# The geometry and arithmetic of bielliptic Picard curves

Jef Laga  Department of Mathematics, Princeton University, Princeton, NJ
08544, USA<EMAIL_ADDRESS>and  Ari Shnidman  Einstein Institute of
Mathematics, Hebrew University of Jerusalem, Israel
<EMAIL_ADDRESS>
###### Abstract.

We study the geometry and arithmetic of the curves $C\colon
y^{3}=x^{4}+ax^{2}+b$ and their associated Prym abelian surfaces $P$. We prove
a Torelli theorem in this context and give a geometric proof of the fact that
$P$ has quaternionic multiplication (QM) by the quaternion order of
discriminant $6$. This allows us to describe the Galois action on the
geometric endomorphism algebra of $P$. As an application, we classify the
torsion subgroups of the Mordell-Weil groups $P({\rm\mathbb{Q}})$, as both
abelian groups and ${\rm End}(P)$-modules.

###### Contents

  1. 1 Introduction
  2. 2 Bielliptic Picard curves and their Pryms
  3. 3 A Torelli theorem
  4. 4 Shimura curves and quaternionic multiplication
  5. 5 Identifying bielliptic Picard curves in the pencil
  6. 6 Explicit quaternionic multiplication
  7. 7 $6$-torsion points in the Prym variety
  8. 8 Classifying rational torsion in Prym varieties
  9. 9 Classifying rational torsion in Pryms of ${\rm GL}_{2}$-type

##  1\. Introduction

Let $k$ be a field of characteristic neither $2$ nor $3$. A bielliptic Picard
curve over $k$ is a smooth projective curve $C$ with an affine model of the
form

(1.1) $\displaystyle y^{3}=x^{4}+ax^{2}+b$

for some $a,b\in k$. Such a curve is equipped with both a $\mu_{3}$-action
$(x,y)\mapsto(x,\omega y)$ and a commuting involution
$\tau\colon(x,y)\mapsto(-x,y)$. The induced involution $\tau^{*}$ on the
Jacobian variety $J=\operatorname{Jac}_{C}$ allows us to define the Prym
variety $P=\ker(1+\tau^{*})$. The abelian surface $P/k$ inherits a
$\mu_{3}$-action and carries a $(1,2)$-polarization; it need not be
principally polarizable over $k$.

The goal of this paper is to explore the remarkably rich geometry and
arithmetic of bielliptic Picard curves and their associated Pryms.

###  1.1. Basic results

We first prove some foundational results for bielliptic Picard curves. We

  1. (1)

prove a Torelli theorem for the association $C\mapsto P$, once suitable data
on both sides is fixed (Theorem 3.13);

  2. (2)

show ${\rm End}(P_{\bar{k}})$ contains a maximal order $\mathcal{O}$ in a
discriminant $6$ quaternion algebra (Proposition 6.1), in other words
$P_{\bar{k}}$ has quaternionic multiplication (QM) by $\mathcal{O}$;

  3. (3)

give an explicit description of the Artin representation $\rho_{{\rm
End}}\colon{\rm Gal}_{k}\to{\rm Aut}(\mathcal{O})\hookrightarrow{\rm
GL}_{4}({\rm\mathbb{Q}})$ describing the Galois action on the endomorphism
ring $\mathcal{O}$ (Corollary 6.6);

  4. (4)

relate the moduli space of bielliptic Picard curves to a unitary Shimura curve
$Y/{\rm\mathbb{Q}}$ and a quaternionic Shimura curve (Section 4.3);

  5. (5)

determine the finitely many $\bar{\rm\mathbb{Q}}$-isomorphism classes of Pryms
$P/{\rm\mathbb{Q}}$ which are geometrically non-simple (Proposition 6.18),
corresponding to the CM points on $Y$.

Our Torelli result (1) is perhaps surprising because Barth [4] has shown that
the Prym construction on all bielliptic genus $3$ curves does not satisfy a
Torelli theorem, but has one-dimensional fibres. There are other instances of
algebraic families of abelian surfaces with QM in the literature [3, 21], but
(2) is interesting because of the very simple nature of this family. Moreover,
contrary to previous approaches, our proof constructs the quaternionic action
explicitly and geometrically, using the automorphisms and the polarization of
$P$. Part (3) gives an interesting source of abelian surfaces with large
endomorphism field and, in the notation of [17], produces the first published
examples of geometrically simple abelian surfaces with Sato–Tate group
$J(E_{3})$ and $J(E_{6})$ (Example 6.7). Part (4) and Shimura reciprocity
allow us to calculate all geometrically non-simple Prym varieties in (5).

###  1.2. Rational torsion subgroups

To further illustrate the rich and accessible nature of these curves, we
classify the finite torsion subgroups that arise in the Mordell-Weil groups
$P({\rm\mathbb{Q}})$ of Pryms of bielliptic Picard curves over
${\rm\mathbb{Q}}$.

######  Theorem 1.1.

Let $P/{\rm\mathbb{Q}}$ be the Prym surface of a bielliptic Picard curve
$C/{\rm\mathbb{Q}}$. Then

$P({\rm\mathbb{Q}})_{\mathrm{tors}}\simeq\begin{cases}{\rm\mathbb{Z}}/n{\rm\mathbb{Z}}&\mbox{
for some }n\in\\{1,2,3,6\\},\mbox{ or }\\\
{\rm\mathbb{Z}}/n{\rm\mathbb{Z}}\times{\rm\mathbb{Z}}/n{\rm\mathbb{Z}}&\mbox{
for some }n\in\\{2,3\\},\mbox{ or }\\\
{\rm\mathbb{Z}}/6{\rm\mathbb{Z}}\times{\rm\mathbb{Z}}/3{\rm\mathbb{Z}}.&\\\
\end{cases}$

Conversely, for each finite abelian group $G$ above, there exist infinitely
many $\bar{{\rm\mathbb{Q}}}$-isomorphism classes of bielliptic Picard Prym
surfaces $P/{\rm\mathbb{Q}}$ with $P({\rm\mathbb{Q}})_{\mathrm{tors}}\simeq
G$.

Theorem 1.1 is the analogue of Mazur’s classification of rational torsion
points of elliptic curves [33] in our setting. As with elliptic curves, each
finite group allowed by Theorem 1.1 can be realized by an explicit family (in
fact, sometimes multiple families) of Pryms whose coefficients can be
rationally parameterized. This is in accordance with the philosophy of Mazur
and Ogg, namely that torsion should only occur as a consequence of the ambient
geometry of the relevant moduli space; see [34] for a recent survey. Theorem
1.1 is significantly easier to prove than Mazur’s theorem, essentially because
$P$ has everywhere potentially good reduction. Nonetheless, it appears to be
the first classification of rational torsion on a universal abelian variety
over a Shimura variety which is not a modular curve.

######  Remark 1.2.

In a separate work with Schembri and Voight [28], we significantly extend some
of our arguments here to show that for any abelian surface $A/{\rm\mathbb{Q}}$
such that ${\rm End}(A_{\bar{{\rm\mathbb{Q}}}})$ is a maximal order in a non-
split quaternion algebra over ${\rm\mathbb{Q}}$, there is the uniform bound
$|A({\rm\mathbb{Q}})_{\mathrm{tors}}|\leq 18$. Theorem 1.1 shows that this
bound is sharp. Contrary to [28, Theorem 1.3], Theorem 1.1 is a complete
classification, which we achieve using arguments specific to the geometry of
the abelian surfaces considered here.

######  Example 1.3.

By Proposition 7.16 below, for each $t\in{\rm\mathbb{Q}}\setminus\\{0,1,-1\\}$
the Prym $P_{t}$ associated to the curve

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

satisfies
$P_{t}({\rm\mathbb{Q}})_{\mathrm{tors}}\simeq{\rm\mathbb{Z}}/3{\rm\mathbb{Z}}\times{\rm\mathbb{Z}}/6{\rm\mathbb{Z}}$,
achieving the maximum order allowed by Theorem 1.1.

######  Remark 1.4.

Theorem 1.1 implies that the rational torsion subgroup
$J({\rm\mathbb{Q}})_{\mathrm{tors}}$ of the Jacobian of a bielliptic Picard
curve is $12$-torsion. A complete classification of the groups
$J({\rm\mathbb{Q}})_{\mathrm{tors}}$ would require parameterizing points of
order $4$ in $J({\rm\mathbb{Q}})$.

It is natural to ask how the torsion subgroup of $P({\rm\mathbb{Q}})$ is
affected by the presence of endomorphisms defined over ${\rm\mathbb{Q}}$.
Generically, we have ${\rm End}(P)={\rm\mathbb{Z}}$, but it can also happen
that ${\rm End}(P)$ is an order in a quadratic field. Such $P$ are said to be
of ${\rm GL}_{2}$-type, and are modular by work of Khare-Wintenberger [25].
After proving explicit conditions on $a$ and $b$ for $P$ to be of ${\rm
GL}_{2}$-type (see §6.3), we classify all the possible ${\rm End}(P)$-modules
that arise as $P({\rm\mathbb{Q}})_{\mathrm{tors}}$.

######  Theorem 1.5.

Let $P/{\rm\mathbb{Q}}$ be the Prym surface of a non-CM bielliptic Picard
curve over ${\rm\mathbb{Q}}$, with ${\rm End}(P)\neq{\rm\mathbb{Z}}$. Then
${\rm End}(P)\simeq{\rm\mathbb{Z}}[\sqrt{D}]$ for some $D\in\\{2,6\\}$.
Furthermore, there is an isomorphism of ${\rm\mathbb{Z}}[\sqrt{D}]$-modules

$P({\rm\mathbb{Q}})_{\mathrm{tors}}\simeq\begin{cases}\\{0\\}&\mbox{ or }\\\
{\rm\mathbb{Z}}[\sqrt{D}]/\mathfrak{a_{p}}&\mbox{ for some $p\in\\{2,3\\}$
},\end{cases}$

where $\mathfrak{a}_{p}$ is the unique prime ideal of
${\rm\mathbb{Z}}[\sqrt{D}]$ above $p$. Conversely, for each $D\in\\{2,6\\}$,
and for each of the three cyclic ${\rm\mathbb{Z}}[\sqrt{D}]$-modules $G$ that
appear above, there are infinitely many
$\overline{{\rm\mathbb{Q}}}$-isomorphism classes of Pryms $P/{\rm\mathbb{Q}}$
such that ${\rm End}(P)\simeq{\rm\mathbb{Z}}[\sqrt{D}]$ and
$P({\rm\mathbb{Q}})_{\mathrm{tors}}\simeq G$ as
${\rm\mathbb{Z}}[\sqrt{D}]$-modules.

When $D=2$ and $p=3$, Theorem 1.5 gives examples of ${\rm GL}_{2}$-type Pryms
such that $P({\rm\mathbb{Q}})_{\mathrm{tors}}\simeq{\rm\mathbb{F}}_{9}$,
showing that the upper bound proven in [28, Theorem 1.4] for the order of the
rational torsion subgroup of a QM abelian surface of ${\rm GL}_{2}$-type over
${\rm\mathbb{Q}}$ is in fact sharp.

###  1.3. Methods

The geometry of $(1,2)$-polarized surfaces was analyzed in detail by Barth [4]
and his results are crucial to our study of $C$ and $P$. The key player in the
story is the bigonal dual curve $\widehat{C}\colon
y^{3}=x^{4}+8ax^{2}+16(a^{2}-4b)$, which is the intersection of $P$ with a
well chosen theta divisor on $J$. The corresponding line bundle
$\mathscr{M}=\mathcal{O}_{P}(\widehat{C})$ represents the $(1,2)$-polarization
on $P$. The linear system $|\mathscr{M}|$ is a pencil of genus $3$ curves, and
the $\mu_{3}$-action on $C$ induces a $\mu_{3}$-action on $P$ and on the
pencil. The technical heart of this paper is to analyze this $\mu_{3}$-action
on the pencil $|\mathscr{M}|$, both theoretically and explicitly. Our Torelli
theorem (1) boils down to showing that $\widehat{C}$ is the unique bielliptic
Picard curve in $|\mathscr{M}|$ of the correct signature. For the QM property
(2), we give two different proofs. The first is indirect, relating the moduli
of bielliptic Picard curves to a unitary Shimura curve (in other words, we
first prove (4) and then deduce (2)). The second proof constructs the QM
explicitly, by showing that a specific sextic twist of the original curve $C$
also lives in $|\mathscr{M}|$, but carrying the opposite signature. This
second proof allows us to prove (3) as well.

To prove Theorems 1.1 and 1.5 we must first exhibit explicit families with
specified torsion subgroups, and then we must rule out all other finite
abelian groups. To exhibit groups, we describe elements of $P[2]$ and $P[3]$
geometrically, as coming from the fixed points for the $\mu_{6}$-action. To
rule out groups which are not $6$-torsion, we first use general arguments for
QM abelian surfaces which are developed in greater generality in [28]. Instead
of quoting the main results there, we give a simpler proof (tailored to our
special family of surfaces), which also allows us to handle the geometrically
non-simple case not treated in [28].111For those interested in reading that
paper as well, it is perhaps instructive to first read §8 of this paper
(consulting [28] for the details of a few key results), before reading the
more elaborate proofs in [28], where nothing is assumed about the geometry of
the abelian surfaces involved. However, these general arguments only go so
far; for example, they are not enough to rule out $4$-torsion in general QM
abelian surfaces $A/{\rm\mathbb{Q}}$. To eliminate groups such as
${\rm\mathbb{Z}}/4{\rm\mathbb{Z}}$ and
$({\rm\mathbb{Z}}/2{\rm\mathbb{Z}})^{2}\times({\rm\mathbb{Z}}/3{\rm\mathbb{Z}})$,
we use arguments that are very much specific to the geometry of bielliptic
Picard curves (see Propositions 8.5 and 8.6).

###  1.4. Previous results

Petkova–Shiga [36] considered bielliptic Picard curves from a complex analytic
viewpoint. Using period matrix computations, they show that their Pryms over
${\rm\mathbb{C}}$ have quaternionic multiplication. Hashimoto-Murabayashi [21]
and Baba-Granath [3] have studied genus two curves whose Jacobians
$J^{\prime}$ have QM by $\mathcal{O}$. Over $\overline{k}$, each surface
$J^{\prime}$ becomes isomorphic to a bielliptic Picard Prym $P$, but generally
not over $k$. There is also work of Bonfanti-van Geemen [6] which gives an
algebraic description of the embedding of the Shimura curve inside the moduli
space of $(1,2)$-polarized abelian surfaces.

###  1.5. Future work

There are many other avenues of research related to bielliptic Picard curves
and their Pryms that are ripe for investigation. In [29], we use our
description of the ${\rm Gal}_{k}$-module ${\rm
End}(P_{k^{\operatorname{sep}}})$ to characterize the bielliptic Picard curves
with torsion Ceresa cycle. One can also try to study the average rank of the
Mordell-Weil group $P({\rm\mathbb{Q}})$; the average rank of
$P({\rm\mathbb{Q}})$ in cubic and sextic twist families was already considered
in [1] and [42]. To help explore other arithmetic questions, it would be
worthwhile to develop more robust algorithms that are tailored towards these
curves, such as explicit descent algorithms, explicit addition laws, an
analogue of Tate’s algorithm, $L$-functions, etc. We hope this work encourages
others to study some of these topics.

###  1.6. Structure of paper

We start with some generalities on bielliptic Picard curves and their (dual)
Prym varieties in §2. In particular, we discuss bigonal duality in §2.7. We
prove our Torelli theorem in §3. In §4 we make the connection with unitary and
quaternionic Shimura curves. In §5 we perform some explicit calculations with
a pencil of curves on the Prym variety. These calculations are used in §6 to
explicitly construct the quaternionic multiplication. In §7 we give explicit
descriptions of the $2$ and $3$-torsion subgroups of Pryms of bielliptic
Picard curves. In §8 and §9 we classify rational torsion subgroups and prove
Theorems 1.1 and 1.5.

###  1.7. Acknowledgements

We thank Adam Morgan, Ciaran Schembri, Michael Stoll, and John Voight for
helpful conversations and remarks. The second author was funded by the
European Research Council (ERC, CurveArithmetic, 101078157). Part of this
research was carried out while the first author was visiting the second author
in Jerusalem. We thank the Hebrew University of Jerusalem and the Einstein
Institute of Mathematics for their hospitality.

###  1.8. Notation and conventions

  * •

Our base field will typically be denoted by $k$, with choice of separable and
algebraic closure $k^{\operatorname{sep}}\subset\bar{k}$ and absolute Galois
group ${\rm Gal}_{k}={\rm Gal}(k^{\operatorname{sep}}/k)$. All fields in this
paper will be assumed to have characteristic $\neq 2,3$ unless explicitly
stated otherwise. Galois actions will typically be right actions.

  * •

A variety over a field $k$ is a separated finite type scheme over $k$. A
variety is called nice if it is smooth, projective and geometrically integral.

  * •

If $X,Y/k$ are varieties, $f\colon X_{k^{\operatorname{sep}}}\rightarrow
Y_{k^{\operatorname{sep}}}$ is a morphism of
$k^{\operatorname{sep}}$-varieties and $\sigma\in{\rm Gal}_{k}$, then
$f^{\sigma}$ denotes the $k^{\operatorname{sep}}$-morphism $x\mapsto
f(x^{\sigma^{-1}})^{\sigma}$.

  * •

For a nice variety $X/k$, denote by ${\rm Pic}(X)$ its Picard group and by
${\rm Pic}_{X}$ its Picard scheme.

  * •

If $X/k$ is a nice curve, let ${\rm Pic}^{n}(X)\subset{\rm Pic}(X)$ be the
subset of line bundles of degree $n$, let $\operatorname{Jac}_{X}={\rm
Pic}^{0}_{X}$ denote its Jacobian, let ${\rm Aut}(X)$ denote the
$k$-automorphism group of $X$ and let $\mathbf{Aut}(X)$ be the _scheme_ of
automorphisms of $X$, with $\mathbf{Aut}(X)(K)={\rm Aut}(X_{K})$ for every
field extension $K/k$.

  * •

For every integer $n\geq 1$ we define the group scheme $\mu_{n}={\rm
Spec}\left(k[t]/(t^{n}-1)\right)$. If $X/k$ a variety, we define a
$\mu_{n}$-action on $X$ to be a morphism of $k$-schemes
$\mu_{n}\times_{k}X\rightarrow X$ satisfying the axioms of a (left) group
action. If $m,n$ are coprime, giving a $\mu_{mn}$-action is the same as giving
commuting $\mu_{m}$ and $\mu_{n}$-actions, using the isomorphism
$\mu_{m}\times\mu_{n}\rightarrow\mu_{mn}$ induced by the inclusion maps.

  * •

We will typically write $\omega\in\bar{k}$ for a choice of primitive third
root of unity and $k(\omega)$ for the smallest field extension of $k$
containing such $\omega$.

  * •

If $V$ is a vector space over a field $k$, we let
${\rm\mathbb{P}}(V)=\operatorname{Proj}({\rm Sym}^{\bullet}(V^{\vee}))$ be the
projective space parametrizing lines in $V$. If $X/k$ is variety, a morphism
$X\rightarrow{\rm\mathbb{P}}(V)$ is the same as a pair $(\mathscr{L},\phi)$,
where $\mathscr{L}$ is a line bundle on $X$ and $\phi\colon
V^{\vee}\otimes_{k}\mathcal{O}_{X}\rightarrow\mathscr{L}$ is a surjection (up
to a suitable notion of isomorphism).

  * •

If $\mathscr{L}$ is a line bundle on a variety $X/k$ we denote by
$|\mathscr{L}|:={\rm\mathbb{P}}(\operatorname{H}^{0}(X,\mathscr{L})^{\vee})$
the linear system of effective divisors in $\mathscr{L}$. If $D$ is a Cartier
divisor on $X$ we also write $|D|$ for $|\mathcal{O}_{X}(D)|$. If
$\mathscr{L}$ has no base points, we thus get a natural morphism
$\phi_{\mathscr{L}}\colon X\rightarrow|\mathscr{L}|$.

  * •

If $A/k$ is an abelian variety, let $A^{\vee}={\rm Pic}^{0}_{A}$ be its dual
abelian variety. If $\mathscr{L}\in{\rm Pic}(A)$, denote by
$\lambda_{\mathscr{L}}\colon A\to A^{\vee}$ the homomorphism $x\mapsto
t_{x}^{*}\mathscr{L}\otimes\mathscr{L}^{-1}$.

##  2\. Bielliptic Picard curves and their Pryms

###  2.1. Basic definitions

Let $k$ be a field (always assumed of characteristic $\neq 2,3$).

######  Definition 2.1.

Let $C/k$ be a nice genus $3$ curve. We call a faithful $\mu_{6}$-action
$\gamma\colon\mu_{6}\hookrightarrow\mathbf{Aut}(C)$ on $C$ bielliptic if the
quotient $C/\mu_{2}$ is a genus one curve. We say $C$ is a bielliptic Picard
curve over $k$ if it admits a bielliptic $\mu_{6}$-action.

