(\textup{Jac}_{X})$ of the Jacobian variety
of $X/K$, a curve being acted upon by $G$.

######  Definition 2.6 ([13] Definition 3.3).

Let $G$ be a finite group which acts on a curve $X/K$ via $K$-automorphisms.
We say that $\Theta$ is a pseudo Brauer relation for $G$ and $X$ if $\Theta$
is a pseudo Brauer relation relative to $V_{\ell}(\textup{Jac}_{X})$.

As already mentioned, any pseudo Brauer relation for $G$ and $X$ induces an
isogeny between Jacobians. This isogeny is then utilised by Theorem 1.14 to
extract certain rank parities. This can be accomplished by first finding a
subset of self-dual $\mathbb{Q}_{p}[G]$-representations $S_{\Theta,p}$, and
then the multiplicity of these representations in
$\mathcal{X}_{p}(\textup{Jac}_{X})$.

######  Definition 2.7 (cf. [13] Definition 2.9).

We write $\mathcal{R}_{G}$ to denote the set of all self-dual
$\mathbb{Q}_{p}[G]$-representations which are either irreducible or of the
form $\tau\oplus\tau^{*}$ for some irreducible $\tau$. Then, we define the
subsets $S_{\Theta,p}(X)\subseteq\mathcal{R}_{G}$ to be

$S_{\Theta,p}(X)=\\{\rho\in\mathcal{R}_{G}\ |\
\langle\rho,\mathcal{X}_{p}(\textup{Jac}_{X})\rangle>0\ \textup{and}\
\textup{ord}_{p}\ \mathcal{C}_{\Theta}(\rho)\equiv 1\quad\textup{mod }2\\},$

where $\Theta$ is a pseudo Brauer relation for $G$ and $X$. Where no confusion
lurks, we will simply write $S_{\Theta,p}$ to denote this set.

We end this subsection by recording some instances where the regulator
constant $\mathcal{C}_{\Theta}(\mathcal{V})$ is known to be trivial in which
case $\mathcal{V}$ cannot lie in $S_{\Theta,p}$.

######  Lemma 2.8 (cf. [10] Corollary 2.25, Lemma 2.26).

Let $\mathcal{V}$ be a self-dual $L[G]$-representation, and let
$\Theta=\sum_{i}H_{i}-\sum_{j}H_{j}^{\prime}$ be a pseudo Brauer relation with
respect to $\mathcal{V}$. In addition, suppose that either

  1. (1)

$\langle\mathcal{V},L[G/H_{i}]\rangle=\langle\mathcal{V},L[G/H_{j}^{\prime}]\rangle=0$
for all $i,j$,

  2. (2)

$\mathcal{V}\otimes\bar{L}$ is a symplectic $\bar{L}[G]$-representation,

  3. (3)

$\mathcal{V}\otimes\bar{L}\cong\tau\oplus\tau$ for some
$\bar{L}[G]$-representation $\tau$

  4. (4)

all $\bar{L}$-irreducible constituents of $\mathcal{V}\otimes\bar{L}$ are not-
self dual

Then, $\mathcal{C}_{\Theta}(\mathcal{V})=1$.

###  2.2. The local invariant $\Lambda_{\Theta}(X/\mathcal{K})$

We recall the definition of the local arithmetic invariant
$\Lambda_{\Theta}(X/\mathcal{K})$ attached to a curve $X$ which is defined
over a local field $\mathcal{K}$. This appears on the right hand side of
Theorem 1.14.

######  Notation 2.9.

We continue to write $\Theta=\sum_{i}H_{i}-\sum_{j}H_{j}^{\prime}$ to denote a
pseudo Brauer relation for $G$ and $X$. In addition, we fix bases
$\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ for
$\Omega^{1}(\prod_{i}\operatorname{Jac}_{X/H_{i}})$ and
$\Omega^{1}(\prod_{j}\operatorname{Jac}_{X/H_{j}^{\prime}})$, respectively and
write $\omega(\mathcal{B}_{1})$ to denote the exterior form on
$\prod_{i}\operatorname{Jac}_{X/H_{i}}$ given by the wedge product of elements
in $\mathcal{B}_{1}$, and similarly for $\omega(\mathcal{B}_{2})$.

######  Definition 2.10.

Suppose that $\Omega^{1}(\operatorname{Jac}_{X})$ is a
$\mathcal{K}[G]$-representation which is realisable over $\mathbb{Q}$. For any
choice of bases $\mathcal{B}_{1},\mathcal{B}_{2}$ for
$\Omega^{1}(\prod_{i}\operatorname{Jac}_{X/H_{i}})$ and
$\Omega^{1}(\prod_{j}\operatorname{Jac}_{X/H_{j}^{\prime}})$, we define

$\Lambda_{\Theta}(X/{\mathcal{K}})=\frac{C(\prod_{i}\operatorname{Jac}_{X/H_{i}},\omega(\mathcal{B}_{1}))\
}{C(\prod_{j}\operatorname{Jac}_{X/H_{j}^{\prime}},\omega(\mathcal{B}_{2}))}\cdot\frac{\prod_{i}\mu({X/H_{i}})}{\prod_{j}\mu({X/H_{j}^{\prime}})}\cdot\
\Biggl{|}\sqrt{\frac{\mathcal{C}_{\Theta}^{\textup{sf}}(\Omega^{1}(\operatorname{Jac}_{X}))}{\mathcal{C}_{\Theta}^{\mathcal{B}_{1},\mathcal{B}_{2}}(\Omega^{1}(\operatorname{Jac}_{X}))}}\Biggr{|}_{\mathcal{K}},$

where $C$ and $\mu$ are as in §1.5, and
${\mathcal{C}_{\Theta}^{\textup{sf}}(\Omega^{1}(\operatorname{Jac}_{X}))}$ is
the unique, square-free integer satisfying
${\mathcal{C}_{\Theta}^{\textup{sf}}(\Omega^{1}(\operatorname{Jac}_{X}))}\equiv\mathcal{C}_{\Theta}(\Omega^{1}(\operatorname{Jac}_{X}))\quad\textup{mod
}\mathbb{Q}^{\times 2}$.

######  Remark 2.11.

For the purposes of the present paper, it suffices to consider the definition
of $\Lambda_{\Theta}(X/\mathcal{K})$ specialised to the case where
$\Omega^{1}$ is realisable over $\mathbb{Q}$. For a slightly more general
definition of $\Lambda_{\Theta}(X/\mathcal{K})$ allowing $\Omega^{1}$ to be
self-dual, see [13, §5].

###  2.3. Galois closures, monodromy and Riemann-Hurwitz

We now record some well-known results concerning monodromy. These tools will
be of great importance when constructing Galois covers using a down-to-top
approach by taking the Galois closure of a branched cover of curves.

######  Warning.

For the remainder of this subsection only, our primary assumption will be that
all curves are geometrically connected and, unless otherwise stated, defined
over $\mathbb{C}$. We note that this deviates from the ongoing conventions in
§1.6.

We adopt a similar notation to the one used in [24] to record the ramification
in the fibre of a branch point and the cycle type of a permutation.

######  Definition 2.12.

Let $f:Z\to\mathbb{P}^{1}$ be a morphism of complex curves, and let
$\mathcal{B}_{f}=\\{b_{i}\\}_{i=1}^{r}\subset\mathbb{P}^{1}(\mathbb{C})$ be
its branch locus. Suppose that the fibre $f^{-1}(b_{i})$ contains $n_{j}$
points each with ramification index $e_{j}>1$, and suppose that $j$ takes
values in $\\{1,\dots,k\\}$. Then, we write
$R_{b_{i}}=\prod_{j=1}^{k}(n_{j})^{e_{j}}$ to denote the ramification
structure of $f$ at the point $b_{i}$. The ramification structure of $f$ is
the multi-set given by $\\{R_{b_{i}}\\}_{i=1}^{r}$.

######  Definition 2.13.

Let $\sigma$ be a permutation in $S_{n}$ expressed as a product of disjoint
cycles. Suppose that $\sigma$ contains $m_{j}$ cycles each of length $t_{j}>1$
where $j$ takes values in $\\{1,\dots,k\\}$. Then, we write
$r_{\sigma}=\prod_{j=1}^{k}(t_{j})^{m_{j}}$ to denote the cycle type of
$\sigma$.

######  Example 2.14.

Let $\sigma\in S_{7}$ be the element $(1,2)(3,4)(5,6,7)$. Then,
$r_{\sigma}=(2)^{2}(3)^{1}$.

######  Theorem 2.15 ([26]).

Let $f:Z\to\mathbb{P}^{1}$ be a degree $n$ cover of complex curves branched at
$r$ points $\\{b_{i}\\}_{i=1}^{r}$, and write $F:X_{Z}\to\mathbb{P}^{1}$ to
denote its Galois closure. Then, there exists an $r$-tuple
$(\sigma_{1},\dots,\sigma_{r})$ of elements in $S_{n}$ such that the
ramification structure at $b_{i}$ matches the cycle type of $\sigma_{i}$. In
addition, the Galois group of $X_{Z}\to\mathbb{P}^{1}$ is a transitive
subgroup of $S_{n}$ generated by the $\sigma_{i}$.

The tuple $(\sigma_{1},\dots,\sigma_{r})$ afforded by Theorem 2.15 is usually
called the monodromy tuple of $F$.

######  Remark 2.16.

As detailed in [24, Proposition 3.18 & Remark 3.19], the cyclic group
$\langle\sigma_{i}\rangle$ afforded by Theorem 2.15 coincides (up to
conjugacy) with the decomposition group of any point in the fibre
$F^{-1}(b_{i})$.

The following theorem gives a description for the $G$-module structure of the
$\ell$-adic Tate module of the Jacobian variety of a complex curve $X$ in
terms of the of monodromy of the branched cover $X\to Y$. Given an element
$\sigma\in G$, we write $\mathbf{1}_{\sigma}^{*}$ to denote the permutation
representation $\textup{Ind}_{\langle\sigma\rangle}^{G}\mathbf{1}$.

######  Theorem 2.17 ([17] Proposition 1.1).

Let $F:X\to Y$ be a Galois cover of complex curves which is branched at
$\\{b_{i}\\}_{i=1}^{r}$, and write $G$ to denote its Galois group. In
addition, suppose that $g_{i}$ is a generator for the decomposition group of
any point in $F^{-1}(b_{i})$. Then, for any prime $\ell$, the following
$G$-representations

$V_{\ell}(\textup{Jac}_{X})\ \ \textup{and}\ \ \mathbf{1}^{\oplus
2}\oplus(\mathbf{1}_{e}^{*})^{\oplus(2g(Y)-2)}\oplus\bigoplus_{i=1}^{r}(\mathbf{1}_{e}^{*}\ominus\mathbf{1}_{g_{i}}^{*}),$

become isomorphic after extending scalars to $\mathbb{C}$, where $g(Y)$
denotes the genus of the curve $Y$.

For the remainder of this section, we write $K$ to denote a number field or a
local field of characteristic $0$.

######  Corollary 2.18.

Let $X/K$ be a geometrically connected curve. Let $G$ be a subgroup of
$K$-automorphisms of $X$. Then, for any prime $\ell$, Theorem 2.17 holds for
$X/K$.

