# Modularity and effective Mordell I.

Levent Alpöge

###### Abstract.

We give an effective proof of Faltings’ theorem for curves mapping to Hilbert
modular stacks over odd-degree totally real fields.

We do this by giving an effective proof of the Shafarevich conjecture for
abelian varieties of $\mathrm{GL}_{2}$-type over an odd-degree totally real
field.

We deduce for example an effective height bound for $K$-points on the curves
$C_{a}:x^{6}+4y^{3}=a^{2}$ ($a\in K^{\times}$) when $K$ is odd-degree totally
real.

(Over $\overline{\mathbb{Q}}$ all hyperbolic hyperelliptic curves admit an
étale cover dominating $C_{1}$.)

##  1 Introduction.

Faltings’ theorem is one of the classic ineffective results in mathematics.
His (first) method of proof is roughly to realize, following Parshin, a given
curve inside a moduli space, and then to prove a finiteness result for
integral points on that moduli space (compactness provides integrality). He is
forced to work with the entire moduli space of principally polarized abelian
varieties because of a construction of Kodaira that he invokes.

This forces him to consider quite general Galois representations. We will deal
only with Galois representations valued in $\mathrm{GL}_{2}$ by working only
with curves mapping to Hilbert modular varieties111We will use this
terminology for the fine moduli spaces, which are, without level structures
imposed, not varieties but stacks. (or, slightly more generally, curves over
which there is a non-isotrivial family of $\mathrm{GL}_{2}$-type abelian
varieties). We will further put ourselves in a situation where it is known
that there are motives attached to the conjecturally corresponding automorphic
forms by working only over odd-degree totally real fields.

In that situation we will prove that all relevant abelian varieties become
modular over a computable222We remind the reader that to say that a quantity
is computable, or, equivalently, effectively computable, is to say that there
is a Turing machine that terminates on all inputs and which, on input the
relevant parameters — in this case the base number field and bounds on the
dimensions and conductors of the relevant abelian varieties — outputs said
quantity. finite list of odd-degree totally real extensions, and then deduce a
height bound on all such abelian varieties using the usual construction of an
abelian variety associated to a parallel weight two Hilbert modular
eigencuspform over an odd-degree totally real field.

The technique of course further gives an effective determination of the
$S$-integral $K$-points on a Hilbert modular variety (or indeed, interpreted
suitably, all Hilbert modular varieties of given dimension together) when $K$
is totally real of odd degree. It also generalizes naturally assuming the
usual Galois-to-automorphic-to-motivic conjectures.

We conclude by giving an example application, namely to the curves
$x^{6}+4y^{3}=a^{2}$ ($a\in\overline{\mathbb{Q}}^{\times}$ odd-degree totally
real), over each of which we use a particular hypergeometric family.

###  1.1 Main theorems.

We prove the following.

######  Theorem 1.1.

There is a finite-time algorithm that, on input $(g,K,S)$, with
$g\in\mathbb{Z}^{+}$, $K/\mathbb{Q}$ totally real of odd degree, and $S$ a
finite set of places of $K$, outputs the $g$-dimensional abelian varieties
$A/K$ which are of $\mathrm{GL}_{2}$-type over $K$ and which have good
reduction outside $S$.

Let $\mathfrak{o}$ be an order in a totally real field $F/\mathbb{Q}$. Write
$H_{\mathfrak{o}}/\mathbb{Q}$ for the333We will use this sloppy notation (one
needs slightly more than just the data of the ring $\mathfrak{o}$ to produce a
Hilbert modular variety) for convenience. corresponding
$\mathop{\mathrm{rk}}_{\mathbb{Z}}{\mathfrak{o}}=[F:\mathbb{Q}]$-dimensional
Hilbert modular variety and $\mathcal{H}_{\mathfrak{o}}$ for its canonical
integral model.444We are free to increase $S$ to avoid subtleties with the
latter. It follows that, at least for $K/\mathbb{Q}$ totally real of odd
degree, we may compute $\mathcal{H}_{\mathfrak{o}}(\mathfrak{o}_{K,S})$ in
finite time.555We are implicitly invoking the polarization estimates of
Gaudron-Rémond (Theorem $1.1$ of their [14]) and the endomorphism estimates of
Masser-Wüstholz (the main theorem of their [18], though an explicit such
estimate can be extracted from Lemma $9.2$ of Gaudron-Rémond’s [14]) in order
to deal with the $\mathfrak{o}$-linear polarizations that are hidden by the
notation.

######  Definition 1.2.

Let $C/\overline{\mathbb{Q}}$ be a smooth projective hyperbolic curve.
$C/\overline{\mathbb{Q}}$ is _of type $\mathrm{I}$_ if and only if it admits a
nonconstant map to a Hilbert modular variety.

Let now $K/\mathbb{Q}$ be a totally real field. Let $C/K$ be a smooth
projective hyperbolic curve. $C/K$ is _of type $\dot{\mathrm{I}}$_ if and only
if it admits a nonconstant map defined over a totally real $L/K$ to a Hilbert
modular variety.

Finally let $K/\mathbb{Q}$ be a totally real field of odd degree. Let $C/K$ be
a smooth projective hyperbolic curve. $C/K$ is _of type $\bigiy$_ if and only
if it admits a nonconstant map defined over an odd-degree totally real $L/K$
to a Hilbert modular variety.

Because each $H_{\mathfrak{o}}$ can be compactified to a projective variety by
adding finitely many points, when $\mathfrak{o}\neq\mathbb{Z}$ it follows that
there are infinitely many complete curves with nonconstant maps to
$H_{\mathfrak{o}}$.666(Add level structure to produce curves rather than just
stacky curves from this argument.) Indeed, by e.g. intersecting with general
linear subspaces defined over $\mathbb{Q}$ of codimension
$\mathop{\mathrm{rk}}_{\mathbb{Z}}{\mathfrak{o}}-1$, it follows that there are
curves $C/\mathbb{Q}$ of type $\bigiy$, so that these are nonempty
notions.777There is moreover enough freedom in choosing $\mathfrak{o}$, the
implicit level structure, and the linear subspace to guarantee that
$C(\mathbb{Q})\neq\varnothing$. This comment is necessary because e.g.
(canonical models of) Shimura curves associated to indefinite quaternion
algebras over $\mathbb{Q}$ are tautologically of type $\bigiy$, since they map
to Hilbert modular surfaces, but of course they lack real points, let alone
rational points over odd-degree totally real fields. We are not sure how to
check if e.g. a given $C/\overline{\mathbb{Q}}$ is _not_ of type $\mathrm{I}$.

The definition allows us to state the following.

######  Corollary 1.3.

There is a finite-time algorithm that, on input $(K,C/K)$ with $K/\mathbb{Q}$
totally real of odd degree and $C/K$ of type $\bigiy$, outputs $C(K)$.

This is completely equivalent to the following perhaps clearer reformulation.

######  Corollary 1.4 (Corollary 1.3, height version.).

Let $K/\mathbb{Q}$ be a totally real field of odd degree. Let $C/K$ be a curve
of type $\bigiy$. Then: there is an effectively computable
$\text{\c{s}}_{C,K}\in\mathbb{Z}^{+}$ depending only on $K$ and $C/K$ such
that all $P\in C(K)$ satisfy:

$h(P)\leq\text{\c{s}}_{C,K}.$

In other words, for such $C/K$, $C(L)$ is effectively computable for all odd-
degree totally real extensions $L/K$.

We emphasize that this is the first time such an effective result has been
proven. Even admitting the Chabauty hypothesis (which presumably fails for
sufficiently large such $L/K$), etc., one does not know ahead of time that the
upper bound on $\\#|C(L)|$ computed by $p$-adic methods and suitable sieving
will ever match the lower bound computed by searching for $L$-points of larger
and larger height. On the other hand, it is certainly the case that existing
methods (employing Chabauty, Chabauty-Kim, etc.) are far more practical than
the algorithm implicit in Corollary 1.3 when they work (namely quite often).

We conclude with an example. Using Theorem 1.1 we prove the following.

######  Theorem 1.5.

Let $K/\mathbb{Q}$ be a totally real field of odd degree. Let $a\in
K^{\times}$. Let $C_{a}:x^{6}+4y^{3}=a^{2}$. Then: there is a finite time
algorithm that, on input an odd-degree totally real extension $L/K$, outputs
$C_{a}(L)$.

Again, this has the following completely equivalent reformulation.

######  Theorem 1.6 (Theorem 1.5, height version.).

Let $K/\mathbb{Q}$ be a totally real field of odd degree. Let $a\in
K^{\times}$. Let $C_{a}:x^{6}+4y^{3}=a^{2}$. Then: there is an effectively
computable $\bighardsign_{K,a}\in\mathbb{Z}^{+}$ depending only on $K$ and $a$
such that all $P\in C_{a}(K)$ satisfy:

$h(P)\leq\bighardsign_{K,a}.$

We again emphasize that this is the first time such an effective form of the
Mordell conjecture has been proven (note that, given an odd-degree totally
real $a\in\overline{\mathbb{Q}}$, the theorem bounds the heights of points in
$C_{a}(L)$ for infinitely many number fields $L/\mathbb{Q}$). If one could
remove the phrase "odd-degree totally real" from Theorem 1.5 (i.e. in the case
of the single curve $x^{6}+4y^{3}=1$ if one could allow all number fields
$L/\mathbb{Q}$), by a theorem of Poonen [20] generalizing work of Bogomolov-
Tschinkel [7] one would have an effective form of Faltings’ theorem for all
solvable covers of $\mathbb{P}^{1}$ over all number fields.888See e.g. Chapter
$8$ of the author’s thesis [4]. Over $\mathbb{C}$ the curve $x^{6}+4y^{3}=1$
corresponds to the commutator subgroup (which has index $18$) of the
arithmetic triangle group $\Delta(3,6,6)$. Similarly, and again over
$\mathbb{C}$, the curve $y^{2}=x^{6}+1$ used in the aforementioned theorems of
Poonen/Bogomolov-Tschinkel and in Chapter $8$ of the author’s thesis [4]
corresponds to the commutator subgroup (which has index $12$) of the
arithmetic triangle group $\Delta(2,6,6)$. The point is that $\Delta(3,6,6)$
is an index-$2$ subgroup of $\Delta(2,6,6)$, so that, at least over
$\overline{\mathbb{Q}}$, there is an étale cover of $y^{2}=x^{6}+1$ by
$x^{6}+4y^{3}=1$ of degree $3=\frac{2\cdot 18}{12}$. Explicitly,
$x^{6}+4y^{3}=1$ can be rewritten as
$\left(x^{3}+\frac{2y^{3}}{x^{3}}\right)^{2}=4\cdot\left(\frac{y}{x}\right)^{6}+1$
(just complete the square).

###  1.2 Remarks.

First, when $K=\mathbb{Q}$ and $g=1$, Theorem 1.1 was proven in explicit form
by Murty-Pasten [19]. Independently and simultaneously, von Känel [25] proved
the same theorem with only the restriction that $K=\mathbb{Q}$, again in
explicit form. Were Serre’s conjecture known over all e.g. odd-degree totally
real fields, von Känel’s argument would easily prove Theorem 1.1 in explicit
form. So our key insight in proving Theorem 1.1 is really in how to get away
with using only potential modularity theorems rather than the modularity
theorems von Känel uses over $\mathbb{Q}$.

Though von Känel never observes that one can bound heights of _rational_
points on curves by using his main theorem, his bounds on heights of
$\mathrm{GL}_{2}$-type abelian varieties over $\mathbb{Q}$ of given conductor
do indeed produce (explicit!) height bounds on $P\in C(\mathbb{Q})$ for smooth
projective hyperbolic curves $C/\mathbb{Q}$ admitting a nonconstant map
defined over $\mathbb{Q}$ to a Hilbert modular variety. This observation,
namely that there are complete curves in these moduli spaces, and thus curves
whose rational points are (unconditionally!) controlled by explicit
automorphic forms, was the origin of this work — note, though, that the former
point is also observed in recent independent work of Baldi-Grossi [6] about
obstructions to rational points on these moduli spaces.

We believe our use of hypergeometric families associated to _cocompact_
triangle groups $\Delta(e_{0},e_{1},e_{\infty})$ (our example in Section 6
amounts to such a use for the arithmetic triangle group $\Delta(3,6,6)$) for
these Diophantine questions is new — see Chapter $10$ (and specifically
Section $10.2.1$) of the author’s thesis [4] for a more detailed discussion
which includes the nonarithmetic triangle groups and uses the new potential
modularity theorems [1] over CM fields.

It should be clear that, assuming suitable modularity _and_ existence-of-
abelian-variety conjectures for Galois representations valued in
$\mathrm{GSp}_{2g}(\mathbb{Q}_{\ell})$, one can prove in the same way that
there is a finite-time algorithm that determines the $S$-integral $K$-points
on the Siegel modular variety, i.e. that such conjectures imply an effective
form of the Shafarevich conjecture. In a similar vein, the main result of
[3]999See also Chapter $7$ of [4], which is based on the same work. implies
that the effective Shafarevich conjecture (and thus an effective form of the
Mordell conjecture) also follows from standard motivic conjectures (Fontaine-
Mazur, Hodge, and Tate). In fact making this latter work unconditional in some
cases was the main motivation for this work.

##  2 Acknowledgements.

This article is based on Chapters $9$ and $11$ of the author’s Ph.D. thesis at
Princeton University. I would like to thank both my advisor Manjul Bhargava
and Peter Sarnak for their patience and encouragement. I would especially like
to thank Professor Sarnak for a discussion which left me determined to find an
unconditional result like Theorem 1.1.

I would also like to thank Gurbir Dhillon, Mladen Dimitrov, Tony Feng, Brian
Lawrence, Curtis McMullen, Christopher Skinner, Jacob Tsimerman, Akshay
Venkatesh, Raphael von Känel, and Shou-Wu Zhang for informative discussions.

I thank the National Science Foundation (via their grant DMS-$2002109$) and
Columbia University for their support during the pandemic.

##  3 Preliminaries.

We will use the following mostly standard results in the proofs of the main
theorems.

###  3.1 Faltings’ Lemma.

The first such result is the usual form of Faltings’ Lemma (save perhaps the
observation that it also works for the rings $\mathfrak{o}/\lambda^{N}$).

######  Lemma 3.1 (Faltings, see Satz $5$ of his [13]).

Let $d\in\mathbb{Z}^{+}$. Let $N\in\mathbb{Z}^{+}$. Let $K/\mathbb{Q}$ be a
number field and $S$ a finite set of places of $K$. Let $\mathfrak{o}$ be an
order in the ring of integers of a number field and $\lambda$ be a prime of
$\mathfrak{o}$ with $\mathrm{Nm}\,{\lambda}\leq N$. Let $T_{K,S,d,N}$ be a
finite set of primes of $K$ that are prime to $\mathrm{Nm}\,{\lambda}$ which
is disjoint from $S$ such that, for all Galois extensions $L/K$ that are
unramified outside $S$ and the primes dividing $\mathrm{Nm}\,{\lambda}$ and of
degree $[L:K]\leq N^{2d^{2}}$, the map
$T_{K,S,d,N}\to\mathop{\mathrm{Gal}}(L/K)/\text{conj.}$ via
$\mathfrak{p}\mapsto\mathrm{Frob}_{\mathfrak{p}}$ is surjective. Let
$R:=\mathfrak{o}_{\lambda}$ or $\mathfrak{o}/\lambda^{m}$ for some
$m\in\mathbb{N}$. Let
$\rho,\rho^{\prime}:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)\to\mathrm{GL}_{d}(R)$
be unramified outside $S$ and the primes dividing $\mathrm{Nm}\,{\lambda}$ and
such that
$\mathrm{tr}(\rho(\mathrm{Frob}_{\mathfrak{p}}))=\mathrm{tr}(\rho^{\prime}(\mathrm{Frob}_{\mathfrak{p}}))$
for all $\mathfrak{p}\in T_{K,S,d,N}$.