For $a,b\in k$, consider the projective plane curve $C_{a,b}$ over $k$ with
affine equation

(2.1) $\displaystyle C_{a,b}\colon y^{3}=x^{4}+ax^{2}+b.$

The curve $C_{a,b}$ has a unique ($k$-rational) point $\infty$ at infinity,
and $C_{a,b}$ is smooth if and only if

$\Delta_{a,b}:=16b(a^{2}-4b)$

is nonzero. The $\mu_{2}$-action $\pm 1\cdot(x,y)=(\pm x,y)$ and
$\mu_{3}$-action $\omega\cdot(x,y)=(x,\omega y)$ (where $\omega^{3}=1$)
combine to a faithful bielliptic $\mu_{6}$-action $\gamma_{a,b}$, given by
$\zeta\cdot(x,y)=(\zeta^{3}x,\zeta^{4}y)$, for every
$\zeta\in\mu_{6}(k^{\operatorname{sep}})$. The unique fixed point for this
action is $\infty$.

######  Theorem 2.2.

Let $C/k$ be nice genus $3$ curve with faithful $\mu_{6}$-action $\gamma$.
Then the following conditions are equivalent:

  1. $(1)$

$\gamma$ is a bielliptic $\mu_{6}$-action (and consequently $C$ is a
bielliptic Picard curve).

  2. $(2)$

$\gamma$ has a unique fixed point.

  3. $(3)$

$C/\mu_{3}$ has genus zero.

  4. $(4)$

$C$ is a plane curve.

  5. $(5)$

There is an isomorphism $C\simeq C_{a,b}$ for some $a,b\in k$ such that
$\gamma$ corresponds to $\gamma_{a,b}$ or $\gamma_{a,b}^{-1}$.

###### Proof.

For each $i$ dividing $6$, let $n_{i}$ be the number of $\mu_{i}$-fixed
points. By Riemann-Hurwitz, we have $n_{2}\in\\{0,4,8\\}$,
$n_{3}\in\\{2,5\\}$, and $n_{6}\leq 3$. Moreover, $\mu_{2}$ permutes the
$\mu_{3}$-fixed points and vice versa. So $n_{2}-n_{6}$ is divisible by $3$
and $n_{3}-n_{6}$ is divisible by $2$. There are then three possibilities:

  1. $(a)$

$(n_{2},n_{3},n_{6})=(0,2,0)$, or

  2. $(b)$

$(n_{2},n_{3},n_{6})=(4,5,1)$, or

  3. $(c)$

$(n_{2},n_{3},n_{6})=(8,2,2)$.

The equivalence of $(1)$, $(2)$, and $(3)$ follows immediately from the above
cases. To show the equivalence of $(1)$ and $(4)$, first observe that case
$(c)$ is hyperelliptic because $C/\mu_{2}$ is genus $0$. Case $(a)$ is also
hyperelliptic since an unramified double cover of a genus two curve is again
hyperelliptic [16]. It remains to show that $C$ is non-hyperelliptic in case
$(b)$. In that case, there is only one $\mu_{6}$-fixed point $\infty$, which
is therefore $k$-rational. Moreover, $\infty$ must have different local
monodromy type for the $\mu_{3}$-cover $C\to
C/\mu_{3}\simeq{\rm\mathbb{P}}^{1}$ compared to the other four ramification
points (which are permuted by the $\mu_{2}$-action). By Kummer theory, there
is a model $C\colon y^{3}=f(x)$, for some quartic polynomial $f(x)\in k[x]$,
and in particular $C$ is a plane quartic curve. This shows that $(1)-(4)$ are
equivalent.

It is clear that $(5)$ implies all the rest, so it remains to show the
converse. However, we have seen that such a $C$ has a model $y^{3}=f(x)$ which
we may assume takes the form

(2.2) $y^{3}=x^{4}+ax^{2}+cx+d$

with the $\mu_{3}$-action $\omega\cdot(x,y)=(x,\omega y)$ or
$\omega\cdot(x,y)=(x,\omega^{-1}y)$ for every $\omega\in\mu_{3}$. We see from
this model (as in the proof of [10, Lemma 1.21(a)]) that $C$ admits an
involution commuting with the $\mu_{3}$-action only if $c=0$ and this
involution is given by $(x,y)\mapsto(-x,y)$, proving $(5)$. ∎

######  Remark 2.3.

A Picard curve is a plane quartic curve of the form (2.2). Such a curve is
bielliptic (admitting a $\mu_{3}$-equivariant double cover to an elliptic
curve) precisely when $c=0$, explaining our terminology.

There is one $k^{\operatorname{sep}}$-isomorphism class of bielliptic Picard
curves that behaves so differently that it deserves its own name.

######  Definition 2.4.

A bielliptic Picard curve $C$ is special if it is
$k^{\operatorname{sep}}$-isomorphic to $C_{0,1}:y^{3}=x^{4}+1$.

A bielliptic Picard curve $C$ is special precisely when ${\rm
Aut}(C_{k^{\operatorname{sep}}})$ is larger than
${\rm\mathbb{Z}}/6{\rm\mathbb{Z}}$; in that case, it is a group of order $48$
[30, Theorem 3.1], with GAP label (48,33). We will see in Lemma 2.9 that
$C_{a,b}$ is special if and only if $a=0$.

###  2.2. Marked bielliptic Picard curves

It is often useful to pin down a $\mu_{6}$-action on a bielliptic Picard curve
with specified signature around the (unique, by Theorem 2.2) $\mu_{6}$-fixed
point $\infty$.

######  Definition 2.5.

A marked bielliptic Picard curve is a bielliptic Picard curve $C$ with
bielliptic $\mu_{6}$-action $\gamma$ such that $\mu_{6}$ acts on the tangent
space $T_{\infty}C$ via the identity character $\mu_{6}\rightarrow\mu_{6}$.

An isomorphism of marked bielliptic Picard curves
$(C,\gamma)\rightarrow(C^{\prime},\gamma^{\prime})$ is by definition an
isomorphism of curves $C\rightarrow C^{\prime}$ that is equivariant with
respect to the $\mu_{6}$-actions. A calculation using the generator
$\frac{x}{y}\in(T_{\infty}C_{a,b})^{\vee}$ shows that the pair
$(C_{a,b},\gamma_{a,b})$ defined in §2.1 is a marked bielliptic Picard curve.

######  Lemma 2.6.

Let $C/k$ be a bielliptic Picard curve with bielliptic $\mu_{6}$-action
$\gamma$. Then one of $(C,\gamma^{\pm 1})$ is a marked bielliptic Picard
curve, and if $C$ is not special this marked $\mu_{6}$-action is the unique
one.

###### Proof.

Since $\gamma$ is faithful, it acts on $T_{\infty}C$ via the identity
character or its inverse, so either $\gamma$ or $\gamma^{-1}$ acts via the
identity. If $C$ is not special, $\mathbf{Aut}(C)\simeq\mu_{6}$ by [30,
Theorem 3.1], proving the uniqueness in that case. ∎

###  2.3. Automorphisms

######  Lemma 2.7.

Every isomorphism
$(C_{a,b},\gamma_{a,b})\rightarrow(C_{a^{\prime},b^{\prime}},\gamma_{a^{\prime},b^{\prime}})$
of marked bielliptic Picard curves over $k$ is of the form
$(x,y)\mapsto(\lambda^{3}x,\lambda^{4}y)$ for some $\lambda\in k^{\times}$
such that $(a^{\prime},b^{\prime})=(\lambda^{6}a,\lambda^{12}b)$.

###### Proof.

Let $\phi\colon C_{a,b}\rightarrow C_{a^{\prime},b^{\prime}}$ be an
isomorphism preserving the $\mu_{6}$-actions. Since $C_{a,b}$ and
$C_{a^{\prime},b^{\prime}}$ are canonically embedded, $\phi$ is induced by a
linear isomorphism of the ambient projective space ${\rm\mathbb{P}}^{2}_{k}$.
This isomorphism preserves the point at infinity (being the unique
$\mu_{6}$-fixed point) and the $\mu_{6}$-eigenspaces of
$\operatorname{H}^{0}({\rm\mathbb{P}}^{2},\mathcal{O}(1))=\text{span}\\{1,x,y\\}$.
A short calculation shows that $\phi$ must be of the form above. ∎

######  Lemma 2.8.

Every marked bielliptic Picard curve over $k$ is of the form
$(C_{a,b},\gamma_{a,b})$ for some $a,b\in k$. Two marked bielliptic Picard
curves $(C_{a,b},\gamma_{a,b})$ and
$(C_{a^{\prime},b^{\prime}},\gamma_{a^{\prime},b^{\prime}})$ are isomorphic if
and only if there exists $\lambda\in k^{\times}$ such that
$a^{\prime}=\lambda^{6}a$ and $b^{\prime}=\lambda^{12}b$.

###### Proof.

The first part follows from Theorem 2.2(5). The second part follows from Lemma
2.7. ∎

######  Lemma 2.9.

The bielliptic Picard curve $C_{a,b}$ is special if and only if $a=0$.

###### Proof.

We may assume that $k$ is algebraically closed. Lemma 2.8 shows that $C_{a,b}$
is special if $a=0$. Conversely, suppose that $C_{a,b}$ is special. By [30,
Theorem 3.1], $G={\rm Aut}(C_{a,b})$ is a group of order $48$ with GAP label
(48,33). Moreover, ${\rm Aut}(C_{a,b},\gamma_{a,b})$ is the centralizer in $G$
of a cyclic order $6$ subgroup. A group theory calculation similar to [10,
Lemma 4.1.5(c)] shows that all cyclic order $6$ subgroup of $G$ are conjugate
and have centralizer of order $12$, so ${\rm Aut}(C_{a,b},\gamma_{a,b})$ has
order $12$. On the other hand, Lemma 2.7 shows that ${\rm
Aut}((C_{a,b},\gamma_{a,b}))=\\{\lambda\in
k^{\times}\mid(a,b)=(\lambda^{6}a,\lambda^{12}b)\\}$. If $a\neq 0$, the latter
set has size $6$, so if $C$ is special then $a=0$. ∎

######  Lemma 2.10.

Let $(C,\gamma)$ be a marked bielliptic Picard curve over $k$. Then
$\mathbf{Aut}(C,\gamma)=\mu_{6}$ if $C$ is not special and
$\mathbf{Aut}(C,\gamma)=\mu_{12}$ if $C$ is special.

###### Proof.

Combine Lemmas 2.7 and 2.9. ∎

Lemma 2.8 shows that the moduli stack of marked bielliptic Picard curves can
be identified with the weighted projective stack ${\rm\mathbb{P}}(6,12)$ minus
the discriminant locus $\Delta_{a,b}=0$. As with the moduli stack of elliptic
curves, the coarse space has a simpler description. Given a bielliptic Picard
curve $C/k$, its $j$-invariant is defined as

(2.3) $\displaystyle j(C):=j_{a,b}:=\frac{4b-a^{2}}{4b}\in k\setminus\\{0\\},$

where $a,b\in k$ are such that $C\simeq C_{a,b}$. Lemmas 2.6, 2.8 and 2.9 show
that this is well defined and that $C_{k^{\operatorname{sep}}}\simeq
C^{\prime}_{k^{\operatorname{sep}}}$ if and only if $j(C)=j(C^{\prime})$, even
if $C$ or $C^{\prime}$ is special.

###  2.4. Twists

Let $(C,\gamma)$ be a marked bielliptic Picard curve over $k$. By the twisting
principle, every cocycle
$\xi\in\operatorname{H}^{1}(k,\mathbf{Aut}(C,\gamma))$ determines a bielliptic
Picard curve $(C_{\xi},\gamma_{\xi})$ that is
$k^{\operatorname{sep}}$-isomorphic to $(C,\gamma)$. Its $k$-isomorphism class
is characterized by the existence of a $k^{\operatorname{sep}}$-isomorphism
$\phi\colon(C_{\xi},\gamma_{\xi})_{k^{\operatorname{sep}}}\rightarrow(C,\gamma)_{k^{\operatorname{sep}}}$
such that $\sigma\mapsto\phi^{\sigma}\circ\phi^{-1}$ represents the class of
$\xi$.

In particular, since $\gamma$ determines a subgroup
$\mu_{6}\subset\mathbf{Aut}(C,\gamma)$ and since
$\operatorname{H}^{1}(k,\mu_{6})\simeq k^{\times}/k^{\times 6}$, every
$\delta\in k^{\times}$ gives rise to a sextic twist
$(C_{\delta},\gamma_{\delta})$ of $(C,\gamma)$. Concretely, if $C=C_{a,b}$
then the sextic twist of $C_{a,b}$ by $\delta$ is isomorphic to $C_{\delta
a,\delta^{2}b}$. If $C$ is not special, Lemmas 2.6 and 2.7 show that
$\mathbf{Aut}(C,\gamma)=\mathbf{Aut}(C)=\mu_{6}$ and so every twist of $C$ (as
a marked and unmarked curve) is isomorphic to a sextic twist.

###  2.5. The Prym variety $P$

Let $(C,\gamma)$ be a marked bielliptic Picard curve over $k$. We associate to
$C$ an abelian surface $P$ whose study will occupy much of the rest of this
paper.

Restricting the $\mu_{6}$-action to $\mu_{2}$ defines an involution
$\tau\colon C\rightarrow C$. The quotient $\pi\colon C\rightarrow E:=C/\tau$
is a genus $1$ curve. When endowed with the $k$-point $\pi(\infty)$ it has the
structure of an elliptic curve, so we can identify $\operatorname{Jac}_{E}$
with $E$. Let $J=\operatorname{Jac}_{C}$ be the Jacobian of $C$.

######  Definition 2.11.

The Prym variety of $C$ is defined as $P:=\ker(1+\tau^{*}\colon J\rightarrow
J)$.

This is an abelian surface. We refer to [27, §3.3] for more details concerning
Prym varieties of bielliptic genus $3$ curves. The properties we will use are
summarized in the following commutative diagram:

${0}$${{E[2]}}$${E}$${0}$${P}$${J}$${E}$${0}$${A}$${0}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\times
2}$$\scriptstyle{\lambda}$

It depicts two dual exact sequences, where $\pi_{*}$ (resp. $\pi^{*}$) denotes
pushforward (resp. pullback) of divisors. The principal polarization on $J$
restricts to a $(1,2)$-polarization $\lambda\colon P\rightarrow A:=P^{\vee}$
with kernel
$P[\lambda](k^{\operatorname{sep}})\simeq({\rm\mathbb{Z}}/2{\rm\mathbb{Z}})^{2}$.
The intersection of $P$ and $\pi^{*}(E)$ is $\pi^{*}(E[2])=P[\lambda]$. The
sum of the two maps $P\rightarrow J$ and $E\rightarrow J$ is an isogeny
$P\times E\rightarrow J$ with kernel $\\{(\pi^{*}(x),x)\mid x\in E[2]\\}\simeq
E[2]$.

If $C=C_{a,b}$ for some $a,b\in k$, we may write $E_{a,b}$, $J_{a,b}$,
$\lambda_{a,b}\colon P_{a,b}\rightarrow A_{a,b}$ et cetera, but we drop the
subscripts when convenient.

######  Remark 2.12.

$P$ need not be principally polarizable over $k$; indeed we prove in Corollary
6.14 that if $P$ is geometrically simple, then it is principally polarizable
if and only if $16b(a^{2}-4b)$ is a sixth power in $k^{\times}$. However, if
$b=s^{3}$ is a cube in $k^{\times}$, then the Jacobian of the following genus
two curve is in the isogeny class of $P_{a,s^{3}}$:

$-asy^{2}=(x^{2}+2x-2)(s^{3}x^{4}+4s^{3}x^{3}+2dx-d).$

where $d=a^{2}-4s^{3}$; this follows from [39, Theorem 1.1].

We now incorporate the $\mu_{3}$-action on $C$ into the picture. By Albanese
functoriality, there is a unique $\mu_{6}$-action on $J$ such that the
Abel–Jacobi map $\text{AJ}_{\infty}\colon C\hookrightarrow J$ is
$\mu_{6}$-equivariant. This induces a $\mu_{3}$-action on the subvariety $P$
and the quotient $E$. We record the following important signature calculation.

######  Lemma 2.13.

For every primitive third root of unity $\omega\in k^{\operatorname{sep}}$,
the action of $\omega$ on
$\operatorname{H}^{0}(P_{k^{\operatorname{sep}}},\Omega^{1}_{P_{k^{\operatorname{sep}}}})$
has characteristic polynomial $T^{2}+T+1$.

###### Proof.

We may assume that $(C,\gamma)=(C_{a,b},\gamma_{a,b})$ by Lemma 2.8 and that
$k=k^{\operatorname{sep}}$, so fix such a $\omega\in k$. Then $\omega$ acts on
$C_{a,b}$ via $(x,y)\mapsto(x,\omega y)$. The vector space
$\operatorname{H}^{0}(C,\Omega^{1}_{C})$ has basis
$\frac{dx}{y^{2}},x\frac{dx}{y^{2}},\frac{dx}{y}$, hence the $\omega$-action
on this vector space has eigenvalues $\omega,\omega,\omega^{2}$. The vector
space $\operatorname{H}^{0}(E,\Omega^{1}_{E})$ has basis $\frac{dx}{y^{2}}$,
hence $\omega$ has eigenvalue $\omega$. Since the map $P\times E\rightarrow
J,(p,e)\mapsto p+\pi^{*}(e)$ is an isogeny with kernel $E[2]$, it induces a
$\mu_{3}$-equivariant isomorphism
$\operatorname{H}^{0}(J,\Omega^{1}_{J})\simeq\operatorname{H}^{0}(P,\Omega^{1}_{P})\oplus\operatorname{H}^{0}(E,\Omega^{1}_{E})$.
Combining the last three sentences with the isomorphism
$\operatorname{H}^{0}(C,\Omega^{1}_{C})\simeq\operatorname{H}^{0}(J,\Omega_{J}^{1})$
proves the lemma. ∎

######  Lemma 2.14.

The subgroup of $\mu_{3}$-fixed points of $P$ is of size $9$ and contained in
$P[3]$.

###### Proof.

We may assume that $k=k^{\operatorname{sep}}$. So let $\omega\in k$ be a third
root of unity and $\alpha\colon P\rightarrow P$ the corresponding
automorphism. Let $\beta=1-\alpha$. By Lemma 2.13, $\beta$ induces an
isomorphism on differentials so $\beta$ is an isogeny. Since
$\beta\circ(\alpha^{2}+\alpha+1)=0$ and $\beta$ is surjective,

(2.4) $\displaystyle\alpha^{2}+\alpha+1=0.$

Therefore $\beta^{2}=1-2\alpha+\alpha^{2}=-3\alpha$, so $\beta$ has degree $9$
and $P[\beta]\subset P[-3\alpha]=P[3]$. ∎

###  2.6. The dual Prym variety $A$

Keep the notations from §2.5. Since $P[\lambda]\subset P[2]$, there exists a
unique isogeny222We warn the reader that $\widehat{\lambda}$ is not the same
as the dual $\lambda^{\vee}$ of $\lambda$. Being a polarization, $\lambda$ is
in fact self-dual! $\widehat{\lambda}\colon A\rightarrow P$ such that
$\widehat{\lambda}\circ\lambda=[2]$. The next proposition describes this
isogeny geometrically and characterizes $A$ in terms of the curve $C$ with its
involution $\tau\colon C\rightarrow C$.

######  Proposition 2.15.

Let $i\colon C\rightarrow A$ be the composite of the Abel–Jacobi map
$C\rightarrow J$ with respect to $\infty\in C(k)$ and the projection map
$J\rightarrow A$.

  1. (1)

The morphism $i\colon C\hookrightarrow A$ is a closed embedding.

  2. (2)

The divisor $i(C)$ is ample and induces the polarization $\widehat{\lambda}$.

  3. (3)

If $B/k$ is an abelian surface and $j\colon C\hookrightarrow B$ a closed
embedding mapping $\infty$ to $0$ and such that $[-1]\circ j=j\circ\tau$, then
there exists a unique isomorphism of abelian surfaces $\phi\colon A\rightarrow
B$ such that $\phi\circ i=j$.

###### Proof.

This is due to Barth [4]; see [27, Proposition 3.5] for a detailed proof. ∎

Let ${\rm Sym}^{2}C$ be the symmetric square of $C$, a nice surface over $k$
parameterizing effective divisors of degree $2$ on $C$. In analogy with
Jacobians of genus $2$ curves, we will describe the fibres of the surjective
map $i^{(2)}\colon{\rm Sym}^{2}C\rightarrow A$ given by $p+p^{\prime}\mapsto
i(p)+i(p^{\prime})$; this is due to Ikeda [23].

To this end, we define two involutions on ${\rm Sym}^{2}C$. The first is
$\tau^{(2)}(p+p^{\prime})=\tau(p)+\tau(p^{\prime})$. The second involution
$\kappa$ sends $p+p^{\prime}$ to the unique effective degree $2$ divisor
linearly equivalent to $4\infty-p-p^{\prime}$; this is well defined by
Riemann–Roch and the fact that $C$ is not hyperelliptic. Since
$\tau(\infty)=\infty$, the involutions $\tau^{(2)}$ and $\kappa$ on ${\rm
Sym}^{2}C$ commute. Finally, the double cover $\pi\colon C\rightarrow E$
induces a map $\pi^{*}\colon E\rightarrow{\rm Sym}^{2}C$, giving an embedding
$E\simeq\pi^{*}(E)\hookrightarrow{\rm Sym}^{2}C$.