###### Proof.

Base-changing along a field embedding $K\hookrightarrow{}\mathbb{C}$, it
suffices to prove the result for $K$ replaced with $\mathbb{C}$ and $X$ a
complex curve. This is precisely Theorem 2.17. ∎

######  Corollary 2.19.

Let $f:Z\to\mathbb{P}^{1}$ be a $K$-rational morphism of geometrically
connected curves. We write $\mathcal{B}_{f}=\\{b_{i}\\}_{i=1}^{r}$ to denote
its branch locus. Let $F:X_{Z}\to\mathbb{P}^{1}$ be the Galois closure of $f$,
and in addition suppose that $X_{Z}/K$ is geometrically connected. Then, for
any prime $\ell$, the $G$-representations

$V_{\ell}(\textup{Jac}_{X_{Z}})\ \ \textup{and}\ \ \mathbf{1}^{\oplus
2}\ominus(\mathbf{1}_{e}^{*})^{\oplus
2}\oplus\bigoplus_{i=1}^{r}(\mathbf{1}_{e}^{*}\ominus\mathbf{1}_{\sigma_{i}}^{*}),$

become isomorphic after extending scalars to $\mathbb{C}$, where the
$\sigma_{i}$ coincide with the permutations afforded by Theorem 2.15.

###### Proof.

By [31, Corollary 3.9.3(b)], the branch locus of $Z\to\mathbb{P}^{1}$
coincides with the branch locus of its Galois closure
$X_{Z}\to\mathbb{P}^{1}$, while by Remark 2.16 we deduce that the groups
generated by the $\sigma_{i}$ are (up to conjugacy) equal to the decomposition
at any point in the fibre of $X_{Z}\to\mathbb{P}^{1}$ at $b_{i}$. As a result,
the claim follows by Corollary 2.18.

∎

##  3\. Hyperelliptic Jacobians over $K(\sqrt{d})$

In the following section, we provide a local expression for the
$2^{\infty}$-Selmer rank of the Jacobian of a hyperelliptic curve $Y/K$ base
changed to a quadratic field extension $L=K(\sqrt{d})$. We first prove pseudo
Brauer verifiability for the isogeny

$\textup{Res}_{L/K}\textup{Jac}_{Y/L}\to\textup{Jac}_{Y}\times\textup{Jac}_{Y^{d}},$

where $Y^{d}$ denotes the quadratic twist of $Y$ by $d$. We write $Y_{L}$ to
denote $Y\times_{K}L$ the base change to $L$, and as in §1.6, we write
$X={}_{K}(Y_{L})$ to denote the curve obtained from $Y_{L}$ by forgetting the
$L$-structure.

######  Theorem 3.1.

Let $Y/K$ be a hyperelliptic curve with affine model $y^{2}=f(x)$, where $f$
is a square-free polynomial. Then, the function field inclusions
$K(x)\subseteq K(x,y)\subseteq L(x,y)$ fit into a Galois diagram

${X}$${Y=X/C_{2}^{a}}$${Y^{d}=X/C_{2}^{b}}$${X/C_{2}^{c}}$${\mathbb{P}_{x}^{1}=X/(C_{2}\times
C_{2})}$

where $C_{2}^{a},C_{2}^{b},C_{2}^{c}$ denote all proper subgroups of
$C_{2}\times C_{2}$.

###### Proof.

The function field $K(X)$ coincides with the biquadratic extension of $K(x)$
obtained by adjoining $y$ and $\sqrt{d}$. By [13, Remark A.31], this induces a
$C_{2}\times C_{2}$-action on $X$ via $K$-automorphisms obtained by separately
negating $y$ and $\sqrt{d}$. The quotient curves agree with what is asserted
by the claim in view of [13, Remark A.30]. Indeed, two of the intermediate
field extensions are $K(x,y)/(y^{2}-f(x)),K(x,\sqrt{d}y)/(dy^{2}-f(x))$ which
coincide with the function fields of $Y$ and $Y^{d}$ respectively. ∎

######  Theorem 3.2.

The pseudo Brauer relation $\Theta=\\{e\\}+2(C_{2}\times
C_{2})-C_{2}^{a}-C_{2}^{b}-C_{2}^{c}$ for $C_{2}\times C_{2}$ and $X$ verifies
an isogeny
$\textup{Res}_{L/K}\textup{Jac}_{Y/L}\to\textup{Jac}_{Y/K}\times\textup{Jac}_{Y^{d}/K}$.

###### Proof.

The function fields obtained by taking $C_{2}^{c}$ and $(C_{2}\times
C_{2})$-invariance in Theorem 3.1 are $K(x,\sqrt{d})$ and $K(x)$ respectively.
Therefore, the quotients of $X$ by $C_{2}^{c}$ and $C_{2}\times C_{2}$ are
both curves of genus $0$. Therefore, the Brauer relation
$\Theta=\\{e\\}+2(C_{2}\times C_{2})-C_{2}^{a}-C_{2}^{b}-C_{2}^{c}$ for
$C_{2}\times C_{2}$ induces an isogeny
$\textup{Jac}_{X/K}\to\textup{Jac}_{Y/K}\times\textup{Jac}_{Y^{d}/K}$. By [13,
Lemma A.22], it follows
$\textup{Jac}_{X/K}=\textup{Res}_{L/K}\textup{Jac}_{Y/L}$ from which the
result follows. ∎

######  Theorem 3.3.

Let $Y/K$ be a hyperelliptic curve and $Y^{d}/K$ the twist of $Y$ by
$L=K(\sqrt{d})$. Then,

$\textup{rk}_{2}\ \textup{Jac}_{Y}/K+\textup{rk}_{2}\
\textup{Jac}_{Y^{d}}/K\equiv\sum_{v\ \textup{place of}\ K}\textup{ord}_{2}\
\Lambda_{\Theta}(X/K_{v})\quad\textup{mod }2,$

where ${\Theta}$ is the pseudo Brauer relation for $C_{2}\times C_{2}$ and $X$
afforded by Theorem 3.2.

###### Proof.

We write $\mathcal{X}_{2}$ to denote $\mathcal{X}_{2}(\textup{Jac}_{X})$. We
write $\epsilon_{a}$ and $\epsilon_{b}$ to denote non-identity characters of
$C_{2}\times C_{2}$ on which $C_{2}^{a}$ and $C_{2}^{b}$ act trivially. It
follows that
$\mathcal{X}_{2}\cong\epsilon_{a}^{\textup{rk}_{2}\textup{Jac}_{Y}}\oplus\epsilon_{b}^{\textup{rk}_{2}\textup{Jac}_{Y^{d}}}$.
Therefore, $S_{\Theta,2}\subseteq\\{\epsilon_{a},\epsilon_{b}\\}$ (see
Definition 2.7). An elementary calculation reveals that
$\mathcal{C}_{\Theta}(\epsilon_{a})\equiv\mathcal{C}_{\Theta}(\epsilon_{b})\equiv
2\quad\textup{mod }\mathbb{Q}^{\times 2}$ from which we deduce
$S_{\Theta,2}=\\{\epsilon_{a},\epsilon_{b}\\}.$ The local expression asserted
by the claim follows upon applying Theorem 1.14 to $\Theta$ afforded by
Theorem 3.2 with $p=2$. ∎

##  4\. Richelot Isogeny

In this section, we use Brauer relations and regulator constants to deduce a
local expression for the rank parity of the Jacobian of a genus $2$ curve
$\textup{Jac}_{X}$ admitting a Richelot isogeny under some mild assumption on
the Galois group of its $2$-torsion field (see Definition 4.2 below). We first
collect some useful yet well-known facts concerning Richelot isogenies, see
for example [2] and [30] for details.

######  Lemma 4.1.

Let $Z/K$ be genus $2$ curve with affine Weierstrass model $y^{2}=F(x)$, where
$F(x)$ is a square-free polynomial of degree $6$ admitting a factorisation

$F(x)=F_{1}(x)F_{2}(x)F_{3}(x)$

into 3 coprime quadratics subject to the condition that the set
$\\{F_{1},F_{2},F_{3}\\}$ is stable under the action of
$\textup{Gal}(\bar{K}/K)$. Then, there exists an isogeny
$\phi:\textup{Jac}_{Z}\to A$, where $A$ is a principally polarised abelian
variety such that $\phi^{\vee}\circ\phi=[2]_{\textup{Jac}_{Z}}.$

###### Proof.

See [30, Proposition 8.2.3] ∎

We say that $\\{F_{1},F_{2},F_{3}\\}$ is a quadratic splitting for $Z$. In
view of [2, Lemma 4.1], a Jacobian of a genus $2$ curve admits a
$(2,2)$-isogeny only when $Z$ admits a quadratic splitting. For the remainder
of this section, we will assume that $\textup{Gal}(K(\textup{Jax}_{Z}[2])/K)$
is a $2$-group. Since
$\textup{Gal}(K(\textup{Jax}_{Z}[2])/K)=\textup{Gal}(F)$, this assumption is
equivalent to $\textup{Gal}(F)\leq D_{8}\times C_{2}$, where $D_{8}$ denotes
the dihedral group of order $8$. Following the terminology introduced in [14],
we refer to these curves as C2D4 curves (albeit acknowledging the discrepancy
in notation regarding dihedral groups.).

######  Definition 4.2.

Let $Z/K$ be a genus $2$ curve admitting a quadratic splitting
$\\{F_{1},F_{2},F_{3}\\}$. We say that $Z/K$ is a C2D4 curve if
$\textup{Gal}(F_{1}F_{2}F_{3})\subseteq D_{8}\times C_{2}$ as a permutation
group on $6$ roots, in which case the two factors $D_{8}$ and $C_{2}$ act
separately on $4$ and $2$ roots respectively.

######  Convention.

We note that the individual quadratic factors $F_{i}$ need not be defined over
$K$. In particular, under the assumption that $Z$ is a $C2D4$ curve, then two
of the quadratic factors could be Galois conjugate over a quadratic field
extension $L=K(\sqrt{d})$. When this happens, we adhere to the convention that
$F_{1}$ and $F_{2}$ are Galois conjugate over $K(\sqrt{d})$, while $F_{3}$ is
necessarily defined over $K$.

######  Notation 4.3.

For the remainder of this section, for any $C2D4$ curve $Z$ we fix the
following constant $d=d(Z)\in K^{\times}$. If $Z$ admits a quadratic splitting
in which $2$ quadratic factors are Galois conjugate over a quadratic extension
$L/K$, then we fix $d\in K^{\times}$ to be any generator for $L=K(\sqrt{d})$.
If all quadratic factors are defined over $K$, then we fix $d=1$.

If we assume that the quadratic splitting is non-singular in the sense of [30,
Definition 8.2.4], then [2, Proposition 4.3] asserts that the target of this
isogeny is the Jacobian of a $C2D4$ curve with an affine model
$\tilde{Z}_{d}=\\{dy^{2}=G_{1}(x)G_{2}(x)G_{3}(x)\\}$. The polynomials $G_{i}$
are defined as follows.