Then: $\mathrm{tr}\circ\rho=\mathrm{tr}\circ\rho^{\prime}$ on
$\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)$.

###### Proof.

The standard proof extends upon noting that, for $M\subseteq R^{\oplus n}$,
$\\#|M/\lambda|\leq\\#|R/\lambda|^{n}$, as can be seen101010Brian Lawrence
points out a far more elegant proof: take the preimage $\tilde{M}$ of $M$
under the evident surjection $\mathfrak{o}_{\lambda}^{\oplus
n}\twoheadrightarrow R^{\oplus n}$ and then run the usual argument:
$\tilde{M}$ is a free $\mathfrak{o}_{\lambda}$-module of rank $\leq n$, now
reduce mod $\lambda$, QED. by e.g. computing the $\mathrm{Tor}_{1}$.

Namely, it evidently suffices to show that the $R$-span of
$\mathrm{im}\,{(\rho\oplus\rho^{\prime}:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)\to\mathrm{GL}_{d}(R)^{\times
2})}$ inside $M_{d}(R)^{\times 2}$ is in fact spanned by
$\bigcup_{\mathfrak{p}\in
T_{K,S,d,N}}(\rho\oplus\rho^{\prime})(\mathrm{Frob}_{\mathfrak{p}})$, where
$\mathrm{Frob}_{\mathfrak{p}}\subseteq\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)$
is the Frobenius conjugacy class of $\mathfrak{p}$. To do this one uses
Nakayama to reduce mod $\lambda$, after which it follows from the hypothesis
on $T_{K,S,d,N}$. ∎

Note that the usual explicit form of the Chebotarev density theorem gives an
explicit $T_{K,S,d,N}$ satisfying the above — thus e.g. we may take
$T_{K,S,d,N}$ satisfying the above and so that all $\mathfrak{p}\in
T_{K,S,d,N}$ satisfy $\mathrm{Nm}\,{\mathfrak{p}}\ll_{K,S,d,N}1$, where the
implied constant is explicit.

###  3.2 An explicit bound on the height of an isogeny factor of an abelian
variety.

In this section we will prove the following lemma.

######  Lemma 3.2.

Let $g\in\mathbb{Z}^{+}$. Let $H\in\mathbb{Z}^{+}$. Let $N\in\mathbb{Z}^{+}$.
Let $K/\mathbb{Q}$ be a number field. Then: there is an explicit (thus
effectively computable) constant $C_{g,K,N,H}\in\mathbb{Z}^{+}$, depending
explicitly and only on $g$, $K$, $N$, and $H$, such that the following holds:

  * •

Let $B/K$ be an abelian variety with $\dim{B}\leq g$, $h(B)\leq H$, and with
good reduction outside the primes of $K$ dividing $N$. Let $A/K$ be an abelian
variety with good reduction outside the primes of $K$ dividing $N$ and such
that $A/\overline{\mathbb{Q}}$ is a $\overline{\mathbb{Q}}$-isogeny factor of
$B/\overline{\mathbb{Q}}$. Then: $h(A)\leq C_{g,K,N,H}$.

The lemma will follow easily from combining three standard facts. We note that
the argument we give already occurs in von Känel’s [25].

The first standard fact is the Masser-Wüstholz isogeny estimate, in the
explicit form given in Gaudron-Rémond’s [14] (Raynaud’s isogeny estimate [22]
would work just as well).

######  Theorem 3.3 (Gaudron-Rémond, see Theorem $1.4$ of their [14]).

Let $A,A^{\prime}/K$ be $K$-isogenous abelian varieties over a number field
$K$. Write $g:=\dim{A}$. Then: there is a $K$-isogeny $\varphi:A\to
A^{\prime}$ of degree

$\deg{\varphi}\leq\left((14g)^{64g^{2}}\cdot[K:\mathbb{Q}]\cdot\max(h(A),\log{[K:\mathbb{Q}]},1)^{2}\right)^{2^{10}g^{3}}=:\kappa(A),$

where $h(A)$ is the Faltings height of $A$ using Faltings’ original
normalization.

Consequently $|h(A^{\prime})-h(A)|\leq\frac{1}{2}\log{\kappa(A)}$.

The second standard fact is Bost’s lower bound for the Faltings height of an
abelian variety in terms of its dimension.

######  Theorem 3.4 (Bost, Gaudron-Rémond (see Corollary $8.4$ and then
paragraph $2.3$ of their [15])).

Let $A/\overline{\mathbb{Q}}$ be an abelian variety. Then:

$h(A)\geq-\frac{\log{(2\pi^{2})}}{2}\cdot\dim{A}.$

The third standard fact is a theorem of Silverberg producing an explicit
extension over which all endomorphisms of a given abelian variety (and,
consequently, its geometric isogeny decomposition) are defined.

######  Theorem 3.5 (Silverberg, Grothendieck).

Let $g\in\mathbb{Z}^{+}$. Let $K/\mathbb{Q}$ be a number field. Let $S$ be a
finite set of places of $K$. Then: there is an explicit finite Galois
extension $K^{\prime}/K$ depending only on $g$, $K$, and $S$ such that the
following holds:

  * •

Let $A/K$ be a $g$-dimensional abelian variety over $K$ with good reduction
outside $S$. Then: $A/K^{\prime}$ is split semistable and
$\mathrm{End}_{K^{\prime}}(A)=\mathrm{End}_{\overline{\mathbb{Q}}}(A)$.

Consequently its $K^{\prime}$-isogeny decomposition
$A\sim_{K^{\prime}}\prod_{i}B_{i}^{\times n_{i}}$ into $K^{\prime}$-simple
pairwise non-$K^{\prime}$-isogenous $B_{i}/K^{\prime}$ is such that all
$B_{i}/\overline{\mathbb{Q}}$ are $\overline{\mathbb{Q}}$-simple and pairwise
non-$\overline{\mathbb{Q}}$-isogenous.

(The statement about split semistability, which is the only reason the above
carries Grothendieck’s name, is not necessary for us here — we include it only
for the reader’s convenience.)

###### Proof.

Let $K^{\prime\prime}/K$ be the compositum of all extensions $L/K$ of degree
$[L:K]\leq 10^{10^{10}\cdot g^{10^{10}}}$ that are unramified outside primes
dividing $10^{10}!\cdot\prod_{\mathfrak{p}\in S}\mathrm{Nm}\,{\mathfrak{p}}$.
That $K^{\prime\prime}/K$ is an explicit finite extension follows from
Minkowski’s proof of the Hermite-Minkowski theorem.

Now, all $g$-dimensional $A/K$ with good reduction outside $S$ have full
$10^{10}!$-torsion defined over $K^{\prime\prime}$, since
$K(A[10^{10}!])\hookrightarrow K^{\prime\prime}$. Thus by the usual explicit
form of Grothendieck’s semistable reduction theorem we see that
$A/K^{\prime\prime}$ is semistable. Since the property is local, by passing to
an explicit finite Galois extension $K^{\prime}/K$ containing
$K^{\prime\prime}$ and depending only on $g$, $K$, and $S$, we ensure that
$A/K^{\prime}$ is split semistable.

Finally it similarly follows from a standard observation of Silverberg (see
Theorem $2.4$ of her [23]) that
$\mathrm{End}_{K^{\prime\prime}}(A)=\mathrm{End}_{\overline{\mathbb{Q}}}(A)=\mathrm{End}_{K^{\prime}}(A)$.
∎

Now let us prove Lemma 3.2.

###### Proof of Lemma 3.2..

Replacing $K$ by the explicit $K^{\prime}/K$ produced by Theorem 3.5 applied
to $A\times B$, by Theorem 3.5 it suffices without loss of generality to upper
bound $h(A)$ for $A/K$ a $K$-isogeny factor of $B/K$. By Poincaré complete
reducibility, there is an abelian variety $C/K$ such that $B\sim_{K}A\times
C$. By Theorem 3.3, $h(A)+h(C)=h(A\times C)\ll_{h(B),[K:\mathbb{Q}],\dim{B}}1$
with explicit implied constant. By Theorem 3.4, $h(C)\gg-\dim{C}\geq-\dim{B}$.
Thus $h(A)\ll_{h(B),[K:\mathbb{Q}],\dim{B}}1$ with explicit implied constant.

∎

###  3.3 Decomposition of $\mathrm{GL}_{2}$-type abelian varieties over the
reals.

The following lemma arises because the abelian varieties $A/K$ we search for
may not necessarily be $K$-simple, and modularity only implies that a
$K$-simple such abelian variety is a quotient of the Jacobian of a suitable
Shimura curve. The lemma shows that this is not an issue because we produce a
$d$-th root as such a quotient for some $d\leq\dim{A}$, so we can (and do)
just replace the Jacobian with its $(\dim{A})$-th power in the argument.

######  Lemma 3.6.

Let $K/\mathbb{Q}$ be a number field with a real place
$K\hookrightarrow\mathbb{R}$. Let $A/K$ be an abelian variety of
$\mathrm{GL}_{2}$-type over $K$. Then: there is a $K$-simple abelian variety
$B/K$ of $\mathrm{GL}_{2}$-type over $K$ such that
$A\sim_{K}B^{\times\frac{\dim{A}}{\dim{B}}}$.

In fact we will prove the following more general lemma.

######  Lemma 3.7.

Let $K/\mathbb{Q}$ be a number field. Let $A/K$ be an abelian variety which is
of $\mathrm{GL}_{2}$-type over $K$. Then: either there is a $K$-simple abelian
variety $B/K$ of $\mathrm{GL}_{2}$-type over $K$ such that
$A\sim_{K}B^{\times\frac{\dim{A}}{\dim{B}}}$, or else there are two $K$-simple
abelian varieties $B_{1},B_{2}/K$ which are non-$K$-isogenous and admit
sufficiently many complex multiplications over $K$ such that
$A\sim_{K}B_{1}^{\times\frac{\dim{A}}{2\dim{B_{1}}}}\times
B_{2}^{\times\frac{\dim{A}}{2\dim{B_{2}}}}$.

###### Proof.

Let $E/\mathbb{Q}$ be a CM field of degree $g:=[E:\mathbb{Q}]=\dim{A}$
admitting an embedding $E\hookrightarrow\mathrm{End}_{K}^{0}(A)$. Write, via
Poincaré complete reducibility, $A\sim_{K}\prod_{i=1}^{m}B_{i}^{\times n_{i}}$
with $B_{i}/K$ $K$-simple and pairwise non-$K$-isogenous. Write
$D_{i}:=\mathrm{End}_{K}^{0}(B_{i})$, $F_{i}/\mathbb{Q}$ for the centre of the
division algebra $D_{i}$, and $d_{i}$ for the index of $D_{i}$ over $F_{i}$.
Note that by the Albert classification $F_{i}/\mathbb{Q}$ is CM, and either
$d_{i}=1$ (i.e. $D_{i}=F_{i}$) and
$[F_{i}^{+}:\mathbb{Q}]\,\big{|}\dim{B_{i}}$, or else $d_{i}\geq 2$ and
$\frac{d_{i}^{2}\cdot[F_{i}:\mathbb{Q}]}{2}\,\big{|}\dim{B_{i}}$ (moreover
$d_{i}\leq 2$ if $F_{i}/\mathbb{Q}$ is totally real). Let $j$ be such that
$n_{j}\dim{B_{j}}\leq n_{i}\dim{B_{i}}$ for all $i$ — thus
$n_{j}\dim{B_{j}}\leq\frac{g}{m}$. Note that
$E\hookrightarrow\mathrm{End}_{K}^{0}(B_{i}^{\times n_{i}})=M_{n_{i}}(D_{i})$
for all $i$ (since $E$ is a field and the identity is in the image). After
tensoring up to $\mathbb{C}$ (i.e. considering
$E\otimes_{\mathbb{Q}}\mathbb{C}\cong\mathbb{C}^{\oplus g}\hookrightarrow
M_{n_{i}}(D_{i}\otimes_{\mathbb{Q}}\mathbb{C})\cong
M_{d_{i}n_{i}}(\mathbb{C})^{\oplus[F_{i}:\mathbb{Q}]}$) we find that $g\leq
n_{i}d_{i}\cdot[F_{i}:\mathbb{Q}]$ — this forces $g\leq n_{i}\dim{B_{i}}$ if
either $F_{i}/\mathbb{Q}$ is totally real or if $d_{i}\geq 2$, and $g\leq
2n_{i}\dim{B_{i}}$ otherwise. It follows that $m\leq 2$. If $m=1$ then
$g=n_{1}\dim{B_{1}}$ so $\dim{B_{1}}\leq d_{1}\cdot[F_{1}:\mathbb{Q}]$, whence
$d_{1}\leq 2$ since
$\frac{d_{1}^{2}\cdot[F_{1}:\mathbb{Q}]}{2}\,\big{|}\dim{B_{1}}$ if $d_{1}\geq
2$, and we are done because quaternion algebras contain many CM quadratic
extensions. Otherwise $m=2$, whence, for all $i$, $g=2n_{i}\dim{B_{i}}$ and
moreover $D_{i}=F_{i}$ is imaginary CM of degree
$[F_{i}:\mathbb{Q}]=2\dim{B_{i}}=\frac{g}{n_{i}}$. In other words both
$B_{1}/K$ and $B_{2}/K$ admit sufficiently many complex multiplications over
$K$. ∎

Thus to prove Lemma 3.6 we must rule out CM factors — we will do so via the
following.

######  Lemma 3.8.

Let $K/\mathbb{Q}$ be a number field with a real place
$K\hookrightarrow\mathbb{R}$. Let $L/\mathbb{Q}$ be an imaginary CM field. Let
$\Phi\subseteq\mathop{\text{Hom}}_{\mathbb{Q}\text{-alg.}}(L,\mathbb{C})$ be a
CM type of $L$. Let $A/K$ be an abelian variety admitting a map
$L\hookrightarrow\mathrm{End}_{K}^{0}(A)$. Write

$\mathrm{Lie}(A/\mathbb{C})\cong\bigoplus_{\sigma\in\Phi}\sigma^{a_{\sigma}}\oplus\overline{\sigma}^{b_{\sigma}}$

as complex representations of $L$ (the base change of $A/K$ taken along
$K\hookrightarrow\mathbb{R}\hookrightarrow\mathbb{C}$) — thus
$a_{\sigma},b_{\sigma}\in\mathbb{N}$. Then: $a_{\sigma}=b_{\sigma}$ for all
$\sigma\in\Phi$.

###### Proof.

$L$ preserves $\mathrm{Lie}(A/K)$ by hypothesis, so that the traces of
$L\curvearrowright\mathrm{Lie}(A/\mathbb{C})$ all lie in $K$. For each
$\sigma\in\Phi$, use Minkowski to produce an $x\in\mathfrak{o}_{L}$ with
$\sigma(x)$ large and almost purely imaginary, and $\tau(x)$ tiny for every
other $\sigma\neq\tau\in\Phi$. Since
$\sum_{\tau\in\Phi}a_{\tau}\cdot\tau(x)+b_{\tau}\cdot\overline{\tau(x)}\in
K\hookrightarrow\mathbb{R}$, it follows that $a_{\sigma}=b_{\sigma}$. ∎

Lemma 3.6 follows immediately.