######  Proposition 2.16 (Ikeda).

The map $i^{(2)}\colon{\rm Sym}^{2}C\rightarrow A$ contracts $\pi^{*}(E)$ to
the origin. Two points $D,D^{\prime}\in{\rm Sym}^{2}C$ not in $\pi^{*}(E)$ map
to the same point of $A$ if and only if $D^{\prime}=\kappa(\tau^{(2)}(D))$,
i.e. $D^{\prime}+\tau(D)\sim 4\infty$.

###### Proof.

This follows from the proof of [23, Lemma 3.1]. ∎

We see that $A$ is obtained from ${\rm
Sym}^{2}C/\langle\tau^{(2)}\circ\kappa\rangle$ by contracting the rational
curve $\pi^{*}(E)/\langle-1\rangle$. This description will be useful in
Section 7.5, where we relate the bitangents of $C$ to the group $A[2]$.

###  2.7. Bigonal duality

Given a marked bielliptic Picard curve $(C,\gamma)$, it turns out that we can
define another marked bielliptic Picard curve $(\widehat{C},\hat{\gamma})$
such that the role of the Prym variety and its dual are reversed: the Prym
variety of $\widehat{C}$ is isomorphic to $A=P^{\vee}$. This is called the
bigonal dual of $C$, originally defined by Pantazis [35] (inspired by ideas of
Donagi [12]) and analyzed by Barth [4] in this specific situation; see [27,
§3.4] for more details. Let $\Theta_{2\infty}\subset J$ be the image of the
Abel–Jacobi map ${\rm Sym}^{2}C\rightarrow J$ sending $P+P^{\prime}$ to
$P+P^{\prime}-2\infty$.

######  Definition 2.17.

The bigonal dual of $C$ is defined by $\widehat{C}:=P\cap\Theta_{2\infty}$.

The $\mu_{3}$-action on $P$ restricts to a $\mu_{3}$-action on $\widehat{C}$.
Inversion $[-1]$ on $P$ restricts to an involution $\hat{\tau}$ on
$\widehat{C}$. Using the isomorphism $\mu_{2}\times\mu_{3}\rightarrow\mu_{6}$
induced by the two inclusions, we obtain a $\mu_{6}$-action $\hat{\gamma}$ on
$\widehat{C}$.

######  Lemma 2.18.

The pair $(\widehat{C},\hat{\gamma})$ is a marked bielliptic Picard curve.

###### Proof.

By [27, Lemma 3.5], $\widehat{C}$ is a nice genus $3$ curve and
$\widehat{C}/\mu_{2}$ is a genus $1$ curve. Since the $\mu_{3}$-action on $P$
has only isolated fixed points (Lemma 2.14), the $\mu_{3}$-action on
$\widehat{C}$ is nontrivial. Therefore $\hat{\gamma}$ is a faithful bielliptic
action and $\widehat{C}$ is a bielliptic Picard curve. The origin $0\in P(k)$
is a $\mu_{6}$-fixed point of $\widehat{C}$, which is unique by Theorem 2.2.

It remains to show that $(\widehat{C},\widehat{\gamma})$ is a marked
bielliptic Picard curve, in other words that $\mu_{6}$ acts on
$T_{0}\widehat{C}$ via the identity character. Equivalently, we will show that
$\mu_{6}$ acts on $(T_{0}\widehat{C})^{\vee}$ via the _inverse_ of the
identity character. Choose a uniformizer $t\in\mathcal{O}_{C,\infty}$. This
induces a $k$-basis $\\{t_{1},t_{2}\\}$ of $(T_{(\infty,\infty)}(C\times
C))^{\vee}=(T_{\infty}C)^{\vee}\oplus(T_{\infty}C)^{\vee}$ and an isomorphism
of completed local rings $\widehat{\mathcal{O}}_{C\times
C,(\infty,\infty)}\simeq k[[t_{1},t_{2}]]$. In turn, this induces an
isomorphism $\widehat{\mathcal{O}}_{{\rm Sym}^{2}C,2\infty}\simeq k[[u,v]]$,
where $u=t_{1}+t_{2}$ and $v=t_{1}t_{2}$. It follows that $(T_{2\infty}{\rm
Sym}^{2}C)^{\vee}\simeq(T_{0}\Theta_{2\infty})^{\vee}$ has basis $\\{u,v\\}$
and $[-1]$ sends $u,v$ to $-u,v$ respectively. Therefore
$T_{0}\widehat{C}=T_{0}\Theta_{2\infty}\cap T_{0}P$ has basis $\\{u\\}$ which
indeed has the correct character. ∎

It follows that the objects defined in §2.5 also apply to
$(\widehat{C},\hat{\gamma})$, giving
$\widehat{J}=\operatorname{Jac}_{\widehat{C}}$, $\widehat{P}$, $\widehat{A}$,
$\hat{\pi}\colon\widehat{C}\rightarrow\widehat{E}$ in this context. The
inclusion $\widehat{C}\hookrightarrow P$ induces a homomorphism
$\widehat{J}\rightarrow P$.

######  Proposition 2.19.

The homomorphism $\widehat{J}\rightarrow P$ factors through the projection
$\widehat{J}\rightarrow\widehat{A}$ and induces an isomorphism
$\widehat{A}\xrightarrow{\sim}P$ of $(1,2)$-polarized surfaces.

###### Proof.

See [27, Proposition 3.6]. The fact that the isomorphism preserves the
polarizations means that the maps $A\rightarrow P$ and
$\widehat{P}\rightarrow\widehat{A}$ that are both denoted by
$\widehat{\lambda}$ are the same under these identifications. ∎

Choosing an isomorphism $(C,\gamma)\simeq(C_{a,b},\gamma_{a,b})$ for some
$a,b\in k$ using Lemma 2.8, we can write down equations for $\widehat{C}$ [27,
§3.4 Eq (3.7), (3.8)]:

(2.5) $\displaystyle C_{a,b}$ $\displaystyle:y^{3}=x^{4}+ax^{2}+b,$ (2.6)
$\displaystyle E_{a,b}$ $\displaystyle:y^{2}=x^{3}+16(a^{2}-4b),$ (2.7)
$\displaystyle\widehat{C}_{a,b}$
$\displaystyle:y^{3}=x^{4}+8ax^{2}+16(a^{2}-4b),$ (2.8)
$\displaystyle\widehat{E}_{a,b}$ $\displaystyle:y^{2}=x^{3}+b.$

The coordinates on $E_{a,b}$ and $\widehat{E}_{a,b}$ are chosen so that the
double covers $C_{a,b}\rightarrow E_{a,b}$ and
$\widehat{C}_{a,b}\rightarrow\widehat{E}_{a,b}$ are given by
$(x,y)\mapsto(4y,8x^{2}+4a)$ and $(x,y)\mapsto(y/4,x^{2}/8+a/2)$ respectively.

The third equation (2.7) shows that bigonal duality takes the following
elegant form on the $j$-invariant (2.3):

######  Lemma 2.20.

Let $C$ be a bielliptic Picard curve over $k$. Then $j(\widehat{C})=1/j(C)$.

The quartic polynomial $\hat{f}:=x^{4}+8ax^{2}+16(a^{2}-4b)$ defining the
bigonal dual curve can also be constructed from the Galois theory of the
original quartic $f:=x^{4}+ax^{2}+b$. A calculation shows:

######  Lemma 2.21.

Suppose that $f$ has roots $\\{\pm\alpha,\pm\beta\\}$ in
$k^{\operatorname{sep}}$. Then $\hat{f}$ has roots $\\{\pm 2\alpha\pm
2\beta\\}$.

This description of $\hat{f}$ will be useful when analyzing $3$-torsion in $P$
in §7.2.

##  3\. A Torelli theorem

A bielliptic Picard curve $C$ gives rise to two $(1,2)$-polarized abelian
surfaces with $\mu_{3}$-actions: the Prym surface $P$ and its dual $A$. We
show that one can recover $C$ from $A$, together with its polarization and
$\mu_{3}$-action; in other words, we prove a Torelli-type theorem (Theorem
3.13). We do this by studying the pencil of genus three curves in the linear
system $|C|$ on $A$. In fact, we will prove that _any_ indecomposable
$(1,2)$-polarized abelian surface with compatible $\mu_{3}$-action is the dual
Prym variety of a bielliptic Picard curve, so we work in this generality
below.

###  3.1. Generalities on $(1,2)$-polarized surfaces

We recall and complement some results of Barth [4]. Barth works over
${\rm\mathbb{C}}$, but the proofs of the basic results quoted here are valid
in arbitrary characteristic $\neq 2$.

Let $(A,\lambda)$ be a $(1,2)$-polarized abelian surface over $k$. We assume
that $(A,\lambda)$ is indecomposable, that is, not isomorphic to a product of
two elliptic curves $(E,O)\times(E^{\prime},2O^{\prime})$ as polarized abelian
varieties. Recall our notations concerning linear systems and abelian
varieties in §1.8.

######  Lemma 3.1.

Let $\mathscr{L}\in{\rm Pic}(A_{k^{\operatorname{sep}}})$ be a line bundle
representing the polarization: $\lambda_{\mathscr{L}}=\lambda$.

  1. (1)

$\dim_{k}\operatorname{H}^{0}(A,\mathscr{L})=2$.

  2. (2)

The linear system $|\mathscr{L}|$ has four base-points on which
$A[\lambda](k^{\operatorname{sep}})$ acts simply transitively.

  3. (3)

At most finitely many members of the pencil are singular (arithmetic) genus
$3$ curves, in which case they are curves of geometric genus $2$ with one node
or the union of two genus $1$ curves $X_{1},X_{2}$ with intersection number
$(X_{1}\cdot X_{2})=2$.

###### Proof.

See [4, §1.2]. ∎

######  Lemma 3.2.

There exists a unique $\mathscr{L}\in{\rm Pic}(A_{k^{\operatorname{sep}}})$
representing $\lambda$ whose base locus is $A[\lambda]$. This line bundle is
symmetric: $[-1]^{*}\mathscr{L}\simeq\mathscr{L}$. Moreover, for every divisor
$D\in|\mathscr{L}|$ we have $[-1]^{*}D=D$.

###### Proof.

Denote the base locus of a line bundle $\mathscr{L}$ by
$\text{BL}(\mathscr{L})$. Given $a\in A(k^{\operatorname{sep}})$, let
$t_{a}\colon A\rightarrow A$ be the translation map and $\lambda(a)\in
A^{\vee}(k^{\operatorname{sep}})={\rm Pic}^{0}(A_{k^{\operatorname{sep}}})$ be
its image under $\lambda$. Then for any $\mathscr{L}$ representing $\lambda$,
the following identity holds:

(3.1)
$\displaystyle\text{BL}(\lambda(a)\otimes\mathscr{L})=t_{a}^{*}\text{BL}(\mathscr{L}).$

Indeed, by definition $\lambda(a)\otimes\mathscr{L}=t_{a}^{*}\mathscr{L}$. Let
$D,D^{\prime}$ be two distinct members of the pencil $|\mathscr{L}|$. Then
$t_{a}^{*}D,t_{a}^{*}D^{\prime}$ are two distinct members of
$|t_{a}^{*}\mathscr{L}|$. So $\text{BL}(t_{a}^{*}\mathscr{L})=t_{a}^{*}D\cap
t_{a}^{*}D^{\prime}=t_{a}^{*}(D\cap
D^{\prime})=t_{a}^{*}\text{BL}(\mathscr{L})$, proving (3.1). This identity,
together with Lemma 3.1(2), proves the existence and uniqueness of
$\mathscr{L}$. The remainder of the lemma follows from [4, Proposition 1.6]. ∎

For the remainder of this section fix $\mathscr{L}\in{\rm
Pic}(A_{k^{\operatorname{sep}}})$ satisfying the properties of Lemma 3.2.
Since there is a unique such line bundle up to isomorphism and $A$ has a
$k$-point, $\mathscr{L}$ is automatically Galois invariant and descends to a
unique line bundle on $A$ that we will continue to denote by $\mathscr{L}$.

######  Remark 3.3.

This marks an interesting contrast with principal polarizations on abelian
surfaces, which may not be represented by a $k$-rational line bundle. Indeed,
this is the case for a positive proportion of genus 2 Jacobians over
${\rm\mathbb{Q}}$ ordered by height [37, Theorem 23].

The next proposition shows that every smooth member of the pencil
$|\mathscr{L}|$ realizes $A$ as the dual Prym variety of a bielliptic genus
$3$ curve, as in Proposition 2.15. A bielliptic genus $3$ curve $(X,\tau)$ is
a nice genus $3$ curve together with an involution $\tau$ on $X$ with $4$
fixed points. For such a curve, let
$\operatorname{Prym}(X,\tau)=\ker(1+\tau^{*}\colon\operatorname{Jac}_{X}\rightarrow\operatorname{Jac}_{X})$
be its Prym variety.

######  Proposition 3.4.

Suppose that $X\in|\mathscr{L}|(k)$ is smooth, and let $\tau$ be the
restriction of $[-1]$ to $X$. Then $(X,\tau)$ is a bielliptic genus $3$ curve
and the inclusion $X\rightarrow A$ induces an isomorphism
$\operatorname{Prym}(X,\tau)^{\vee}\simeq A$ of $(1,2)$-polarized surfaces.

###### Proof.

See [4, Proposition (1.8) and Theorem (1.12)]. ∎

The abelian surface $A$ can also be reconstructed from the singular members of
the pencil. By Lemma 3.1(3), there are two cases to consider.

For the first case, let $X\in|\mathscr{L}|(k)$ be a curve with one node, whose
normalization $\tilde{X}\rightarrow X$ is a genus $2$ curve. Let $p\in X(k)$
be the node and $q_{1},q_{2}$ be its preimages in
$\tilde{X}(k^{\operatorname{sep}})$. The morphism $\tilde{X}\rightarrow
X\hookrightarrow A$ induces a homomorphism of abelian varieties
$\phi\colon\operatorname{Jac}_{\tilde{X}}\rightarrow A$, sending a degree zero
divisor class on $\tilde{X}$ to its sum in $A$.

######  Proposition 3.5.

The homomorphism $\phi\colon\operatorname{Jac}_{\tilde{X}}\rightarrow A$ is an
isogeny with kernel $\\{0,q_{1}-q_{2}\\}$.

###### Proof.

The image of $\phi$ is an abelian subvariety containing $X$, so $\phi$ is
surjective. By construction, $\phi(q_{1}-q_{2})=0$. It suffices to prove
$\phi$ has degree $2$. This follows from an intersection number calculation.
Given divisors $D,E$ on a nice surface, denote by $[D],[E]$ their class in the
Picard group and their intersection number by $(D\cdot E)$. The origin $0\in
A(k)$ lifts to a unique point $\infty\in\tilde{X}(k)$ and we identify
$\tilde{X}$ with its image in $\operatorname{Jac}_{\tilde{X}}$ using the point
$\infty$. Then $\phi^{*}([X])=(\deg\phi)[\tilde{X}]$ and by the projection
formula

$\displaystyle(\deg\phi)(X,X)=(\phi^{*}X,\phi^{*}X)=(\deg\phi)^{2}(\tilde{X},\tilde{X}).$

The adjunction formula tells us that $(X,X)=2\cdot 3-2=4$ and
$(\tilde{X},\tilde{X})=2$. Therefore $\deg\phi=2$, as claimed. ∎

For the second case, let $X=X_{1}\cup X_{2}\in|\mathscr{L}|(k)$ be a pair of
genus $1$ curves with intersection number $2$ and let $\tilde{X}=X_{1}\sqcup
X_{2}\rightarrow X$ be its normalization. The map $\tilde{X}\rightarrow
X\hookrightarrow A$ induces a homomorphism
$\phi\colon\operatorname{Jac}_{\tilde{X}}:=\operatorname{Jac}_{X_{1}}\times\operatorname{Jac}_{X_{2}}\rightarrow
A$.

######  Proposition 3.6.

The map $\phi$ is an isogeny of degree $2$.

###### Proof.

The proof is very similar to that of Proposition 3.5; we omit the details. ∎

###  3.2. Compatible $\mu_{3}$-actions on $(1,2)$-polarized abelian surfaces

######  Definition 3.7.

A $\mu_{3}$-abelian surface over $k$ is a triple $(A,\lambda,\alpha)$, where
$(A,\lambda)$ is a $(1,2)$-polarized abelian surface and
$\alpha\colon\mu_{3}\rightarrow{\rm Aut}(A_{k^{\operatorname{sep}}})$ is a
${\rm Gal}_{k}$-equivariant group homomorphism such that for every primitive
third root of unity $\omega\in k^{\operatorname{sep}}$:

  1. (1)

$\alpha(\omega)^{\vee}\circ\lambda\circ\alpha(\omega)=\lambda$;

  2. (2)

$\alpha(\omega)$ has characteristic polynomial $X^{2}+X+1$ on
$T_{0}A=\operatorname{H}^{0}(A,\Omega^{1}_{A})^{\vee}$.

The signature condition (2) is equivalent to
$\alpha(\omega)^{2}+\alpha(\omega)+1=0$, and is motivated by Lemma 2.13.
Condition (1) can be reformulated as follows: after extending
$\mu_{3}\hookrightarrow{\rm Aut}(A_{k^{\operatorname{sep}}})$ to a
${\rm\mathbb{Q}}$-algebra homomorphism
${\rm\mathbb{Q}}(\omega)\hookrightarrow{\rm
End}(A_{k^{\operatorname{sep}}})\otimes{\rm\mathbb{Q}}$, the Rosati involution
associated to $\lambda$ restricts to the conjugation involution
$a+b\omega\mapsto a+b\bar{\omega}$ on ${\rm\mathbb{Q}}(\omega)$.

######  Remark 3.8.

It would be more accurate to call such triples ‘$(1,2)$-polarized abelian
surfaces with compatible $\mu_{3}$-action of signature $(1,1)$’, but we stick
with $\mu_{3}$-abelian surfaces for brevity.

Let $(A,\lambda,\alpha)$ be a $\mu_{3}$-abelian surface over $k$. Assume that
$(A,\lambda)$ is indecomposable, so the results of §3.1 apply. Let
$\mathscr{L}$ be the unique line bundle on $A$ representing $\lambda$ with
base locus $A[\lambda]$. The compatibility of $\alpha$ and $\lambda$ and the
uniqueness of $\mathscr{L}$ show that
$\alpha(\omega)^{*}\mathscr{L}\simeq\mathscr{L}$ for all
$\omega\in\mu_{3}(k^{\operatorname{sep}})$. Each such isomorphism is unique up
to $k^{\times}$-scaling so determines a _canonical_ isomorphism
$|\alpha(\omega)^{*}\mathscr{L}|\simeq|\mathscr{L}|$ Via pullback this induces
a $\mu_{3}$-action on the pencil $|\mathscr{L}|$.

######  Proposition 3.9.

Every $\mu_{3}$-fixed point of $|\mathscr{L}|$ is a smooth curve.

###### Proof.

We may assume that $k=k^{\operatorname{sep}}$. Let $\omega\in k$ be a third
root of unity and for ease of notation denote $\alpha(\omega)\colon
A\rightarrow A$ by $\alpha$. Let $X\in|\mathscr{L}|$ be fixed by $\alpha$.
Assume for the sake of contradiction that $X$ is singular, with normalization
$\tilde{X}\rightarrow X$. By Lemma 3.1(3) and Propositions 3.5 and 3.6, the
map $\tilde{X}\rightarrow A$ induces an isogeny
$\phi\colon\operatorname{Jac}_{\tilde{X}}\rightarrow A$ whose kernel
$\\{0,D\\}$ has order $2$. Since $\alpha$ preserves $X$, it induces an
automorphism $\tilde{\alpha}$ of $\operatorname{Jac}_{\tilde{X}}$ and $\phi$
is equivariant with respect to $\tilde{\alpha}$ and $\alpha$. It follows that
$\tilde{\alpha}$ fixes $D$. Since $\alpha^{2}+\alpha+1=0$ (Condition (2) in
Definition 3.7) and $\phi$ is an isogeny,
$\tilde{\alpha}^{2}+\tilde{\alpha}+1=0$. Therefore the degree of
$1-\tilde{\alpha}$ is a power of $3$, contradicting the fact that
${\rm\mathbb{Z}}/2{\rm\mathbb{Z}}\simeq\\{0,D\\}\subset\operatorname{Jac}_{\tilde{X}}[1-\tilde{\alpha}]$.
∎

######  Corollary 3.10.

Assume that $k=k^{\operatorname{sep}}$. Then there are exactly $2$ curves in
$|\mathscr{L}|$ that are preserved by the $\mu_{3}$-action.