######  Definition 4.4.

Let $\\{F_{1},F_{2},F_{3}\\}$ be a non-singular quadratic splitting for a
genus $2$ curve, and let $F_{i}(x)=\sum_{j=0}^{2}f_{i,j}x^{j}$. Then, for
$(i,j,k)=(1,2,3),(2,3,1),(3,1,2)$, we define

$G_{i}({x})=\det\begin{pmatrix}f_{1,0}&f_{1,1}&f_{1,2}\\\
f_{2,0}&f_{2,1}&f_{2,2}\\\
f_{3,0}&f_{3,1}&f_{3,2}\end{pmatrix}^{-1}\cdot\bigg{(}F_{k}(x)\frac{\mathrm{d}}{\mathrm{d}x}F_{j}(x)-F_{j}(x)\frac{\mathrm{d}}{\mathrm{d}x}F_{k}(x)\bigg{)}.$

A classical way of deducing the existence of an isogeny
$\textup{Jac}_{Z}\to\textup{Jac}_{\tilde{Z}_{d}}$ is via a
$(2,2)$-correspondence $\Gamma_{d}$ between $Z$ and $\tilde{Z}_{d}$. This is
defined as follows.

######  Definition 4.5.

Let $Z/K$ be a genus $2$ curve admitting a non-singular quadratic splitting
$\\{F_{1},F_{2},F_{3}\\}$. By expressing $\tilde{Z}_{d}$ in terms of
$(\tilde{x},\tilde{y})$ instead of $(x,y)$, the Richelot correspondence on $Z$
and $\tilde{Z}_{d}$ is the curve $\subseteq Z\times\tilde{Z}_{d}$ given by the
equations

$\Gamma_{d}:\begin{cases}\hfill
F_{1}(x)G_{1}(\tilde{x})+F_{2}(x)G_{2}(\tilde{x})&=0\\\ \hfill
F_{1}(x)G_{1}(\tilde{x})(x-\tilde{x})&=\sqrt{d}\tilde{y}y\\\ \hfill
F_{2}(x)G_{2}(\tilde{x})(x-\tilde{x})&=-\sqrt{d}\tilde{y}y\\\ \end{cases}$

######  Remark 4.6.

The projection morphisms

$\displaystyle\pi_{1}:\Gamma_{d}$ $\displaystyle\to Z\hskip
42.67912pt\textup{and}\hskip 42.67912pt\pi_{2}:\Gamma_{d}$
$\displaystyle\to\tilde{Z}_{d}$ $\displaystyle(x,y,\tilde{x},\tilde{y})$
$\displaystyle\mapsto(x,y)\hskip 85.35826pt(x,y,\tilde{x},\tilde{y})$
$\displaystyle\mapsto(\tilde{x},\tilde{y})$

induce an isogeny $\textup{Jac}_{Z}\to\textup{Jac}_{\tilde{Z}_{d}}$ via the
composition $(\pi_{2})_{*}\pi_{1}^{*}$. It is important to observe that the
specific choice of $d$ as indicated in Notation 4.3 ensures that the induced
isogeny is defined over $K$. This is because both $\Gamma_{d}$ and the
$\pi_{i}$ are defined over $K$ when this choice is made. This need not be true
for an arbitrary choice of $d$.

In the following Proposition, we utilise the Richelot correspondence to prove
pseudo Brauer verifiability for a Richelot isogeny between $C2D4$ curves. As
always, we write $\mathbb{P}^{1}_{x}$ to denote $\mathbb{P}^{1}$ together with
a choice of $x$ which generates its function field over $K$.

######  Proposition 4.7.

Let $Z/K$ be a $C2D4$ curve, and suppose that $\\{F_{1},F_{2},F_{3}\\}$ is a
non-singular quadratic splitting for it. Suppose (as we may) that $F_{1}$ and
$F_{2}$ are monic polynomials given by $F_{i}(x)=x^{2}+f_{i,1}x+f_{i,2}$.
Then, define

${T}(x)=\begin{cases}\frac{-f_{1,2}f_{2,1}+f_{1,1}f_{2,2}+(f_{1,1}-f_{2,1})x^{2}}{(f_{1,2}-f_{2,2})+(f_{1,1}-f_{2,1})x},&\text{if
}f_{1,1}\neq f_{2,1},\\\ x(f_{1,1}+x),&\text{if }f_{1,1}=f_{2,1}.\end{cases}$

Then, it follows that

  1. (1)

$T(x)$ is a $K$-rational function,

  2. (2)

The function field extension $K(T)\subseteq K(x)\subseteq K(Z)\subseteq
K(\Gamma_{d})$ is Galois with Galois group $D_{8}=\langle s,r\ |\
r^{4}=s^{2}=1,rs=r^{3}s\rangle$,

  3. (3)

Assuming (as we may) that $Z=X/\langle s\rangle$, then these function field
inclusions fit into a Galois diagram

${\Gamma_{d}}$${Z=X/\langle s\rangle}$${\Gamma_{d}/\langle
r^{2}\rangle}$${\tilde{Z}_{d}=\Gamma_{d}/\langle
sr\rangle}$${\mathbb{P}^{1}_{x}=\Gamma_{d}/\langle
s,r^{2}\rangle}$${\Gamma_{d}/\langle
r\rangle}$${\mathbb{P}^{1}_{\tilde{x}}={\Gamma_{d}/\langle
sr,r^{2}\rangle}}$${\mathbb{P}^{1}_{{T}}=\Gamma_{d}/D_{8}}$

###### Proof.

If the individual $F_{i}$ are defined over $K$, then (1) is clear. If $F_{1}$
and $F_{2}$ are quadratic conjugate over $K(\sqrt{d})$, then the non-trivial
element $\sigma\in\textup{Gal}(K(\sqrt{d})/K)$ acts on the coefficients by
permuting $f_{1,j}\leftrightarrow f_{2,j}$. Since $T(x)$ is $\sigma$-stable,
then $T(x)$ is $K$-rational. This completes the proof of $(1)$.

In order to prove $(2)$, we first fix explicit generators for the function
field of $\Gamma_{d}$. We fix $\tilde{y}\in\overline{K(\tilde{x})}$ by letting
$\tilde{y}^{2}=G_{1}(\tilde{x})G_{2}(\tilde{x})G_{3}(\tilde{x})$. Then, by the
description of $\Gamma_{d}$ given in Definition 4.5, we deduce that
$K(\Gamma_{d})=K(x,y,\tilde{x},\sqrt{d}\tilde{y})$. It follows that all $3$
function fields $K(x,y)=K(Z),K(x,\tilde{x})$ and $\
K(\tilde{x},\sqrt{d}\tilde{y})=K(\tilde{Z}_{d})$ admit degree $2$ embeddings
into $K(\Gamma_{d})$. These correspond to three involutions on $\Gamma_{d}$,
say $g_{1},g_{2}$ and $g_{3}$ respectively, and let $H=\langle
g_{1},g_{2},g_{3}\rangle$. Since $g_{1}$ and $g_{2}$ fix $x$, then they fix
$T(x)$. Adding on, one checks that

$T(x)=\begin{cases}\frac{f_{2,1}{F_{1}(x)}-f_{1,1}{F_{2}(x)}}{{F_{2}(x)}-{F_{1}(x)}},&\text{if
}f_{1,1}\neq f_{2,1}\\\
\frac{f_{2,2}{F_{1}(x)}-f_{1,2}{F_{2}(x)}}{{F_{2}(x)}-{F_{1}(x)}},&\text{if
}f_{1,1}=f_{2,1}.\end{cases}$

Since $\frac{F_{1}(x)}{F_{2}(x)}=\frac{-G_{2}(\tilde{x})}{G_{1}(\tilde{x})}$,
we deduce that $T$ is also a function in $\tilde{x}$, and since $g_{3}$ fixes
$\tilde{x}$, then it must fix $T$. This shows that $K(T)\subseteq
K(\Gamma_{d})^{H}$, and since $[K(\Gamma_{d}):K(T)]=8$, we deduce that
$|H|\leq 8$. Since $K(x)$ isn’t fixed by $H$ (since $g_{3}$ doesn’t fix $x$),
we deduce that $|H|>4$. Therefore $|H|=8$ and in addition
$K(\Gamma_{d})^{H}=K(T)$.

The claim in (2) will follow upon showing that the Galois closure of
$Z\to\mathbb{P}^{1}_{x}\to\mathbb{P}^{1}_{T}$ is $D_{8}$. We write $W$ to
denote the discriminant curve (see [13, Lemma A.7] or [15, §2.2]) of
$Z\to\mathbb{P}^{1}_{T}$. It suffices to show that $W\to\mathbb{P}^{1}_{T}$ is
not the trivial cover. By using [15, Lemma 2.3], we deduce that
$W\to\mathbb{P}^{1}_{T}$ is branched at $2$ distinct points, namely $T(w_{1})$
and $T(w_{2})$ where $w_{1}$ and $w_{2}$ are the two roots of the remaining
quadratic factor $F_{3}(x)$. It’s easy to check that $T(w_{1})\neq T(w_{2})$
(as otherwise either $Z$ itself or the quadratic splitting would be singular).
This completes the proof of (2). Claim (3) follows from the proof of (2). ∎

######  Notation 4.8.

We write $C_{2}^{a}$ and $C_{2}^{b}$ to denote the two non-central subgroups
of $D_{8}$ of order $2$. In addition, write $(C_{2}^{2})^{a}$ and
$(C_{2}^{2})^{b}$ to denote the two subgroups of $D_{8}$ which contain
$C_{2}^{a}$ and $C_{2}^{b}$ respectively.

######  Theorem 4.9.

Let $Z/K$ be a $C2D4$ curve which admits a non-singular, quadratic splitting.
Then, the pseudo Brauer relation
$C_{2}^{a}-C_{2}^{b}+(C_{2}^{2})^{b}-(C_{2}^{2})^{a}$ for $D_{8}$ and
$\Gamma_{d}$ verifies a Richelot isogeny
$\textup{Jac}_{Z}\to\textup{Jac}_{\tilde{Z}_{d}}$.

###### Proof.

We note that $C_{2}^{a}-C_{2}^{b}+(C_{2}^{2})^{b}-(C_{2}^{2})^{a}$ is a Brauer
relation for the group $D_{8}$. Following the notation from Proposition 4.7,
we deduce that $(C_{2}^{2})^{a}$ and $(C_{2}^{2})^{b}$ coincide with the
subgroups $\langle s,r^{2}\rangle$ and $\langle sr,r^{2}\rangle.$ By
Proposition 4.7(3), the quotients $\Gamma_{d}/(C_{2}^{2})^{a}$ and
$\Gamma_{d}/(C_{2}^{2})^{b}$ are curves of genus $0$. This gives the required
result. ∎

We write $\epsilon_{r}$ the denote the non-identity, $1$-dimensional character
of $D_{8}$ on which $r$ acts trivially and $\rho$ for the irreducible
$D_{8}$-representation of dimension $2$.

######  Lemma 4.10.

The $D_{8}$-representations

$\mathcal{X}_{2}(\textup{Jac}_{\Gamma_{d}})\ \ \textup{and}\ \
\epsilon_{r}^{\textup{rk}_{2}\textup{Jac}_{\Gamma_{d}/\langle
r^{2}\rangle}}\oplus\rho^{\textup{rk}_{2}\textup{Jac}_{Z}}$

become isomorphic after extending scalars to $\mathbb{C}$.