###### Proof of Lemma 3.6..

By Lemma 3.7 it suffices to show that there are no abelian varieties $B/K$
with sufficiently many complex multiplications over $K$. But this follows from
Lemma 3.8 (for such abelian varieties $a_{\sigma}+b_{\sigma}=1$ for all
$\sigma\in\Phi$). ∎

###  3.4 The residual images of relevant compatible families of two-
dimensional Galois representations.

The following is an effectivization and generalization of a theorem of
Dimitrov. We note that it is also an improvement on Lemma $7.1.3$ of the ten-
author paper [1], which takes as input a compatible family and concludes the
same for a Dirichlet density $1$ subset of the primes. It is also a key input
in [2], which proves a "height-free" isogeny estimate for
$\mathrm{GL}_{2}$-type abelian varieties.

######  Lemma 3.9 (Cf. Propositions $3.1$ and $3.8$ of Dimitrov’s [12], and
Lemma $7.1.3$ of the ten-author paper [1].).

Let $d\in\mathbb{Z}^{+}$. Let $K/\mathbb{Q}$ be a number field. Let
$N\in\mathbb{Z}^{+}$. Let $\ell$ be a prime of $\mathbb{Z}$ which is prime to
$N$.

Then: there is an explicit (thus effectively computable) constant
$C_{d,K,N,\ell}\in\mathbb{Z}^{+}$, depending explicitly and only on $d$, $K$,
$N$, and $\ell$, such that the following holds. Let $p\geq C_{d,K,N,\ell}$ be
a prime of $\mathbb{Z}$. Let $E/\mathbb{Q}$ be a number field of degree
$[E:\mathbb{Q}]\leq d$. Let $\lambda,\mathfrak{p}\subseteq\mathfrak{o}_{E}$ be
primes of $\mathfrak{o}_{E}$ with $\lambda|(\ell)$ and $\mathfrak{p}|(p)$. Let
$\rho_{\lambda},\rho_{\mathfrak{p}}$ be representations

$\displaystyle\rho_{\lambda}:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)$
$\displaystyle\to\mathrm{GL}_{2}(\mathfrak{o}_{E,\lambda})$
$\displaystyle\rho_{\mathfrak{p}}:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)$
$\displaystyle\to\mathrm{GL}_{2}(\mathfrak{o}_{E,\mathfrak{p}})$

satisfying the following:

  * •

for all $v|\infty$ and primes $\mathfrak{m}\subseteq\mathfrak{o}_{K}$ of
$\mathfrak{o}_{K}$ with $\mathfrak{m}\nmid(N\ell p)$ and such that
$\mathrm{Nm}\,{\mathfrak{m}}\leq C_{d,K,N,\ell}$,

$\mathrm{tr}(\rho_{\lambda}(\mathrm{Frob}_{\mathfrak{m}}))=\mathrm{tr}(\rho_{\mathfrak{p}}(\mathrm{Frob}_{\mathfrak{m}}))\in\mathfrak{o}_{E}$

and

$|\mathrm{tr}(\rho_{\lambda}(\mathrm{Frob}_{\mathfrak{m}}))|_{v}\leq
10^{10}\cdot(\mathrm{Nm}\,{\mathfrak{m}})^{10^{10}},$

  * •

$\rho_{\lambda}\otimes_{\mathfrak{o}_{E,\lambda}}\overline{\mathbb{Q}}_{\ell}$
is irreducible, unramified at primes not dividing $N\ell$, and not of the form
$\mathrm{Ind}_{\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/L)}^{\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)}{(\chi)}$
with $L/K$ quadratic and
$\chi:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/L)\to\overline{\mathbb{Q}}_{\ell}^{\times}$
a character,

  * •

$\rho_{\mathfrak{p}}\otimes_{\mathfrak{o}_{E,\mathfrak{p}}}E_{\mathfrak{p}}$
has conductor dividing $(N)\cdot(p)^{\infty}$ and is crystalline with Hodge-
Tate weights $\\{0,-1\\}$ under all embeddings
$K\hookrightarrow\overline{\mathbb{Q}}_{p}$.

Then: writing the mod-$\mathfrak{p}$ residual representation as
$\overline{\rho}_{\mathfrak{p}}:=\rho_{\mathfrak{p}}\otimes_{\mathfrak{o}_{E,\mathfrak{p}}}\mathfrak{o}_{E,\mathfrak{p}}/\mathfrak{p}$,
there is a $g\in\mathrm{GL}_{2}(\mathfrak{o}_{E}/\mathfrak{p})$ and a subfield
$\mathbb{F}_{q}\subseteq\mathfrak{o}_{E}/\mathfrak{p}$ such that:

$g\cdot\mathrm{SL}_{2}(\mathbb{F}_{q})\cdot
g^{-1}\subseteq\overline{\rho}_{\mathfrak{p}}(\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K))\subseteq(\mathfrak{o}_{E}/\mathfrak{p})^{\times}\cdot(g\cdot\mathrm{GL}_{2}(\mathbb{F}_{q})\cdot
g^{-1}).$

(Thus in particular the Taylor-Wiles hypothesis at $\mathfrak{p}$, namely that

$(\overline{\rho}_{\mathfrak{p}}\otimes_{\mathfrak{o}_{E}/\mathfrak{p}}\overline{\mathbb{F}}_{p})|_{\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K(\zeta_{p}))}:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K(\zeta_{p}))\to\mathrm{GL}_{2}(\overline{\mathbb{F}}_{p})$

is irreducible, holds.)

The last line follows because $\mathrm{SL}_{2}(\mathbb{F}_{p})$ is perfect, so
that the composition

$g\cdot\mathrm{SL}_{2}(\mathbb{F}_{p})\cdot
g^{-1}\hookrightarrow\overline{\rho}_{\mathfrak{p}}(\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K))\twoheadrightarrow\overline{\rho}_{\mathfrak{p}}(\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K))/\overline{\rho}_{\mathfrak{p}}(\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K(\zeta_{p}))$

is trivial (since the latter is surjected upon by
$\mathop{\mathrm{Gal}}(K(\zeta_{p})/K)\hookrightarrow\mathbb{F}_{p}^{\times}$),
and
$\mathrm{SL}_{2}(\mathbb{F}_{p})\hookrightarrow\mathrm{GL}_{2}(\overline{\mathbb{F}}_{p})$
is evidently irreducible (consider the torus and then the unipotents).

When $K/\mathbb{Q}$ is totally real our argument largely parallels Dimitrov’s
except for a trick we introduce, which informally amounts to "moving a
congruence in a compatible family until it becomes an equality on a Faltings-
Serre set".

###### Proof.

Because $\mathrm{SL}_{2}(\mathbb{F}_{q})$ is its own commutator subgroup, it
suffices to show that there is a subfield
$\mathbb{F}_{q}\subseteq\mathfrak{o}_{E}/\mathfrak{p}$ such that the
corresponding projective representation
$\mathbb{P}(\overline{\rho}_{\mathfrak{p}}):\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)\to\mathrm{PGL}_{2}(\mathfrak{o}_{E}/\mathfrak{p})$
has image conjugate to a subgroup of
$\mathrm{PGL}_{2}(\mathfrak{o}_{E}/\mathfrak{p})$ which contains
$\mathrm{PSL}_{2}(\mathbb{F}_{q})$ and which is contained in
$\mathrm{PGL}_{2}(\mathbb{F}_{q})$. By a classical theorem of Dickson [11, see
pages $285$-$286$ of Ch. XII], it then suffices to show that
$\overline{\rho}_{\mathfrak{p}}$ is absolutely irreducible, is (absolutely)
not induced, and that $\mathbb{P}(\overline{\rho}_{\mathfrak{p}})$ does not
have image of size $\ll 1$.

Because we are going to eventually reduce showing that
$\overline{\rho}_{\mathfrak{p}}$ is not induced to showing irreducibility over
one of an explicit finite set of quadratic extensions of $K$, for clarity we
introduce a second parameter $C_{d,K,N,\ell}^{\prime}\leq C_{d,K,N,\ell}$
which will be taken to be explicitly sufficiently large.

Let $L/\mathbb{Q}$ be a Galois closure of $K/\mathbb{Q}$, let
$\widetilde{E}/\mathbb{Q}$ denote a compositum of $L$ and a Galois closure of
$E/\mathbb{Q}$, and let
$\widetilde{\mathfrak{p}}\subseteq\mathfrak{o}_{\widetilde{E}}$ be a prime of
$\mathfrak{o}_{\widetilde{E}}$ with $\widetilde{\mathfrak{p}}|\mathfrak{p}$.
(Thus e.g.
$[\widetilde{E}:\mathbb{Q}]\leq[E:\mathbb{Q}]!\cdot[K:\mathbb{Q}]!$.)

Now let us note a few consequences of crystallineness of
$\rho_{\mathfrak{p}}\otimes_{\mathfrak{o}_{E,\mathfrak{p}}}E_{\mathfrak{p}}$.
First, because $p$ is large, $\rho_{\mathfrak{p}}$ is Fontaine-Laffaille with
Hodge-Tate weights $\\{0,-1\\}$ under all embeddings
$K\hookrightarrow\overline{\mathbb{Q}}_{p}$. Thus, because this property
passes to subquotients, and because the Fontaine-Laffaille-module functor is
exact, it follows that $\overline{\rho}_{\mathfrak{p}}$ is Fontaine-Laffaille
with the given weights.

Let $\mathfrak{q}\subseteq\mathfrak{o}_{K}$ with $\mathfrak{q}|(p)$ be a prime
of $\mathfrak{o}_{K}$ above $p$. Write
$I_{\mathfrak{q}}\subseteq\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)$ for
the corresponding inertia subgroup (via a chosen
$K_{\mathfrak{q}}\hookrightarrow\overline{\mathbb{Q}}_{p}$). If
$0\to\alpha\to\overline{\rho}_{\mathfrak{p}}\otimes_{\mathfrak{o}_{E}/\mathfrak{p}}\overline{\mathbb{F}}_{p}\to\beta\to
0$ with
$\alpha,\beta:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)\to\overline{\mathbb{F}}_{p}^{\times}$
characters, then, because $\alpha$ and $\beta$ are subquotients of
$\overline{\rho}_{\mathfrak{p}}\otimes_{\mathfrak{o}_{E}/\mathfrak{p}}\overline{\mathbb{F}}_{p}$,
they too are Fontaine-Laffaille. It moreover follows that there is a partition
$\mathop{\text{Hom}}_{\mathbb{Q}\text{-alg.}}(K_{\mathfrak{q}},\widetilde{E}_{\widetilde{\mathfrak{p}}})=S_{\alpha}\bigsqcup
S_{\beta}$ such that $\alpha|_{I_{\mathfrak{q}}}$ has weight $-1$ at an
embedding
$\sigma:K_{\mathfrak{q}}\hookrightarrow\widetilde{E}_{\widetilde{\mathfrak{p}}}$
if and only if $\sigma\in S_{\alpha}$.

Because they are Fontaine-Laffaille characters (or for example by twisting by
Lubin-Tate characters, or else via Theorem $3.4.3$ of Raynaud’s [21]), it
follows that $\alpha$ and $\beta$ are products of the fundamental characters
corresponding to the respective subsets $S_{\alpha}$ and $S_{\beta}$:

$\displaystyle\alpha|_{I_{\mathfrak{q}}}$ $\displaystyle\equiv\prod_{\sigma\in
S_{\alpha}}\sigma\mkern 4.0mu({\operator@font mod}\mkern
6.0mu\widetilde{\mathfrak{p}}),$ $\displaystyle\beta|_{I_{\mathfrak{q}}}$
$\displaystyle\equiv\prod_{\sigma\in S_{\beta}}\sigma\mkern
4.0mu({\operator@font mod}\mkern 6.0mu\widetilde{\mathfrak{p}}),$

where we are implicitly identifying the inertia subgroup
$I_{\mathfrak{q}}^{\text{ab.}}$ of
$\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}_{p}/K_{\mathfrak{q}})^{\text{ab.}}$
with $\mathfrak{o}_{K,\mathfrak{q}}^{\times}$ via local class field theory.

Now let us go through the cases in Dickson’s classification one by one.

The third case is easily dealt with: if
$\mathbb{P}(\overline{\rho}_{\mathfrak{p}})$ has image of size $\ll 1$, then,
writing
$K^{\prime}:=\overline{\mathbb{Q}}^{\ker{\mathbb{P}(\overline{\rho}_{\mathfrak{p}})}}$
(thus $[K^{\prime}:K]\ll 1$), there is a character
$\alpha:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K^{\prime})\to(\mathfrak{o}_{E}/\mathfrak{p})^{\times}$
such that
$\overline{\rho}_{\mathfrak{p}}(g)\equiv\alpha(g)\cdot\mathop{\mathrm{id}}\mkern
4.0mu({\operator@font mod}\mkern 6.0mu\mathfrak{p})$ for all
$g\in\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K^{\prime})\subseteq\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)$
— thus in particular
$\det{\overline{\rho}_{\mathfrak{p}}(g)}\equiv\alpha(g)^{2}\mkern
4.0mu({\operator@font mod}\mkern 6.0mu\mathfrak{p})$. In other words, letting
$\mathfrak{q}^{\prime}\subseteq\mathfrak{o}_{K^{\prime}}$ with
$\mathfrak{q}^{\prime}|(p)$ be a prime above $p$, because
$e(\mathfrak{q}^{\prime}/p)\leq[K^{\prime}:K]\ll 1$ (and so still $p\gg
e(\mathfrak{q}^{\prime}/p)$) the above reasoning gives that
$\alpha|_{I_{\mathfrak{q}^{\prime}}}$ is a product of fundamental characters
with multiplicities at most $e(\mathfrak{q}^{\prime}/p)\ll 1$, contradicting
$\chi_{p}|_{I_{\mathfrak{q}^{\prime}}}\equiv\alpha^{2}|_{I_{\mathfrak{q}^{\prime}}}\mkern
4.0mu({\operator@font mod}\mkern 6.0mu\mathfrak{p})$. So the third case
follows.

Now let us show absolute irreducibility. Suppose for the sake of contradiction
that
$\overline{\rho}_{\mathfrak{p}}\otimes_{\mathfrak{o}_{E}/\mathfrak{p}}\overline{\mathbb{F}}_{p}$
is reducible. Then there is a number field $E^{\prime}/E$ of degree
$[E^{\prime}:E]\leq 2$, a prime
$\mathfrak{p}^{\prime}\subseteq\mathfrak{o}_{E^{\prime}}$ of
$\mathfrak{o}_{E^{\prime}}$ with $\mathfrak{p}^{\prime}|\mathfrak{p}$, and
abelian characters
$\psi,\psi^{\prime}:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)\to(\mathfrak{o}_{E^{\prime}}/\mathfrak{p}^{\prime})^{\times}$
such that

$\mathrm{tr}\circ\rho_{\mathfrak{p}}\equiv\psi+\psi^{\prime}\mkern
4.0mu({\operator@font mod}\mkern 6.0mu\mathfrak{p}^{\prime}).$

Let $E^{\prime\prime}/\mathbb{Q}$ be a compositum of $E^{\prime}/\mathbb{Q}$
and $\widetilde{E}/\mathbb{Q}$, and let
$\mathfrak{p}^{\prime\prime}\subseteq\mathfrak{o}_{E^{\prime\prime}}$ be a
prime of $\mathfrak{o}_{E^{\prime\prime}}$ with
$\mathfrak{p}^{\prime\prime}|\mathfrak{p}^{\prime}$.