###### Proof.

Let $\omega\in k$ be a third root of unity. By Proposition 3.9, every fixed
point of $\alpha(\omega)$ on $|\mathscr{L}|$ is smooth. Since $|\mathscr{L}|$
always contains singular curves [4, §1.2], this implies that $\alpha(\omega)$
acts nontrivially on $|\mathscr{L}|$. By the classification of order $3$
elements of ${\rm PGL}_{2}(k)$, we may choose coordinates on
${\rm\mathbb{P}}^{1}$ so that $\alpha(\omega)$ is given by
$(x:y)\mapsto(\omega x:y)$. This map clearly has $2$ fixed points. ∎

Let $X_{1},X_{2}\in|\mathscr{L}_{k^{\operatorname{sep}}}|$ be the members of
the pencil that are preserved by the $\mu_{3}$-action $\alpha$. By Lemma 3.2,
the involution $[-1]$ also preserves $X_{1}$ and $X_{2}$. These combine to a
$\mu_{6}$-action $\gamma_{i}$ on $X_{i}$ for each $i=1,2$. The next
proposition shows that each $X_{i}$ is a bielliptic Picard curve and the
$\mu_{6}$-actions have ‘opposite’ signature.

######  Proposition 3.11.

Reordering $X_{1}$ and $X_{2}$ if necessary, the pairs $(X_{1},\gamma_{1})$
and $(X_{2},\gamma_{2}^{-1})$ are marked bielliptic Picard curves. Moreover,
these pairs are defined over $k$.

###### Proof.

The second sentence follows from the first, since ${\rm Gal}_{k}$ preserves
the $\mu_{3}$-fixed points of $|\mathscr{L}_{k^{\operatorname{sep}}}|$ and the
signatures of the $\mu_{6}$-action. To show that $(X_{1},\gamma_{1})$ and
$(X_{2},\gamma_{2})$ are bielliptic Picard curves of opposite signature, we
may assume that $k=k^{\operatorname{sep}}$.

Let $(X,\gamma)=(X_{i},\gamma_{i})$ for some $i=1,2$. Proposition 3.9 shows
that $X$ is smooth, and Proposition 3.4 shows that $(X,[-1]|_{X})$ is a
bielliptic genus 3 curve. Since $\alpha^{2}+\alpha+1=0$, the $\mu_{3}$-action
on $A$ has isolated fixed points, so the induced $\mu_{3}$-action on $X$ is
faithful. It follows that $\gamma$ is a faithful bielliptic action and hence
$X$ is a bielliptic Picard curve. Note that $0\in X(k)\subset A(k)$ is the
unique $\mu_{6}$-fixed point.

To prove the claim about signatures, it suffices to show that the
$\mu_{3}$-action on $T_{0}X_{1}$ and $T_{0}X_{2}$ act via inverse characters.
Since $\mathscr{L}$ has self-intersection $4$ and $|\mathscr{L}|$ has four
base points (Lemma 3.1), $X_{1}$ and $X_{2}$ intersect transversally at all
these base points. Since $0\in A(k)$ is such a base point,

(3.2) $\displaystyle T_{0}A=T_{0}X_{1}\oplus T_{0}X_{2}.$

Condition (2) of Definition 3.7 says that the $\mu_{3}$-action on $T_{0}A$ is
a direct sum of the two nontrivial characters of $\mu_{3}$. Since the
decomposition (3.2) is $\mu_{3}$-stable, the proposition follows. ∎

###  3.3. A Torelli theorem for marked bielliptic Picard curves

Let $(C,\gamma)$ be a marked bielliptic Picard curve over $k$. The
$\mu_{6}$-action on $C$ induces a $\mu_{6}$-action on
$J=\operatorname{Jac}_{C}$ such that the Abel–Jacobi map $C\hookrightarrow J$
is $\mu_{6}$-equivariant. This induces $\mu_{3}$-actions $\alpha$ on $P$ and
$\hat{\alpha}$ on333A calculation shows that the $\mu_{3}$-actions
$\hat{\alpha}$ (making $J\rightarrow A$ $\mu_{3}$-equivariant) and
$\alpha^{\vee}$ (the dual of the $\mu_{3}$-action on $P$) on $A$ are inverse
to each other. $A$.

######  Lemma 3.12.

$\operatorname{Prym}(C,\gamma):=(P,\lambda,\alpha)$ and
$\operatorname{Prym}(C,\gamma)^{\vee}:=(A,\widehat{\lambda},\hat{\alpha})$ are
indecomposable $\mu_{3}$-abelian surfaces over $k$.

###### Proof.

The first condition of Definition 3.7 for
$\operatorname{Prym}(C,\gamma)^{\vee}$ follows from Proposition 2.15 and the
fact that $\hat{\alpha}$ preserves $C\hookrightarrow A$. Moreover, since $C$
is irreducible, $(A,\widehat{\lambda})$ is indecomposable. The second
condition for $\operatorname{Prym}(C,\gamma)$ follows from Lemma 2.13. The
lemma now follows from bigonal duality (Proposition 2.19). ∎

The next theorem is a strong Torelli statement for the association
$(C,\gamma)\mapsto\operatorname{Prym}(C,\gamma)^{\vee}$. To state it, we
upgrade the set of marked bielliptic Picard curves (respectively
$\mu_{3}$-abelian surfaces) to a category, where the morphisms are
isomorphisms of curves (respectively polarized abelian surfaces) respecting
the $\mu_{6}$-structure (respectively $\mu_{3}$-structure).

######  Theorem 3.13.

The association $(C,\gamma)\mapsto\operatorname{Prym}(C,\gamma)^{\vee}$
induces an equivalence of categories:

(3.3) $\displaystyle\begin{Bmatrix}\text{Marked bielliptic}\\\ \text{Picard
curves over }k\end{Bmatrix}\rightarrow\begin{Bmatrix}\text{Indecomposable}\\\
\mu_{3}\text{-abelian surfaces over }k\end{Bmatrix}.$

###### Proof.

This association is well defined by Lemma 3.12. It is functorial: an
isomorphism of marked bielliptic Picard curves
$(C,\gamma)\rightarrow(C^{\prime},\gamma^{\prime})$ induces an isomorphism
$\operatorname{Prym}(C,\gamma)\rightarrow\operatorname{Prym}(C^{\prime},\gamma^{\prime})$
of marked $\mu_{3}$-abelian surfaces. We explicitly construct a quasi-inverse.
To a triple $(A,\lambda,\alpha)$, in the notation of Proposition 3.11, we
associate the marked bielliptic Picard curve $(X_{1},\gamma_{1})$. Since the
constructions of §3.1 and §3.2 are canonical, this association is also
functorial. The fact that this is a quasi-inverse follows from Propositions
2.15 and 3.4. ∎

By Lemma 2.8, the left hand side of (3.3) is in bijection with the set of
equivalence classes $\\{(a,b)\in k\times k\mid b(a^{2}-4b)\neq 0\\}/\sim$,
where $(a,b)\sim(a^{\prime},b^{\prime})$ if
$(a,b)=(\lambda^{6}a^{\prime},\lambda^{12}b^{\prime})$ for some $\lambda\in
k^{\times}$. If $k=k^{\operatorname{sep}}$, the $j$-invariant (2.3) maps this
set bijectively to the ‘$j$-line’ $k\setminus\\{0\\}$. Theorem 3.13 thus shows
that this description also applies to the right hand side of (3.3). In
particular, we may define the $j$-invariant of a $\mu_{3}$-abelian surface
$(A,\lambda,\alpha)$ as $j((C,\gamma))$, where $(C,\gamma)$ is any bielliptic
Picard curve with $\operatorname{Prym}(C,\gamma)^{\vee}=(A,\lambda,\alpha)$.

######  Remark 3.14.

Since
$\operatorname{Prym}(C,\gamma)\simeq\operatorname{Prym}(\widehat{C},\hat{\gamma})^{\vee}$
(Proposition 2.19) and bigonal duality induces an autoequivalence on the
category of marked bielliptic Picard curves, Theorem 3.13 implies that the
association $(C,\gamma)\mapsto\operatorname{Prym}(C,\gamma)$ also induces an
equivalence of categories of the form (3.3). On the other hand, the
association $(A,\lambda,\alpha)\mapsto(A,\lambda,\alpha^{-1})$ is again an
auto-equivalence on the category of marked $\mu_{3}$-abelian surfaces. In
Corollary 5.5 we will prove a precise relationship between these equivalences.

######  Remark 3.15.

Barth has shown that Torelli fails for the family of all bielliptic genus $3$
curves $(X,\tau)$. Suppose that $k$ is algebraically closed of characteristic
zero and $A$ is an indecomposable $(1,2)$-polarized abelian surface
$(A,\lambda)/k$. Then there exist many bielliptic genus $3$ curves $(X,\tau)$
over $k$ with dual Prym variety isomorphic to $(A,\lambda)$; by Proposition
3.4 any smooth member of the pencil $|\mathscr{L}|$ on $A$ will do. Theorem
3.13 shows that we can correct this when $C$ is a bielliptic Picard curve and
we fix the $\mu_{6}$-action on $C$ and the $\mu_{3}$-action on $A$.

######  Remark 3.16.

Let $S$ be a ${\rm\mathbb{Z}}[1/6]$-scheme. A marked bielliptic Picard curve
over $S$ is a smooth proper morphism $C\rightarrow S$ together with a
$\mu_{6}$-action $\gamma\colon\mu_{6}\times_{S}C\rightarrow C$ such that for
every geometric point $\bar{s}\in S$, $(C_{\bar{s}},\gamma_{\bar{s}})$ is a
marked bielliptic Picard curve. With the evident notion of isomorphisms, this
defines the stack $\mathfrak{M}$ of (marked) bielliptic Picard curves over
${\rm\mathbb{Z}}[1/6]$ (in the fppf topology), isomorphic to an open substack
of the weighted projective stack ${\rm\mathbb{P}}(6,12)$. A $\mu_{3}$-abelian
surface over $S$ is a $(1,2)$-polarized abelian scheme $(A,\lambda)$ over $S$
together with a $\mu_{3}$-action $\alpha\colon\mu_{3}\times_{S}A\rightarrow A$
such that for every geometric point $\bar{s}\in S$,
$(A_{\bar{s}},\lambda_{\bar{s}},\alpha_{\bar{s}})$ is a $\mu_{3}$-abelian
surface. This defines the stack $\mathfrak{Y}$ of $\mu_{3}$-abelian surfaces
over ${\rm\mathbb{Z}}[1/6]$. There exists an open substack
$\mathfrak{Y}^{\text{indec}}\subset\mathfrak{Y}$ of those surfaces such that
each geometric fiber is indecomposable. Theorem 3.13 then generalizes to the
statement that the association
$(C,\gamma)\mapsto\operatorname{Prym}(C,\gamma)^{\vee}$ induces an isomorphism
of stacks $\mathfrak{M}\xrightarrow{\sim}\mathfrak{Y}^{\text{indec}}$.

##  4\. Shimura curves and quaternionic multiplication

We relate the moduli space of bielliptic Picard curves to certain unitary and
quaternionic Shimura curves, which will allow us to deduce that these
bielliptic Picard Pryms surfaces $P$ have quaternionic multiplication over
$\overline{k}$ (Corollary 4.8). This will also help us study which $P$ are
geometrically non-simple in §6.4, since these correspond to CM points on the
Shimura curve side.

###  4.1. The quaternion order $\mathcal{O}$

Let $B$ be the quaternion algebra $(-3,2)_{{\rm\mathbb{Q}}}$ with
${\rm\mathbb{Q}}$-basis $\\{1,i,j,ij\\}$ and relations $i^{2}=-3$, $j^{2}=2$,
$ij=-ji$. This is an indefinite quaternion algebra of discriminant $6$. Let
$\omega:=\frac{1}{2}(-1+i)$, so that $\omega^{2}+\omega+1=0$. Let
$\mathcal{O}:={\rm\mathbb{Z}}[\omega,j]={\rm\mathbb{Z}}+{\rm\mathbb{Z}}\omega+{\rm\mathbb{Z}}j+{\rm\mathbb{Z}}\omega
j$. A discriminant calculation shows that $\mathcal{O}$ is a maximal order in
$B$. As an associative ring, $\mathcal{O}$ has generators $\omega,j$ and
relations $\omega^{2}+\omega+1=0$, $j^{2}=2$ and $j\omega=\omega^{-1}j$.

Let ${\rm Aut}(\mathcal{O})$ denote the set of ring automorphisms of
$\mathcal{O}$, which we will think of as acting on the right on $\mathcal{O}$.
By the Skolem–Noether theorem, every element of ${\rm Aut}(\mathcal{O})$ is
given by conjugating by an element of $b\in B^{\times}$ (uniquely defined up
to ${\rm\mathbb{Q}}^{\times}$) normalizing $\mathcal{O}$; denote the
conjugation action of such an element $b$ by $[b]\in{\rm Aut}(\mathcal{O})$.

For reasons that will become clear in §6.2, we define the groups

$\displaystyle{\rm Aut}_{i}^{+}(\mathcal{O})$
$\displaystyle:=\\{\varphi\in{\rm Aut}(\mathcal{O})\mid i^{\varphi}=i\\},$
$\displaystyle{\rm Aut}_{i}(\mathcal{O})$ $\displaystyle:=\\{\varphi\in{\rm
Aut}(\mathcal{O})\mid i^{\varphi}=\pm i\\}.$

######  Lemma 4.1.

${\rm Aut}_{i}^{+}(\mathcal{O})=\langle[1-\omega]\rangle$ is cyclic of order
$6$ and ${\rm Aut}_{i}(\mathcal{O})=\langle[1-\omega],[j]\rangle$ is dihedral
of order $12$.

###### Proof.

A direct calculation shows that $\langle[1-\omega],[j]\rangle\subset{\rm
Aut}_{i}(\mathcal{O})$, that this subgroup is dihedral of order $12$ and that
$\langle[1-\omega],[j]\rangle\cap{\rm
Aut}_{i}^{+}(\mathcal{O})=\langle[1-\omega]\rangle$. Conversely, suppose
$[b]\in{\rm Aut}_{i}(\mathcal{O})$. After possibly multplying by $j$, we may
assume $i^{[b]}=b^{-1}ib=i$. In that case $b$ centralizes $i$, so
$b\in{\rm\mathbb{Q}}(i)$. By [28, Lemma 2.2.1], $[b]$ can be represented by an
element of $\mathcal{O}\cap{\rm\mathbb{Q}}(i)={\rm\mathbb{Z}}[\omega]$ of norm
dividing $6$. Therefore $[b]\in\langle[1-\omega]\rangle$, as desired. ∎

######  Lemma 4.2.

Let $G\leq{\rm Aut}_{i}(\mathcal{O})$ be a subgroup. Then the subring
$\mathcal{O}^{G}$ fixed by $G$ is: $\mathcal{O}$ if $G=\\{1\\}$;
${\rm\mathbb{Z}}[\omega]$ if $\\{1\\}\neq G\subset\langle[1-\omega]\rangle$;
isomorphic to ${\rm\mathbb{Z}}[\sqrt{2}]$ if $G$ is conjugate to
$\langle[j]\rangle$; isomorphic to ${\rm\mathbb{Z}}[\sqrt{6}]$ if $G$ is
conjugate to $\langle[ij]\rangle$. In all other cases, $G$ is dihedral of
order $6$ or $12$ and $\mathcal{O}^{G}={\rm\mathbb{Z}}$.

###### Proof.

This is a direct calculation using the explicit description of ${\rm
Aut}_{i}(\mathcal{O})$ from Lemma 4.1. ∎

###  4.2. Moduli of marked $\mu_{3}$-abelian surfaces

In Remark 3.16, we have introduced the stack $\mathfrak{Y}$ of
$\mu_{3}$-abelian surfaces $(A,\lambda,\alpha)$ over ${\rm\mathbb{Z}}[1/6]$,
which we now view as a stack over ${\rm\mathbb{Q}}$. This is a Deligne–Mumford
stack; let $Y/{\rm\mathbb{Q}}$ be its coarse space. By a variant of [26,
Proposition 2.1] for $(1,2)$-polarizations, $Y$ is smooth and purely one-
dimensional. The next lemma shows that $Y$ may be viewed as a compactification
of the $j$-line ${\rm\mathbb{P}}^{1}\setminus\\{0,\infty\\}$ from §2.3.

######  Lemma 4.3.

There exists a unique isomorphism $j\colon
Y\xrightarrow{\sim}{\rm\mathbb{P}}^{1}_{{\rm\mathbb{Q}}}$ with the property
that $j([\operatorname{Prym}(C,\gamma)^{\vee}])=j(C)$ for all marked
bielliptic Picard curves $(C,\gamma)$ over $\bar{{\rm\mathbb{Q}}}$.

###### Proof.

The open substack $\mathfrak{Y}^{\text{indec}}\subset\mathfrak{Y}$
parametrizing those triples $(A,\lambda,\alpha)$ for which $(A,\lambda)$ is
indecomposable has coarse space an open subset $Y^{\text{indec}}\subset Y$.
Every decomposable $\mu_{3}$-abelian surface $(A,\lambda,\alpha)$ over
${\rm\mathbb{C}}$ is isomorphic to $(E\times E,\lambda_{O}\times
2\lambda_{O},\alpha_{1}\times\alpha_{2})$, where $E:y^{2}=x^{3}+1$,
$\lambda_{O}$ is the unique principal polarization on $E$ and $\alpha_{i}$ are
mutually inverse nontrivial $\mu_{3}$-actions on $E$. Since there are two
choices for the pair $(\alpha_{1},\alpha_{2})$, even up to isomorphism,
$(Y\setminus Y^{\text{indec}})({\rm\mathbb{C}})$ has size $2$. Since $Y$ is
purely one-dimensional, $Y^{\text{indec}}$ is dense in $Y$. Theorem 3.13 and
Lemma 2.8 show that the morphism
${\rm\mathbb{A}}^{2}_{{\rm\mathbb{Q}}}\setminus\\{\Delta_{a,b}=0\\}\rightarrow
Y^{\text{indec}},(a,b)\mapsto[\operatorname{Prym}(C_{a,b},\gamma_{a,b})^{\vee}]$
is surjective, factors through the $j$-invariant morphism
${\rm\mathbb{A}}^{2}_{{\rm\mathbb{Q}}}\setminus\\{\Delta_{a,b}=0\\}\rightarrow{\rm\mathbb{P}}^{1}_{{\rm\mathbb{Q}}}\setminus\\{0,\infty\\},(a,b)\mapsto
j_{a,b}$ of (2.3), and induces an isomorphism
${\rm\mathbb{P}}^{1}_{{\rm\mathbb{Q}}}\setminus\\{0,\infty\\}\simeq
Y^{\text{indec}},j\mapsto[\operatorname{Prym}(C_{a,b},\gamma_{a,b})^{\vee}]$,
where $a,b$ are any elements with $j=j_{a,b}$. Putting everything together,
$Y$ is a purely one-dimensional smooth variety having an open subset
isomorphic to ${\rm\mathbb{P}}^{1}\setminus\\{0,\infty\\}$ whose complement
has size $2$. We conclude that $Y\simeq{\rm\mathbb{P}}^{1}_{{\rm\mathbb{Q}}}$
and that the isomorphism $j\colon
Y^{\text{indec}}\rightarrow{\rm\mathbb{P}}_{{\rm\mathbb{Q}}}^{1}\setminus\\{0,\infty\\}$
uniquely extends to an isomorphism
$Y\rightarrow{\rm\mathbb{P}}_{{\rm\mathbb{Q}}}^{1}$. ∎

######  Remark 4.4.

The curve $Y_{{\rm\mathbb{C}}}$ is a disjoint union of Shimura varieties for
groups of type $\mathrm{GU}(1,1)$. Similarly to [26, Proposition 3.1], the
connected components of $Y_{{\rm\mathbb{C}}}$ are parametrized by isometry
classes of rank $2$ Hermitian ${\rm\mathbb{Z}}[\omega]$-lattices of
discriminant $2$. Lemma 4.3 shows that there is precisely one such Hermitian
lattice, although this is certainly not the easiest way to see that.

######  Remark 4.5.

The isomorphism $j\colon Y\rightarrow{\rm\mathbb{P}}^{1}$ identifies
$Y^{\text{indec}}$ with ${\rm\mathbb{P}}^{1}\setminus\\{0,\infty\\}$. The two
decomposable points $0,\infty$ might be considered ‘cusps’.

###  4.3. Comparison with a quaternionic Shimura curve

We now compare $Y$ to a quaternionic Shimura curve, in the same spirit as [22,
§4.2]. However, our analysis is a little different since we must consider
twists and non-principal polarizations. We refer the reader to [45, Chapter
43] for background on Shimura curves.

######  Definition 4.6.

Let $X$ be the (quaternionic) Shimura curve of discriminant $6$ over
${\rm\mathbb{Q}}$.

The nice curve $X$ is the coarse space of the moduli stack of triples
$(A,\lambda,\iota)$, where $(A,\lambda)$ is a $(1,2)$-polarized abelian
surface and $\iota\colon\mathcal{O}\rightarrow{\rm End}(A)$ is an embedding
such that the Rosati involution restricts to $b\mapsto i\bar{b}i^{-1}$, where
we recall from §4.1 that $i=1+2\omega$. (The moduli problem is typically
formulated for principal polarizations, but the results of [45, §43.6] have
analogues for the polarized order $(\mathcal{O},i)$, see [9, §12].) To such a
triple $(A,\lambda,\iota)$ we would like to associate a $\mu_{3}$-abelian
surface by restricting $\iota$ to
${\rm\mathbb{Z}}/3{\rm\mathbb{Z}}\simeq\langle\omega\rangle\subset{\rm\mathbb{Z}}[\omega]^{\times}\subset\mathcal{O}^{\times}$.
However, this defines a ${\rm\mathbb{Z}}/3{\rm\mathbb{Z}}$-action on $A$; to
get a $\mu_{3}$-action we need to suitably twist the situation.