###### Proof.

The set of irreducible $\mathbb{C}$-representations for $D_{8}$ are
$\\{\mathbf{1},\epsilon_{s},\epsilon_{r},\epsilon_{sr},\rho\\}$. Let
$(C_{2}^{2})^{a}=\langle s,r^{2}\rangle$. Then, by Frobenius reciprocity,
$\langle\mathcal{X}_{2},\mathbb{C}[D_{8}/(C_{2}^{2})^{a}]\rangle=\langle\mathcal{X}_{2},\mathbf{1}\oplus\epsilon_{s}\rangle$
coincides with the dimension of $(C_{2}^{2})^{a}$-invariant vectors in
$\mathcal{X}_{2}$. This coincides with the $2^{\infty}$-Selmer rank of the
Jacobian of the quotient of $\Gamma_{d}$ by $(C_{2}^{2})^{a}$ (see [13, Remark
A.29]), which must be $0$ by Proposition 4.7(3). Therefore,
$\langle\mathcal{X}_{2},\mathbf{1}\rangle=\langle\mathcal{X}_{2},\epsilon_{s}\rangle=0$.
The multiplicities of the remaining irreducible representations in
$\mathcal{X}_{2}$ are found in a similar way. ∎

######  Theorem 4.11.

Let $Z/K$ be a $C2D4$ genus $2$ curve which admits a non-singular quadratic
splitting. Then,

$\textup{rk}_{2}(\textup{Jac}_{Z}/K)=\sum_{v\ \textup{place of}\
K}\textup{ord}_{2}\ \Lambda_{\Theta}(\Gamma_{d}/K_{v})\quad\textup{mod }2,$

where $\Theta$ is the pseudo Brauer relation for $D_{8}$ and $\Gamma_{d}$
afforded by Theorem 4.9.

###### Proof.

We write $\mathcal{X}_{2}$ to denote
$\mathcal{X}_{2}(\textup{Jac}_{\Gamma_{d}}).$ An elementary computation
reveals that $\mathcal{C}_{\Theta}(\rho)\equiv 2$, while
$\mathcal{C}_{\Theta}(\epsilon_{r})\equiv 1$. Therefore,
$S_{\Theta,2}=\\{\rho\\}$ (see Definition 2.7). By Lemma 4.10, we deduce that
$\sum_{\tau\in\mathcal{S}_{\Theta,2}}\langle\tau,\mathcal{X}_{2}\rangle=2^{\textup{rk}_{2}\textup{Jac}_{Z}}.$
The result follows upon applying Theorem 1.14 to the pseudo Brauer relation
$\Theta$ for $D_{8}$ and $\Gamma_{d}$ afforded by Theorem 4.9 with $p=2$. ∎

##  5\. Prym varieties of trigonal curves

We let $Z/K$ be a geometrically connected curve of genus $g$, and in addition
we suppose $\textup{Jac}_{Z}[2](k)$ has a non-trivial torsion point, or
equivalently there exists an unramified double cover $\pi:\tilde{Z}\to Z$
defined over $K$. The Prym variety denoted $\textup{Prym}(\tilde{Z}/Z)$ of
$\pi$ is the abelian variety given by the connected component of
$\textup{Ker}(\pi_{*})$ containing the identity. It is classically well-known
that there exists a $K$-rational isogeny

$\textup{Jac}(\tilde{Z})\to\textup{Jac}(Z)\times\textup{Prym}(\tilde{Z}/Z),$

It is worthy of note that $\textup{Prym}(\tilde{Z}/Z)$ need not be a Jacobian,
but it necessarily is principally polarised. The trigonal construction [15,
§2.4] establishes that for trigonal curves $Z$ defined over algebraically
closed fields, the Prym variety is a Jacobian of a tetragonal curve. A
generalisation of this in the case of non-algebraically closed fields can be
found in [3] where the authors prove that the Prym is necessarily a Jacobian
upon taking a quadratic twist of the cover $\pi:\tilde{Z}\to Z$ if needed.

######  Definition 5.1.

Let $\pi:\tilde{Z}\to Z$ be a degree $2$ cover. We say that
$\pi_{d}:\tilde{Z}_{d}\to Z$ is a quadratic twist of $\pi$ by $d\in
K^{\times}$ if there exists an isomorphism
$\psi:\tilde{Z}_{d}\xrightarrow{\sim}\tilde{Z}$ defined over $K(\sqrt{d})$
such that $\pi_{d}=\pi\circ\psi.$

As can be deduced from [3, Proposition 2.3], for an appropriate value of $d\in
K^{\times}$, there exists an isogeny

$\textup{Jac}_{\tilde{Z}_{d}}\to\textup{Jac}_{Z}\times\textup{Prym}(\tilde{Z}_{d}/Z),$

and, at this instance, $\textup{Prym}(\tilde{Z}_{d}/Z)$ is a Jacobian of a
tetragonal curve. We note that we allow $d$ to lie in $K^{\times 2}$ in which
case $\tilde{Z}_{d}\cong_{K}\tilde{Z}$ is the trivial twist. We write
$X_{\tilde{Z}}\to\mathbb{P}^{1}$ to denote the Galois closure of $\tilde{Z}\to
Z\to\mathbb{P}^{1}$, and we define $X_{\tilde{Z}_{d}}$ similarly.

######  Theorem 5.2.

Let $\pi:\tilde{Z}\to Z$ be an unramified degree $2$ cover. Then, there exists
$d\in K^{\times}$, and a quadratic twist $\pi_{d}:\tilde{Z}_{d}\to Z$ such
that $S_{4}$ acts on $X_{\tilde{Z}_{d}}$. In addition,
$\Theta=C_{2}^{2}-D_{8}-S_{3}+S_{4}$ is a pseudo Brauer relation for $S_{4}$
and $X_{\tilde{Z}_{d}}$ which verifies an isogeny
$\textup{Jac}_{\tilde{Z}_{d}}\to\textup{Jac}_{Z}\times\textup{Prym}(\tilde{Z}_{d}/Z)$.

###### Proof.

In general, $\textup{Gal}(X_{\tilde{Z}}\to\mathbb{P}^{1})\cong
C_{2}^{3}\rtimes S_{3}\cong C_{2}\times S_{4}$. As can be deduced from [15],
over $\overline{K}$, we have
$\textup{Gal}(X_{\tilde{Z}}/\overline{K}\to\mathbb{P}^{1}/\overline{K})\cong
S_{4}$. Therefore, we deduce that $X_{\tilde{Z}}$ need not be geometrically
connected, and its geometric components are (at worst) defined over a
quadratic extension $K(\sqrt{d})$. Upon twisting $\pi$ by $d\in K^{\times}$
however, we have that $\textup{Gal}(X_{\tilde{Z}_{d}}/\mathbb{P}^{1})=S_{4}$
in which case $X_{\tilde{Z}_{d}}$ is geometrically connected. By replacing
$X_{\tilde{Z}_{d}}$ with the $S_{4}$-closure of $\tilde{Z}_{d}\to
Z\to\mathbb{P}^{1}$ in the sense of [13, Definition A.16] we can assume that
the $\textup{Gal}(X_{\tilde{Z}_{d}}/\mathbb{P}^{1})$ is always $S_{4}$. Then,
by construction, we have that

(5.3) $X/S_{4}=\mathbb{P}^{1},X_{\tilde{Z}_{d}}/D_{8}=Z\ \textup{while}\
X_{\tilde{Z}_{d}}/(C_{2}^{2})=\tilde{Z}_{d}.$

We have that $C_{2}^{2}-D_{8}-S_{3}+S_{4}$ is a Brauer relation for the group
$S_{4}$. Therefore, it’s a pseudo Brauer relation for $S_{4}$ acting on
$X_{\tilde{Z}_{d}}$ and in view of the quotients given in (5.3), we can deduce
an isogeny
$\textup{Jac}_{\tilde{Z}_{d}}\to\textup{Jac}_{Z}\times\textup{Jac}_{X_{\tilde{Z}_{d}/S_{3}}}$.
As detailed in [3, §2.2] and [15, Theorem 2.11], the Jacobian of the quotient
$X_{\tilde{Z}_{d}}/S_{3}$ is the Prym variety of $\pi_{d}$. This completes the
proof. ∎

For the remainder of this section, we write $\sigma$ to denote the standard
$3$-dimensional irreducible representation of $S_{4}$, and $\tau$ for the
irreducible $2$-dimensional representation lifted from the $S_{3}$ quotient of
$S_{4}$.

######  Lemma 5.4.

For any prime $p$, the multiplicities of $\sigma$ and $\tau$ in
$\mathcal{X}_{p}(\textup{Jac}_{X_{\tilde{Z}_{d}}})$ satisfy

$\langle\mathcal{X}_{p}(\textup{Jac}_{X_{\tilde{Z}_{d}}}),\sigma\rangle=\textup{rk}_{p}\
\textup{Prym}(\tilde{Z}_{d}/Z)\ \ \textup{and}\ \
\langle\mathcal{X}_{p}(\textup{Jac}_{X_{\tilde{Z}_{d}}}),\tau\rangle=\textup{rk}_{p}\
\textup{Jac}_{Z}.$

###### Proof.

Follows in exactly the same way as Lemma 4.10. ∎

######  Theorem 5.5.

Let $\pi:\tilde{Z}\to Z$ be an unramified,double cover of a trigonal curve
$Z$. Then, there exists a $d\in K^{\times}$ and a quadratic twist
$\pi_{d}:\tilde{Z}_{d}\to Z$ of $\pi$ such that

$\textup{rk}_{2}\ \textup{Jac}_{Z}+\textup{rk}_{2}\
\textup{Prym}(\tilde{Z}_{d}/Z)\equiv\sum_{v\ \textup{place of}\
K}\textup{ord}_{2}\ \Lambda_{\Theta}(X_{\tilde{Z}_{d}}/K_{v})\quad\textup{mod
}2,$

where $\Theta$ is the pseudo Brauer relation for $S_{4}$ and
$X_{\tilde{Z}_{d}}$ afforded by Theorem 5.2.

###### Proof.

We write $\mathcal{X}_{2}$ to denote
$\mathcal{X}_{2}(\textup{Jac}_{X_{\tilde{Z}_{d}}})$. Then, we claim that
$S_{\Theta,2}=\\{\tau,\sigma\\}.$ Indeed, by combining the proof of Theorem
5.2 with Lemma 2.8(1), we deduce that
$S_{\Theta,2}\subseteq\\{\tau,\sigma\\}$. An elementary computation reveals
that $\mathcal{C}_{\Theta}(\tau)\equiv\mathcal{C}_{\Theta}(\sigma)\equiv 2$,
and therefore $S_{\Theta,2}=\\{\tau,\sigma\\}$. In view of Lemma 5.4, the
local formula asserted by the claim follows upon applying Theorem 1.14 to
$\Theta$ afforded by Theorem 5.2 with $p=2$. ∎

##  6\. Elliptic curves with cyclic isogeny

In this section, we deduce a local expression for the $p^{\infty}$-Selmer rank
of an elliptic curve that admits a cyclic isogeny of prime degree $p$ using
pseudo Brauer relations and regulator constants. For the remainder of this
section, we fix $E/K$ to be an elliptic curve, and we let $P\in E[p]$ be a
generator for the kernel of this isogeny. Finally, we write $K(P)$ to denote
the field acquired by adjoining the coordinates of $P$ to $K$ and $\tau_{P}$
for the translation-by-$P$ automorphism on $E$.