By virtue of being subquotients of $\overline{\rho}_{\mathfrak{p}}$, the
characters $\psi$ and $\psi^{\prime}$ have conductors dividing
$(\Delta_{N})\cdot(p)^{\infty}$ with
$\Delta_{N}:=N^{(10^{10}\cdot[K:\mathbb{Q}]\cdot[E:\mathbb{Q}])!}$. Therefore
by class field theory we may also regard them as characters of the narrow ray
class group
$\mathrm{Cl}^{+}_{(\Delta_{N})\cdot(p)^{\infty}}(K):=\varprojlim_{k\in\mathbb{Z}^{+}}\mathrm{Cl}^{+}_{(\Delta_{N})\cdot(p)^{k}}(K)$
associated to the conductor $(\Delta_{N})\cdot(p)^{\infty}$. Again by class
field theory we have the exact sequence:

$0\to\left(\prod_{\mathfrak{q}|(p)}\mathfrak{o}_{K,\mathfrak{q}}^{\times}\right)/\overline{\\{\eta\in\mathfrak{o}_{K}^{\times}:\eta\equiv
1\mkern 4.0mu({\operator@font mod}\mkern 6.0mu\Delta_{N}),\eta\text{ tot.\
pos.}\\}}\to\mathrm{Cl}^{+}_{(\Delta_{N})\cdot(p)^{\infty}}(K)\to\mathrm{Cl}^{+}_{(\Delta_{N})}(K)\to
0,$

where $\overline{\\{\eta\in\mathfrak{o}_{K}^{\times}:\eta\equiv 1\mkern
4.0mu({\operator@font mod}\mkern 6.0mu\Delta_{N}),\eta\text{ tot.\ pos.}\\}}$
denotes the closure of $\\{\eta\in\mathfrak{o}_{K}^{\times}:\eta\equiv 1\mkern
4.0mu({\operator@font mod}\mkern 6.0mu\Delta_{N}),\eta\text{ tot.\
pos.}\\}\subseteq\prod_{\mathfrak{q}|(p)}\mathfrak{o}_{K,\mathfrak{q}}^{\times}$
under the diagonal embedding and in the product topology.

As we saw in our analysis of the consequences of the Fontaine-Laffaille
assumption, for every prime $\mathfrak{q}\subseteq\mathfrak{o}_{K}$ of
$\mathfrak{o}_{K}$ with $\mathfrak{q}|(p)$, there is a partition
$Z_{\mathfrak{q}}\bigsqcup
Z_{\mathfrak{q}}^{\prime}=\mathop{\text{Hom}}_{\mathbb{Q}\text{-alg.}}(K_{\mathfrak{q}},E^{\prime\prime}_{\mathfrak{p}^{\prime\prime}})$
of
$\mathop{\text{Hom}}_{\mathbb{Q}\text{-alg.}}(K_{\mathfrak{q}},E^{\prime\prime}_{\mathfrak{p}^{\prime\prime}})\subseteq\mathop{\text{Hom}}_{\mathbb{Q}\text{-alg.}}(K,E^{\prime\prime})=\mathop{\text{Hom}}_{\mathbb{Q}\text{-alg.}}(K,L)$
such that $\psi|_{I_{\mathfrak{q}}}\equiv\prod_{\sigma\in
Z_{\mathfrak{q}}}\sigma\mkern 4.0mu({\operator@font mod}\mkern
6.0mu\mathfrak{p}^{\prime\prime})$ and
$\psi^{\prime}|_{I_{\mathfrak{q}}}\equiv\prod_{\sigma\in
Z_{\mathfrak{q}}^{\prime}}\sigma\mkern 4.0mu({\operator@font mod}\mkern
6.0mu\mathfrak{p}^{\prime\prime})$ (after implicitly composing with the local
class field theory isomorphism
$I_{\mathfrak{q}}^{\text{ab.}}\simeq\mathfrak{o}_{K,\mathfrak{q}}^{\times}$).

Let then
$Z:=\bigsqcup_{\mathfrak{q}|(p)}Z_{\mathfrak{q}}\subseteq\mathop{\text{Hom}}_{\mathbb{Q}\text{-alg.}}(K,L)$
and $Z^{\prime}:=\bigsqcup_{\mathfrak{q}|(p)}Z_{\mathfrak{q}}^{\prime}$.

Regarding $\psi$ as a character of
$\mathrm{Cl}^{+}_{(\Delta_{N})\cdot(p)^{\infty}}(K)$ and evaluating at a
totally positive unit $\eta\in\mathfrak{o}_{K}^{\times}$ with $\eta\equiv
1\mkern 4.0mu({\operator@font mod}\mkern 6.0mu\Delta_{N})$, we find that

$1\equiv\prod_{\sigma\in Z}\sigma(\eta)\mkern 4.0mu({\operator@font mod}\mkern
6.0mu\mathfrak{p}^{\prime\prime}).$

Now, by Minkowski the finite-index subgroup
$\\{\eta\in\mathfrak{o}_{K}^{\times}:\eta\equiv 1\mkern 4.0mu({\operator@font
mod}\mkern 6.0mu\Delta_{N}),\eta\text{ tot.\
pos.}\\}\subseteq\mathfrak{o}_{K}^{\times}$ is generated by elements of
explicitly bounded height (in terms of $K$ and $N$). Thus so long as
$C_{d,K,N,\ell}^{\prime}$ is explicitly sufficiently large in terms of $K$ and
$N$, which we may and will assume, at least on the generators the congruence
$1\equiv\prod_{\sigma\in Z}\sigma(\eta)\mkern 4.0mu({\operator@font mod}\mkern
6.0mu\mathfrak{p}^{\prime\prime})$ implies an equality $1=\prod_{\sigma\in
Z}\sigma(\eta)$, and then this equality extends to the whole of
$\\{\eta\in\mathfrak{o}_{K}^{\times}:\eta\equiv 1\mkern 4.0mu({\operator@font
mod}\mkern 6.0mu\Delta_{N}),\eta\text{ tot.\ pos.}\\}$ by multiplicativity.
Therefore the homomorphism
$\mathfrak{o}_{K}^{\times}\to\mathfrak{o}_{L}^{\times}$ via
$\eta\mapsto\prod_{\sigma\in Z}\sigma(\eta)$ has finite image, and thus its
image lies in the roots of unity of $L$.

Choosing an embedding $i:L\hookrightarrow\mathbb{C}$, we see that the
homomorphism $\mathfrak{o}_{K}^{\times}\to\mathbb{R}^{+}$ via
$\eta\mapsto\prod_{\sigma\in Z}|(i\circ\sigma)(\eta)|$ is therefore trivial.
So let us redo the analysis that one does when classifying algebraic Hecke
characters.

Recall that Dirichlet’s unit theorem may be rephrased as the statement that:
on any torsion-free finite-index subgroup
$G\subseteq\mathfrak{o}_{K}^{\times}$, the only relations among the
multiplicative characters $G\to\mathbb{R}^{+}$ corresponding to embeddings
$i\circ\sigma:K\hookrightarrow\mathbb{C}$ via
$\eta\mapsto|(i\circ\sigma)(\eta)|$ are those generated by the norm relation
$\prod_{\sigma:K\hookrightarrow\mathbb{C}}|(i\circ\sigma)|_{G}|=1$ and the
"trivial" relations $|(i\circ\sigma)|_{G}|=|(i\circ\sigma^{\prime})|_{G}|$
when $i\circ\sigma,i\circ\sigma^{\prime}:K\hookrightarrow\mathbb{C}$ are
complex-conjugate embeddings.

It follows that: either $Z=\varnothing$,
$Z=\mathop{\text{Hom}}_{\mathbb{Q}\text{-alg.}}(K,L)$, or else
$Z^{\prime}=Z^{\text{conj.}}$, where
$Z^{\text{conj.}}\subseteq\mathop{\text{Hom}}_{\mathbb{Q}\text{-alg.}}(K,L)$
is the set of
$\sigma^{\prime}\in\mathop{\text{Hom}}_{\mathbb{Q}\text{-alg.}}(K,L)$ for
which there is a $\sigma\in Z$ for which $i\circ\sigma$ and
$i\circ\sigma^{\prime}$ are complex-conjugate embeddings
$K\hookrightarrow\mathbb{C}$.

Let us discuss the last of these possibilities, which we will call the "CM
case", for a moment. Note that the CM case is independent of the choice of $i$
(by considering $\\#|Z|$, for example). Writing $K^{\text{CM}}\subseteq K$ for
the maximal CM subfield of $K/\mathbb{Q}$, $L^{\text{CM}}\subseteq L$ for the
maximal CM subfield of $L/\mathbb{Q}$, and
$H:=\mathop{\mathrm{Gal}}(L/L^{\text{CM}})\subseteq\mathop{\mathrm{Gal}}(L/\mathbb{Q})$
for the normal subgroup (generated by products of pairs of complex
conjugations) corresponding to $L^{\text{CM}}\subseteq L$, we see that $H$
fixes $Z$ as a set and that $K^{\text{CM}}/\mathbb{Q}$ and
$L^{\text{CM}}/\mathbb{Q}$ are imaginary CM. Further choosing a $\sigma\in Z$,
it follows that the set $Z$ is the set of embeddings $K\hookrightarrow L$
extending an embedding in the uniquely determined111111$\Phi$ is just the set
of those embeddings that are restrictions of embeddings in $Z$. CM type
$\Phi\subseteq\mathop{\text{Hom}}_{\mathbb{Q}\text{-alg.}}(K^{\text{CM}},L)$
of $K^{\text{CM}}=\sigma^{-1}(L^{\text{CM}}\cap\sigma(K))$.

We conclude that in all three cases there are weight $\leq 1$ algebraic Hecke
characters $\chi$ and $\chi^{\prime}$ of $K$ valued in $L$ and thus
$E^{\prime\prime}$ such that the finite-order Galois characters
$\psi\cdot\chi^{-1}\mkern 4.0mu({\operator@font mod}\mkern
6.0mu\mathfrak{p}^{\prime\prime})$ and
$\psi^{\prime}\cdot\chi^{\prime-1}\mkern 4.0mu({\operator@font mod}\mkern
6.0mu\mathfrak{p}^{\prime\prime})$ are unramified outside primes dividing $N$.
Indeed: when $Z=\varnothing$ we take $\chi=\mathop{\mathrm{triv}}$ and
$\chi^{\prime}$ to be the cyclotomic character, when
$Z=\mathop{\text{Hom}}_{\mathbb{Q}\text{-alg.}}(K,L)$ we take $\chi$ to be the
cyclotomic character and $\chi^{\prime}$ to be trivial, and in the CM case we
take $\chi$, respectively $\chi^{\prime}$ to be the algebraic Hecke character
on $K$ corresponding to the CM type $\Phi$, respectively its complement, on
$K^{\text{CM}}$. In other words (absorbing the finite-order characters into
the notation) we have concluded that there are weight $\leq 1$ algebraic Hecke
characters $\widetilde{\chi}$ and $\widetilde{\chi}^{\prime}$ of conductor
dividing $(\Delta_{N})$ and valued in $E^{\prime\prime}$ such that

$\mathrm{tr}\circ\overline{\rho}_{\mathfrak{p}}\equiv\widetilde{\chi}+\widetilde{\chi}^{\prime}\mkern
4.0mu({\operator@font mod}\mkern 6.0mu\mathfrak{p}^{\prime\prime}),$

where we have abused notation by also using $\widetilde{\chi}$ and
$\widetilde{\chi}^{\prime}$ for the corresponding
$\mathfrak{p}^{\prime\prime}$-adic Galois characters.

Now it is time for our trick. It follows that for all primes
$\mathfrak{m}\subseteq\mathfrak{o}_{K}$ of $\mathfrak{o}_{K}$ such that
$\mathfrak{m}\nmid(N\ell)$ and such that $\mathrm{Nm}\,{\mathfrak{m}}\leq
C_{d,K,N,\ell}^{\prime}$,

$\displaystyle\mathrm{tr}(\rho_{\lambda}(\mathrm{Frob}_{\mathfrak{m}}))$
$\displaystyle=\mathrm{tr}(\rho_{\mathfrak{p}}(\mathrm{Frob}_{\mathfrak{m}}))$
$\displaystyle\equiv\widetilde{\chi}(\mathfrak{m})+\widetilde{\chi}^{\prime}(\mathfrak{m})\mkern
4.0mu({\operator@font mod}\mkern 6.0mu\mathfrak{p}^{\prime\prime}),$

so that by comparing sizes of the left- and right-hand sides we see that
$\mathrm{tr}(\rho_{\lambda}(\mathrm{Frob}_{\mathfrak{m}}))=\widetilde{\chi}(\mathfrak{m})+\widetilde{\chi}^{\prime}(\mathfrak{m})$
for all such $\mathfrak{m}$. By Lemma 3.1, upon choosing
$C_{d,K,N,\ell}^{\prime}$ suitably it follows that the representation
$\rho_{\lambda}$ is reducible (into the direct sum of the $\ell$-adic Galois
characters corresponding to $\widetilde{\chi}$ and
$\widetilde{\chi}^{\prime}$), a contradiction. We conclude that
$\overline{\rho}_{\mathfrak{p}}$ is indeed absolutely irreducible.

It remains only to show that $\overline{\rho}_{\mathfrak{p}}$ is not
absolutely induced. We will do this by reducing to the same argument over an
explicit finite extension $L/K$ depending only on $K$ and $N$. Suppose for the
sake of contradiction that $\overline{\rho}_{\mathfrak{p}}$ is absolutely
induced. Write $L/K$ for the corresponding quadratic extension, and
$\chi:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/L)\to\overline{\mathbb{F}}_{p}^{\times}$
for the corresponding character (so that
$\overline{\rho}_{\mathfrak{p}}\otimes_{\mathfrak{o}_{E}/\mathfrak{p}}\overline{\mathbb{F}}_{p}\cong\mathrm{Ind}_{\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/L)}^{\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)}(\chi)$).