To this end, recall the action of the Atkin–Lehner group
$W=\\{1,w_{2},w_{3},w_{6}\\}\simeq{\rm\mathbb{Z}}/2{\rm\mathbb{Z}}\times{\rm\mathbb{Z}}/2{\rm\mathbb{Z}}$
on $X$: if $b\in\mathcal{O}$ is any element whose (reduced) norm has absolute
value $d\in\\{2,3,6\\}$, then
$w_{d}([A,\lambda,\iota])=[A,\lambda,\iota\circ[b]]$, where $[b]\in{\rm
Aut}(\mathcal{O})$ denotes conjugation by $b$. The following curves will be
important:

  * •

Let $X/w_{3}$ be the quotient of $X$ by $w_{3}$. The $W$-action on $X$ induces
an action of $W/\langle w_{3}\rangle=\\{1,\bar{w}\\}$ on $X/w_{3}$.

  * •

Let $(X/w_{3})^{\bar{w}}$ be the twist of $X/w_{3}$ along the involution
$\bar{w}$ and extension ${\rm\mathbb{Q}}(\omega)/{\rm\mathbb{Q}}$.

  * •

For each $d\in\\{2,6\\}$, let $X^{w_{d}}$ be the twist of $X$ along the
involution $w_{d}$ and extension ${\rm\mathbb{Q}}(\omega)/{\rm\mathbb{Q}}$.

We recall that the twist of a nice curve $C/k$ along an involution $\tau$ and
quadratic extension $K/k$ corresponds to the cocycle given by the image of the
nontrivial element under $\operatorname{H}^{1}({\rm
Gal}(K/k),\langle\tau\rangle)\rightarrow\operatorname{H}^{1}(k,\mathbf{Aut}(C))$.
Note that $W$ acts on $X^{w_{2}}$ and $X^{w_{6}}$, $\bar{w}$ acts on
$(X/w_{3})^{\bar{w}}$, and we have canonical isomorphisms
$X^{w_{2}}/w_{3}\simeq X^{w_{6}}/w_{3}\simeq(X/w_{3})^{\bar{w}}$.

To compare $X$ and $Y$, we will construct morphisms $\pi_{d}\colon
X^{w_{d}}\rightarrow Y$ for each $d\in\\{2,6\\}$. In the following paragraph,
for a field $k/{\rm\mathbb{Q}}$ we write
$K:=k\otimes_{{\rm\mathbb{Q}}}{\rm\mathbb{Q}}(\omega)$ and ${\rm
Aut}_{k}(K)=\\{1,\sigma\\}$. The curve $X^{w_{2}}$ is the coarse space of the
moduli stack whose $k$-points parametrizes triples $(A,\lambda,\iota)$, where
$(A,\lambda)$ is a $(1,2)$-polarized abelian surface over $k$ and
$\iota\colon\mathcal{O}\rightarrow{\rm End}(A_{K})$ is an embedding such that
the Rosati involution restricts to $b\mapsto i\bar{b}i^{-1}$ and such that
$\iota^{\sigma}=\iota\circ[j]$. We claim that such a triple naturally gives
rise to a $\mu_{3}$-abelian surface
$(A,\lambda,\iota|_{\langle\omega\rangle})$ over $k$. Indeed, restricting
$\iota$ to
$\langle\omega\rangle\subset{\rm\mathbb{Z}}[\omega]^{\times}\subset\mathcal{O}^{\times}$
defines homomorphism ${\rm\mathbb{Z}}/3{\rm\mathbb{Z}}\rightarrow{\rm
End}(A_{K})$ which due to the twisting descends to a $\mu_{3}$-action on $A$.
The triple $(A,\lambda,\iota|_{\langle\omega\rangle})$ satisfies first
condition of Definition 3.7 since the Rosati involution sends $\iota(\omega)$
to $\iota(\omega^{-1})$. The second condition follows from the identity
$\omega^{2}+\omega+1=0$. The association
$(A,\lambda,\iota)\mapsto(A,\lambda,\iota|_{\langle\omega\rangle})$ also works
on the level of ${\rm\mathbb{Q}}$-schemes, hence it induces a morphism
$\pi_{2}\colon X^{w_{2}}\rightarrow Y$. Repeating the above discussion but
insisting that $\iota^{\sigma}=\iota\circ[ij]$ instead of
$\iota^{\sigma}=\iota\circ[j]$, we obtain a morphism $\pi_{6}\colon
X^{w_{6}}\rightarrow Y$.

######  Proposition 4.7.

For each $d\in\\{2,6\\}$, $\pi_{d}\colon X^{w_{d}}\rightarrow Y$ induces an
isomorphism $\phi_{d}\colon X^{w_{d}}/w_{3}\xrightarrow{\sim}Y$. Under the
canonical isomorphisms $X^{w_{2}}/w_{3}\simeq
X^{w_{6}}/w_{3}\simeq(X/w_{3})^{\bar{w}}$, $\phi_{2}$ and $\phi_{6}$
correspond to the same isomorphism $(X/w_{3})^{\bar{w}}\simeq Y$.

###### Proof.

It suffices to check these claims over ${\rm\mathbb{C}}$, so let
$\pi=\pi_{2,{\rm\mathbb{C}}}=\pi_{6,{\rm\mathbb{C}}}$. The morphism $\pi$ is
given by mapping an isomorphism class of triples $[A,\lambda,\iota]$ to
$[A,\lambda,\iota|_{\langle\omega\rangle}]$. We have $\pi\circ w_{3}=\pi$,
since $w_{3}([A,\lambda,\iota])=[A,\lambda,\iota\circ[i]]$ and
$\iota|_{\langle\omega\rangle}=(\iota\circ[i])|_{\langle\omega\rangle}$.
(Recall that $i=1+2\omega$.)

Since $\pi$ is a morphism between nice curves, it is either constant or a
finite covering map. To prove the proposition, it suffices to show that $\pi$
has degree $2$, equivalently infinitely many fibres $\pi^{-1}(y)$ have size
$2$. Let $x=[A,\lambda,\iota]\in X({\rm\mathbb{C}})$ be a non-CM point and
$y=\pi(x)=[A,\lambda,\iota|_{\langle\omega\rangle}]$; there are infinitely
many (in fact uncountably many) such points $x$ by the theory of Shimura
curves [13, §2.4]. The set $\pi^{-1}(y)$ is in bijection with the set of
isomorphism classes of triples $(A^{\prime},\lambda^{\prime},\iota^{\prime})$
such that
$(A,\lambda,\iota|_{\langle\omega\rangle})\simeq(A^{\prime},\lambda^{\prime},\iota^{\prime}|_{\langle\omega\rangle})$.
Replacing $(A^{\prime},\lambda^{\prime},\iota^{\prime})$ by an isomorphic
triple, we may assume that $(A^{\prime},\lambda^{\prime})=(A,\lambda)$ and
$\iota^{\prime}|_{{\rm\mathbb{Z}}[\omega]}=\iota|_{{\rm\mathbb{Z}}[\omega]}$.
Therefore $\pi^{-1}(y)$ is in bijection with the orbit space

(4.1) $\displaystyle\\{\iota^{\prime}\colon\mathcal{O}\rightarrow{\rm
End}(A)\mid\iota^{\prime}\text{ compatible with Rosati involution and
}\iota^{\prime}|_{{\rm\mathbb{Z}}[\omega]}=\iota|_{{\rm\mathbb{Z}}[\omega]}\\}/{\rm
Aut}(A,\lambda).$

Since $x$ is assumed to be a non-CM point and $\mathcal{O}$ is maximal, ${\rm
End}(A)\simeq\mathcal{O}$. It follows that every $\iota^{\prime}$ of (4.1) is
of the form $\iota\circ\alpha$ for some $\alpha\in{\rm Aut}(\mathcal{O})$.
Since
$\iota^{\prime}|_{{\rm\mathbb{Z}}[\omega]}=\iota|_{{\rm\mathbb{Z}}[\omega]}$,
$\alpha$ lies in the subgroup ${\rm
Aut}_{i}^{+}(\mathcal{O})=\langle[1-\omega]\rangle$ of Lemma 4.1 (using the
notation of §4.1). Conversely, every element $[b]\in\langle[1-\omega]\rangle$
defines an element of (4.1), and two elements $[b],[b^{\prime}]$ lie in the
same ${\rm Aut}(A,\lambda)$-orbit if and only if
$b^{\prime}b^{-1}\in{\rm\mathbb{Q}}^{\times}\mathcal{O}^{\times}$. It follows
that this orbit space has size
$|\pi^{-1}(y)|=|\langle[1-\omega]\rangle/\langle[\omega]\rangle|=2$, as
desired. ∎

######  Corollary 4.8.

Every $\mu_{3}$-abelian surface $(A,\lambda,\alpha)$ over a field $k$ of
characteristic $0$ admits an embedding $\mathcal{O}\hookrightarrow{\rm
End}(A_{k^{\operatorname{sep}}})$.

###### Proof.

Proposition 4.7 shows that the point $[A,\lambda,\alpha]\in Y(k)$ lifts to a
$k^{\operatorname{sep}}$-point of $X^{w_{2}}$ under $\pi_{2}$. The corollary
follows from the moduli interpretation of $X$. ∎

######  Remark 4.9.

Corollary 4.8 was proved by Petkova and Shiga over ${\rm\mathbb{C}}$ using
period matrix calculations [36, Proposition 7.1].

Our proof of Corollary 4.8 shows that there exists an action of $\mathcal{O}$
on $P_{k^{\operatorname{sep}}}$, but it sheds no light on what this action
actually is. In Proposition 6.1 we will give a more geometric proof of the
quaternionic multiplication of $P_{k^{\operatorname{sep}}}$, which works in
all characteristics $\neq 2,3$ and gives finer arithmetic information, such as
the field of definition of the quaternionic multiplication, see §6.2.

##  5\. Identifying bielliptic Picard curves in the pencil

Let $(C,\gamma)$ be a marked bielliptic Picard curve. Its Prym variety
$\operatorname{Prym}(C,\gamma)=(P,\lambda,\alpha)$ is an indecomposable
$\mu_{3}$-abelian surface (Lemma 3.12). Therefore the results of §3.1-3.2
apply, and the pencil of curves on $P$ contains exactly two bielliptic Picard
curves by Proposition 3.11. We have already seen in §2.7 that this pencil
contains the bigonal dual curve $\widehat{C}$. In this section we will
determine the other bielliptic Picard curve in the pencil: it turns out to be
a sextic twist of the original curve $C$! This calculation is a crucial
ingredient for the results of §6.

Let $\widehat{C}\subset P$ be the dual curve of $C$ from §2.7. Then
$\mathscr{M}:=\mathcal{O}_{P}(\widehat{C})$ is the unique line bundle on $P$
representing the $(1,2)$-polarization with base locus $P[\lambda]$
(Proposition 3.2). By Lemma 2.18, $\widehat{C}$ is a bielliptic Picard curve
preserved by the $\mu_{3}$-action $\alpha$. By Proposition 3.11, there is
exactly one other curve in $|\mathscr{M}|$ that is preserved by the
$\mu_{3}$-action; call this curve $C_{\text{twist}}$. We will explicitly
describe $C_{\text{twist}}$ using the work of Ikeda [23].

We will use the notations from §2.5. If $p\in E(k)$, let $\Theta_{p}$ be the
image of the map ${\rm Sym}^{2}C\rightarrow J$ sending $Q+Q^{\prime}$ to
$Q+Q^{\prime}-\pi^{*}(p)$. Then $D_{p}:=\Theta_{p}\cap P$ is a divisor on $P$.
Note that $D_{\pi(\infty)}=\widehat{C}$. Recall that we view $E$ as an
elliptic curve with origin $O_{E}:=\pi(\infty)\in E(k)$.

######  Lemma 5.1.

  1. (1)

For every $p\in E(k)$, $D_{p}\in|\mathscr{M}|(k)$.

  2. (2)

If $k=k^{\operatorname{sep}}$, every element of $|\mathscr{M}|$ is of the form
$D_{p}$ for some $p\in E(k)$.

  3. (3)

For every $p\in E(k)$, $D_{\iota(p)}=D_{p}$, where $\iota(p)$ denotes the
inverse of $p$ under the group law.

###### Proof.

This is [23, Lemma 3.4] and [23, Remark 3.5]. ∎

Therefore the map $E\rightarrow|\mathscr{M}|,p\mapsto D_{p}$ factors through
the map $\phi\colon E\rightarrow|2O_{E}|$ and gives rise to a
$\mu_{3}$-equivariant map $|2O_{E}|\rightarrow|\mathscr{M}|,y\mapsto D_{y}$.
We note that if $y$ is $k$-rational, then so is $D_{y}$.

By Lemma 2.8, we may and do assume that $(C,\gamma)=(C_{a,b},\gamma_{a,b})$
for some $a,b\in k$. Then $E_{a,b}$ has equation $y^{3}=x^{2}+ax+b$ and $y$
induces an isomorphism $|2O_{E}|\simeq{\rm\mathbb{P}}^{1}$. There are two
fixed points for the $\mu_{3}$-action on $|2O_{E}|$, corresponding to
$0,\infty\in{\rm\mathbb{P}}^{1}(k)$; call them $y_{0},y_{\infty}\in|2O_{E}|$.
Then $D_{y_{\infty}}=D_{O_{E}}=\widehat{C}$ and $D_{y_{0}}=C_{\text{twist}}$.

In order to compute equations for $D_{y_{0}}$, we will give a different
description of the family of divisors $\\{D_{p}\\}_{p\in E}$. For ease of
notation, write $C^{(2)}={\rm Sym}^{2}C$. For each $\xi\in{\rm Pic}^{2}(E)$,
consider the closed subscheme

$\displaystyle F_{\xi}:=\\{Q+Q^{\prime}\in
C^{(2)}\mid\pi(Q)+\pi(Q^{\prime})\sim\xi\\}\subset C^{(2)}.$

Since $C$ is not hyperelliptic, the map $Q+Q^{\prime}-\pi^{*}(p)\mapsto
Q+Q^{\prime}$ induces an isomorphism $\phi_{p}\colon
D_{p}\xrightarrow{\sim}F_{2p}$. Consider the morphism

$\displaystyle\psi\colon C^{(2)}$
$\displaystyle\longrightarrow{\rm\mathbb{P}}^{2}$
$\displaystyle([x:y:z],[x^{\prime}:y^{\prime}:z^{\prime}])$
$\displaystyle\mapsto[yz^{\prime}-y^{\prime}z:xz^{\prime}-x^{\prime}z:xy^{\prime}-x^{\prime}y],$

which sends a pair of distinct points on $C$ to the line it spans in the
canonical embedding $C\subset{\rm\mathbb{P}}^{2}$. Ikeda has shown that for
every $\xi\in{\rm Pic}^{2}(E)$ such that $F_{\xi}$ is smooth, $\psi$ maps
$F_{\xi}$ isomorphically onto a plane quartic in ${\rm\mathbb{P}}^{2}$ and
determined this quartic explicitly [23, Lemma 3.12]. (In fact, he has shown
this for any bielliptic genus $3$ curve.) We will use his calculation to
determine $C_{\text{twist}}$, together with a descent argument along a
quadratic extension.

Let $\alpha,\alpha^{\prime}$ be the roots of $x^{2}+ax+b$. Let $p=(\alpha,0)$
and $p^{\prime}=(\alpha^{\prime},0)$. Let
$K=k(\alpha,\alpha^{\prime})=k(\sqrt{a^{2}-4b})$. Then $p,p^{\prime}\in E(K)$
and $C_{\text{twist}}=D_{y_{0}}=D_{p}=D_{p^{\prime}}$. Both $p$ and
$p^{\prime}$ give rise to $K$-isomorphisms $\phi_{p}\colon D_{p}\rightarrow
F_{2p}$ and $\phi_{p^{\prime}}\colon D_{p^{\prime}}\rightarrow
F_{2p^{\prime}}$. Let $[u:v:1]$ be affine coordinates for the target of the
morphism $\psi\colon{\rm Sym}^{2}C\rightarrow{\rm\mathbb{P}}^{2}$.

######  Lemma 5.2.

$\psi(F_{2p})=\psi(F_{2p^{\prime}})\subset{\rm\mathbb{P}}^{2}_{K}$ is cut out
by the projective closure of the equation

(5.1) $\displaystyle(2a^{2}-8b)v^{3}=bu^{4}+au^{2}+1.$

###### Proof.

This is a special case [23, Lemma 3.12]444The results of loc. cit. assume that
$k=\mathbb{C}$, but the proof of the lemma works over any field of
characteristic $\neq 2$.. For the reader’s convenience, we provide the
translation to Ikeda’s notation. In his notation: $C$ has equation
$(z^{2}+\frac{a}{2}y^{2})^{2}=x^{3}y+\left(\frac{1}{4}a^{2}-b\right)y^{4}$,
$S(x,y)=\frac{1}{4}a^{2}y^{2}$, $T(x,y)=x^{3}y+(\frac{1}{4}a^{2}-b)y^{4}$, and
$p=[0:\alpha:1]$. ∎

Let $\mathcal{Q}=\psi(F_{2p})\subset{\rm\mathbb{P}}^{2}_{K}$. Lemma 5.2 shows
that $\mathcal{Q}=\psi(F_{2p^{\prime}})$ and that this quartic is defined over
$k$. The substitution $(u,v)\mapsto((2a^{2}-b)^{3}bu,(2a^{2}-b)^{2}bv)$ shows
that $\mathcal{Q}$ is $k$-isomorphic to the curve
$y^{3}=x^{4}+dax^{2}+d^{2}b$, where $d=16b(a^{2}-4b)^{4}$. Therefore
$\mathcal{Q}$ and $C$ are sextic twists in the sense of §2.4 and thus
isomorphic over $k^{\operatorname{sep}}$. However, these curves are not
necessarily isomorphic over $k$, since the isomorphisms
$C_{\text{twist},K}=D_{p}\xrightarrow{\phi_{p}}F_{2p}$ and
$F_{2p}\xrightarrow{\psi}\mathcal{Q}_{K}$ are only defined over the (at most
quadratic) extension $K/k$. We descend down to $k$ by determining the Galois
action on these isomorphisms; see §1.8 for our conventions on how ${\rm
Gal}_{k}$ acts on morphisms.

######  Lemma 5.3.

Suppose that $K\neq k$ and let $\sigma\in{\rm Gal}(K/k)$ be the unique
nontrivial element. Let $\chi=\psi|_{F_{2p}}\circ\phi_{p}\colon
C_{\text{twist},K}\xrightarrow{\sim}\mathcal{Q}_{K}$. Then
$\chi^{\sigma}\circ\chi^{-1}\colon\mathcal{Q}_{K}\rightarrow\mathcal{Q}_{K}$
equals the involution $(u,v)\mapsto(-u,v)$.

###### Proof.

Let $(u,v)\in\mathcal{Q}_{K}$ and let $Q_{1}+Q_{2}\in F_{2p}$ be the unique
element mapping to $(u,v)$ under $\psi|_{F_{2p}}$. Since
$p^{\sigma}=p^{\prime}$, a calculation shows that
$\chi^{\sigma}\circ\chi^{-1}$ maps $(u,v)$ to $\psi(R_{1}+R_{2})$, where
$R_{1}+R_{2}$ is the unique effective divisor on $C$ such that
$R_{1}+R_{2}-\pi^{*}(p^{\prime})\sim Q_{1}+Q_{2}-\pi^{*}(p)$. We interpret
this geometrically: let $\ell=\psi(Q_{1}+Q_{2})$ be the line spanned by the
points $Q_{1},Q_{2}$ in the embedding $C=C_{a,b}\subset{\rm\mathbb{P}}^{2}$.
The condition $Q_{1}+Q_{2}\in F_{2p}$ implies that
$Q_{1}+Q_{2}+\tau(Q_{1})+\tau(Q_{2})\sim 2\pi^{*}(p)$, where we recall that
$\tau\colon C\rightarrow C$ denotes the bielliptic involution. Combining the
above two linear equivalences shows that
$\tau(Q_{1})+\tau(Q_{2})+R_{1}+R_{2}\sim\pi^{*}(p)+\pi^{*}(p^{\prime})\sim
4\infty$. Since $4\infty$ is canonical, it follows that
$\tau(Q_{1}),\tau(Q_{2}),R_{1},R_{2}$ are collinear and
$\psi(R_{1}+R_{2})=\psi(\tau(Q_{1})+\tau(Q_{2}))=\tau(\ell)$. In the
coordinates $(u,v)$, this shows that $\psi(R_{1}+R_{2})=(-u,v)$, proving the
lemma. ∎

######  Theorem 5.4.

Let $C$ be a bielliptic Picard curve of the form $C_{a,b}$ for some $a,b\in
k$. Then $C_{\text{twist}}$ is again a bielliptic Picard curve, isomorphic to
$C_{\delta a,\delta^{2}b}$ where $\delta=\Delta_{a,b}=16b(a^{2}-4b)$. In
particular, $C$ and $C_{\text{twist}}$ are isomorphic over
$k^{\operatorname{sep}}$.