######  Lemma 6.1.

The field extension $K(P)/K$ is Galois, and the homomorphism

$\begin{split}\textup{Gal}{(}K(P)|K{)}&\xrightarrow{\psi}\textup{Aut}(\langle\tau_{P}\rangle)\\\
\sigma&\mapsto(\tau_{P}\mapsto\tau_{P^{\sigma}})\end{split}$

defines a faithful action on $\langle\tau_{P}\rangle$.

###### Proof.

Upon fixing a basis for $E[p]=\langle P,Q\rangle$, we can identify
$\textup{Gal}(K(E[p])/K)$ with a subgroup of $GL_{2}(\mathbb{F}_{p})$. Then,
$K=K(E[p])^{B}$ while $K(P)=K(E[p])^{B^{\prime}}$, where $B$ denotes the
subgroup of upper triangular matrices, while $B^{\prime}$ denotes the subgroup
of $B$ with top, left entry equal to $1$. Since $B^{\prime}$ is a normal
subgroup of $B$, then $K(P)/K$ is Galois. It suffices to prove that $\psi$ is
injective. Indeed, if $\psi(\sigma_{1})=\psi(\sigma_{2})$, then
$\sigma_{1}\sigma_{2}^{-1}$ fixes $P$ pointwise and therefore
$\sigma_{1}=\sigma_{2}$. ∎

######  Notation 6.2.

For the remainder of this section, we write $M=K(P)$. Then,
$\textup{Gal}(M/K)$ acts on $M(E)$ (on the left) via its action on constant
functions $M$. For $f\in M(E)$, we denote this action exponentially on the
left by $f\mapsto{}^{\sigma}f$. By Lemma 6.1, $\textup{Gal}(M/K)\cong C_{m}$
is cyclic of order dividing $p-1$, and in addition we can form the semi-direct
product $\langle\tau_{P}\rangle\rtimes\textup{Gal}(M/K)=C_{p}\rtimes C_{m}$.
Finally, we write ${}_{K}(E_{M})$ to denote the curve obtained from
$E_{M}=E\times_{K}M$ by forgetting the $M$-structure (see §1.6).

######  Proposition 6.3.

The semi-direct product $G=C_{p}\rtimes C_{m}$ acts on ${}_{K}(E_{M})$ via
$K$-automorphisms. In addition,

(6.4) ${}_{K}(E_{M})/C_{m}=E/K,\ \
{}_{K}(E_{M})/C_{p}={}_{K}(E^{\prime}_{M}),\ \ {}_{K}(E_{M})/G=E^{\prime}/K,$

where $E^{\prime}=E/\langle P\rangle$ denotes the $p$-isogenous elliptic
curve.

###### Proof.

The group $\langle\tau_{P}\rangle\rtimes\textup{Gal}(M/K)=C_{p}\rtimes C_{m}$
acts on functions in $M(E)$ via

$f\xrightarrow{(\tau,\sigma)}{}^{\sigma}f\circ\tau.$

By using [7, Exercise 7.8.2], this fits into a Galois diagram

$K(E^{\prime})$$M(E^{\prime})$$K(E)$$M(E)$$\textup{Gal}(M/K)$$\langle\tau_{P}\rangle$

By [13, Remark A.31], we deduce that the group $C_{p}\rtimes C_{m}$ acts on
${}_{K}(E_{M})$ via $K$-automorphisms as required. In addition, for any
subgroup $H\leq C_{p}\rtimes C_{m}$, the field of $H$-invariant functions in
the function field of ${}_{K}(E_{M})$ coincides with the function fields of
the quotient curves ${}_{K}(E_{M})/H$ in view of [13, Remark A.30]. The claim
regarding the various quotients follows by combining this observation with the
Galois diagram above. ∎

######  Remark 6.5.

For a given subgroup $D\leq C_{p}\rtimes C_{m}$, the geometric components of
the quotient curve ${}_{K}(E_{M})/D$ are defined over $M^{\pi(D)}$ where
$\pi:C_{p}\rtimes C_{m}\to C_{m}$ denotes the natural map. Indeed, given an
element $(\tau,\sigma)$ in $C_{p}\rtimes C_{m}$, then $\tau$ has no effect on
constant functions, while by construction $\sigma$ acts only on constant
functions $M$ in $M(E)$. Therefore, $M(E)^{D}\cap\overline{K}=M(E)^{D}\cap
M=M^{\pi(D)}$. From this, we deduce that the geometric components of
${}_{K}(E_{M})/D$ are defined over $M^{\pi(D)}$.

######  Theorem 6.6.

Let $G=C_{p}\rtimes C_{m}$ be the subgroup of $K$-automorphisms of
${}_{K}(E_{M})$ afforded by Proposition 6.3. Then, $\Theta=C_{m}-G$ is a
pseudo Brauer relation for $C_{p}\rtimes C_{m}$ and ${}_{K}(E_{M})$ verifying
an isogeny $E\to E^{\prime}$.

###### Proof.

We write $V_{\ell}$ to denote the $G$-representation of the $\ell$-adic Tate
module of the Jacobian of ${}_{K}(E_{M})$. We decompose
$\mathbb{C}[G/C_{m}]\cong\mathbb{C}[G/G]\oplus T,$ where $T$ is some
complementary $G$-representation. Then, $\langle V_{\ell},T\rangle$ satisfies

(6.7) $\langle V_{\ell},T\rangle\underset{\mathrm{Frob.rec.}}{=}\textup{dim}\
V_{\ell}^{C_{m}}-\textup{dim}\
V_{\ell}^{G}\underset{\eqref{quotients_elliptic_curves}}{=}\textup{dim}\
V_{\ell}(E)-\textup{dim}\ V_{\ell}(E^{\prime})=0.$

Therefore, $\Theta=C_{m}-G$ is a pseudo Brauer relation for $C_{p}\rtimes
C_{m}$ and ${}_{K}(E_{M})$. By invoking (6.4) again, the claim regarding
pseudo Brauer verifiability follows. ∎

We now find a description for the $G$-module structure on $\Omega^{1}$ of the
Jacobian of ${}_{K}(E_{M})$. Using [13, Lemma A.22], we deduce that
$\textup{Jac}_{{}_{K}(E_{M})}=\textup{Res}_{M/K}E_{M}$.

######  Lemma 6.8.

There exists an isomorphism of $C_{m}=\textup{Gal}(M/K)$-modules,

$\Omega^{1}(\textup{Res}_{M/K}E_{M})\otimes\mathbb{C}\cong\textup{Ind}_{\\{e\\}}^{C_{m}}\mathbf{1}.$

###### Proof.

As an abelian variety over $M$, $\textup{Res}_{M/K}E_{M}$ decomposes into a
product $\prod_{g}E^{g},$ where $g$ runs over the elements of
$\textup{Gal}(M/K).$ Then, $\textup{Gal}(M/K)$ acts on
$\textup{Res}_{M/K}E_{M}$, and the corresponding action on $\prod_{g}E^{g}$
permutes the components of the product. In particular, $h\in\textup{Gal}(M/K)$
acts via mapping $E^{g}\to E^{gh}$. Therefore,
$\Omega^{1}(\textup{Res}_{M/K}E_{M})\otimes
M\cong\textup{Ind}_{\\{e\\}}^{C_{m}}\mathbf{1}$. ∎

######  Proposition 6.9.

For any prime number $\ell$, the $G$ representations

$V_{\ell}(\textup{Res}_{M/K}{E_{M}})\ \ \textup{and}\ \
(\textup{Ind}_{C_{p}}^{G}\mathbf{1})^{\oplus 2}$

are isomorphic representations after extending scalars to $\mathbb{C}$. In
addition,

$\Omega^{1}(\textup{Res}_{M/K}{E_{M}})\otimes\mathbb{C}\cong\textup{Ind}_{C_{p}}^{G}\mathbf{1}.$

###### Proof.

For brevity, we write $A$ to denote $\textup{Res}_{M/K}E_{M}$ and $V_{\ell}$
(resp. $\Omega^{1}$) for the G-representations obtained from $V_{\ell}(A)$
(resp. $\Omega^{1}(A)$) after extending scalars to $\mathbb{C}$. By [13, Lemma
3.18], $V_{\ell}$ satisfies
$\textup{dim}(V_{\ell})^{D}=2\cdot\textup{dim}(\textup{Jac}_{{}_{K}(E_{M})/D})$
for any subgroup $D$ of $G$. By remodelling the argument presented in [12,
Theorem 3.3], we deduce that $V_{\ell}$ is the unique representation
satisfying the following.

  1. (1)

all character values of $G$ on $V_{\ell}$ are rational

  2. (2)

and
$\textup{dim}(V_{\ell})^{D}=2\cdot\textup{dim}(\textup{Jac}_{{}_{K}(E_{M})/D})$.

The dimension of $D$-invariant vectors of $\textup{Ind}_{C_{p}}^{G}\mathbf{1}$
is

$\displaystyle\textup{dim}\ (\textup{Ind}_{C_{p}}^{G}\mathbf{1})^{D}$
$\displaystyle=\langle\textup{Ind}_{C_{p}}^{G}\mathbf{1},\textup{Ind}_{D}^{G}\mathbf{1}\rangle=\langle\mathbf{1},\textup{Res}_{C_{p}}^{G}\textup{Ind}_{D}^{G}\mathbf{1}\rangle=\\#D\backslash
G/C_{p}.$

In addition, we deduce that $\\#D\backslash
G/C_{p}=\frac{\\#\textup{Gal}(M/K)}{\\#\pi(D)}=[M^{\pi(D)}:K]$ which agrees
with the dimension of the Jacobian of ${}_{K}(E_{M})/D$ in view of Remark 6.5.
Since $(\textup{Ind}_{C_{p}}^{G}\mathbf{1})^{\oplus 2}$ is realisable over
$\mathbb{Q}$, both conditions listed are satisfied for
$(\textup{Ind}_{C_{p}}^{G}\mathbf{1})^{\oplus 2}$. By uniqueness,
$V_{\ell}\cong(\textup{Ind}_{C_{p}}^{G}\mathbf{1})^{\oplus 2}$ as required.
For the claim regarding $\Omega^{1}$, we begin by decomposing $V_{\ell}$ into
irreducible representations, say

$V_{\ell}\cong\bigoplus_{i=1}^{m}\tilde{\chi}_{i}^{\oplus 2},$

where each $\tilde{\chi}_{i}$ is a lift of a $1$-dimensional character
$\chi_{i}$ from the $G/C_{p}\cong C_{m}$ quotient. By [13, Lemma 5.1(1)], we
deduce that $\Omega^{1}$ is a subrepresentation of $V_{\ell}$ and therefore,
we can decompose $\Omega^{1}\cong\bigoplus_{i\in I}\tilde{\chi_{i}}^{a_{i}}$,
for a subset $I\subseteq\\{1,2,\dots,m\\}$ and $a_{i}\in\\{0,1,2\\}$. By Lemma
6.8,
$\textup{Res}_{C_{m}}^{G}\Omega^{1}\cong\textup{Ind}_{\\{e\\}}^{C_{m}}\mathbf{1}\cong\bigoplus_{i=1}^{m}\chi_{i}$.
Since $\textup{Res}_{C_{m}}^{G}\tilde{\chi}_{i}=\chi_{i}$ we deduce that
$I=\\{1,2,\dots,m\\}$, and $a_{i}=1$ for all $i\in I$. Therefore,
$\Omega^{1}\cong\oplus_{i=1}^{m}\tilde{\chi}_{i}\cong\textup{Ind}_{C_{p}}^{G}\mathbf{1}$
as required. ∎

######  Theorem 6.10.

Let $E/K$ be an elliptic curve which admits a $K$-rational cyclic isogeny of
prime degree $p$. Then,

$\textup{rk}_{p}(E/K)=\sum_{v\ \textup{place of}\ K}\textup{ord}_{p}\
\Lambda_{\Theta}({}_{K}(E_{M})/K_{v})\quad\textup{mod }2,$

where $\Theta$ is the pseudo Brauer relation for $C_{p}\rtimes C_{m}$ and
${}_{K}(E_{M})$ afforded by Theorem 6.6.