Note that $L/K$ is automatically unramified outside primes dividing $Np$. We
claim that it is in fact unramified at primes above $p$ as well. Indeed, on
the one hand we have already seen that, for all primes
$\mathfrak{q}\subseteq\mathfrak{o}_{K}$ of $\mathfrak{o}_{K}$ with
$\mathfrak{q}|(p)$, $\overline{\rho}_{\mathfrak{p}}|_{I_{\mathfrak{q}}}$ is
reducible, i.e. it admits a stable line, or equivalently
$\mathbb{P}(\overline{\rho}_{\mathfrak{p}}(I_{\mathfrak{q}}))\subseteq\mathrm{PGL}_{2}(\mathfrak{o}_{E}/\mathfrak{p})$
has a fixed point in $\mathbb{P}^{1}(\mathfrak{o}_{E}/\mathfrak{p})$. On the
other hand, were $L/K$ ramified at $\mathfrak{q}$ then, writing
$\mathfrak{q}^{\prime}|\mathfrak{q}$ for the prime of $\mathfrak{o}_{L}$ above
$\mathfrak{q}$, the injection
$I_{\mathfrak{q}}/I_{\mathfrak{q}^{\prime}}\hookrightarrow\mathop{\mathrm{Gal}}(L/K)$
would be an isomorphism. Thus
$\mathrm{Ind}_{\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/L)}^{\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)}(\chi)\big{|}_{I_{\mathfrak{q}}}\simeq\mathrm{Ind}_{I_{\mathfrak{q}^{\prime}}}^{I_{\mathfrak{q}}}(\chi|_{I_{\mathfrak{q}}^{\prime}})$
(for example by Mackey). Thus in the canonical basis there is a lift
$\sigma\in I_{\mathfrak{q}}$ of the nontrivial element of
$\mathop{\mathrm{Gal}}(L/K)$ which acts by
$\left(\begin{array}[]{cc}0&\chi(\sigma^{2})\\\ 1&0\end{array}\right)$, while
a $g\in I_{\mathfrak{q}^{\prime}}$ acts by
$\left(\begin{array}[]{cc}\chi(g)&0\\\ 0&\chi(\sigma^{-1}\cdot
g\cdot\sigma)\end{array}\right)$, and for these to have a common fixed point
in $\mathbb{P}^{1}(\overline{\mathbb{F}}_{p})$ we must have that
$\chi(g)=\chi(\sigma^{-1}\cdot g\cdot\sigma)$ for all $g\in
I_{\mathfrak{q}^{\prime}}$. But then
$\mathbb{P}(\overline{\rho}_{\mathfrak{p}})|_{I_{\mathfrak{q}}}$ would be
trivial on all of $I_{\mathfrak{q}^{\prime}}$, and thus have image of order at
most $2$. Consequently we would be able to repeat the argument (which only
used that the image of
$\mathbb{P}(\overline{\rho}_{\mathfrak{p}})|_{I_{\mathfrak{q}}}$ had bounded
exponent) we used to deal with the third case of the Dickson classification to
deduce that $p\ll 1$, a contradiction. So $L/K$ is indeed unramified at
$\mathfrak{q}$ for all primes $\mathfrak{q}\subseteq\mathfrak{o}_{K}$ of
$\mathfrak{o}_{K}$ with $\mathfrak{q}|(p)$.

So we see that $L/K$ is a quadratic extension which is unramified outside
primes dividing $N$. By inspection or by Minkowski’s proof of the Hermite-
Minkowski theorem it follows that there is an explicit finite set
$\mathcal{F}$ of quadratic extensions of $K$ depending only on $K$ and $N$
such that $L\in\mathcal{F}$. We conclude that, were
$\overline{\rho}_{\mathfrak{p}}$ induced, then there would be an
$L\in\mathcal{F}$ such that
$\overline{\rho}_{\mathfrak{p}}|_{\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/L)}$
would be absolutely reducible. Therefore so long as we choose
$C_{d,K,N,\ell}:=1+\max_{L\in\mathcal{F}}(C_{d,L,N,\ell}^{\prime})^{2}$ (the
square because the relevant $L/K$ are quadratic), i.e. so long as the above
absolute irreducibility argument works not only for $K$ but for all
$L\in\mathcal{F}$ as well, it follows that $\overline{\rho}_{\mathfrak{p}}$ is
not absolutely induced. ∎

##  4 Proof of Theorem 1.1.

We will now prove Theorem 1.1. Rather than writing out an algorithm in
pseudocode and proving its correctness and that it always terminates as in
Chapter $9$ of [4], we will instead just describe the technique in words in
the course of the proof.

For the duration of this section, and for the sake of brevity, we will call an
$A/K$ "relevant" if and only if it is a $g$-dimensional abelian variety over
$K$ with good reduction outside $S$ and which is of $\mathrm{GL}_{2}$-type
over $K$.

First let us show that it suffices to produce an effectively computable finite
set $\mathcal{F}$ of odd-degree totally real extensions such that all relevant
$A/K$ are modular over at least one $L\in\mathcal{F}$.

######  Lemma 4.1.

Let $\mathcal{F}$ be a finite set of odd-degree totally real extensions of $K$
such that for all relevant $A/K$ there is an $L\in\mathcal{F}$ such that $A/L$
is modular. Then: there is an explicit (thus effectively computable) constant
$C_{g,K,S,\mathcal{F}}\in\mathbb{Z}^{+}$ depending explicitly and only on $g$,
$K$, $S$, and $\mathcal{F}$ such that, for all relevant $A/K$, $h(A)\leq
C_{g,K,S,\mathcal{F}}$.

###### Proof.

Without loss of generality we may assume $\mathcal{F}$ is a singleton, say
$\mathcal{F}=\\{L\\}$ (just take a maximum). We may further without loss of
generality assume $L=K$ since the absolute Faltings height is invariant under
base change — in other words we need only prove such a bound for those
relevant $A/K$ which are modular over $K$ itself.

To do this we will show that all relevant $A/K$ which are modular over $K$ are
$K$-isogeny factors of the $g$-th power of the Jacobian of an explicit Shimura
curve with level structure, though the notation will make the evident proof
(modularity produces an "$f$", Hida produces an $A_{f}$ with matching
$L$-function as such a quotient, so since $A$ and $A_{f}$ have matching
degree-two $L$-functions by Faltings there is a nontrivial $K$-map $A_{f}\to
A$ whence $A$ is a $K$-quotient of $B^{\dim{A}}$) quite clumsy.

Let $A/K$ be relevant and modular over $K$. Let $\mathfrak{o}$ be an order in
a CM field $F/\mathbb{Q}$ such that $[F:\mathbb{Q}]=g$ and
$\mathfrak{o}\hookrightarrow\mathrm{End}_{K}(A)$ (one exists by hypothesis).
Let $\ell\in\mathbb{Z}^{+}$ be an explicitly sufficiently large prime of
$\mathbb{Z}$ (so that e.g. $\ell$ is prime to the discriminant of
$\mathfrak{o}$ and all the norms of primes in $S$).

Let, via Lemma 3.6, $\widetilde{A}/K$ be $K$-simple, of $\mathrm{GL}_{2}$-type
over $K$, and such that
$A\sim_{K}\widetilde{A}^{\times(\dim{A}/\dim{\widetilde{A}})}$.

For each prime $\lambda\subseteq\mathfrak{o}$ of $\mathfrak{o}$ with
$\lambda|(\ell)$, write
$\rho_{A,\lambda}:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)\to\mathrm{GL}_{2}(\mathfrak{o}_{\lambda})$
for the representation corresponding to the $\lambda$-adic Tate module
$T_{\lambda}(A):=\varprojlim_{n}A[\lambda^{n}]$ of $A/K$. Write also
$\rho_{A,\ell}:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)\to\mathrm{GL}_{2g}(\mathbb{Z}_{\ell})$
for the representation corresponding to the $\ell$-adic Tate module
$T_{\ell}(A):=\varprojlim_{n}A[\ell^{n}]$ of $A/K$. Thus e.g.
$\rho_{A,\ell}\otimes_{\mathbb{Z}_{\ell}}\overline{\mathbb{Q}}_{\ell}\simeq\bigoplus_{\lambda|(\ell),\sigma:F_{\lambda}\hookrightarrow\overline{\mathbb{Q}}_{\ell}}\rho_{A,\lambda}\otimes_{\mathfrak{o}_{\lambda},\sigma}\overline{\mathbb{Q}}_{\ell}$.

Modularity entails the existence of a parallel weight $2$ Hilbert modular
eigencuspform $f$ over $K$ and a $\lambda\subseteq\mathfrak{o}$ with
$\lambda|(\ell)$ such that $L(s,\rho_{\lambda})=L(s,f)$. Said another way,
modularity entails the existence of a parallel weight $2$ Hilbert modular
eigencuspform $f$ such that $L(s,f)$ matches one of the degree-two
$L$-functions of $A/K$. Let $\mathfrak{n}\subseteq\mathfrak{o}_{K}$ be the
conductor of $A/K$ (thus $\mathrm{Nm}\,{\mathfrak{n}}\ll_{[K:\mathbb{Q}],S}1$
with explicit implied constant). By the equality of $\varepsilon$-factors in
Section $0.5$ of Carayol’s [8], it follows that the level of $f$ is
$\mathfrak{n}$.

Now by a standard construction of Hida (combine Theorem $4.4$ of Hida’s [16]
with Jacquet-Langlands transfer) there is an explicit abelian variety $B/K$,
namely the Jacobian of an explicit Shimura curve with level structure
depending explicitly on $\mathfrak{n}$, such that there is a $K$-isogeny
factor $A_{f}/K$ of $B/K$ which is of $\mathrm{GL}_{2}$-type over $K$ and such
that $L(s,f)$ matches one of the degree-two $L$-functions of $L(s,A_{f})$. It
follows that
$T_{\ell}(A)\otimes_{\mathbb{Z}_{\ell}}\overline{\mathbb{Q}}_{\ell}$ and
$T_{\ell}(A_{f})\otimes_{\mathbb{Z}_{\ell}}\overline{\mathbb{Q}}_{\ell}$ share
a nonzero summand, whence
$\mathop{\text{Hom}}_{K}(A,A_{f})\otimes_{\mathbb{Z}}\overline{\mathbb{Q}}_{\ell}\simeq\mathop{\text{Hom}}_{\mathbb{Z}_{\ell}[\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)]}(T_{\ell}(A),T_{\ell}(A_{f}))\otimes_{\mathbb{Z}_{\ell}}\overline{\mathbb{Q}}_{\ell}\neq
0$, where we have used Faltings’ proof of the Tate conjecture for
homomorphisms of abelian varieties. Thus $A/K$ and $A_{f}/K$ share a nonzero
$K$-isogeny factor (automatically of the form $\widetilde{A}^{\times k}$ for
some $k\in\mathbb{Z}^{+}$), whence $A$ is a $K$-isogeny factor of
$A_{f}^{\times g}$, whence also a $K$-isogeny factor of $B^{\times g}$.

Finally an application of Lemma 3.2 provides the desired explicit height
bound. ∎

So to prove Theorem 1.1 it suffices to prove that there is an effectively
computable finite set $\mathcal{F}$ of odd-degree totally real extensions of
$K$ such that all relevant $A/K$ are modular over some $L\in\mathcal{F}$. This
we do by explaining how to effectively compute the extension produced by an
argument of Snowden proving Taylor’s potential modularity theorem.

###### Proof of Theorem 1.1..

Let us now follow Snowden’s [24]. The first thing to note is that Proposition
$5.3.1$ of his [24] can evidently be improved to a statement asserting the
existence of a finite-time algorithm computing, in his notation, such a pair
$(F^{\prime}/F,A/F^{\prime})$ given an input
$(F,K,M,\mathfrak{p},\mathfrak{l},\rho_{\mathfrak{p}},\rho_{\mathfrak{l}},t_{\mathfrak{p}},t_{\mathfrak{l}})$
as in the statement of the proposition. Indeed, the moduli space $X$ he
produces is a quasiprojective variety (via the usual Satake/Baily-Borel
compactification), and he invokes Moret-Bailly’s theorem to prove $X$ has a
totally real point with certain properties that can easily be checked in
finite time. Because $X$ is quasiprojective, say via
$X\hookrightarrow\mathbb{P}^{N}$, we may simply enumerate the totally real
points of $\mathbb{P}^{N}$ of larger and larger height and degree, and in
finite time we will find a suitable point on $X$ by his proof of his
Proposition $5.3.1$.121212One should of course be able to be more explicit,
but this would require giving an explicit proof of Moret-Bailly’s theorem at
least in this case. So Snowden’s Proposition $5.3.1$ may be replaced by the
stronger statement that there is an effectively computable pair
$(F^{\prime}/F,A/F^{\prime})$ (corresponding to an effectively computable
totally real point on $X$, and effectively computable in terms of the input
$(F,K,M,\mathfrak{p},\mathfrak{l},\rho_{\mathfrak{p}},\rho_{\mathfrak{l}},t_{\mathfrak{p}},t_{\mathfrak{l}})$)
satisfying the conclusions of the proposition.

Now let $N:=10^{10}!\cdot\prod_{\mathfrak{q}\in
S}(\mathrm{Nm}\,{\mathfrak{q}})^{(10^{10}\cdot[K:\mathbb{Q}])!}$. Let
$\ell\in\mathbb{Z}^{+}$ be a prime of $\mathbb{Z}$ with $\ell>N$ and $\ell\ll
N$ with explicit implied constant. Let $p\in\mathbb{Z}^{+}$ be a prime of
$\mathbb{Z}$ produced by Lemma 3.9 on input $(g,K,N,\ell)$ and such that
$p\ll_{g,K,N,\ell}1$ with explicit implied constant.

Let $q:=p^{(10^{10}\cdot g)!}$. Let $\Phi$ be the explicit finite set of
(isomorphism classes of) representations
$\overline{\rho}:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)\to\mathrm{GL}_{2}(\mathbb{F}_{q})$
which are odd, unramified outside primes dividing $Np$, are such that
$\det{\overline{\rho}}\cdot\chi_{p}^{-1}$ is unramified outside primes
dividing $N$, and which have image containing a conjugate of
$\mathrm{SL}_{2}(\mathbb{F}_{p})\subseteq\mathrm{GL}_{2}(\mathbb{F}_{q})$.
(This is an explicit finite set of representations by Minkowski’s proof of the
Hermite-Minkowski theorem.) Note that all such $\overline{\rho}\in\Phi$
satisfy Snowden’s hypothesis "(A1)" (and of course his hypothesis "(A2)" since
$p$ is large).

Lemma 3.9 amounts to the statement that, for all relevant $A/K$, the mod-$p$
two-dimensional residual representation
$\overline{\rho}_{A,\overline{\mathbb{F}}_{p}}:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K)\to\mathrm{GL}_{2}(\overline{\mathbb{F}}_{p})$
corresponding to $A/K$ is an element of $\Phi$, in the sense that there is a
$\overline{\rho}\in\Phi$ with
$\overline{\rho}_{A,\overline{\mathbb{F}}_{p}}\cong\overline{\rho}\otimes_{\mathbb{F}_{q}}\overline{\mathbb{F}}_{p}$.

It is clear from Snowden’s proof (see especially his $(5.5)$) of Theorem
$5.1.1$ of his [24] that, upon replacing his Proposition $5.3.1$ with the
strengthening given above, one produces an effectively computable output
$(M^{\prime},F^{\prime})$ given the input $(\overline{\rho},\psi,M,t)$, to use
the notation of his Theorem $5.1.1$.

Let then $\mathcal{F}^{\prime}$ be the effectively computable finite set of
totally real Galois extensions $F^{\prime}$ produced by (the algorithm
implicit in) said improvement of Theorem $5.1.1$ applied to all tuples
$(\overline{\rho},\overline{\mathbb{Q}}^{\ker{\overline{\rho}}},t)$ with
$\overline{\rho}\in\Phi$ and (using Snowden’s terminology) $t$ a definite type
function on the primes of $K$ above $p$.

We claim that, for all relevant $A/K$, there is an $L\in\mathcal{F}^{\prime}$
such that $A/L$ is modular over $L$, and indeed this follows immediately from
$(5.6)$ of Snowden’s [24]. Indeed, by construction there is a
$\overline{\rho}\in\Phi$ such that
$\overline{\rho}_{A,\overline{\mathbb{F}}_{p}}\cong\overline{\rho}\otimes_{\mathbb{F}_{q}}\overline{\mathbb{F}}_{p}$,
and so by construction of $\mathcal{F}^{\prime}$ there is an
$L\in\mathcal{F}^{\prime}$ such that there is a Hilbert modular eigencuspform
$\widetilde{f}$ over $L$ with mod-$p$ residual representation
$\overline{\rho}$ and type function matching that of $A/L$ at primes of $L$
above $p$ (the type function of an abelian variety is definite by purity). By
Snowden’s Theorem $4.4.2$ we conclude that $A/L$ is modular.