###### Proof.

By Lemma 5.2, $\mathcal{Q}\simeq C_{da,d^{2}b}$, where $d=16b(a^{2}-4b)^{4}$.
Suppose first that $K=k$. Then the isomorphism
$\psi|_{F_{2p}}\circ\phi_{p}\colon C_{\text{twist}}\rightarrow\mathcal{Q}$ are
defined over $k$, hence $C_{\text{twist}}\simeq C_{da,d^{2}b}$ too. Since
$K=k(\sqrt{a^{2}-4b})$, $a^{2}-4b$ is a square in $k^{\times}$. Therefore $d$
and $\delta$ have the same image in $k^{\times}/k^{\times 6}$, so
$C_{\text{twist}}\simeq C_{da,d^{2}b}\simeq C_{\delta a,\delta^{2}b}$ by Lemma
2.8.

Suppose now that $K/k$ is a quadratic extension and let $\sigma\in{\rm
Gal}(K/k)$ be the nontrivial element. Lemma 5.3 shows that $C_{\text{twist}}$
is isomorphic to the quadratic twist of $\mathcal{Q}$ along the extension
$K/k$ and involution $(u,v)\mapsto(-u,v)$. A computation (using §2.4) shows
that this twist is exactly $C_{\delta a,\delta^{2}b}$. ∎

Theorem 5.4 has consequences for the Prym variety of $C$ and its dual. If
$(X,\mu,\beta)$ is a $\mu_{3}$-abelian surface in the sense of Definition 3.7,
the $\mu_{3}$-action $\beta$ and $\mu_{2}$-action $[-1]$ combine to a
$\mu_{6}$-action. Therefore, just as in §2.4, every $\delta\in
k^{\times}/k^{\times 6}$ gives rise to a sextic twist
$(X_{\delta},\mu_{\delta},\beta_{\delta})=(X,\mu,\beta)_{\delta}$ of
$(X,\mu,\beta)$. The formation of the $\mu_{3}$-abelian surfaces
$\operatorname{Prym}(C,\gamma)$ and $\operatorname{Prym}(C,\gamma)^{\vee}$
from §3.3 is compatible with sextic twisting:
$\operatorname{Prym}((C,\gamma)_{\delta})\simeq\operatorname{Prym}(C,\gamma)_{\delta}$
and
$\operatorname{Prym}((C,\gamma)_{\delta})^{\vee}\simeq\operatorname{Prym}(C,\gamma)^{\vee}_{\delta}$.

######  Corollary 5.5.

Let $(C,\gamma)=(C_{a,b},\gamma_{a,b})$ be a marked bielliptic Picard curve
over $k$. Let $\delta=16b(a^{2}-4b)$. Let
$\operatorname{Prym}(C,\gamma)=(P,\lambda,\alpha)$ and
$\operatorname{Prym}(C,\gamma)^{\vee}=(A,\widehat{\lambda},\hat{\alpha})$.
Then
$(P,\lambda,\alpha)\simeq(A,\widehat{\lambda},\hat{\alpha}^{-1})_{\delta}$. In
particular, $P_{k^{\operatorname{sep}}}$ and $A_{k^{\operatorname{sep}}}$ are
isomorphic as $(1,2)$-polarized abelian surfaces and
$P_{k^{\operatorname{sep}}}$ is self-dual.

###### Proof.

The $\mu_{3}$-action $\alpha$ and the involution $[-1]$ on $P$ combine to a
$\mu_{6}$-action that restricts to $\mu_{6}$-actions $\hat{\gamma}$ and
$\gamma_{\text{twist}}$ on $\widehat{C}$ and $C_{\text{twist}}$ respectively.
Lemma 2.18 shows that $(\widehat{C},\hat{\gamma})$ is a marked bielliptic
Picard curve; so is $(C_{\text{twist}},\gamma_{\text{twist}}^{-1})$ by
Proposition 3.11. Therefore
$(P,\lambda,\alpha)\simeq\operatorname{Prym}(\widehat{C},\hat{\gamma})^{\vee}$
and
$(P,\lambda,\alpha^{-1})\simeq\operatorname{Prym}(C_{\text{twist}},\gamma_{\text{twist}}^{-1})^{\vee}$
by Proposition 3.4.

It remains to show that
$(P,\lambda,\alpha)\simeq(A,\widehat{\lambda},\hat{\alpha}^{-1})_{\delta}$, or
equivalently that
$\operatorname{Prym}(C_{\text{twist}},\gamma_{\text{twist}}^{-1})^{\vee}\simeq(A,\widehat{\lambda},\hat{\alpha})_{\delta}$.
Recall that $(C_{\delta},\gamma_{\delta})$ denotes the sextic twist of $C$
along $\delta$. Theorem 5.4 shows that
$(C_{\text{twist}},\gamma_{\text{twist}}^{-1})\simeq(C_{\delta},\gamma_{\delta})$.
Therefore

$\operatorname{Prym}(C_{\text{twist}},\gamma_{\text{twist}}^{-1})^{\vee}\simeq\operatorname{Prym}(C_{\delta},\gamma_{\delta})^{\vee}\simeq(A,\hat{\lambda},\hat{\alpha})_{\delta},$

as desired. ∎

######  Remark 5.6.

In contrast to the results of §2.6, 2.7, 3.1, Corollary 5.5 crucially uses
properties specific to bielliptic Picard curves. Indeed, the last sentence of
the corollary might fail for a general bielliptic genus $3$ curve.

######  Remark 5.7.

The isomorphism $A\simeq P_{\delta^{-1}}$ of Corollary 5.5 can be chosen
‘universally’. More precisely, let
$R:={\rm\mathbb{Z}}[1/6,a,b,\Delta_{a,b}^{-1}]$ and $S:={\rm Spec}(R)$.
Equation (2.1) defines the universal marked bielliptic Picard curve (in the
sense of Remark 3.16) with Prym variety $\mathcal{P}\rightarrow S$ and dual
$\mathcal{A}\rightarrow S$. Then there exists an isomorphism
$\Phi\colon\mathcal{A}\rightarrow\mathcal{P}_{\delta^{-1}}$ that restricts to
an isomorphism of Corollary 5.5 for every geometric point of $S$. (This can be
seen, for example, by spreading out an isomorphism over the generic point to
all of $S$ using [15, §2, Lemma 1].)

As a first application of Corollary 5.5, we show that the Atkin–Lehner
involution on $X/w_{3}$ corresponds to inversion on the $j$-line
$Y\simeq{\rm\mathbb{P}}^{1}_{j}$.

######  Lemma 5.8.

Under the isomorphisms $(X/w_{3})^{\bar{w}}\simeq
Y\simeq{\rm\mathbb{P}}^{1}_{j}$ of Proposition 4.7 and Lemma 4.3, the
involution $\bar{w}$ on $(X/w_{3})^{\bar{w}}$ corresponds to the involution
$j\mapsto 1/j$ on ${\rm\mathbb{P}}^{1}_{j}$.

###### Proof.

We use the notations of §4.2, 4.3. It suffices to check the claim over
${\rm\mathbb{C}}$ on the locus of indecomposable $\mu_{3}$-abelian surfaces
$Y^{\text{indec}}\subset Y$, so let $y=[A,\lambda,\alpha]\in
Y^{\text{indec}}({\rm\mathbb{C}})$ be such a point. Let
$x=[A,\lambda,\iota]\in X^{w_{2}}({\rm\mathbb{C}})=X({\rm\mathbb{C}})$ be a
lift of $y$ under $\pi_{2}$, so $\alpha=\iota|_{\langle\omega\rangle}$. Then
$\bar{w}(y)=\pi_{2}(w_{2}(x))$. Since $j$ has norm $2$,
$w_{2}(x)=[A,\lambda,\iota\circ[j]]$. Moreover $[j]$ maps $\omega$ to
$\omega^{-1}$, so $\iota\circ[j]$ restricts to $\alpha^{-1}$. It follows that
$\pi_{2}(w_{2}(x))=[A,\lambda,\alpha^{-1}]$. On the other hand, if
$(A,\lambda,\alpha)\simeq\operatorname{Prym}(C,\gamma)^{\vee}$, then Corollary
5.5 shows that $(A,\lambda,\alpha^{-1})\simeq\operatorname{Prym}(C,\gamma)$,
which is itself isomorphic to
$\operatorname{Prym}(\widehat{C},\hat{\gamma})^{\vee}$, where
$(\widehat{C},\hat{\gamma})$ is the bigonal dual curve from §2.7. Since the
$j$-invariant of $\widehat{C}$ equals the inverse of the $j$-invariant of $C$
(Lemma 2.20), the lemma follows from the explicit description of the
isomorphism $Y\simeq{\rm\mathbb{P}}^{1}_{j}$. ∎

##  6\. Explicit quaternionic multiplication

In Corollary 4.8 we proved that there exists an action of $\mathcal{O}$ on
$P_{k^{\operatorname{sep}}}$. Next we use Theorem 5.4 to describe this action
explicitly. For the remainder of this paper, we fix a universal isomorphism
$\Phi\colon\mathcal{A}\rightarrow\mathcal{P}_{\delta^{-1}}$ as in Remark 5.7.

###  6.1. Explicit quaternionic multiplication

Let $(C,\gamma)=(C_{a,b},\gamma_{a,b})$ be a marked bielliptic Picard curve
over $k$. Let $\delta=\Delta_{a,b}=16b(a^{2}-4b)$, which is nonzero since $C$
is smooth. Consider the ${\rm Gal}_{k}$-set:

(6.1) $\displaystyle\mathcal{S}:=\\{(\zeta,\varepsilon)\in
k^{\operatorname{sep}}\times k^{\operatorname{sep}}\mid\zeta\text{ is a
primitive sixth root of unity and }\varepsilon^{6}=\delta\\}.$

Consider $\mu_{3}$-abelian surfaces
$\operatorname{Prym}(C,\gamma)=(P,\lambda,\alpha)$ and
$\operatorname{Prym}(C,\gamma)^{\vee}=(A,\widehat{\lambda},\hat{\alpha})$. For
every $(\zeta,\varepsilon)\in\mathcal{S}$, we will define elements
$r_{\zeta},s_{\varepsilon}\in{\rm End}(P_{k^{\operatorname{sep}}})$. Since
$\zeta^{2}$ is a third root of unity, we may set
$r_{\zeta}:=\alpha(\zeta^{2})\in{\rm End}(P_{k^{\operatorname{sep}}})$.

To define $s_{\varepsilon}$, we use Corollary 5.5. By that corollary and
Remark 5.7, the specialization of the universal isomorphism $\Phi$ at the
$k$-point $(a,b)\in S(k)$ is an isomorphism
$\phi\colon(A,\widehat{\lambda},\hat{\alpha}^{-1})\xrightarrow{\sim}(P,\lambda,\alpha)_{\delta^{-1}}$.
The element $\varepsilon$ determines an isomorphism
$C_{\delta^{-1},k^{\operatorname{sep}}}\rightarrow
C_{k^{\operatorname{sep}}},(x,y)\mapsto(\varepsilon^{3}x,\varepsilon^{4}y)$,
hence an isomorphism
$\chi_{\varepsilon}\colon(P,\lambda,\alpha)_{\delta^{-1},{k^{\operatorname{sep}}}}\rightarrow(P,\lambda,\alpha)_{k^{\operatorname{sep}}}$
between their Prym varieties. Let $s_{\varepsilon}$ be the composition:

(6.2) $\displaystyle
P_{k^{\operatorname{sep}}}\xrightarrow{\lambda}A_{k^{\operatorname{sep}}}\xrightarrow{\phi}(P_{\delta^{-1}})_{k^{\operatorname{sep}}}\xrightarrow{\chi_{\varepsilon}}P_{k^{\operatorname{sep}}}.$

######  Proposition 6.1.

For every $x=(\zeta,\varepsilon)\in\mathcal{S}$ the assignment $\omega\mapsto
r_{\zeta},j\mapsto s_{\varepsilon}$ extends to an embedding

(6.3) $\displaystyle\iota_{x}\colon\mathcal{O}\rightarrow{\rm
End}(P_{k^{\operatorname{sep}}})={\rm End}(P_{\bar{k}}).$

###### Proof.

We may assume $k=k^{\operatorname{sep}}$. Fix
$x=(\zeta,\varepsilon)\in\mathcal{S}$ and write $r=r_{\zeta}$,
$s=s_{\varepsilon}$. We will show that $r^{2}+r+1=0$, $s^{2}=2$ and
$sr=r^{-1}s$. Since the associative ring $\mathcal{O}$ from §4.1 is generated
by $\omega,j$ and satisfies the same three relations, this will imply that
there exists a unique ring homomorphism
$\iota_{x}\colon\mathcal{O}\rightarrow{\rm End}(P_{k^{\operatorname{sep}}})$
with $\iota_{x}(\omega)=r$ and $\iota_{x}(j)=s$. Such a homomorphism is
automatically an embedding since $\mathcal{O}\otimes{\rm\mathbb{Q}}$ is a
simple algebra.

Equation (2.4) of Lemma 2.14 shows that $r^{2}+r+1=0$. To check that
$sr=r^{-1}s$, note that the surfaces $P,P_{\delta^{-1}},A$ have
$\mu_{3}$-actions $\alpha,\alpha_{\delta^{-1}},\hat{\alpha}$ respectively. The
morphisms $\lambda$ and $\chi_{\varepsilon}$ from (6.2) are equivariant with
respect to these $\mu_{3}$-actions. On the other hand, $\phi$ intertwines the
$\mu_{3}$-action on the domain with the _inverse_ of the $\mu_{3}$-action on
the target. The same must therefore be true for their composition $s$, so
$sr=r^{-1}s$.

It suffices to prove that $s^{2}=2$; we will use the results of §4.3 to
achieve this. Write $\psi=\chi_{\varepsilon}\circ\phi$. The morphism $\psi$ is
compatible with the polarizations since $\phi$ and $\chi_{\varepsilon}$ are,
so $\psi^{\vee}\circ\lambda\circ\psi=\widehat{\lambda}$. Therefore

(6.4) $\displaystyle
s^{2}=\psi\lambda\psi\lambda=(\psi(\psi^{\vee})^{-1})(\psi^{\vee}\lambda\psi\lambda)=(\psi(\psi^{\vee})^{-1})(\widehat{\lambda}\lambda)=2(\psi(\psi^{\vee})^{-1}).$

It follows that $s^{2}=2$ if and only if $\psi^{\vee}=\psi$. To prove the
latter statement, we will first show that $(\psi^{\vee})^{-1}\circ\psi\in{\rm
Aut}(A,\widehat{\lambda},\hat{\alpha})$. The identity
$\psi^{\vee}\circ\lambda=\widehat{\lambda}\circ\psi$ implies that
$\lambda\circ(\psi\circ\lambda\circ\psi^{\vee}-\widehat{\lambda})\circ\lambda=0$.
Since $\lambda$ is an isogeny, the inner expression must be zero, so
$\psi\circ\lambda\circ\psi^{\vee}=\widehat{\lambda}$. In other words
$\psi^{\vee}$ is compatible with the polarizations. Moreover, $\psi$
intertwines the $\mu_{3}$-action $\hat{\alpha}$ on $A$ with the _inverse_ of
the $\mu_{3}$-action $\alpha$ on $P$:
$\psi(\hat{\alpha}(\omega)(x))=\alpha(\omega)^{-1}(\psi(x))$ for all $x\in A$
and $\omega\in\mu_{3}$. Applying duality to this identity and using that
$\alpha^{\vee}(\omega)=\hat{\alpha}(\omega^{-1})$, $\psi^{\vee}$ again
intertwines the $\mu_{3}$-action with the inverse of the other. We conclude
that $(\psi^{\vee})^{-1}\circ\psi\in{\rm Aut}(A,\lambda,\alpha)$.

To show that this automorphism is the identity, we will use that its
construction can be carried out in families. More precisely, let
$R={\rm\mathbb{Z}}[1/6,a,b,\Delta_{a,b}^{-1}]$ and
$\tilde{R}=R[\varepsilon]/(\varepsilon^{6}-\Delta_{a,b})$. Equation (2.1)
defines a universal marked bielliptic Picard curve $(\mathcal{C},\gamma)$ over
$S={\rm Spec}(R)\subset{\rm\mathbb{A}}^{2}_{{\rm\mathbb{Z}}[1/6]}$ with Prym
variety $(\mathcal{P},\lambda,\alpha)$ and dual
$(\mathcal{A},\widehat{\lambda},\hat{\alpha})$. On the cover $\tilde{S}={\rm
Spec}(\tilde{R})\rightarrow S$ given by adjoining a sixth root of
$\Delta_{a,b}$, there exists an isomorphism
$\Psi\colon\mathcal{A}_{\tilde{S}}\rightarrow\mathcal{P}_{\tilde{S}}$ with the
property that $\Psi$ specializes to $\psi$ for all morphisms ${\rm
Spec}(k)\rightarrow\tilde{S}$ and that $(\Psi^{\vee})^{-1}\circ\Psi\in{\rm
Aut}_{\tilde{S}}(\mathcal{A},\lambda,\alpha)$. Since the moduli stack of
$(1,2)$-polarized abelian surfaces is separated and Deligne–Mumford,
$\mathbf{Aut}_{\tilde{S}}(\mathcal{A},\lambda,\alpha)\rightarrow\tilde{S}$ is
finite and unramified. Since finite unramified morphisms are étale locally on
the target disjoint unions of closed immersions [44, Tag 04HJ] and the base
$\tilde{S}$ is connected, it suffices to show that $(\Psi^{\vee})^{-1}\Psi$ is
the identity for a _single_ specialization $s\in\tilde{S}$.

The specialization $(a_{0},b_{0},\varepsilon_{0})=(1,1/8,1)$ defines a
${\rm\mathbb{Q}}$-point $s_{0}$ of $\tilde{S}$. Theorem 3.13 and Lemma 2.10
shows that $\mathbf{Aut}((\mathcal{A},\lambda,\alpha)_{s_{0}})=\mu_{6}$. Since
$\mu_{6}({\rm\mathbb{Q}})=\\{\pm 1\\}$, either $(\Psi^{\vee})^{-1}\Psi=1$ or
$(\Psi^{\vee})^{-1}\Psi=-1$. Assume for the sake of contradiction that
$(\Psi^{\vee})^{-1}\Psi=-1$. Then (6.4) shows that $s^{2}=-2$. This implies
that $r$ and $s$ determine an embedding of the _definite_ quaternion algebra
$(-3,-2)_{{\rm\mathbb{Q}}}$ into ${\rm
End}(\mathcal{P}_{s})\otimes{\rm\mathbb{Q}}$ for every specialization
$s\in\tilde{S}({\rm\mathbb{C}})$. By the classification of endomorphism
algebras of complex abelian surfaces [5, Proposition 5.5.7, Exercise 9.10(1)
and Exercise 9.10(4)], this implies that $\mathcal{P}_{s}$ is isogenous to the
square of a CM elliptic curve. This is a contradiction, since there are
(uncountably) many specializations $s$ for which $\mathcal{P}_{s}$ is simple,
using Proposition 4.7 and CM theory for the quaternionic Shimura curve $X$. We
conclude that $\psi^{\vee}=\psi$ and hence by (6.4) that $s^{2}=2$, as
desired. ∎

If $P_{\bar{k}}$ is simple, then $\mathcal{O}$ is the full ring of
endomorphisms of $P_{\bar{k}}$:

######  Lemma 6.2.

Let $C/k$ be a bielliptic Picard curve with Prym variety $P$ and suppose that
$k$ is algebraically closed. Then $P$ is not simple $\Leftrightarrow$
$\mathcal{O}\not\simeq{\rm End}(P)$ $\Leftrightarrow$ $P$ is isogenous to the
square of a CM elliptic curve. Consequently, if $P$ is simple then ${\rm
End}(P)=\iota_{x}(\mathcal{O})$ for every $x\in\mathcal{S}$.

###### Proof.

This is a direct consequence of the Albert classification [5, Proposition
5.5.7, Exercise 9.10(1) and Exercise 9.10(4)]. ∎

######  Remark 6.3.

If $\mathrm{char}(k)=0$ or if $k=\overline{{\rm\mathbb{F}}}_{p}$ with $p>3$,
then these conditions are also equivalent to the condition that $P$ is
isomorphic to a product of elliptic curves. See [40] for the former case and
[38, Theorem 3.1] for the latter.

We will study geometrically split Prym varieties in more detail in §6.4.

###  6.2. Galois action on the endomorphisms

Keep the notations of the previous section. We proceed to determine how the
various embeddings $\iota_{x}\colon\mathcal{O}\rightarrow{\rm
End}(P_{k^{\operatorname{sep}}})$ are related and deduce an explicit
description of the Galois action on $\mathcal{O}$.

Let ${\rm Sym}(\mathcal{S})$ be the group of bijections
$\mathcal{S}\rightarrow\mathcal{S}$ acting on the right on the ${\rm
Gal}_{k}$-set $\mathcal{S}$ of (6.1). Define the subgroup $D_{6}\subset{\rm
Sym}(\mathcal{S})$ generated by $\rho,\sigma$, where
$(\zeta,\varepsilon)^{\rho}=(\zeta,\zeta^{-1}\varepsilon)$ and
$(\zeta,\varepsilon)^{\tau}\mapsto(\zeta^{-1},\varepsilon)$. Then $D_{6}$ is a
dihedral group of order $12$ and $D_{6}$ acts simply transitively on
$\mathcal{S}$.

