# Dwork Motives, Monodromy and Potential Automorphy

Lambert A’Campo

###### Abstract.

In this paper we study certain families of motives, which arise as direct
summands of the cohomology of the Dwork family. We computationally find
examples of interesting families with the following three properties. Firstly,
their geometric monodromy group is Zariski dense in $\operatorname{SL}_{n}$.
Secondly, they realise many different unipotent operators as the monodromy
operator at $t=\infty$. Thirdly, all their Hodge numbers are $\leq 1$.

This has consequences for Galois representations. Namely, if a nilpotent
operator $N$ appears as the monodromy at $t=\infty$ in one of our families, we
can construct potentially automorphic representations with $\ell$-adic
monodromy given by $N$ at a fixed prime $p$. As another application, we obtain
a new proof of some cases of the recent local-global compatibility theorem of
Matsumoto.

###### Contents

  1. 1 Introduction
    1. 1.1 Families of Motives
    2. 1.2 Galois Representations
  2. 2 The Dwork Family
    1. 2.1 Statements
    2. 2.2 The Method of Griffiths
    3. 2.3 The Cohomology at $t=0$
    4. 2.4 The Picard–Fuchs Equation
    5. 2.5 The Geometric Monodromy
    6. 2.6 The Arithmetic Monodromy
  3. 3 Applications to Galois Representations
    1. 3.1 Preliminaries on Weil–Deligne Representations
    2. 3.2 The Potential Automorphy Theorem
    3. 3.3 Local-Global Compatibility
  4. 4 Searching Hypergeometric Parameters
    1. 4.1 The Computational Check
    2. 4.2 Results of the Program

##  1\. Introduction

Let $d\geq 3$ be an integer and consider the Dwork family of projective
hypersurfaces

$Y_{t}:x_{0}^{d}+x_{1}^{d}+\dots+x_{d-1}^{d}=dtx_{0}x_{1}\cdots x_{d-1}.$

In this paper we study certain families of motives $M_{t}$, which are
$\mathbb{Z}[\zeta_{d}]$-linear direct summands of
$H^{d-2}(Y_{t},\mathbb{Z}[\frac{1}{d},\zeta_{d}])$, where $\zeta_{d}$ is a
primitive $d$th root of unity. Let $N$ be a $n\times n$ nilpotent matrix. For
$n\leq 5$ and all possible choices of $N$, we have computationally found a
value of $d$ and a direct summand $M_{t}$ of dimension $n$, such that the
following special properties are satisfied (see Proposition 2.1.9 for a
precise statement)

  1. (BM)

Big monodromy: The Zariski closure of the geometric monodromy group
$G_{geom}\subset\operatorname{GL}(M_{B,t}\otimes\mathbb{C})$ contains
$\operatorname{SL}_{n}$, where $M_{B,t}$ is the Betti realisation of $M_{t}$.

  2. (R)

Regular Hodge–Tate weights: The Hodge numbers of
$M_{dR,t}\otimes_{\iota}\mathbb{C}$ are $\leq 1$ for every embedding
$\iota:\mathbb{Q}(\zeta_{d})\to\mathbb{C}$, where $M_{dR,t}$ is the de Rham
realisation of $M_{t}$.

  3. (UM)

Unipotent monodromy at infinity: The monodromy operator at $t=\infty$ acting
on $M_{B,t}$ is conjugate to $\exp(N)$.

When $N$ has maximal rank, such families are known for all $n$ by the work of
Barnet-Lamb [BL13], [BL10]. For general $N$ it turns out to be (at least
computationally) more difficult to find direct summands which simultaneously
satisfy (BM), (R) and (UM). We have not succeeded in finding a general recipe.
However, when a family $M_{t}$ with these properties exists for a given $N$,
we can apply the method of [HSBT10] to $M_{t}$ to prove a potential automorphy
theorem with additional control on local monodromy. This theorem and its
consequences will be introduced in §1.2 below.

###  1.1. Families of Motives

Let $0<n<d$ be an integer. We define a hypergeometric parameter of dimension
$n$ modulo $d$ as a tuple a tuple
$(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ of elements of
$\mathbb{Z}/d\mathbb{Z}$ such that

  * •

$\sum_{i}\alpha_{i}-\sum_{i}\beta_{i}=\binom{d}{2}$,

  * •

$\alpha_{i}\neq\beta_{j}$ for $i,j\in\\{1,\dots,n\\}$,

  * •

$\beta_{i}\neq\beta_{j}$ for $i\neq j$.

Associated with such a parameter, one obtains an $n$-dimensional family of
motives

$M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})_{t}$

which is a $\mathbb{Z}[\zeta_{d}]$-linear direct summand of the cohomology of
the Dwork family of degree $d$. Conversely, any irreducible summand of
$H^{d-2}(Y_{t},\mathbb{Z}[\frac{1}{d},\zeta_{d}])$ is of this form.

######  1.1.1

In §2 we explain how to describe properties of the family of motives
$M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})_{t}$ in terms of
simple combinatorics of the tuple
$(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$. For instance, the
Jordan normal form of the monodromy operator at $t=\infty$ has block sizes
$\\#\\{i\in\\{1,\dots,n\\}:\alpha_{i}=\alpha\\}$ for
$\alpha\in\mathbb{Z}/d\mathbb{Z}$. This treats property (UM). We also state
explicit combinatorial criteria for properties (BM) and (R).