So we have proven that there is a finite set of totally real Galois extensions
$\mathcal{F}^{\prime}$ such that all relevant $A/K$ are modular over some
$L\in\mathcal{F}^{\prime}$. The only thing left to note is that, just as in
Snowden’s proof of his Proposition $9.4.1$, if an abelian variety $A/K$ is
modular over an $L\in\mathcal{F}^{\prime}$, then, letting
$H\subseteq\mathop{\mathrm{Gal}}(L/K)$ be a $2$-Sylow subgroup (thus
solvable), by Langlands’ solvable descent for $\mathrm{GL}_{2}$ [17] it
follows that $A/K$ is modular over $L^{H}$.

So we let $\mathcal{F}$ be the effectively computable finite set of extensions
$L^{H}$ with $L\in\mathcal{F}^{\prime}$ and
$H\subseteq\mathop{\mathrm{Gal}}(L/K)$ a $2$-Sylow subgroup. We have therefore
shown that all relevant $A/K$ are modular over some $L\in\mathcal{F}$, so we
are done by Lemma 4.1. ∎

##  5 Proof of Corollary 1.3.

The proof of Corollary 1.3 is exactly the same as Parshin’s reduction of the
Mordell conjecture to the Shafarevich conjecture on abelian varieties via
Kodaira’s families: compactness provides an a priori estimate on the
conductors of abelian varieties occurring in a given family over a proper
curve.

###### Proof of Corollary 1.3..

Let $H_{\mathfrak{o}}$ be a Hilbert modular variety and $f:C\to
H_{\mathfrak{o}}$ a map, without loss of generality defined over $K$ itself,
witnessing the fact that $C/K$ is of type $\bigiy$ (such an $H_{\mathfrak{o}}$
and such a map $C\to H_{\mathfrak{o}}$ may be found in finite time by brute
force search).

Let $\mathscr{C}/\mathfrak{o}_{K}$ be the minimal proper regular model of
$C/K$, and let $T$ be an explicit finite set of primes containing $S$ such
that $\mathscr{C}/\mathfrak{o}_{K,T}$ and
$\mathcal{H}_{\mathfrak{o}}/\mathfrak{o}_{K,T}$ (the canonical integral model)
are smooth and such that the map $C\to H_{\mathfrak{o}}$ extends to
$\mathscr{C}\to\mathcal{H}_{\mathfrak{o}}$ over $\mathfrak{o}_{K,T}$.

Thus
$C(K)\hookrightarrow\mathscr{C}(\mathfrak{o}_{K,T})\to\mathcal{H}_{\mathfrak{o}}(\mathfrak{o}_{K,T})$
is finite-to-one, where we have used properness to produce the first map of
sets — thus e.g. for all $P\in C(K)$ the abelian variety $A_{P}/K$
corresponding to the image of $P$ under $C\to H_{\mathfrak{o}}$ has good
reduction outside $T$.

The usual explicit estimate for the difference between the Faltings height and
the modular height, aka the height pulled back from the Satake/Baily-Borel
compactification, implies that there is an explicit very ample divisor (more
properly, height data) $D$ of $C$ which is defined over $K$ and such that, for
all $P\in C(K)$, $h_{D}(P)$ is explicitly bounded in terms of $h(A_{P})$,
where $h_{D}$ is the height of $P\in C(K)$ with respect to $D$.

Thus by Theorem 1.1 we are done. ∎

##  6 Proof of Theorem 1.5.

We have already explained why there are infinitely many curves of type
$\bigiy$, but we found it difficult to produce explicit examples of such
curves by following that argument (which is explicit, though evidently only in
principle) because we were unable to find sufficiently explicit presentations
of Hilbert modular varieties (with suitable level structure) as subvarieties
of projective space in the literature.

So we will proceed in a different manner. The following family of examples was
reverse-engineered from formula $(17)$ of Deines-Fuselier-Long-Swisher-Tu’s
[10].

###### Proof of Theorem 1.5..

Let, for $a\in\overline{\mathbb{Q}}^{\times}$ totally real of odd degree,
$C_{a}:x^{6}+4y^{3}=a^{2}$.

Let, for $\lambda\neq 0,1,\infty$, $X_{\lambda}/\mathbb{Q}(\lambda)$ be the
desingularization of the curve $y^{6}=x^{4}(1-x)^{3}(1-\lambda\cdot x)$. Note
that each fibre of this family $X\to\mathbb{P}^{1}-\\{0,1,\infty\\}$ is a
genus $3$ curve. Let $\mathrm{Jac}\,{X_{\lambda}}/\mathbb{Q}(\lambda)$ be the
corresponding family of Jacobians, and let $A_{\lambda}/\mathbb{Q}(\lambda)$
be the family of Prym varieties corresponding to the map to the
desingularization $E_{\lambda}:y^{2}=x^{3}+16\lambda^{2}$ of the curve
$y^{3}=x^{4}(1-x)^{3}(1-\lambda\cdot x)$. In other words, let
$A_{\lambda}/\mathbb{Q}(\lambda)$ be the connected component of the identity
in the kernel of the map on Jacobians induced by the map $X_{\lambda}\to
E_{\lambda}$ arising from $(x,y)\mapsto(x,y^{2})$ taking
$y^{6}=x^{4}(1-x)^{3}(1-\lambda\cdot x)$ to
$y^{3}=x^{4}(1-x)^{3}(1-\lambda\cdot x)$.

Thanks to the automorphism $(x,y)\mapsto(x,\zeta_{6}\cdot y)$ of
$y^{6}=x^{4}(1-x)^{3}(1-\lambda\cdot x)$, it follows that
$\mathbb{Z}[\zeta_{3}]\hookrightarrow\mathrm{End}_{\mathbb{Q}(\zeta_{3},\lambda)}(A_{\lambda})$.
Thus each abelian surface $A_{\lambda}/\mathbb{Q}(\lambda)$ is of
$\mathrm{GL}_{2}(\mathbb{Z}[\zeta_{3}])$-type over
$\mathbb{Q}(\zeta_{3},\lambda)$.

Because the relevant triangle group, namely $\Delta(3,6,6)$, is arithmetic, we
will get even more. Specifically, we will see that the base change
$A_{\lambda}/\overline{\mathbb{Q}(\lambda)}$ admits quaternionic
multiplication by an order in the indefinite quaternion algebra over
$\mathbb{Q}$ of discriminant $6$. In fact we will further see that the
quaternionic multiplication is defined over
$\mathbb{Q}\left(\zeta_{3},\lambda,(\lambda\cdot\left(\frac{1-\lambda}{4}\right)^{2})^{\frac{1}{6}}\right)$,
but let us leave this aside for a moment.

Let us first explain how to form a family of abelian surfaces over our
$C_{a}/\mathbb{Q}(a^{2})$ using this construction.

Write, for $P=:(x,y)\in C_{a}$, $f_{a}(P):=\frac{x^{6}}{a^{2}}$. We claim that
the pullback family $P\mapsto A_{f_{a}(P)}$, which is a priori defined over
$C_{a}-f_{a}^{-1}\left(\\{0,1,\infty\\}\\}\right)$, extends to a family over
all of $C_{a}$. This is essentially checked in a number of places (see e.g.
Section $3.1$ of Cohen-Wolfart’s [9]), but we will prove this in Appendix A in
order to be thorough and because the relevant period calculations are quite
enjoyable. The point is that, because $P\mapsto f_{a}(P)$, respectively
$P\mapsto 1-f_{a}(P)$, has a holomorphic sixth root, respectively a
holomorphic cube root, on $C_{a}(\mathbb{C})$, the particular Schwarz triangle
functions that arise in the period lattice of $X_{f_{a}(P)}$ may be
holomorphically continued to their singular points.

So the extended family $A\to C_{a}$ is a non-isotrivial family of abelian
surfaces over $C_{a}$ which (by considering the family over
$C_{a}-f_{a}^{-1}(\\{0,1,\infty\\})$) is defined over $\mathbb{Q}(a^{2})$.
Moreover we see that
$\mathbb{Z}[\zeta_{3}]\hookrightarrow\mathrm{End}_{\mathbb{Q}(\zeta_{3},f_{a}(P))}(A_{f_{a}(P)})$
for all $P\in C_{a}$.

Just as in the proof of Corollary 1.3, we deduce that there is an explicit
finite set $S$ such that, for all $P\in C_{a}(K)$, the abelian surface
$A_{f_{a}(P)}/K$ has good reduction outside $S$.131313For example we show in
Appendix A that there is a map $C_{a}\to A_{3}$ such that, writing the induced
family of abelian threefolds as $J\to C_{a}$, we have that $A_{f_{a}(P)}$ is
an isogeny factor of $J_{P}$. Now one can argue exactly as in the proof of
Corollary 1.3, with the moduli space $A_{3}$ replacing $H_{\mathfrak{o}}$.

Now let us show that, at least for $P\in C_{a}(K)$ with
$f_{a}(P)\not\in\\{0,1,\infty\\}$, in fact $A_{f_{a}(P)}/K$ is of
$\mathrm{GL}_{2}$-type over $K$. Of course the relevant endomorphism ring is
not
$\mathbb{Z}[\zeta_{3}]\hookrightarrow\mathrm{End}_{K(\zeta_{3})}(A_{f_{a}(P)})$,
but thanks to said endomorphism ring for $p$ large we do find that, letting
$\mathfrak{p}\subseteq\mathbb{Z}[\zeta_{3}]$ with $\mathfrak{p}|(p)$ be a
prime of $\mathbb{Z}[\zeta_{3}]$ above $p$ and writing
$\rho_{P,\mathfrak{p}}:\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K(\zeta_{3}))\to\mathrm{GL}_{2}(\mathbb{Z}[\zeta_{3}]_{\mathfrak{p}})$
for the representation corresponding to
$T_{\mathfrak{p}}(A_{f_{a}(P)}):=\varprojlim_{n}A[\mathfrak{p}^{n}]$,
$\rho_{A_{f_{a}(P)},p}\simeq\rho_{P,\mathfrak{p}}\oplus\sigma\circ\rho_{P,\mathfrak{p}}$,
where $\sigma\in\mathop{\mathrm{Gal}}(\mathbb{Q}(\zeta_{3})/\mathbb{Q})$ is
complex conjugation (acting at the level of coefficients) and as usual
$\rho_{A_{f_{a}(P)},p}$ is the representation corresponding to
$T_{p}(A_{f_{a}(P)})$. (This decomposition corresponds to and arises from the
decomposition
$\mathbb{Z}[\zeta_{3}]\otimes_{\mathbb{Z}}\mathbb{Z}_{p}\simeq\bigoplus_{\mathfrak{q}|(p)}\mathbb{Z}[\zeta_{3}]_{\mathfrak{q}}$,
specifically via the action of the corresponding idempotents.) So far we have
simply expressed that $A_{f_{a}(P)}/K(\zeta_{3})$ is of $\mathrm{GL}_{2}$-type
over $K(\zeta_{3})$.

Now it is time to see the arithmeticity of $\Delta(3,6,6)$ at the level of
Galois representations. We claim that
$(\sigma\circ\rho_{P,\mathfrak{p}})\otimes_{\mathbb{Z}}\mathbb{Q}\cong\rho_{P,\mathfrak{p}}\otimes_{\mathbb{Z}}\mathbb{Q}$,
or in other words that the Frobenius traces of $\rho_{P,\mathfrak{p}}$ at
large primes are rational integers.

By counting points on $X_{\lambda}$ over finite fields in the usual way one
sees that the Frobenius traces of $\rho_{P,\mathfrak{p}}$ are given by finite
field hypergeometric functions — specifically, by Proposition $13$ of Deines-
Fuselier-Long-Swisher-Tu’s [10], for each large prime
$\mathfrak{q}\subseteq\mathfrak{o}_{K(\zeta_{3})}$ of
$\mathfrak{o}_{K(\zeta_{3})}$, writing $q:=\mathrm{Nm}\,{\mathfrak{q}}$ and
$\eta:(\mathfrak{o}_{K(\zeta_{3})}/\mathfrak{q})^{\times}\twoheadrightarrow\mu_{6}\subseteq\mathbb{Z}[\zeta_{3}]^{\times}\subseteq\mathfrak{o}_{K(\zeta_{3})}^{\times}$
for the sextic character such that
$\eta(\alpha)\equiv\alpha^{\frac{q-1}{6}}\mkern 4.0mu({\operator@font
mod}\mkern 6.0mu\mathfrak{q})$,

$\displaystyle\left\\{\mathrm{tr}(\rho_{P,\mathfrak{p}}(\mathrm{Frob}_{\mathfrak{q}})),\sigma(\mathrm{tr}(\rho_{P,\mathfrak{p}}(\mathrm{Frob}_{\mathfrak{q}})))\right\\}$
$\displaystyle=\left\\{-\eta(-1)\cdot
q\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\eta&\eta^{2}\\\
&\eta^{5}\end{array}\right|f_{a}(P)\right)_{q},-\eta^{-1}(-1)\cdot
q\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\eta^{-1}&\eta^{-2}\\\
&\eta^{-5}\end{array}\right|f_{a}(P)\right)_{q}\right\\}$

as sets.

By Proposition $9$ of Deines-Fuselier-Long-Swisher-Tu’s [10], we see that

$\,{}_{2}F_{1}\left(\left.\begin{array}[]{cc}\eta&\eta^{2}\\\
&\eta^{5}\end{array}\right|f_{a}(P)\right)_{q}$
$\displaystyle=\eta^{3}(-1)\cdot\eta(-f_{a}(P))\cdot\eta^{2}(1-f_{a}(P))\cdot\frac{J(\eta^{2},\eta^{3})}{J(\eta,\eta^{4})}\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\eta^{-1}&\eta^{-2}\\\
&\eta^{-5}\end{array}\right|f_{a}(P)\right)_{q}$
$\displaystyle=\eta(f_{a}(P))\cdot\eta^{2}(1-f_{a}(P))\cdot\frac{J(\eta^{2},\eta^{3})}{J(\eta,\eta^{4})}\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\eta^{-1}&\eta^{-2}\\\
&\eta^{-5}\end{array}\right|f_{a}(P)\right)_{q},$

where $J(\cdot,\cdot)$ denotes the usual Jacobi sum.

Now let us evaluate the ratio of Jacobi sums. Of course
$\frac{J(\eta^{2},\eta^{3})}{J(\eta,\eta^{4})}=\frac{G(\eta^{2})G(\eta^{3})}{G(\eta)G(\eta^{4})}$
with $G(\cdot)$ the usual Gauss sum (with respect to a chosen additive
character $\mathfrak{o}_{K(\zeta_{3})}/\mathfrak{q}\to\mu_{q}(\mathbb{C})$).
Now we apply the Hasse-Davenport relation with "base" multiplicative character
$\eta$ and "shift" multiplicative character $\eta^{3}$. We find
$G(\eta)G(\eta^{4})=-\eta^{-2}(2)G(\eta^{2})G(\mathop{\mathrm{triv}})G(\eta^{3})$,
where as usual $G(\mathop{\mathrm{triv}})=-1$. So we conclude that

$\,{}_{2}F_{1}\left(\left.\begin{array}[]{cc}\eta&\eta^{2}\\\
&\eta^{5}\end{array}\right|f_{a}(P)\right)_{q}$
$\displaystyle=\eta\left(f_{a}(P)\cdot\left(\frac{1-f_{a}(P)}{4}\right)^{2}\right)\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\eta^{-1}&\eta^{-2}\\\
&\eta^{-5}\end{array}\right|f_{a}(P)\right)_{q}.$

Finally
$f_{a}(P)\cdot\left(\frac{1-f_{a}(P)}{4}\right)^{2}=\frac{X^{6}}{a^{2}}\cdot\left(\frac{Y^{3}}{a^{2}}\right)^{2}=\left(\frac{XY}{a}\right)^{6}$
is a sixth power in $K$, thus in $K(\zeta_{3})$, thus since $\mathfrak{q}$ is
large $\eta\left(f_{a}(P)\cdot\left(\frac{1-f_{a}(P)}{4}\right)^{2}\right)=1$.
So we see by Chebotarev and Brauer-Nesbitt (and Faltings’ proof of
semisimplicity) that
$\rho_{P,\mathfrak{p}}\otimes_{\mathbb{Z}}\mathbb{Q}\cong(\sigma\circ\rho_{P,\mathfrak{p}})\otimes_{\mathbb{Z}}\mathbb{Q}$
as claimed.