Recall the subgroup ${\rm Aut}_{i}(\mathcal{O})$ from Lemma 4.1. The
assignment $\rho\mapsto[1-\omega]$, $\tau\mapsto[j]$ induces an isomorphism
$\varphi\colon D_{6}\xrightarrow{\sim}{\rm Aut}_{i}(\mathcal{O})$.

######  Lemma 6.4.

For all $x\in\mathcal{S}$, $g\in D_{6}$ and $b\in\mathcal{O}$,
$\iota_{x^{g}}(b^{\varphi(g)})=\iota_{x}(b)$

###### Proof.

It suffices to prove the claimed identity when $g\in\\{\tau,\rho\\}$ and
$b\in\\{\omega,j\\}$, for which it can be checked using routine but intricate
calculations. Write $x=(\zeta,\varepsilon)$. Suppose first that $g=\tau$, then
$\iota_{x^{g}}(b^{\varphi(g)})=\iota_{(\zeta^{-1},\varepsilon)}(j^{-1}bj)$. If
$b=\omega$, this equals $\iota_{(\zeta^{-1},\varepsilon)}(j^{-1}\omega
j)=\iota_{(\zeta^{-1},\varepsilon)}(\omega^{-1})=r_{\zeta^{-1}}^{-1}=r_{\zeta}=\iota_{x}(\omega)$.
If $b=j$, this equals
$\iota_{(\zeta^{-1},\varepsilon)}(j)=s_{\varepsilon}=\iota_{x}(j)$.

Suppose next that $g=\rho$. If $b=\omega$, both sides are simply $r_{\zeta}$.
If $b=j$, then

$\displaystyle\iota_{x^{g}}(b^{\varphi(g)})=\iota_{x^{g}}(j^{[1-\omega]})=\iota_{x^{g}}(-\omega^{-1}j)=-\iota_{(\zeta,\zeta^{-1}\varepsilon)}(\omega^{-1})\iota_{(\zeta,\zeta^{-1}\varepsilon)}(j)=-r_{\zeta}^{-1}s_{\zeta^{-1}\varepsilon}.$

The definition of $s_{\varepsilon}$ in (6.2) shows that
$s_{\zeta^{-1}\varepsilon}$ is given by postcomposing $s_{\varepsilon}$ with
the automorphism of $P$ induced by $\gamma(\zeta^{-1})$. A simple calculation
shows that this is exactly $-r_{\zeta}$, so
$s_{\zeta^{-1}\varepsilon}=-r_{\zeta}s_{\varepsilon}$. Therefore
$\iota_{x^{g}}(j^{g})=-r_{\zeta}^{-1}(-r_{\zeta}s_{\varepsilon})=s_{\varepsilon}=\iota_{x}(b)$.
∎

The Galois action on $\mathcal{S}$ determines a homomorphism ${\rm
Gal}_{k}\rightarrow{\rm Sym}(\mathcal{S})$. This homomorphism factors through
an injection ${\rm Gal}(L/k)\hookrightarrow D_{6}$, where
$L=k(\omega,\sqrt[6]{\delta})$ is the splitting field of $f(T)=T^{6}-\delta$.
Denote the composition ${\rm Gal}_{k}\rightarrow
D_{6}\xrightarrow{\varphi}{\rm Aut}_{i}(\mathcal{O})$ again by $\varphi$.
Recall from §1.8 that if $f\colon X_{k^{\operatorname{sep}}}\rightarrow
Y_{k^{\operatorname{sep}}}$ is a $k^{\operatorname{sep}}$-morphism between
$k$-varieties and $\sigma\in{\rm Gal}_{k}$, then $f^{\sigma}$ denotes the
$k^{\operatorname{sep}}$-morphism $x\mapsto f(x^{\sigma^{-1}})^{\sigma}$.

######  Theorem 6.5.

For all $x\in\mathcal{S}$, $\sigma\in{\rm Gal}_{k}$ and $b\in\mathcal{O}$,
$\iota_{x}(b)^{\sigma}=\iota_{x}(b^{\varphi(\sigma^{-1})})$. Consequently,
there exists an embedding $\mathcal{O}\subset{\rm
End}(P_{k^{\operatorname{sep}}})$ such that
$f^{\sigma}=f^{\varphi(\sigma^{-1})}$ for all $f\in\mathcal{O}$ and
$\sigma\in{\rm Gal}_{k}$. In particular, $\mathcal{O}$ is ${\rm
Gal}_{k}$-stable and defined over $L=k(\omega,\sqrt[6]{\delta})$.

###### Proof.

Write $x=(\zeta,\varepsilon)$. The explicit construction of $r_{\zeta}$ and
$s_{\varepsilon}$ shows that $r_{\zeta}^{\sigma}=r_{\zeta^{\sigma}}$ and
$s_{\varepsilon}^{\sigma}=s_{\varepsilon^{\sigma}}$. Therefore
$\iota_{x}(b)^{\sigma}=\iota_{x^{\sigma}}(b)$. By the previous lemma,
$\iota_{x}(b)^{\sigma}=\iota_{x^{\sigma}}(b)=\iota_{x}(b^{\varphi(\sigma^{-1})})$.
∎

The endomorphism field of an abelian variety is the smallest field extension
over which all endomorphisms are defined [20]. If $P$ is geometrically simple,
then ${\rm End}(P_{\bar{k}})=\mathcal{O}$ by Lemma 6.2. Since the action of
$D_{6}$ on $\mathcal{O}$ is faithful, Theorem 6.5 has the following immediate
corollary.

######  Corollary 6.6.

Let $C=C_{a,b}$ be a bielliptic Picard curve over $k$. Suppose that
$P_{\bar{k}}$ is simple. Then the endomorphism field of $P/k$ is
$k(\omega,\sqrt[6]{16b(a^{2}-4b)})$.

The endomorphism field is invariant under isogeny, so the corollary holds
verbatim for the dual Prym $A/k$.

######  Example 6.7.

Corollary 6.6 gives many examples of abelian surfaces with large endomorphism
fields. For example, for the curve $C=C_{3,4}:y^{3}=x^{4}+3x^{2}+4$ over
${\rm\mathbb{Q}}$, the Prym $P=P_{3,4}$ is geometrically simple (see
Proposition 6.18) and its endomorphism field is
${\rm\mathbb{Q}}(\sqrt{-3},\sqrt[6]{-7})$, a dihedral degree $12$ extension of
${\rm\mathbb{Q}}$. In the notation of [17], this seems to be the first
published example of a geometrically simple abelian surface over
${\rm\mathbb{Q}}$ with Sato–Tate group $J(E_{6})$. Similarly, $P_{-4,2}$ is a
geometrically simple abelian surface with endomorphism field
${\rm\mathbb{Q}}(\sqrt{-3},\sqrt[3]{2})$ and Sato–Tate group $J(E_{3})$.

We also record a simple description of the Artin representations ${\rm
End}^{0}(P_{k^{\mathrm{sep}}})$ and ${\rm
NS}(P_{k^{\mathrm{sep}}})\otimes{\rm\mathbb{Q}}$. Recall that ${\rm
NS}(P_{k^{\mathrm{sep}}})$ is the group of line bundles on
$P_{k^{\mathrm{sep}}}$ modulo algebraic equivalence. Write $G={\rm Gal}(L/k)$
and view it as a subgroup of $D_{6}$. Let $\chi\colon D_{6}\rightarrow\\{\pm
1\\}$ be the unique homomorphism with kernel $C_{6}=\langle\rho\rangle$ and
let $\mathrm{std}\colon D_{6}\rightarrow{\rm GL}_{2}({\rm\mathbb{Q}})$ be a
model for the unique $2$-dimensional irreducible faithful representation of
$D_{6}$.

######  Corollary 6.8.

Suppose that $P_{\bar{k}}$ is simple. Then there are isomorphisms of
$G$-representations ${\rm
End}^{0}(P_{k^{\mathrm{sep}}})\simeq(1\oplus\chi\oplus\mathrm{std})|_{G}$ and
${\rm
NS}(P_{k^{\mathrm{sep}}})\otimes{\rm\mathbb{Q}}\simeq(1\oplus\mathrm{std})|_{G}$.

###### Proof.

The first part immediately follows from Theorem 6.5: $1\in B$ generates the
trivial representation, $i\in B$ generates $\chi$ and $\\{j,ij\\}$ generate
$\mathrm{std}$. To prove the second isomorphism, recall that the ${\rm
Gal}_{k}$-equivariant map $L\mapsto\lambda^{-1}\phi_{L}$ gives an isomorphism
between ${\rm NS}(P_{k^{\mathrm{sep}}})\otimes{\rm\mathbb{Q}}$ and the
subgroup of ${\rm End}^{0}(P_{k^{\mathrm{sep}}})$ fixed by the Rosati
involution. The Rosati involution on ${\rm
End}^{0}(P_{k^{\mathrm{sep}}})\simeq B$ is given by $b\mapsto i\bar{b}i^{-1}$,
the unique positive involution on $B$ that induces complex conjugation on
${\rm\mathbb{Q}}(i)={\rm\mathbb{Q}}(\omega)$. It follows that ${\rm
NS}(P_{k^{\mathrm{sep}}})\otimes{\rm\mathbb{Q}}\simeq\\{f\in B\mid
i\bar{f}i^{-1}=f\\}=\text{span}\\{1\\}\oplus\text{span}\\{j,ij\\}\simeq(1\oplus\mathrm{std})|_{G}$.
∎

###  6.3. Prym surfaces of ${\rm GL}_{2}$-type

Let $C=C_{a,b}$ be a bielliptic Picard curve over $k$ with Prym variety $P$,
and let $\iota\colon\mathcal{O}\hookrightarrow{\rm
End}(P_{k^{\operatorname{sep}}})$ be an embedding satisfying the conclusion of
Theorem 6.5. Recall that $k(\omega)$ denotes the smallest field extension of
$k$ containing a primitive third root of unity.

######  Corollary 6.9.

The following conditions are equivalent:

  1. (1)

$16b(a^{2}-4b)$ is a sixth power in $k(\omega)$;

  2. (2)

$\iota(\mathcal{O})\subset{\rm End}(P_{k(\omega)})$;

  3. (3)

$\iota(\mathcal{O})\cap{\rm End}(P)$ is either an order in a real quadratic
field or all of $\iota(\mathcal{O})$.

###### Proof.

$(1)$ and $(2)$ are equivalent by Theorem 6.5. To prove the equivalence
between $(2)$ and $(3)$, let $G$ be the image of the homomorphism
$\varphi\colon{\rm Gal}_{k}\rightarrow{\rm Aut}_{i}(\mathcal{O})$ of Theorem
6.5, and let $H=\text{Im}({\rm Gal}_{k(\omega)}\rightarrow{\rm
Aut}_{i}^{+}(\mathcal{O}))$. Then $H=G\cap{\rm Aut}_{i}^{+}(\mathcal{O})$,
where ${\rm Aut}_{i}^{+}(\mathcal{O})=\langle[1-\omega]\rangle$. A group
theory calculation using Lemma 4.2 shows that $H=\\{1\\}$ (in other words,
$(2)$ holds) if and only if $\mathcal{O}^{G}$ contains an order in a real
quadratic field (in other words $(3)$ holds). ∎

######  Remark 6.10.

Recall that an abelian surface $X/k$ is of ${\rm GL}_{2}$-type if ${\rm
End}(X)$ contains a quadratic ring. If $P$ is geometrically simple and $k\neq
k(\omega)$ then $\iota(\mathcal{O})\cap{\rm End}(P)={\rm End}(P)$ (Lemma 6.2)
and ${\rm End}(P)$ is either ${\rm\mathbb{Z}}$ or an order in a real quadratic
field by Lemma 4.2, so the conditions above are equivalent to $P$ being of
${\rm GL}_{2}$-type. If $k=k(\omega)$ then
${\rm\mathbb{Z}}[\omega]\hookrightarrow{\rm End}(P)$, so $P$ is always of
${\rm GL}_{2}$-type.

We can be more precise about the ring of endomorphisms ${\rm End}(P)$. A
calculation shows that an element $\delta\in k^{\times}$ is a sixth power in
$k(\omega)$ if and only if either $\delta$ or $-27\delta$ is a sixth power in
$k$. This dichotomy breaks up Corollary 6.9 into the following two cases.

######  Corollary 6.11.

Assume that $j(C)\neq 1$. Then the following conditions are equivalent.

  1. (1)

$\Delta_{a,b}=16b(a^{2}-4b)$ is a sixth power in $k$.

  2. (2)

${\rm End}(P)\cap\iota(\mathcal{O})$ contains a subring isomorphic to
${\rm\mathbb{Z}}[\sqrt{2}]$.

  3. (3)

There exist $t,d\in k^{\times}$ such that
$(a,b)=(2(t^{2}+1)^{2}td^{3},(t^{2}+1)^{3}t^{2}d^{6})$.

If these conditions are satisfied then $P$ and $A=P^{\vee}$ are
$k$-isomorphic. A sextic twist of $P$ satisfies the above conditions if and
only if $j(C)=-t^{2}$ for some $t\in k^{\times}$.

###### Proof.

$(1)$ implies $P\simeq A$ by Corollary 5.5. To show $(1)\Leftrightarrow(2)$,
let $H\leq G$ be the subgroups of ${\rm Aut}_{i}(\mathcal{O})$ defined in the
proof of Corollary 6.9. Then $\mathcal{O}^{G}$ contains a subring isomorphic
to ${\rm\mathbb{Z}}[\sqrt{2}]$ if and only if $G$ is conjugate to a subgroup
of $\langle[j]\rangle$ by Lemma 4.2. Theorem 6.5 and Galois theory show that
this is equivalent to $T^{6}-\delta$ having a $k$-rational root. This proves
that $(1)\Leftrightarrow(2)$. The equivalence of $(1)$ and $(3)$ as well as
the last claim of the corollary are routine algebra. ∎

######  Corollary 6.12.

Assume that $j(C)\neq 1$. Then the following conditions are equivalent.

  1. (1)

$-27\Delta_{a,b}=-432b(a^{2}-4b)$ is a sixth power in $k$.

  2. (2)

${\rm End}(P)\cap\iota(\mathcal{O})$ contains a subring isomorphic to
${\rm\mathbb{Z}}[\sqrt{6}]$.

  3. (3)

There exist $t,d\in k^{\times}$ such that
$(a,b)=(18d^{3}t(1-3t^{2})^{2},3^{4}d^{6}t^{2}(1-3t^{2})^{3})$.

If these conditions hold then $P\simeq A_{-27}$, i.e. $P$ is isomorphic to the
quadratic twist of $A$ along $k(\omega)$. A sextic twist of $P$ satisfies the
above conditions if and only if $j(C)=3t^{2}$ for some $t\in k^{\times}$.

###### Proof.

The proof is similar to that of Corollary 6.11 and is omitted. ∎

######  Remark 6.13.

Corollaries 6.11 and 6.12 show that a $j$-invariant $j\in
Y({\rm\mathbb{Q}})={\rm\mathbb{P}}^{1}({\rm\mathbb{Q}})$ lifts to a
${\rm\mathbb{Q}}$-rational point under the morphism $\pi_{2}$ (respectively
$\pi_{6}$) from §4.3 if and only if $j\in-{\rm\mathbb{Q}}^{\times 2}$
(respectively $j\in 3{\rm\mathbb{Q}}^{\times 2}$). We also note that the
equivalence of $(1)$ and $(2)$ holds even when $j(C)=1$.

######  Corollary 6.14.

Suppose $k\neq k(\omega)$ and $P$ is geometrically simple. Then

  1. (1)

${\rm End}(P)$ is isomorphic to either ${\rm\mathbb{Z}}$,
${\rm\mathbb{Z}}[\sqrt{2}]$ or ${\rm\mathbb{Z}}[\sqrt{6}]$.

  2. (2)

$P$ carries a principal polarization if and only if ${\rm
End}(P)\simeq{\rm\mathbb{Z}}[\sqrt{2}]$, and in this case the unique principal
polarization is $\frac{1}{2}\lambda(2-\sqrt{2})$.

###### Proof.

$(1)$ follows from Lemma 4.2. For $(2)$, if ${\rm End}(P)={\rm\mathbb{Z}}$,
then ${\rm Hom}(P,A)={\rm\mathbb{Z}}\lambda$ so cannot contain a principal
polarization. Otherwise, we use the fact that any abelian surface with
${\rm\mathbb{Z}}[\sqrt{2}]$-RM is principally polarized, whereas any
$(1,2)$-polarized abelian surface $P$ with ${\rm
End}(P)\simeq{\rm\mathbb{Z}}[\sqrt{6}]$ is not, by [19, Proposition 3.11]. If
${\rm End}(P)={\rm\mathbb{Z}}[\sqrt{2}]$, the unique principal polarization
$\lambda^{\prime}\colon P\to A$ is $\frac{1}{2}\lambda(2-\sqrt{2})$; see [19,
Proposition 2.1]. Equivalently, $\lambda^{\prime}=\lambda-\psi^{-1}$, in the
notation of the proof of Proposition 6.1. ∎

###  6.4. Geometrically split Prym varieties

In this subsection, we assume that $k$ has characteristic zero. Let $C/k$ be a
bielliptic Picard curve with Prym variety $P$. If any of the equivalent
conditions of Lemma 6.2 is satisfied for $P_{\bar{k}}$ we call $C$ (or $P$)
CM. Being CM only depends on the $j$-invariant (2.3) of $C$. Choose an
embedding $\iota\colon\mathcal{O}\hookrightarrow{\rm End}(P_{\bar{k}})$ of the
form $\iota_{x}$ for some $x\in\mathcal{S}$ as in §6.1. Let

(6.5) $\displaystyle{\rm End}_{\mathcal{O}}(P_{\bar{k}}):=\\{f\in{\rm
End}(P_{\bar{k}})\mid f\circ\iota(b)=\iota(b)\circ f\text{ for all
}b\in\mathcal{O}\\}.$

This subring does not depend on the choice of $x\in\mathcal{S}$. If $P$ is not
CM, then ${\rm End}_{\mathcal{O}}(P_{\bar{k}})={\rm\mathbb{Z}}$; if $P$ is CM,
then ${\rm End}_{\mathcal{O}}(P_{\bar{k}})$ is an order $R$ in an imaginary
quadratic field, and we say that $P$ has CM by the order $R$.

######  Example 6.15.

The Prym variety $P$ of the special bielliptic Picard curve $C:y^{3}=x^{4}+1$
has CM by the order ${\rm\mathbb{Z}}[i]$, hence $j(C)=1$ is a CM
$j$-invariant. (In this example alone, the symbol $i$ means $\sqrt{-1}$.)
Indeed, over $\bar{{\rm\mathbb{Q}}}$, the curve $C$ has an automorphism
$\beta(x,y)=(ix,y)$ commuting with the $\mu_{6}$-action on $C$. A signature
calculation similar to Lemma 2.13 shows that $\beta$ induces an order $4$
automorphism $\beta_{*}$ on $P$ and has characteristic polynomial $X^{2}+1$.
In the notation of §6.1, choose a pair $(\zeta,\varepsilon)\in\mathcal{S}$. By
construction, the elements $r_{\zeta}$ and $s_{\varepsilon}$ commute with
$\beta_{*}$, so ${\rm\mathbb{Z}}[\beta_{*}]={\rm\mathbb{Z}}[i]\subset{\rm
End}_{\mathcal{O}}(P_{\bar{k}})$.

We now state some basic properties of the set of CM $j$-invariants, completely
analogous to the elliptic curve setting, using the results of §4.3. In the
notation of that subsection, a $j$-invariant $j\in Y({\rm\mathbb{C}})$ is CM
if and only if it lifts to a CM point of $X$ under the map $\pi_{2}$.

######  Lemma 6.16.

Every CM $j$-invariant is algebraic over ${\rm\mathbb{Q}}$. Moreover, for
every $d\in{\rm\mathbb{Z}}_{\geq 1}$, there are only finitely many CM
$j$-invariants $j$ with $[{\rm\mathbb{Q}}(j):{\rm\mathbb{Q}}]\leq d$.

###### Proof.

Follows immediately from Proposition 4.7 and the corresponding facts
concerning CM points Shimura curves [13, §2.4]. ∎

We conclude by determining all ${\rm\mathbb{Q}}$-rational CM $j$-invariants,
equivalently all geometrically split $\operatorname{Prym}(C,\gamma)$ that can
be defined over ${\rm\mathbb{Q}}$. To this end, we will use Elkies’
computations of all rational CM points on the full Atkin–Lehner quotient
$X^{*}:=X/W\simeq{\rm\mathbb{P}}^{1}_{{\rm\mathbb{Q}}}$. Following Elkies [13,
§3.1], let $t$ be the unique coordinate
$X^{*}\xrightarrow{\sim}{\rm\mathbb{P}}^{1}_{t}$ with the property that
$t=0,1,\infty$ corresponds to the unique CM point by the order
${\rm\mathbb{Z}}[\sqrt{-6}]$, ${\rm\mathbb{Z}}[i]$, ${\rm\mathbb{Z}}[\omega]$
respectively. Recall from Lemma 4.3 the isomorphism $j\colon
Y\rightarrow{\rm\mathbb{P}}^{1}$. The isomorphism $Y\simeq(X/w_{3})^{\bar{w}}$
of Proposition 4.7 determines a quotient map $\Pi\colon
Y\simeq(X/w_{3})^{\bar{w}}\rightarrow((X/w_{3})^{\bar{w}})/\langle\bar{w}\rangle=X^{*}$.