###### Proof.

We write $\mathcal{X}_{p}$ to denote
$\mathcal{X}_{p}(\textup{Res}_{M/K}E_{M})$. We show that
$S_{\Theta,p}=\\{\mathbf{1}\\}$. By choosing the trivial pairing on
$\mathbf{1}$, we deduce that
$\mathcal{C}_{\Theta}(\mathbf{1})\equiv\frac{m}{pm}\equiv p\quad\textup{mod
}\mathbb{Q}^{\times 2}$, from which we deduce $\\{1\\}\subseteq S_{\Theta,p}$.
By Lemma 2.8(1), showing equality will follow upon showing that
$\mathcal{X}_{p}$ and $\mathbb{C}[G/C_{m}]$ share no common irreducible
constituents other than $\mathbf{1}$. This follows as in (6.7). Finally, the
local expression asserted by the claim follows upon applying Theorem 1.14 to
$\Theta$ afforded by Theorem 6.6. We note that Theorem 6.8 is applicable in
view of Proposition 6.9 which proves that $\Omega^{1}$ is self-dual as a
$G$-representation. ∎

##  7\. Genus $2$ curves with elliptic subcovers

###  7.1. Genus $2$ curves admitting a map to an elliptic curve

In this section, we provide a local expression for the parity of the
$p^{\infty}$-Selmer rank for the Jacobian of a genus $2$ curve $Y/K$ which
admits an elliptic subcover of prime degree $p$, equivalently a $K$-rational
morphism $\phi:Y\to E$ to an elliptic curve $E/K$. As a result, its Jacobian,
$\textup{Jac}_{Y}$, is isogenous to $E\times E^{\prime}$ for some
complementary elliptic curve $E^{\prime}$. As noted in [29], the complimentary
elliptic curve $E^{\prime}$ is not uniquely determined.

This section is designed to accomplish three distinct goals. Firstly, we
establish a canonical choice for the complementary elliptic curve, referred to
as $E^{\prime}$, under specific generality assumptions regarding the parameter
$\phi$

Under some generality assumptions on $\phi$ (see Definition 7.2 below), we
provide a canonical choice for the complementary elliptic curve $E^{\prime}$,
then we prove pseudo Brauer verifiability for the isogeny $\textup{Jac}_{Y}\to
E\times E^{\prime}$, and finally we use regulator constants to obtain a local
expression for the parity of the $p^{\infty}$-Selmer rank of
$\textup{Jac}_{Y}$. A central part of our argument is the following
commutative diagram

${Y}$${E}$${\mathbb{P}_{x}^{1}}$${\mathbb{P}_{u}^{1}}$$\scriptstyle{\phi}$$\scriptstyle{\pi_{Y}}$$\scriptstyle{\pi_{E}}$$\scriptstyle{\Phi}$

where $\pi_{Y}$ and $\pi_{E}$ are the natural hyperelliptic covers of $Y$ and
$E$, and $\Phi$ is the unique map making this diagram commute. As in §1.5, we
write $\mathbb{P}^{1}_{x}$ and $\mathbb{P}^{1}_{u}$ to indicate the choice of
a parameter $x$ and $u$ on $\mathbb{P}^{1}/K$.

###  7.2. Elliptic subcovers of even prime degree

For ease of exposition, we first illustrate how we utilise this commutative
diagram in the case when $p=2$.

######  Proposition 7.1.

Let $\phi:Y\to E$ be a degree $2$ morphism induced by the involution $r$ on
$Y$. Writing $h_{Y}$ to denote the hyperelliptic involution, then $C_{2}\times
C_{2}\cong\langle r,h_{Y}\rangle$ acts on $Y$ via $K$-automorphisms. This fits
into a Galois diagram

${Y}$${E=Y/\langle r\rangle}$${Y/\langle
rh_{Y}\rangle}$${\mathbb{P}^{1}=Y/\langle
h_{Y}\rangle}$${{\mathbb{P}^{1}=Y/\langle
r,h_{Y}\rangle}}$$\scriptstyle{\phi}$$\scriptstyle{\pi_{Y}}$$\scriptstyle{\pi_{E}}$$\scriptstyle{\Phi}$

In addition, $\\{e\\}+2(C_{2}\times C_{2})-\langle r\rangle-\langle
rh_{Y}\rangle-\langle h_{Y}\rangle$ is a pseudo Brauer relation for
$C_{2}\times C_{2}$ and $Y$ which verifies an isogeny

$\textup{Jac}_{Y}\to E\times\textup{Jac}_{Y/\langle rh_{Y}\rangle}.$

Finally, we have

$\textup{rk}_{2}(\textup{Jac}_{Y})\equiv\sum_{v\ \textup{place of}\
K}\textup{ord}_{2}\ \Lambda_{\Theta}(Y/K_{v})\quad\textup{mod }2.$

###### Proof.

Since $h_{Y}$ lies in the centre of $\textup{Aut}_{K}(Y)$ we deduce that $r$
and $h_{Y}$ commute. Therefore, $\langle r,h_{Y}\rangle\cong C_{2}\times
C_{2}$, and in addition $E=Y/\langle r\rangle$. Since $\\{e\\}+2(C_{2}\times
C_{2})-\langle r\rangle-\langle rh_{Y}\rangle-\langle h_{Y}\rangle$ is a
Brauer relation for the group $C_{2}\times C_{2}$, and $Y/\langle
r,h_{Y}\rangle$ and $Y/\langle h_{Y}\rangle$ are both curves of genus $0$,
then the result concerning pseudo Brauer verifiability follows. The local
formula follows in exactly the same way as Theorem 3.3. ∎

In view of Proposition 7.1, we deduce that a canonical choice for the
complimentary elliptic curve $E^{\prime}$ is $Y/\langle rh_{Y}\rangle$ in the
case when $p=2$. It is worthy of note that Theorem 7.8 below provides a
generalisation of Proposition 7.1.

###  7.3. Elliptic subcovers of odd prime degree

In the following subsection, we acquire a Galois cover of curves
$X_{Y}\to\mathbb{P}_{u}^{1}$ in the presence of an elliptic subcover of odd
prime degree. We only give the construction in the case where $\phi$ is
generic (following Kuhn’s terminology [23]).

######  Definition 7.2.

We say that an elliptic subcover $\phi:Y\to E$ is generic if it is unramified
over $E[2]$, i.e. the Weierstrass points of $E$.

######  Proposition 7.3.

Let $Y/K$ be a genus $2$ curve with a $K$-rational elliptic subcover
$\phi:Y\to E$ of odd prime degree $p$. In addition, suppose $\phi$ is generic.
Then, the Galois closure $X_{Y}\to\mathbb{P}_{u}^{1}$ of
$Y\xrightarrow{\phi}E\xrightarrow{\pi_{E}}\mathbb{P}_{u}^{1}$ is geometrically
connected and has Galois group $S_{p}\times C_{2}$. In addition, these fit
into a Galois diagram

$\mathbb{P}_{u}^{1}=X_{Y}/(S_{p}\times
C_{2})$$Y=X_{Y}/(S_{p-1}\times\\{e\\})$$E=X_{Y}/(S_{p}\times\\{e\\})$$X_{Y}$$\mathbb{P}_{x}^{1}=X_{Y}/(S_{p-1}\times
C_{2})$$\pi_{E}$$\pi_{Y}$$\Phi$$\phi$

###### Proof.

We write $\\{d_{1},d_{2},d_{3},d_{4}\\}$ to denote the branch locus of
$E\to\mathbb{P}^{1}_{u}$, i.e. $u(E[2])$. By [23, §1], the branch locus of
$\Phi:\mathbb{P}_{x}^{1}\to\mathbb{P}_{u}^{1}$ consists of
$\\{d_{1},d_{2},d_{3},d_{4},d_{5}\\}$ when $p\geq 5$ and
$\\{d_{1},d_{2},d_{3},d_{5}\\}$ where $d_{4}\neq d_{5}$ when $p=3$. Write
$X_{\mathbb{P}^{1}}\to\mathbb{P}^{1}_{u}$ to denote the Galois closure of
$\mathbb{P}^{1}_{x}\to\mathbb{P}^{1}_{u}$. By [23, §1], the ramification
structure of $\Phi$ (see Definition 2.12) in the generic case is
$\\{(2)^{\frac{p-1}{2}},(2)^{\frac{p-1}{2}},(2)^{\frac{p-1}{2}},(2)^{\frac{p-3}{2}},(2)^{1}\\}$
for $p\geq 3$. We now fix an arbitrary embedding $K\hookrightarrow\mathbb{C}$.
Following the notation from Theorem 2.15, we write $\sigma_{i}$ to denote the
monodromy of $X_{\mathbb{P}^{1}}\to\mathbb{P}^{1}_{u}$ at $d_{i}$. It follows
that $\sigma_{5}$ is a transposition, and therefore the Galois group of
$X_{\mathbb{P}^{1}}\to\mathbb{P}^{1}_{u}$ viewed as a morphism of complex
curves is the whole of $S_{p}$. Therefore, the corresponding $K$-rational
morphism $X_{\mathbb{P}^{1}}\to\mathbb{P}^{1}_{u}$ must have Galois group
$S_{p}$.

Showing that the Galois group of $X_{Y}\to\mathbb{P}^{1}$ is $S_{p}\times
C_{2}$ (as opposed to $S_{p})$ will follow upon showing that the discriminant
curve $W={X}_{\mathbb{P}^{1}}/A_{p}$ (see [13, Lemma A.7]) isn’t isomorphic to
$E$ over $\bar{K}$. It suffices to show that the branch points of
$W\to\mathbb{P}^{1}_{u}$ do not coincide with $u(E[2]).$ In particular, if
$p\equiv 1\quad\textup{mod }4$, then $\sigma_{1},\sigma_{2},\sigma_{3}$ are
even permutations, while $\sigma_{4}$ and $\sigma_{5}$ are odd permutations.
Therefore, the branch locus of $D\to\mathbb{P}_{u}^{1}$ is
$\\{d_{4},d_{5}\\}$. Similarly, if $p\equiv 3\quad\textup{mod }4$, then the
corresponding branch locus $W\to\mathbb{P}^{1}$ is
$\\{d_{1},d_{2},d_{3},d_{5}\\}$. Therefore, $W\not\cong E$ as required.