Consequently the inclusion
$\mathbb{Q}(\zeta_{3})\otimes_{\mathbb{Q}}\mathbb{Q}_{p}\hookrightarrow\mathrm{End}_{\mathbb{Q}_{p}[\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/K(\zeta_{3}))]}(\rho_{A_{f_{a}(P),p}}\otimes_{\mathbb{Z}}\mathbb{Q})$
is _not_ an isomorphism. Consequently, by Faltings’ proof of the Tate
conjecture for endomorphisms of abelian varieties, the inclusion
$\mathbb{Q}(\zeta_{3})\hookrightarrow\mathrm{End}_{K(\zeta_{3})}^{0}(A_{f_{a}(P)})$
is not an isomorphism. But because the $\mathbb{Q}$-subalgebra of
$\mathrm{End}_{K(\zeta_{3})}^{0}(A_{f_{a}(P)})$ fixed by complex conjugation
must be of dimension at least
$\frac{1}{2}\dim_{\mathbb{Q}}(\mathrm{End}_{K(\zeta_{3})}^{0}(A_{f_{a}(P)}))$
(since said involution respects products and can be diagonalized), we see that
$\dim_{\mathbb{Q}}(\mathrm{End}_{K}^{0}(A_{f_{a}(P)}))\geq 2$, where we have
used that a $K(\zeta_{3})$-endomorphism is fixed by complex conjugation if and
only if it descends to a $K$-endomorphism (consider the graph).

We conclude that, for $P\in C_{a}(K)$ with $f_{a}(P)\not\in\\{0,1,\infty\\}$,
$A_{f_{a}(P)}/K$ is of $\mathrm{GL}_{2}$-type over $K$. We have moreover
already seen that there is an explicit finite set $S$ such that each
$A_{f_{a}(P)}/K$ has good reduction outside $S$. We are done by Theorem 1.1. ∎

##  Appendix A Explicit extension of the hypergeometric family over the
singular points.

In this section we will explain why the family of abelian surfaces over
$C_{a}-f_{a}^{-1}(\\{0,1,\infty\\})$ which we explicitly constructed in
Section 6 extends to a family of abelian surfaces over the whole of $C_{a}$.
Again, this is more or less checked or asserted in a number of references, but
we could not find a treatment to our liking in the literature and anyway the
calculations are really a treat. Though we treat the particular case of
abelian surfaces of interest to us here, it should be clear how to generalize
the below calculations to an arbitrary hypergeometric family associated to the
triangle group $\Delta(p,q,r)$ with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$
and $p,q,r\in\mathbb{Z}^{+}$.

We will reuse all notation from Section 6. To see the claim it suffices to
work over $\mathbb{C}$. Since the family $\lambda\mapsto E_{\lambda}$ is
isotrivial, it further suffices to show that the pullback family
$P\mapsto\mathrm{Jac}\,{X_{f_{a}(P)}}$ extends to all of $C_{a}$. This is a
map from a punctured curve to the moduli space $A_{3}$. By the Borel extension
theorem (here basically Riemann extension) it extends to a map to the
Satake/Baily-Borel compactification $C_{a}\to A_{3}^{\mathrm{B.B.}}$. Let us
now check at the level of periods that the family does not degenerate and
indeed extends holomorphically over those $P$ for which
$f_{a}(P)\in\\{0,1,\infty\\}$.

As in Remark $12$ of Archinard’s [5], we see that the holomorphic
differentials on $X_{\lambda}$ are given by:

$\displaystyle\omega_{1}$
$\displaystyle:=\frac{dx}{y}=\frac{dx}{\sqrt[6]{x^{4}(1-x)^{3}(1-\lambda\cdot
x)}},$ $\displaystyle\omega_{4}$
$\displaystyle:=\frac{x^{2}(1-x)^{2}dx}{y^{4}}=\frac{dx}{\sqrt[3]{x^{2}(1-\lambda\cdot
x)^{2}}},$ $\displaystyle\omega_{5}$
$\displaystyle:=\frac{x^{3}(1-x)^{2}dx}{y^{5}}=\frac{dx}{\sqrt[6]{x^{2}(1-x)^{3}(1-\lambda\cdot
x)^{5}}}.$

Now let us discuss how to compute the periods of $X_{\lambda}$. Note that the
map $X_{\lambda}\to\mathbb{P}^{1}$ induced by $(x,y)\mapsto x$ on the singular
model of $X_{\lambda}$ is unramified outside
$\\{0,1,\frac{1}{\lambda},\infty\\}$. We will occasionally regard this map as
$X_{\lambda}\to X_{\lambda}/\mu_{6}\cong\mathbb{P}^{1}$ — note also that the
map $X_{\lambda}\to E_{\lambda}$ via $(x,y)\mapsto(x,y^{2})$ at the level of
singular models may similarly be regarded as $X_{\lambda}\to
X_{\lambda}/\mu_{2}\cong E_{\lambda}$.

In any case, as is classical, $H_{1}(X_{\lambda},\mathbb{Z})$ is generated by
the various lifts of the Pochhammer cycles $\gamma_{0,1}$ and
$\gamma_{\frac{1}{\lambda},\infty}$ between $0$ and $1$ and
$\frac{1}{\lambda}$ and $\infty$, respectively. Writing
$\widetilde{\gamma}_{0,1}$ and $\widetilde{\gamma}_{\frac{1}{\lambda},\infty}$
for the "principal" such lifts (determined by the choice of the principal
branch of the sixth root), all others are given by
$\zeta_{6}^{k}\cdot\widetilde{\gamma}_{0,1}$ and
$\zeta_{6}^{k^{\prime}}\cdot\widetilde{\gamma}_{\frac{1}{\lambda},\infty}$.
Thus we obtain a set of twelve generators of the rank $6$ free
$\mathbb{Z}$-module $H_{1}(X_{\lambda},\mathbb{Z})$.

This makes it straightforward to compute the periods of $X_{\lambda}$. Write,
for $i\in\\{1,4,5\\}$, $\mu_{i}:=\int_{\widetilde{\gamma}_{0,1}}\omega_{i}$
and $\nu_{i}:=\int_{\widetilde{\gamma}_{\frac{1}{\lambda},\infty}}\omega_{i}$.
Write $\vec{\mu}:=\left(\begin{array}[]{c}\mu_{1}\\\ \mu_{4}\\\
\mu_{5}\end{array}\right)$ and similarly for $\vec{\nu}$. Then the periods of
our chosen basis of holomorphic $1$-forms over
$\zeta_{6}^{k}\cdot\widetilde{\gamma}_{0,1}$ satisfy:

$\displaystyle\left(\begin{array}[]{c}\int_{\zeta_{6}^{k}\cdot\widetilde{\gamma}_{0,1}}\omega_{1}\\\
\int_{\zeta_{6}^{k}\cdot\widetilde{\gamma}_{0,1}}\omega_{4}\\\
\int_{\zeta_{6}^{k}\cdot\widetilde{\gamma}_{0,1}}\omega_{5}\end{array}\right)$
$\displaystyle=\mathrm{diag}(\zeta_{6}^{-k},\zeta_{6}^{-4k},\zeta_{6}^{-5k})\cdot\left(\begin{array}[]{c}\int_{\widetilde{\gamma}_{0,1}}\omega_{1}\\\
\int_{\widetilde{\gamma}_{0,1}}\omega_{4}\\\
\int_{\widetilde{\gamma}_{0,1}}\omega_{5}\end{array}\right),$
$\displaystyle=\mathrm{diag}(\zeta_{6}^{-k},\zeta_{6}^{-4k},\zeta_{6}^{-5k})\cdot\vec{\mu}$

and similarly for the periods over
$\zeta_{6}^{k^{\prime}}\cdot\widetilde{\gamma}_{\frac{1}{\lambda},\infty}$.

We next note that the $\mathbb{Z}$-span of the matrices
$\mathrm{diag}(\zeta_{6}^{k},\zeta_{6}^{4k},\zeta_{6}^{5k})$ has basis
$\mathop{\mathrm{id}}$,
$\mathrm{diag}(\zeta_{6},\zeta_{6}^{4},\zeta_{6}^{5})$,
$\mathrm{diag}(0,2,0)$, and $\mathrm{diag}(0,2\zeta_{6},0)$, the latter two
arising from the relations

$\mathrm{diag}(\zeta_{6}^{2},\zeta_{6}^{8},\zeta_{6}^{10})=-\mathop{\mathrm{id}}+\,\mathrm{diag}(\zeta_{6},\zeta_{6}^{4},\zeta_{6}^{5})-\mathrm{diag}(0,2\zeta_{6},0)$

and

$\mathrm{diag}(\zeta_{6}^{3},\zeta_{6}^{12},\zeta_{6}^{15})=-\mathop{\mathrm{id}}+\,\mathrm{diag}(0,2,0).$

Consequently the rank $6$ period lattice of $X_{\lambda}$ with respect to the
chosen basis $(\omega_{1},\omega_{4},\omega_{5})$ of holomorphic $1$-forms on
$X_{\lambda}$ and the chosen generators of $H_{1}(X_{\lambda},\mathbb{Z})$ is
the $\mathbb{Z}$-span inside $\mathbb{C}^{3}$ of $\vec{\mu}$, $\vec{\nu}$,
$\mathrm{diag}(\zeta_{6},\zeta_{6}^{4},\zeta_{6}^{5})\cdot\vec{\mu}$,
$\mathrm{diag}(\zeta_{6},\zeta_{6}^{4},\zeta_{6}^{5})\cdot\vec{\nu}$,
$\mathrm{diag}(0,2,0)\cdot\vec{\mu}$, $\mathrm{diag}(0,2,0)\cdot\vec{\nu}$,
$\mathrm{diag}(0,2\zeta_{6},0)\cdot\vec{\mu}$, and
$\mathrm{diag}(0,2\zeta_{6},0)\cdot\vec{\nu}$. We of course recognize the
$\mathbb{Z}$-span of the last four generators as a copy of the period lattice
of the CM elliptic curve $E_{\lambda}/\mathbb{C}$ — this also provides the
required two relations, namely that141414We will be more precise since the
calculations are so pleasant.
$\mu_{4},\zeta_{6}\cdot\mu_{4}\in\mathbb{Z}[\zeta_{6}]\cdot\nu_{4}$, among the
eight generators of this rank $6$ lattice.

So we finally find the following $\mathbb{Z}$-basis for the period lattice of
$X_{\lambda}$: $\vec{\mu}$, $\vec{\nu}$,
$\mathrm{diag}(\zeta_{6},\zeta_{6}^{4},\zeta_{6}^{5})\cdot\vec{\mu}$,
$\mathrm{diag}(\zeta_{6},\zeta_{6}^{4},\zeta_{6}^{5})\cdot\vec{\nu}$,
$\mathrm{diag}(0,2,0)\cdot\vec{\nu}$, and
$\mathrm{diag}(0,2\zeta_{6},0)\cdot\vec{\nu}$.

Now let us actually calculate $\vec{\mu}$ and $\vec{\nu}$ in terms of
hypergeometric functions. To do this it suffices to relate the integrals over
Pochhammer cycles to usual integrals over a path, and then to invoke Euler’s
integral representation of Gauss’s hypergeometric functions. Specifically, we
have the following identities (the $(1-e(\alpha))(1-e(\beta))$ factors are
calculated by expanding the relevant differentials around the relevant
singular points):

$\displaystyle\mu_{1}=\int_{\widetilde{\gamma}_{0,1}}\omega_{1}$
$\displaystyle=\left(1-e\left(\frac{2}{3}\right)\right)\left(1-e\left(\frac{1}{2}\right)\right)\cdot\int_{0}^{1}\omega_{1},$
$\displaystyle\mu_{4}=\int_{\widetilde{\gamma}_{0,1}}\omega_{4}$
$\displaystyle=\left(1-e\left(\frac{2}{3}\right)\right)\left(1-e\left(0\right)\right)\cdot\int_{0}^{1}\omega_{4},$
$\displaystyle\mu_{5}=\int_{\widetilde{\gamma}_{0,1}}\omega_{5}$
$\displaystyle=\left(1-e\left(\frac{1}{3}\right)\right)\left(1-e\left(\frac{1}{2}\right)\right)\cdot\int_{0}^{1}\omega_{5},$
$\displaystyle\nu_{1}=\int_{\widetilde{\gamma}_{\frac{1}{\lambda},\infty}}\omega_{1}$
$\displaystyle=\left(1-e\left(\frac{1}{6}\right)\right)\left(1-e\left(\frac{4}{3}\right)\right)\cdot\int_{\frac{1}{\lambda}}^{\infty}\omega_{1},$
$\displaystyle\nu_{4}=\int_{\widetilde{\gamma}_{\frac{1}{\lambda},\infty}}\omega_{4}$
$\displaystyle=\left(1-e\left(\frac{2}{3}\right)\right)\left(1-e\left(\frac{4}{3}\right)\right)\cdot\int_{\frac{1}{\lambda}}^{\infty}\omega_{4},$
$\displaystyle\nu_{5}=\int_{\widetilde{\gamma}_{\frac{1}{\lambda},\infty}}\omega_{5}$
$\displaystyle=\left(1-e\left(\frac{5}{6}\right)\right)\left(1-e\left(\frac{5}{3}\right)\right)\cdot\int_{\frac{1}{\lambda}}^{\infty}\omega_{5},$

where as usual $e(x):=e^{2\pi ix}$. So we see that e.g.
$\mu_{4}=0$.151515After all, $\omega_{4}$ is the pullback of a differential on
$E_{\lambda}\simeq X_{\lambda}/\mu_{2}$ and the map
$E_{\lambda}\to\mathbb{P}^{1}$ via $(x,y)\mapsto x$ at the level of its
singular model is unramified over $1$. Inserting Euler’s integral
representation $B(b,c-b)\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}a&b\\\
&c\end{array}\right|z\right)=\int_{0}^{1}x^{b-1}(1-x)^{c-a-b-1}(1-z\cdot
x)^{-a}\,dx$ we find:

$\displaystyle\vec{\mu}=\left(\begin{array}[]{c}2(1-\zeta_{3}^{2})\cdot
B\left(\frac{1}{3},\frac{1}{2}\right)\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\frac{1}{6}&\frac{1}{3}\\\
&\frac{5}{6}\end{array}\right|\lambda\right)\\\ 0\\\ 2(1-\zeta_{3})\cdot
B\left(\frac{2}{3},\frac{1}{2}\right)\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\frac{5}{6}&\frac{2}{3}\\\
&\frac{7}{6}\end{array}\right|\lambda\right)\end{array}\right).$

Via the change of variables $x\mapsto\frac{1}{\lambda x}$ and then a suitable
application of Euler’s integral representation (exactly as in $(15)$ of
Deines-Fuselier-Long-Swisher-Tu’s [10]), we similarly find that:

$\displaystyle\vec{\nu}=\left(\begin{array}[]{c}(1-\zeta_{6})(1-\zeta_{3})\cdot(-1)^{-\frac{2}{3}}\cdot
B\left(\frac{1}{3},\frac{5}{6}\right)\cdot\lambda^{\frac{1}{6}}\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\frac{1}{2}&\frac{1}{3}\\\
&\frac{7}{6}\end{array}\right|\lambda\right)\\\
(1-\zeta_{3}^{2})(1-\zeta_{3})\cdot(-1)^{-\frac{2}{3}}\cdot
B\left(\frac{1}{3},\frac{1}{3}\right)\cdot\lambda^{-\frac{1}{3}}\\\
(1-\zeta_{6}^{5})(1-\zeta_{3}^{2})\cdot(-1)^{-\frac{4}{3}}\cdot
B\left(\frac{2}{3},\frac{1}{6}\right)\cdot\lambda^{-\frac{1}{6}}\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\frac{1}{2}&\frac{2}{3}\\\
&\frac{5}{6}\end{array}\right|\lambda\right)\end{array}\right),$

where we have used that
$\,{}_{2}F_{1}\left(\left.\begin{array}[]{cc}0&\frac{1}{3}\\\
&\frac{2}{3}\end{array}\right|\lambda\right)=1$ in the calculation of
$\nu_{4}$.