######  Lemma 6.17.

In the coordinates $j$ and $t$ of $Y$ and $X^{*}$ respectively, the map
$\Pi\colon Y\rightarrow X^{*}$ is given by $j\mapsto\frac{(j+1)^{2}}{4j}$. A
$k$-point $t\in k$ on $X^{*}$ lifts to $Y$ if and only if $t(t-1)$ is a square
in $k$.

###### Proof.

Elkies [13, §3.1, end of p 17] has shown that the biquadratic extension of
function fields ${\rm\mathbb{Q}}(X)/{\rm\mathbb{Q}}(X^{*})$ is given by
adjoining $\sqrt{-t}$ and $\sqrt{3(t-1)}$ to
${\rm\mathbb{Q}}(X^{*})={\rm\mathbb{Q}}(t)$. The three intermediate quadratic
subfields correspond to the three intermediate Atkin–Lehner quotients of $X$;
analyzing the ramification shows that the morphism $X/w_{3}\rightarrow X^{*}$
can be realized as the double cover of $X^{*}={\rm\mathbb{P}}^{1}_{t}$
branched along the function $-3t(t-1)$, so $X/w_{3}$ has equation
$s^{2}=-3t(t-1)$. Moreover, the involution $(t,s)\mapsto(t,-s)$ corresponds to
the involution $\bar{w}$. Proposition 4.7 shows that $Y\rightarrow X^{*}$ is
the quadratic twist of $X/w_{3}$ along
${\rm\mathbb{Q}}(\sqrt{-3})/{\rm\mathbb{Q}}$, so
$Y\rightarrow{\rm\mathbb{P}}_{t}^{1}$ has equation $s^{2}=t(t-1)$. Rationally
parametrizing this conic shows that $Y$ has a rational coordinate $r$ such
that $s=(1-r^{2})/(4r)$, $t=(1+r)^{2}/(4r)$ and the involution $\bar{w}$
corresponds to $r\mapsto 1/r$.

We claim that after possibly replacing $r$ by $1/r$, which does not affect the
expression $t=(1+r)^{2}/(4r)$, we have $j=r$. Analyzing the ramification of
$\Pi$, we see that $Y$ has two CM points by ${\rm\mathbb{Z}}[\omega]$
(corresponding to $r=0,\infty$) and one CM point by ${\rm\mathbb{Z}}[i]$
(corresponding to $r=1$). The points $j=0,\infty$ correspond to decomposable
$\mu_{3}$-abelian surfaces, described in the proof of Lemma 4.3, and have CM
by ${\rm\mathbb{Z}}[\omega]$. Example 6.15 shows that $j=1$ corresponds to the
CM point by ${\rm\mathbb{Z}}[i]$. So $j$ and $r$ (or $1/r$) agree on the three
points $0,1,\infty$, so they must agree everywhere, proving the claim. ∎

######  Proposition 6.18.

Table 1 is a complete list of rational CM $j$-invariants and the discriminants
of the corresponding quadratic orders. Consequently, a bielliptic Picard curve
$C/{\rm\mathbb{Q}}$ has geometrically non-simple Prym variety if and only if
$j(C)$ appears in this table.

###### Proof.

A point in $Y({\rm\mathbb{Q}})$ is CM if and only if its image under
$\Pi\colon Y\rightarrow X^{*}$ is CM. It therefore suffices to find all
rational CM points on $X^{*}$ and determine which ones lift to
$Y({\rm\mathbb{Q}})$. Elkies [13, Table 1] has determined all rational CM
points in his $t$-coordinate. Some of these were only conjecturally CM, but
[14] confirms this table unconditionally. The proposition then follows from
Lemma 6.17 and an elementary calculation. ∎

${\rm Disc}(R)$ | $|{\rm Disc}(R)|$ |  $j$ or $1/j$
---|---|---
$-3$ | $3$ | $0$
$-4$ | $2^{2}$ | $1$
$-24$ | $2^{3}\cdot 3$ | $-1$
$-84$ | $2^{2}\cdot 3\cdot 7$ | $-27$
$-120$ | $2^{3}\cdot 3\cdot 5$ | $27/125$
$-228$ | $2^{2}\cdot 3\cdot 19$ | $15625/729$
$-147$ | $3\cdot 7^{2}$ | $-48384/15625$
$-372$ | $2^{2}\cdot 3\cdot 31$ | $-1771561/421875$
$-408$ | $2^{3}\cdot 3\cdot 17$ | $-11390625/4913$
Table 1. Rational CM $j$-invariants and their discriminants

##  7\. $6$-torsion points in the Prym variety

To prove Theorem 1.1, we must analyze the Galois modules $P[n]$ of bielliptic
Picard Pryms. In this section we study $P[2]$, $P[3]$ and $P[6]$ explicitly
and give various criteria for the existence of rational torsion points.

For the remainder of this section, let $(C,\gamma)=(C_{a,b},\gamma_{a,b})$ be
a marked bielliptic Picard curve over a field $k$ (always assumed of
characteristic $\neq 2,3$).

###  7.1. $2$-torsion

Recall from §2.5-2.6 that $[2]=\widehat{\lambda}\lambda$, giving rise to a
short exact sequence

(7.1) $0\rightarrow P[\lambda]\rightarrow P[2]\rightarrow
A[\widehat{\lambda}]\rightarrow 0.$

The diagram from §2.5 shows that there is a canonical isomorphism
$P[\lambda]\simeq E[2]$ and by bigonal duality we have
$A[\widehat{\lambda}]\simeq\widehat{E}[2]$ as well. We conclude:

######  Lemma 7.1.

There is a short exact sequence of ${\rm Gal}_{k}$-modules

(7.2) $0\to E[2]\to P[2]\to\widehat{E}[2]\to 0.$

The $k$-points of the outer terms in (7.2) are easy to determine.

######  Lemma 7.2.

Let $E=E_{a,b}$ and $\widehat{E}=\widehat{E}_{a,b}$ be as above. Then

  1. (1)

$E[2](k)\neq 0$ if and only if $16(a^{2}-4b)$ is a cube in $k$, in which case
$P[2](k)\neq 0$.

  2. (2)

$\widehat{E}[2](k)\neq 0$ if and only if $b$ is a cube in $k$, in which case
$A[2](k)\neq 0$.

Moreover if $k(\omega)\neq k$, then $|E[2](k)|\leq 2$ and
$|\widehat{E}[2](k)|\leq 2$.

###### Proof.

This can be read off the models $E\colon y^{2}=x^{3}+16(a^{2}-4b)$ and
$\widehat{E}\colon y^{2}=x^{3}+b$ of §2.7. ∎

To determine $P[2](k)$, we study the extension class of $\eqref{eq:ses}$ and
determine when an element of $\widehat{E}[2](k)$ lifts to $P[2](k)$. We will
use the following explicit geometric description of $P[2]$; we thank Adam
Morgan for pointing it out to us. Recall that $E$ is an elliptic curve with
origin $O_{E}=\pi(\infty)$.

######  Proposition 7.3.

Each divisor class in $P[2](\bar{k})$ is represented by a unique divisor of
the form $R+\pi^{*}(T)-3\infty$, where $R\in C(\overline{k})$ is a
ramification point of the map $\pi\colon C\rightarrow E$ and $T\in
E(\overline{k})$ is such that $[2](T)=-\pi(R)$.

###### Proof.

We first show that every such divisor defines an element of $P[2](\bar{k})$,
so let $x=R+\pi^{*}(T)-3\infty$ be such a divisor class. Recall that
$\pi(\infty)=O_{E}$ denotes the origin of $E$. The condition $[2](T)=-\pi(R)$
translates to an equivalence $2T+\pi(R)\sim 3O_{E}$. Pulling this equivalence
back along $\pi$ shows that $2x=0$, so $x\in J[2](\bar{k})$. Since both $R$
and $\pi^{*}(T)$ are fixed by $\tau$, $\tau(x)=x$ so $\tau(x)+x=2x=0$ so $x\in
P[2](\bar{k})$. We now claim that two such divisors $R+\pi^{*}(T)-3\infty$ and
$R^{\prime}+\pi^{*}(T^{\prime})-3\infty$ are linearly equivalent if and only
if $R=R^{\prime}$ and $T=T^{\prime}$. Indeed, Proposition 2.15(1) shows that
$R=R^{\prime}$, and then $T=T^{\prime}$ follows from the fact that
$\pi^{*}\colon E\rightarrow J$ is injective.

Since there are $16$ divisors of this form and $P[2](\bar{k})$ has order $16$
too, every element of $P[2](\bar{k})$ has a unique representative of this
form. ∎

This description of $P[2]$ is compatible with the sequence (7.2): the map
$E[2]\rightarrow P[2]$ sends $T$ to $\pi^{*}(T)-2\infty$; the map
$P[2]\rightarrow\widehat{E}[2]$ sends $R+\pi^{*}(T)-3\infty$ with $R=(0,t)\in
C(\bar{k})$ to $(-t,0)\in\widehat{E}[2](\bar{k})$, using the coordinates of
the equations in §2.7.

######  Theorem 7.4.

An element $(-t,0)\in\widehat{E}[2](k)$ with $t\in k$ lifts to $P[2](k)$ under
(7.2) if and only if $g_{a,t}(z):=z^{4}-6tz^{2}+4az-3t^{2}$ has a $k$-rational
root. Consequently, $(P[2]\setminus P[\lambda])(k)\neq\varnothing$ if and only
if there exists $t\in k$ such that $t^{3}=b$ and such that $g_{a,t}$ has a
$k$-rational root.

###### Proof.

By the above proposition, the point lifts to $P[2](k)$ if and only if the
corresponding point $(4t,4a)\in E(k)$ is divisible by $2$ in $E(k)$. Using the
$2$-descent map $E(k)/2E(k)\hookrightarrow\operatorname{H}^{1}(k,E[2])$ and
its interpretation via binary quartic forms, this is equivalent to
$z^{4}-24tz^{2}+32az-48t^{2}$ having a root in $k$ [2, §2, (11)]. Changing $z$
by $2z$ and dividing by $16$ results in the polynomial $g_{a,t}$. ∎

######  Corollary 7.5.

$(P[2]\setminus P[\lambda])(k)\neq\varnothing$ if and only if
$(a,b)=((4s+3)(4s^{2}-3)d^{3},(4s+3)^{3}d^{6})$ for some $s,d\in k$.

###### Proof.

Theorem 7.4 shows that $P[2]\setminus P[\lambda]$ has a $k$-point if and only
if there exists a $t\in k$ with $t^{3}=b$ such that
$g_{a,t}(z)=z^{4}-6tz^{2}+4az-3t^{2}$ has a $k$-rational root. A calculation
shows that if $(a,b)$ is of the above form for some $s,d\in k$ then $t=(4s+3)$
is such that $g_{a,t}(z)$ has the root $z=-4s-3$ in $k$. Conversely, suppose
that $t\in k$ is a cube root of $b$ such that $g_{a,t}(z)$ has a root in $k$.
If $a=0$, a calculation shows that we may take $s=\pm\sqrt{3}/2$, so we may
assume that $a\neq 0$. Note that $\lambda
g_{a,t}(z)=g_{\lambda^{3}a,\lambda^{2}t}(\lambda z)$ for all $\lambda\in k$.
Choosing $\lambda=t/a$ and setting $v=t^{3}/a^{2}$, it follows that
$g_{v,v}(z)$ has a $k$-rational root. Since $t\neq 0$, this root must be
nonzero. The locus of $(v,z)$ with $g_{v,v}(z)=0$ and $z\neq 0$ is isomorphic
to an open subset of a smooth conic. After rationally parametrizing this conic
we find that $v$ must be of the form $(4s+3)/(4s^{2}-3)^{2}$ for some $s\in
k$. It follows that $(a,b)$ is of the above form with $d=a/t$. ∎

In the following we assume for simplicity that $k(\omega)\neq k$, so that
$E[2](k)\neq E[2]$. For abstract groups $G$ and $G^{\prime}$, we sometimes
write $G\subset G^{\prime}$ as shorthand for “there exists an embedding
$G\hookrightarrow G^{\prime}$”.

######  Proposition 7.6.

Suppose $k(\omega)\neq k$. Then the following conditions are equivalent:

  1. (1)

$({\rm\mathbb{Z}}/2{\rm\mathbb{Z}})^{2}\subset P(k)$;

  2. (2)

$E[2](k)\neq 0$ and there exists a branch point of $C\rightarrow E$ different
from $O_{E}$ lying in $2E(k)$;

  3. (3)

$(a,b)=((16w^{6}+40w^{3}-2)d^{3},\left(8w^{3}+1\right)^{3}d^{6})$ for some
pair $w,d\in k$;

  4. (4)

$({\rm\mathbb{Z}}/2{\rm\mathbb{Z}})^{2}\subset A(k)$.

###### Proof.

That $(1)$ and $(2)$ are equivalent follows from (7.2), Lemma 7.2, the fact
that $k(\omega)\neq k$ and Proposition 7.3. Similarly, Proposition 7.6 shows
that $({\rm\mathbb{Z}}/2{\rm\mathbb{Z}})^{2}\subset P_{a,b}(k)$ if and only if
$(a,b)=((4s+3)(4s^{2}-3)d^{3},(4s+3)^{3}d^{6})$ for some $s,d\in k$ and
$16(a^{2}-4b)=16d^{6}(2s-3)(2s+1)^{3}(4s+3)^{2}$ is a cube. The latter is
equivalent to $(3-2s)/4(4s+3)$ being a cube $w^{3}$ in $k$. Solving for $s$,
plugging in back to $a,b$ and absorbing common factors in $d$ proves the
equivalence between $(1)$ and $(3)$.

A calculation shows that if $(a,b)$ is of the form $(3)$ for some $w,d\in k$,
then $(8a,16(a^{2}-4b))$ is of the form $(3)$ with $w^{\prime}=-1/2w$ and
$d^{\prime}=-(2t)^{2}d$. Therefore $(3)\Leftrightarrow(4)$ by bigonal duality
from §2.7. ∎

Note that when $k(\omega)\neq k$, we have $|P[2](k)|\leq 4$ by
$\eqref{eq:ses}$ and Lemma 7.2. So the conditions in Proposition 7.6 are also
equivalent to the condition
$P[2](k)\simeq({\rm\mathbb{Z}}/2{\rm\mathbb{Z}})^{2}$.

###  7.2. $\sqrt{-3}$-torsion

We use the $\mu_{3}$-action on $P$ to explicitly analyze a ${\rm
Gal}_{k}$-submodule of $P[3]$. Let $\mathfrak{p}$ be the ideal
$(1-\omega)\subset{\rm\mathbb{Z}}[\omega]$, viewed as a subgroup of ${\rm
End}(P_{k^{\operatorname{sep}}})$. Since $\mathfrak{p}$ is ${\rm
Gal}_{k}$-stable, so is the subgroup $A[\mathfrak{p}]$ of points fixed by
$\omega$. We have $\mathfrak{p}=(\sqrt{-3})$, so $P[\mathfrak{p}]$ has order
$9$, and is the kernel of an isogeny $P\to B$ over $k$. In fact, the quotient
$B=P/P[\mathfrak{p}]$ is isomorphic to the sextic twist $P_{-27}$, which is
also the quadratic twist of $P$ by the extension $k(\omega)/k$ [41, Rem 2.8].
We therefore have an exact sequence

(7.3) $0\to P[\mathfrak{p}]\to P[3]\to P_{-27}[\mathfrak{p}]\to 0$

Of course, $A=P^{\vee}$ also has a $\mu_{3}$-action, so we can similarly
define $A[\mathfrak{p}]\subset A[3]$, a subgroup of order $9$ that sits in an
analogous short exact sequence.

######  Lemma 7.7.

$A[\mathfrak{p}]\simeq P[\mathfrak{p}]$ as ${\rm Gal}_{k}$-modules.

###### Proof.

The restriction of the polarization $\lambda$ induces such an isomorphism. ∎

Let $f(x):=x^{4}+ax^{2}+b\in k[x]$ and let
$\mathcal{R}=\\{\pm\alpha_{1},\pm\alpha_{2}\\}$ be the $4$ roots of $f$ in
$k^{\operatorname{sep}}$.

######  Lemma 7.8.

The Galois module $P[{\rm\mathfrak{p}}]$ is generated by the classes
$D_{1}:=(\alpha_{1},0)-(-\alpha_{1},0)$ and
$D_{2}:=(\alpha_{2},0)-(-\alpha_{2},0)$.

###### Proof.

These classes live in $P$, are non-trivial, and are annihilated by $1-\omega$
so lie in $P[{\rm\mathfrak{p}}]$. They are linearly independent over
${\rm\mathbb{F}}_{3}$ since otherwise we would have $D_{2}\sim\pm D_{1}$,
which is impossible since $C$ is not hyperelliptic. ∎

Let $D_{4}\subset{\rm Sym}(\mathcal{R})$ the subset of permutations of
$\mathcal{R}$ such that $(-\alpha)^{\sigma}=-\alpha^{\sigma}$ for all
$\alpha\in\mathcal{R}$ and $\sigma\in{\rm Sym}(\mathcal{R})$. This is a
dihedral group of order $8$, generated by its four reflections
$\\{\tau_{1},\tau_{2},\hat{\tau}_{1},\hat{\tau}_{2}\\}$, where $\tau_{i}$ maps
$\alpha_{i}$ to $-\alpha_{i}$ and fixes the other roots, $\hat{\tau}_{1}$
swaps $\alpha_{1}\leftrightarrow\alpha_{2}$ and
$-\alpha_{1}\leftrightarrow-\alpha_{2}$, and $\hat{\tau}_{2}$ swaps
$\alpha_{1}\leftrightarrow-\alpha_{2}$ and
$-\alpha_{1}\leftrightarrow\alpha_{2}$. (The reader is invited to picture
$D_{4}$ acting on a square with labels
$\\{\alpha_{1},\alpha_{2},-\alpha_{1},-\alpha_{2}\\}$.) Let $V$ be the
${\rm\mathbb{F}}_{3}$-vector space with basis
$\\{v_{\alpha}\mid\alpha\in\mathcal{R}\\}$, modulo the relations
$v_{-\alpha}=-v_{\alpha}$. The $D_{4}$-action on $\mathcal{R}$ induces a
linear $D_{4}$-action on $V$. After choosing the basis
$\\{v_{\alpha_{1}},v_{\alpha_{2}}\\}$, this induces a representation
$\varphi\colon D_{4}\rightarrow{\rm GL}_{2}({\rm\mathbb{F}}_{3})$, isomorphic
to the mod $3$ reduction of the reflection representation of $D_{4}$. The
${\rm Gal}_{k}$-action on $\mathcal{R}$ determines a homomorphism
$\chi\colon{\rm Gal}_{k}\rightarrow{\rm GL}_{2}({\rm\mathbb{F}}_{3})$. Lemma
7.8 shows:

######  Proposition 7.9.

The ${\rm Gal}_{k}$-action on $P[\mathfrak{p}](k^{\operatorname{sep}})$ in the
${\rm\mathbb{F}}_{3}$-basis $\\{D_{1},D_{2}\\}$ is given by the composition
${\rm Gal}_{k}\xrightarrow{\chi}D_{4}\xrightarrow{\varphi}{\rm
GL}_{2}({\rm\mathbb{F}}_{3})$.

######  Corollary 7.10.

We have $P[{\rm\mathfrak{p}}](k)\simeq({\rm\mathbb{Z}}/3{\rm\mathbb{Z}})^{2}$
if and only if $f(x)$ splits completely in $k[x]$.

###### Proof.

Use Proposition 7.9 and the fact that $\varphi$ is injective. ∎

######  Corollary 7.11.

The ${\rm\mathbb{F}}_{3}[{\rm Gal}_{k}]$-module $P[{\rm\mathfrak{p}}]$ is
semisimple.

###### Proof.

Use Proposition 7.9 and the fact that $3$ is coprime to the order of $D_{4}$.
∎

Let $\hat{f}:=x^{4}+8ax+16(a^{2}-4b)$.

######  Corollary 7.12.

$P[\mathfrak{p}](k)\neq 0$ if and only if either $f$ or $\hat{f}$ has a root
in $k$.

###### Proof.

Let $G$ be the image of $\chi\colon{\rm Gal}_{k}\rightarrow D_{4}$. Then
$P[\mathfrak{p}](k)\neq 0$ if and only if $G$ fixes some nonzero element of
$V={\rm\mathbb{F}}_{3}^{2}$. A group theory calculation shows that only
reflections have fixed points, so this is equivalent to
$G\subset\langle\tau_{i}\rangle$ or $G\subset\langle\hat{\tau}_{i}\rangle$ for