We show that $X_{Y}$ is geometrically connected. From above,
$X_{\mathbb{P}^{1}}$ must be geometrically connected, while the function field
$K(X_{Y})$ contains $K(X_{\mathbb{P}^{1}})$ as a degree $2$ subfield. Now
aiming for a contradiction, suppose that $X_{Y}$ is not geometrically
connected. Under this assumption, the geometric components of $X_{Y}$ must be
defined over a quadratic field extension of $K$, say $K(\sqrt{m})$. Therefore,
the $3$ quadratic extensions of $K(u)$ inside $K(X_{Y})$ must be
$K(E),K(u,\sqrt{m})$ and $K(E_{m})$, where $E_{m}$ denotes the quadratic twist
of $E$ by $m$. This is impossible, as we know two of these quadratic
extensions, namely $K(E)$ and $K(W)$, and we also know that $K(W)$ is not
isomorphic to $E$ or $E_{m}$ over $\overline{K}$. From this, we deduce that
$X_{Y}$ is geometrically connected.

The existence of the Galois diagram asserted by the claim above is immediate
by construction. This completes the proof. ∎

In the remainder of this subsection, we find the $5$-tuple corresponding to
the monodromy of $X_{Y}\to\mathbb{P}^{1}_{u}$ at its branch points. We use
this to find a description of $V_{\ell}(\textup{Jac}_{X_{Y}})$ as a
$G$-module. We first fix some notation.

######  Notation 7.4.

We let $\pi_{1}:S_{p}\times C_{2}\to S_{p}$ and $\pi_{2}:S_{p}\times C_{2}\to
C_{2}$ be the natural projections to the first and second factors. For the
remainder of this section, we write $\tau_{k}$ to be any element (well-defined
up to conjugacy) in $S_{p}\times C_{2}$ satisfying

  1. (1)

$\pi_{1}(\tau_{k})$ has cycle type $(2)^{k}$ as an element in $S_{p}$,

  2. (2)

and $\pi_{2}(\tau_{k})$ is trivial.

Similarly, we define $\tilde{\tau}_{k}$ analogously, but instead we impose
that $\pi_{2}(\tilde{\tau}_{k})$ is surjective instead.

######  Proposition 7.5.

Let $X_{Y}\to\mathbb{P}^{1}$ be the Galois cover of curves afforded by
Proposition 7.3. Then, $X_{Y}\to\mathbb{P}^{1}_{u}$ is branched at $5$ points,
and in addition the monodromy tuple of $X_{Y}\to\mathbb{P}^{1}_{u}$ is
$S_{p}\times C_{2}$-conjugate to
$(\tilde{\tau}_{\frac{p-1}{2}},\tilde{\tau}_{\frac{p-1}{2}},\tilde{\tau}_{\frac{p-1}{2}},\tilde{\tau}_{\frac{p-3}{2}},\tau_{1})$.

###### Proof.

In view of [23], we deduce that the branch locus of
$Y\xrightarrow{\phi}E\xrightarrow{\pi_{E}}\mathbb{P}_{u}^{1}$ consists of $5$
points. By Proposition 7.3, $X_{Y}\to\mathbb{P}^{1}_{u}$ is the Galois closure
of $Y\xrightarrow{\phi}E\xrightarrow{\pi_{E}}\mathbb{P}_{u}^{1}$, and by using
[31, Corollary 3.9.3(b)], we deduce that the branch locus of
$X_{Y}\to\mathbb{P}^{1}_{u}$ coincides with the branch locus of
$Y\to\mathbb{P}^{1}_{u}$. This proves the first half of the claim.

By Remark 2.16, the elements in the monodromy tuple coincide with the
decomposition groups of any point $y\in X_{Y}$ lying above a branch point,
which in turn coincides with the decomposition group of the corresponding
prime ideal $\mathfrak{p}_{y}$ of $K(X_{Y})$ over $K(u)$. Given any branch
point $d_{i}$, we write $y_{i}$ to denote any point the fibre of $d_{i}$ under
$X_{Y}\to\mathbb{P}^{1}_{u}$, and we set
$\mathfrak{p}_{i}=\mathfrak{p}_{y_{i}}\cap K(X_{\mathbb{P}^{1}})$ and
$\mathfrak{q}_{i}=\mathfrak{p}_{y_{i}}\cap K(E)$. In view of Proposition 7.3,
we deduce that the natural restriction maps induce an isomorphism

$\begin{split}\textup{Gal}(K(X_{Y})/K(u))&\xrightarrow{\sim}\textup{Gal}(K(X_{\mathbb{P}^{1}})/K(u))\times\textup{Gal}(K(E)/K(u))=S_{p}\times
C_{2}\\\
\sigma&\mapsto(\sigma\mid_{K(X_{\mathbb{P}^{1}})},\sigma|_{K(E)}).\end{split}$

In addition, the decomposition group $G_{\mathfrak{p}_{y_{i}}}$ gets
identified with $G_{\mathfrak{p}_{i}}\times G_{\mathfrak{q}_{i}}$ under this
isomorphism. By [23, §1], the ramification structure of
$\Phi:\mathbb{P}^{1}_{x}\to\mathbb{P}^{1}_{u}$ (see Definition 2.12) in the
generic case is
$\\{(2)^{\frac{p-1}{2}},(2)^{\frac{p-1}{2}},(2)^{\frac{p-1}{2}},(2)^{\frac{p-3}{2}},(2)^{1}\\}$,
and therefore the cycle type of $\\{G_{\mathfrak{p}_{i}}\\}_{i=1}^{5}$ matches
this in view of Theorem 2.15. In addition, the first four of these points
correspond to the four branch points of $E\to\mathbb{P}^{1}_{u}$, and
therefore $G_{\mathfrak{q}_{i}}$ is all of $C_{2}$ for $i=1,2,3,4$, while it’s
trivial for $i=5$. This gives the required result. ∎

Following the notation introduced in Theorem 2.17, given $\sigma\in
S_{p}\times C_{2}$, we write $\mathbf{1}_{\sigma}^{*}$ to denote the
permutation representation $\textup{Ind}_{\langle\sigma\rangle}^{S_{p}\times
C_{2}}\mathbf{1}$.

######  Theorem 7.6.

Let $X_{Y}\to\mathbb{P}^{1}$ be the Galois cover of curves afforded by
Proposition 7.3. For any prime number $\ell$, the following
$G$-representations

$V_{\ell}(\textup{Jac}_{X_{Y}})\ \ \textup{and}\ \ \mathbf{1}^{\oplus
2}\oplus(\mathbf{1}_{e}^{*})^{\oplus
3}\ominus\mathbf{1}_{\tau_{1}}^{*}\ominus\mathbf{1}_{\tilde{\tau}_{\frac{p-3}{2}}}^{*}\ominus(\mathbf{1}_{\tilde{\tau}_{\frac{p-1}{2}}}^{*})^{\oplus
3}$

become isomorphic after extending scalars to $\mathbb{C}$.

###### Proof.

The claim follows by combining Proposition 7.5 with Corollary 2.19 (which is
applicable since $X_{Y}/K$ is geometrically connected in view of Proposition
7.3). ∎

###  7.4. Complimentary elliptic curve $E^{\prime}$ and a local expression

The main result in this subsection is Theorem 7.8 below which gives a
canonical choice for the complimentary elliptic curve $E^{\prime}$ as the
Jacobian of the quotient of $X_{Y}$ by a subgroup $H^{\prime}\leq S_{p}\times
C_{2}$. We then prove pseudo Brauer verifiability for the underlying isogeny
$\textup{Jac}_{Y}\to E\times E^{\prime}$, and we end this section by obtaining
a local expression for the parity of the $p^{\infty}$-Selmer rank of
$\textup{Jac}_{Y}$ using regulator constants.

######  Notation 7.7.

We view $S_{p}\times C_{2}$ as a subgroup of $S_{p+2}$ by letting it act on
$\\{1,\dots,p,p+1,p+2\\}$ in the natural way, i.e. the $S_{p}$ factor acts on
$\\{1,\dots,p\\}$ in the natural way, while $C_{2}$ permutes
$p+1\leftrightarrow p+2$. For the remainder of this section, we fix
$H^{\prime}$ to be the subgroup of $S_{p}\times C_{2}$ generated by the
permutations $(1,2,\dots,p-2),(1,2),(p-1,p)(p+1,p+2)$ if $p\geq 5$, while we
write $H^{\prime}$ to denote the subgroup generated by $(p-1,p)(p+1,p+2)$ if
$p\leq 3$. As an abstract group, $H^{\prime}$ is isomorphic to $S_{p-2}\times
C_{2}$. We note that in the case $p=2$, this coincides with the subgroup
generated by $rh_{Y}$ from Proposition 7.1.

For the remainder of this section, we write $\epsilon$ to denote the
$S_{p}\times C_{2}$-representation obtained by lifting the non-trivial
representation from the $C_{2}$ quotient, while $\sigma$ denotes the
representation obtained by lifting the standard representation
$(p-1)$-dimensional representation from the $S_{p}$ quotient.

######  Theorem 7.8.

Let $X_{Y}\to\mathbb{P}^{1}$ be the Galois cover of curves afforded by
Proposition 7.3. Then,
$\Theta=(S_{p-1}\times\\{e\\})-(S_{p}\times\\{e\\})-H^{\prime}$ is a pseudo
Brauer relation for $S_{p}\times C_{2}$ and $X_{Y}$ which verifies an isogeny
$\textup{Jac}_{Y}\to E\times\textup{Jac}_{X_{Y}/H^{\prime}}.$

###### Proof.

The case where $p=2$ follows from Proposition 7.1. For brevity, write
$V_{\ell}$ to denote the $G$-representation
$V_{\ell}(\textup{Jac}_{X_{Y}})\otimes\mathbb{C}$, and we write $A_{1},A_{2}$
and $A_{3}$ to denote the subgroups $S_{p}\times\\{e\\},S_{p-1}\times C_{2}$
and $S_{p-1}\times\\{e\\}$ of $S_{p}\times C_{2}$ (as in Proposition 7.3). In
addition, we write $\mathbf{1}_{H}^{*}$ to denote the representation
$\textup{Ind}_{H}^{S_{p}\times C_{2}}\mathbf{1}$. Then, decomposing into
irreducible representations, we get

$\mathbf{1}_{A_{1}}^{*}=\mathbf{1}\oplus\epsilon,\ \
\mathbf{1}_{A_{2}}^{*}=\mathbf{1}\oplus\sigma,\ \
\mathbf{1}_{A_{3}}^{*}=\mathbf{1}\oplus\epsilon\oplus\sigma\oplus(\sigma\otimes\epsilon),$