So we have calculated the period lattice of $X_{\lambda}$ explicitly.

Now it is time to show that the family of Jacobians does not degenerate as
$f_{a}(P)\to 0$, $1$, or $\infty$. Since we are working over $\mathbb{C}$ it
suffices to show the claim for $C_{1}/\mathbb{C}$ only (via the evident
isomorphism $C_{a}\cong C_{1}$ over $\mathbb{C}$, which also takes
$f_{a}\mapsto f_{1}$). So, writing $C_{1}:X^{6}+4Y^{3}=1$ for clarity, we have
$\lambda=X^{6}$ and $1-\lambda=4Y^{3}$. Consequently (since we do not want to
discuss branches of $z\mapsto\sqrt[6]{z}$) there is an absolute constant
$t\in\mathbb{Z}/6$ such that we may write $\vec{\mu}$ and $\vec{\nu}$ at a
$\lambda=f_{1}(P)$ with $P=:(X,Y)$ as:

$\displaystyle\vec{\mu}$
$\displaystyle=\left(\begin{array}[]{c}2(1-\zeta_{3}^{2})\cdot
B\left(\frac{1}{3},\frac{1}{2}\right)\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\frac{1}{6}&\frac{1}{3}\\\
&\frac{5}{6}\end{array}\right|X^{6}\right)\\\ 0\\\ 2(1-\zeta_{3})\cdot
B\left(\frac{2}{3},\frac{1}{2}\right)\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\frac{5}{6}&\frac{2}{3}\\\
&\frac{7}{6}\end{array}\right|X^{6}\right)\end{array}\right),$
$\displaystyle\vec{\nu}$
$\displaystyle=\left(\begin{array}[]{c}(1-\zeta_{6})(1-\zeta_{3})\cdot(-1)^{-\frac{2}{3}}\cdot
B\left(\frac{1}{3},\frac{5}{6}\right)\cdot\zeta_{6}^{t}\cdot
X\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\frac{1}{2}&\frac{1}{3}\\\
&\frac{7}{6}\end{array}\right|X^{6}\right)\\\
(1-\zeta_{3}^{2})(1-\zeta_{3})\cdot(-1)^{-\frac{2}{3}}\cdot
B\left(\frac{1}{3},\frac{1}{3}\right)\cdot\zeta_{6}^{-2t}\cdot X^{-2}\\\
(1-\zeta_{6}^{5})(1-\zeta_{3}^{2})\cdot(-1)^{-\frac{4}{3}}\cdot
B\left(\frac{2}{3},\frac{1}{6}\right)\cdot\zeta_{6}^{-t}\cdot
X^{-1}\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\frac{1}{2}&\frac{2}{3}\\\
&\frac{5}{6}\end{array}\right|X^{6}\right)\end{array}\right).$

In a punctured neighbourhood around each of the points in
$f_{1}^{-1}(\\{0,1,\infty\\})$ we have that $\mu_{1},\mu_{5},\nu_{4}\neq 0$,
so we may change our basis of holomorphic $1$-forms on $X_{\lambda}$ via
$\mathrm{diag}(\mu_{1}^{-1},\nu_{4}^{-1},\mu_{5}^{-1})\in\mathrm{GL}_{3}(\mathbb{C})$.

This results in the following $\mathbb{Z}$-basis for an equivalent, aka
homothetic, period lattice for $X_{\lambda}$: $(1,0,1)$,
$(\zeta_{6},0,\zeta_{6}^{5})$, $(0,2,0)$, $(0,2\zeta_{6},0)$, and the
following two vectors:

$\displaystyle\left(\begin{array}[]{c}\mathrm{const.}\cdot
s_{\frac{1}{6},\frac{1}{3};\frac{5}{6}}(X^{6})\\\ 1\\\
\widetilde{\mathrm{const.}}\cdot
s_{\frac{5}{6},\frac{2}{3};\frac{7}{6}}(X^{6})\end{array}\right)\text{ and
}\left(\begin{array}[]{c}\zeta_{6}\cdot\mathrm{const.}\cdot
s_{\frac{1}{6},\frac{1}{3};\frac{5}{6}}(X^{6})\\\ \zeta_{6}^{4}\\\
\zeta_{6}^{5}\cdot\widetilde{\mathrm{const.}}\cdot
s_{\frac{5}{6},\frac{2}{3};\frac{7}{6}}(X^{6})\end{array}\right),$

where $\mathrm{const.},\widetilde{\mathrm{const.}}\in\mathbb{C}^{\times}$ are
absolute constants easily calculated from the above, and

$s_{a,b;c}(z):=z^{1-c}\cdot\frac{\,{}_{2}F_{1}\left(\left.\begin{array}[]{cc}a-c+1&b-c+1\\\
&2-c\end{array}\right|z\right)}{\,{}_{2}F_{1}\left(\left.\begin{array}[]{cc}a&b\\\
&c\end{array}\right|z\right)}$

is the classical Schwarz triangle function.

All that is left to note is that the functions
$s_{\frac{1}{6},\frac{1}{3};\frac{5}{6}}(f_{1}(P))$ and
$s_{\frac{5}{6},\frac{2}{3};\frac{7}{6}}(f_{1}(P))$ extend holomorphically to
the whole of $C_{1}(\mathbb{C})$ — indeed this is tautological on
$C_{1}(\mathbb{C})-f_{1}^{-1}([1,\infty])$ since we are implicitly using the
usual principal branch of the Gauss hypergeometric function in using its
definition via Euler’s integral over a Pochhammer contour, and then we extend
to $f_{1}^{-1}([1,\infty])$ via the identity

$\,{}_{2}F_{1}\left(\left.\begin{array}[]{cc}a&b\\\
&c\end{array}\right|z\right)$
$\displaystyle=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}a&b\\\
&a+b+1-c\end{array}\right|1-z\right)$
$\displaystyle+\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}\cdot(1-z)^{c-a-b}\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}c-a&c-b\\\
&1+c-a-b\end{array}\right|1-z\right)$

and the fact that $1-X^{6}=4Y^{3}$ has a holomorphic cube root on
$C_{1}(\mathbb{C})$ (note that $c-a-b\in\frac{1}{3}\cdot\mathbb{Z}$ for
$(a,b,c)\in\left\\{\left(\frac{1}{6},\frac{1}{3},\frac{5}{6}\right),\left(\frac{5}{6},\frac{2}{3},\frac{7}{6}\right)\right\\}$).

So the map $C_{a}\to A_{3}^{\mathrm{B.B.}}$ via
$P\mapsto\mathrm{Jac}\,{X_{f_{a}(P)}}$ indeed lands inside $A_{3}$, whence we
have produced a family $J\to C_{a}$ extending
$\lambda\mapsto\mathrm{Jac}\,{X_{\lambda}}$, and we produce the family $A\to
C_{a}$ extending $\lambda\mapsto A_{\lambda}$ by taking the fibre above $P\in
C_{a}$ to be the connected component of the identity in the kernel of the map
$J_{P}\to E_{P}$, where $E\to C_{a}$ is the evident extension of the
isotrivial family $\lambda\mapsto E_{\lambda}$.

So we are done.

We will end by making transparent the "extra" endomorphism of
$A_{f_{1}(P)}/\mathbb{C}$ at the level of its period lattice, as in Deines-
Fuselier-Long-Swisher-Tu [10] — after all, $C_{1}/\mathbb{C}$ corresponds to
the commutator subgroup of the triangle group $\Delta(3,6,6)$, which is
arithmetic, so $C_{1}/\mathbb{C}$ must have an interpretation as a Shimura
curve with level structure attached to the corresponding quaternion algebra
over $\mathbb{Q}$, namely the indefinite quaternion algebra of discriminant
$6$. (In those terms our family $P\mapsto A_{f_{1}(P)}$ is just the
corresponding universal family.)

We will project the period lattice to $\mathbb{C}^{2}$ via
$\pi:\mathbb{C}^{3}\twoheadrightarrow\mathbb{C}^{2}$ via $(x,y,z)\mapsto(x,z)$
since it suffices to work with the period lattice of an isogenous abelian
surface. We find the following generators for the relevant period lattice:
$\pi(\vec{\mu}),\pi(\vec{\nu}),\mathrm{diag}(\zeta_{6},\zeta_{6}^{5})\cdot\pi(\vec{\mu}),\mathrm{diag}(\zeta_{6},\zeta_{6}^{5})\cdot\pi(\vec{\nu})$.
Via the Euler identity

$\,{}_{2}F_{1}\left(\left.\begin{array}[]{cc}a&b\\\
&c\end{array}\right|z\right)=(1-z)^{c-a-b}\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}a-c+1&b-c+1\\\
&2-c\end{array}\right|z\right),$

we see that there is an absolute constant $t^{\prime}\in\mathbb{Z}/6$ such
that:

$\displaystyle\pi(\vec{\mu})$
$\displaystyle=\left(\begin{array}[]{c}2(1-\zeta_{3}^{2})\cdot
B\left(\frac{1}{3},\frac{1}{2}\right)\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\frac{1}{6}&\frac{1}{3}\\\
&\frac{5}{6}\end{array}\right|X^{6}\right)\\\ 2(1-\zeta_{3})\cdot
B\left(\frac{2}{3},\frac{1}{2}\right)\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\frac{5}{6}&\frac{2}{3}\\\
&\frac{7}{6}\end{array}\right|X^{6}\right)\end{array}\right),$
$\displaystyle\pi(\vec{\nu})$
$\displaystyle=\left(\begin{array}[]{c}(1-\zeta_{6})(1-\zeta_{3})\cdot(-1)^{-\frac{2}{3}}\cdot
B\left(\frac{1}{3},\frac{5}{6}\right)\cdot\zeta_{6}^{t^{\prime}}\cdot
X\cdot(4^{\frac{1}{3}}\cdot
Y)\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\frac{5}{6}&\frac{2}{3}\\\
&\frac{7}{6}\end{array}\right|X^{6}\right)\\\
(1-\zeta_{6}^{5})(1-\zeta_{3}^{2})\cdot(-1)^{-\frac{4}{3}}\cdot
B\left(\frac{2}{3},\frac{1}{6}\right)\cdot\zeta_{6}^{-t^{\prime}}\cdot
X^{-1}\cdot(4^{\frac{1}{3}}\cdot
Y)^{-1}\cdot\,_{2}F_{1}\left(\left.\begin{array}[]{cc}\frac{1}{6}&\frac{1}{3}\\\
&\frac{5}{6}\end{array}\right|X^{6}\right)\end{array}\right).$

We now claim that the transformation

$\displaystyle
M:=\left(\begin{array}[]{cc}0&\frac{(1-\zeta_{6})(1-\zeta_{3})\cdot(-1)^{-\frac{2}{3}}\cdot
B\left(\frac{1}{3},\frac{5}{6}\right)\cdot\zeta_{6}^{t^{\prime}}\cdot
X\cdot(4^{\frac{1}{3}}\cdot Y)}{(1-\zeta_{3})\cdot
B\left(\frac{2}{3},\frac{1}{2}\right)}\\\
\frac{(1-\zeta_{6}^{5})(1-\zeta_{3}^{2})\cdot(-1)^{-\frac{4}{3}}\cdot
B\left(\frac{2}{3},\frac{1}{6}\right)\cdot\zeta_{6}^{-t^{\prime}}\cdot
X^{-1}\cdot(4^{\frac{1}{3}}\cdot Y)^{-1}}{(1-\zeta_{3}^{2})\cdot
B\left(\frac{1}{3},\frac{1}{2}\right)}&0\end{array}\right)\in\mathrm{GL}_{2}(\mathbb{C})$

stabilizes the period lattice — this is the aforementioned "extra"
endomorphism.

Miraculously, $M^{2}=2\cdot\mathop{\mathrm{id}}$ thanks to the equality

$\frac{[(1-\zeta_{6})(1-\zeta_{6}^{5})]\cdot[(1-\zeta_{3})(1-\zeta_{3}^{2})]\cdot
B\left(\frac{2}{3},\frac{1}{6}\right)\cdot
B\left(\frac{1}{3},\frac{5}{6}\right)}{[(1-\zeta_{3})(1-\zeta_{3}^{2})]\cdot
B\left(\frac{1}{3},\frac{1}{2}\right)\cdot
B\left(\frac{2}{3},\frac{1}{2}\right)}=2,$

the miracle of course being that we have calculated constants sufficiently
correctly.

So we may easily check that $M$ stabilizes the period lattice and thus
descends to an endomorphism of an abelian variety isogenous to
$A_{f_{1}(P)}/\mathbb{C}$. Indeed by construction
$M\cdot\pi(\vec{\mu})=2\cdot\pi(\vec{\nu})$, and so
$M\cdot\mathrm{diag}(\zeta_{6},\zeta_{6}^{5})\cdot\pi(\vec{\mu})=2\cdot\mathrm{diag}(\zeta_{6}^{5},\zeta_{6})\cdot\pi(\vec{\nu})\in\mathbb{Z}[\mathrm{diag}(\zeta_{6},\zeta_{6}^{5})]\cdot\pi(\vec{\nu})$.
But then

$M\cdot\pi(\vec{\nu})=\frac{1}{2}\cdot
M^{2}\cdot\pi(\vec{\mu})=\pi(\vec{\mu}),$

and

$M\cdot\mathrm{diag}(\zeta_{6},\zeta_{6}^{5})\cdot\pi(\vec{\nu})=\mathrm{diag}(\zeta_{6}^{5},\zeta_{6})\cdot
M\cdot\pi(\vec{\nu})=\mathrm{diag}(\zeta_{6}^{5},\zeta_{6})\cdot\pi(\vec{\mu})\in\mathbb{Z}[\mathrm{diag}(\zeta_{6},\zeta_{6}^{5})]\cdot\pi(\vec{\mu}).$

Writing $B_{6}/\mathbb{Q}$ for the indefinite quaternion algebra over
$\mathbb{Q}$ of discriminant $6$, we have therefore produced an explicit
inclusion $B_{6}\hookrightarrow\mathrm{End}_{\mathbb{C}}^{0}(A_{f_{1}(P)})$
(namely $i\mapsto 1+2\zeta_{3},j\mapsto M$), concluding this aside and the
appendix.

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