By [13, Lemma 3.18], we have $\langle
V_{\ell},\mathbf{1}_{H}^{*}\rangle=2\cdot\textup{genus}(X_{Z}/H).$ The genera
of the quotients of $X_{Z}$ by $A_{1},A_{2}$ and $A_{3}$ are known in view of
Proposition 7.3 allowing one to deduce that $\langle
V_{\ell},\mathbf{1}\rangle=\langle V_{\ell},\sigma\rangle=0,$ while $\langle
V_{\ell},\epsilon\rangle=\langle V_{\ell},\sigma\otimes\epsilon\rangle=2.$ In
addition, for $p\geq 3$,

$\\#\langle\tau\rangle\backslash(S_{p}\times
C_{2})/H^{\prime}=\begin{cases}p^{2}-3p+3,&\tau=\tau_{1},\\\
\frac{1}{2}(p^{2}-3),&\tau=\tilde{\tau}_{\frac{p-3}{2}},\\\
\frac{1}{2}(p^{2}-1),&\tau=\tilde{\tau}_{\frac{p-1}{2}}.\end{cases}$

Therefore, by invoking Theorem 7.6, we deduce that $\langle
V_{\ell},\mathbf{1}_{H^{\prime}}^{*}\rangle=2$, and therefore
$\textup{genus}(X_{Y}/H^{\prime})=1.$ Moreover,

$\langle\sigma\otimes\epsilon,\mathbf{1}_{H^{\prime}}^{*}\rangle=\langle\mathbf{1}_{A_{3}}^{*}-\mathbf{1}_{A_{2}}^{*}-\mathbf{1}_{A_{1}}^{*}+\mathbf{1},\mathbf{1}_{H^{\prime}}^{*}\rangle=3-2-1+1=1.$

Finally, by writing $\mathbf{1}_{H^{\prime}}^{*}=(\sigma\otimes\epsilon)\oplus
T$, we deduce that $\langle V_{\ell},T\rangle=0$ as we have shown that
$\langle V_{\ell},\sigma\otimes\epsilon\rangle=\langle
V_{\ell},\mathbf{1}_{H^{\prime}}^{*}\rangle=2$. From this we deduce that
$\Theta=(S_{p-1}\times\\{e\\})-(S_{p}\times\\{e\\})-H^{\prime}$ is a pseudo
Brauer relation for $S_{p}\times C_{2}$ and $X_{Y}$. In view of the quotient
curves in Proposition 7.3, $\Theta$ verifies an isogeny $\textup{Jac}_{Y}\to
E\times\textup{Jac}_{X_{Y}/H^{\prime}}.$ ∎

We now use the pseudo Brauer relation from Theorem 7.8 in order to provide a
local expression for $\textup{rk}_{p}\ \textup{Jac}_{Y}$. We first prove some
auxiliary lemmata.

######  Lemma 7.9.

Let $p$ be a rational prime. Then, the multiplicities of $\epsilon$ and
$\sigma\otimes\epsilon$ in $\mathcal{X}_{p}(\textup{Jac}_{X_{Y}})$ satisfy

$\langle\mathcal{X}_{p}(\textup{Jac}_{X_{Y}}),\epsilon\rangle=\textup{rk}_{p}(E)\
\ \ \textup{and}\ \ \
\langle\mathcal{X}_{p}(\textup{Jac}_{X_{Y}}),\sigma\otimes\epsilon\rangle=\textup{rk}_{p}(\textup{Jac}_{X_{Y}/H^{\prime}}),$

###### Proof.

We start by writing $\epsilon=\mathbf{1}_{A_{1}}^{*}-\mathbf{1}$. Then,
$\langle\mathcal{X}_{p},\epsilon\rangle$ satisfies

$\langle\mathcal{X}_{p},\mathbf{1}_{A_{1}}^{*}-\mathbf{1}\rangle\underset{\mathrm{Frob.rec.}}{=}\textup{dim}\
\mathcal{X}_{p}^{A_{1}}-\textup{dim}\ \mathcal{X}_{p}^{S_{p}\times
C_{2}}\underset{\textup{Prop.}\ref{Galois_closure}}{=}\textup{dim}\
\mathcal{X}_{p}(E).$

The second claim follows in the same way after expressing
$\sigma\otimes\epsilon=\mathbf{1}_{A_{3}}^{*}-\mathbf{1}_{A_{2}}^{*}-\mathbf{1}_{A_{1}}^{*}+\mathbf{1}$,
and therefore
$\langle\mathcal{X}_{p},\sigma\otimes\epsilon\rangle=\textup{rk}_{p}(\textup{Jac}_{Y})-0-\textup{rk}_{p}(E)+0=\textup{rk}_{p}(\textup{Jac}_{X_{Y}/H^{\prime}})$
as required. ∎

######  Lemma 7.10.

Let $\Theta=(S_{p-1}\times\\{e\\})-(S_{p}\times\\{e\\})-H^{\prime}$ be the
pseudo Brauer relation for $X_{Y}/K$ afforded by Theorem 7.8. Then,
$\mathcal{C}_{\Theta}(\epsilon)\equiv\mathcal{C}_{\Theta}(\sigma\otimes\epsilon)\equiv
p\quad\textup{mod }\mathbb{Q}^{\times 2}.$

###### Proof.

The calculation for $\mathcal{C}_{\Theta}(\epsilon)$ is straightforward (one
can choose the trivial pairing on $\epsilon$). We fix a basis
$\\{e_{1},\dots,e_{p}\\}$ for $\mathbb{C}^{p}$. Then, $\sigma\otimes\epsilon$
corresponds to the subspace spanned by $v_{i}=e_{i+1}-e_{1}$ for
$i=1,\dots,p-1$ with $(\rho_{1},\rho_{2})\in S_{p}\times C_{2}$ acting via
$(\rho_{1},\rho_{2})\cdot
v_{i}=\textup{sgn}(\rho_{2})(e_{\rho_{1}(i+1)}-e_{\rho_{1}(1)})$. A non-
degenerate, $G$-invariant pairing on $\sigma\otimes\epsilon$ is given by
setting the diagonal entries $\langle\mskip-3.0mu\langle
v_{i},v_{i}\rangle\mskip-3.0mu\rangle=2$ while $\langle\mskip-3.0mu\langle
v_{i},v_{j}\rangle\mskip-3.0mu\rangle=1$ for $i\neq j$. Suppose that the
$S_{p-1}$ factor in $S_{p-1}\times\\{e\\}$ stabilises $1$. Then for
$H\in\\{S_{p-1}\times\\{e\\},S_{p}\times\\{e\\},H^{\prime}\\}$, the space of
$H$-invariant vectors is $1$-dimensional spanned by
$u_{1}=\sum_{i=1}^{p-1}v_{i}$, $0$-dimensional and $1$-dimensional spanned by
$u_{2}=v_{p-2}-v_{p-1}$ respectively. Therefore, $\langle\mskip-3.0mu\langle
u_{1},u_{1}\rangle\mskip-3.0mu\rangle=p(p-1)$, while
$\langle\mskip-3.0mu\langle u_{2},u_{2}\rangle\mskip-3.0mu\rangle=2$. Putting
everything together,

$\mathcal{C}_{\Theta}(\sigma\otimes\epsilon)\equiv\frac{\frac{1}{(p-1)!}\langle\mskip-3.0mu\langle
u_{1},u_{1}\rangle\mskip-3.0mu\rangle}{\frac{1}{2(p-2)!}\langle\mskip-3.0mu\langle
u_{2},u_{2}\rangle\mskip-3.0mu\rangle}\equiv p\quad\textup{mod
}\mathbb{Q}^{\times 2}.\qed$

######  Theorem 7.11.

Let $Y/K$ be a genus $2$ curve which admits a $K$-rational morphism $\phi:Y\to
E$ of prime degree $p$. In addition, if $p$ is an odd prime, suppose that
$\phi$ is generic. Then,

$\textup{rk}_{p}(\textup{Jac}_{Y})\equiv\sum_{v\ \textup{place of}\
K}\textup{ord}_{p}\ \Lambda_{\Theta}(X_{Y}/K_{v})\ \quad\textup{mod }2,$

where $\Theta=(S_{p-1}\times\\{e\\})-(S_{p}\times\\{e\\})-H^{\prime}$ is the
pseudo Brauer relation for $S_{p}\times C_{2}$ and $X_{Y}$ afforded by Theorem
7.8.

###### Proof.

The $p=2$ case is dealt with in Proposition 7.1. We write $\mathcal{X}_{p}$ to
denote $\mathcal{X}_{p}(\textup{Jac}_{X_{Y}})$. Then, we claim that
$S_{\Theta,p}=\\{\epsilon,\sigma\otimes\epsilon\\}.$ Indeed, by combining the
proof of Theorem 7.8 with Lemma 2.8(1), we deduce that
$S_{\Theta,p}\subseteq\\{\epsilon,\sigma\otimes\epsilon\\}$. By Lemma 7.10, it
follows that $S_{\Theta,p}=\\{\epsilon,\sigma\otimes\epsilon\\}$. Since any
$S_{p}\times C_{2}$-representation is self-dual, then Theorem 1.14 is
applicable. In view of Lemma 7.9, the result follows upon applying Theorem
1.14 to $\Theta$ afforded by Theorem 7.8.∎

######  Remark 7.12.

We note that elliptic subcovers occur in pairs $(\phi,\psi)$. In particular,
given a genus $2$ curve admitting an elliptic subcover $\phi:Y\to E$, then
there exists a complimentary elliptic subcover $\psi:Y\to E^{\prime}$ of the
same degree, see [23]. Therefore, upon interchanging the roles of $\phi$ and
$\psi$ and $E$ with $E^{\prime}$ if needed, we note that the conditions for
the validity of Proposition 7.3, Theorem 7.8 and Theorem 7.11 may be relaxed
to apply when $\phi$ or its complementary cover $\psi$ conforms to the
condition of being generic.

######  Remark 7.13.

Drawing from customary techniques, we can derive an analogous local formula in
this case is as follows. We let $\phi:Y\to E$ be a degree $p$ elliptic
subcover, and write $\psi:Y\to E^{\prime}$ for the complimentary elliptic
subcover. Then, $F=\phi^{*}+\psi^{*}$ is a $(p,p)$ isogeny satisfying $F\circ
F^{\vee}=[p]_{\textup{Jac}_{Y}}$. By invoking [32, Theorem 5], we deduce

$\textup{rk}_{p}(\textup{Jac}_{Y})\equiv\sum_{v}\textup{ord}_{p}\ \frac{C_{v}\
(\textup{Jac}_{Y},\phi^{*}\omega_{v}^{0}(E)\wedge\psi^{*}\omega_{v}^{0}(E^{\prime}))}{c_{v}(E)c_{v}({E^{\prime}})}\cdot\theta_{v}\quad\textup{mod
}2,$

where $\omega_{v}^{0}(E)$ and $\omega_{v}^{0}(E^{\prime})$ denote a choice of
minimal differentials on $E/K_{v}$ and ${E^{\prime}}/K_{v}$ respectively,
while $\theta_{v}=\frac{\mu_{v}(Y)}{\mu_{v}(E)\cdot\mu_{v}(E^{\prime})}$ if
$p=2$, and otherwise $\theta_{v}=1$. We note that this method for obtaining
this local expression for $\textup{rk}_{p}(\textup{Jac}_{Y})$ makes no
assumption on $\phi$ or $\psi$ being generic.

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