######  1.1.2

The work of Harris–Shepherd-Barron–Taylor [HSBT10] first introduced the idea
of using the Dwork family for potential automorphy. They considered $d$ odd,
$n=d-1$ and used the hypergeometric parameter

$(0,\dots,0;1,2,\dots,d-1).$

The corresponding family of motives has maximal unipotent monodromy at
$t=\infty$ and the geometric monodromy group is Zariski dense in
$\operatorname{Sp}_{n}$. The Hodge numbers are $h^{i,j}=1$ for $i+j=d-2$ and
$i,j\geq 0$.

######  1.1.3

In later works, Barnet-Lamb [BL13], [BL10] also considered parameters of the
form

$(0,\dots,0;\beta_{1},\dots,\beta_{n})$

for any degree $d$ and suitable choices of
$\beta_{i}\in\mathbb{Z}/d\mathbb{Z}$. This gives rise to monodromy groups
which are dense in $\operatorname{SL}_{n}$, i.e. satisfy (BM). It turns out
that parameters of this form also always satisfy (R). Subsequent variants
[BLGHT11], [Qia22] introduced even more parameters of this form to prove their
potential automorphy theorems. All of these examples share the property that
the monodromy at $t=\infty$ is maximally unipotent, i.e. they do not
necessarily satisfy (UM).

######  1.1.4

To realise nilpotent operators with Jordan block sizes $p_{1},\dots,p_{r}$ in
property (UM), we consider parameters of the form

$(\underbrace{0,\dots,0}_{p_{1}},\underbrace{\gamma_{2},\dots,\gamma_{2}}_{p_{2}},\dots,\underbrace{\gamma_{r},\dots,\gamma_{r}}_{p_{r}};\beta_{1},\dots,\beta_{n}).$

with $\gamma_{i},\beta_{i}\in\mathbb{Z}/d\mathbb{Z}$. Unfortunately, it is
quite rare for such a parameter to have regular Hodge–Tate weights (property
(R)). Although it is easy to check for any given parameter if (R) is
satisfied.

Indeed, Proposition 2.3.2 implies that a parameter $(\alpha_{i};\beta_{j})$
satisfies (R) if and only if for every
$s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$, the following $n$ integers are
pairwise distinct

$\sum_{i}[s\beta_{j}-s\alpha_{i}]-\sum_{i}[s\beta_{j}-s\beta_{i}]\qquad
j=1,\dots,n,$

where $[x]\in\mathbb{Z}$ denotes the unique representative of
$x\in\mathbb{Z}/d\mathbb{Z}$ satisfying $0\leq[x]\leq d-1$. Alternatively,
this condition can be phrased in terms of the cyclic order on
$\mathbb{Z}/d\mathbb{Z}$ as in [RRV22, §5]: For every
$s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$, the zigzag procedure for
$(s\alpha_{i};s\beta_{j})$ should have at most one local maximum and one local
minimum.

Since scaling by $s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$ does not interact
well with the function $[\cdot]$, we do not know a good way of generating
parameters which satisfy (R).

######  1.1.5

For $n\leq 5$ and any nilpotent operator, we have computationally found
parameters satisfying properties (BM), (R), (UM) (see table 2 at the end of
this paper). For example for $n=4$, and

$N=\begin{bmatrix}0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&0\\\ 0&0&0&0\end{bmatrix}$

we found the parameter modulo $18$ given by

$(0,0,0,3;4,11,16,17).$

We also computed some examples with $n=6$. The method of computation is
presented in §4. It would be interesting to improve the brute-force algorithm
or to find more conceptual ways to satisfy the combinatorial properties.

###  1.2. Galois Representations

Applying the method of [HSBT10] to our families of motives, we can prove a
potential automorphy theorem with additional control on the monodromy
operator. To state it we recall a few technical terms.

######  1.2.1

For a number field $F$, we let $\mathbb{A}_{F}$ be the ring of adeles over $F$
and we let $G_{F}$ denote the absolute Galois group of $F$. Moreover, we let
$\mu_{d}$ be the set of $d$th roots of unity in some algebraic closure of $F$
and we let $\chi_{cyc}:G_{F}\to\mathbb{Z}_{\ell}^{\times}$ denote the
$\ell$-adic cyclotomic character. For an isomorphism
$\iota:\overline{\mathbb{Q}}_{\ell}\cong\mathbb{C}$ and a continuous Galois
representation
$\rho:G_{F}\to\operatorname{GL}_{n}(\overline{\mathbb{Q}}_{\ell})$, we refer
to section §3.1 for the definition and properties of the Weil–Deligne
representations $WD_{\iota}(\rho\rvert_{F_{v}})$, where $v$ is a prime of $F$.
Finally, we shall denote by
$\overline{\rho}:G_{F}\to\operatorname{GL}_{n}(\overline{\mathbb{F}}_{\ell})$
the reduction of $\rho$ modulo $\ell$, which is a well-defined semisimple
representation.

######  1.2.2

We say that an automorphic representation $\Pi$ of
$\operatorname{GL}_{n}(\mathbb{A}_{F})$ is cohomological if there exists an
algebraic representation $V$ of $\operatorname{GL}_{n}/F$ such that the
infinitesimal character of $\Pi_{\infty}$ agrees with that of $V$. For an
isomorphism $\iota:\overline{\mathbb{Q}}_{\ell}\cong\mathbb{C}$ and a
cohomological cuspidal automorphic representation, we let
$r_{\iota}(\Pi):G_{F}\to\operatorname{GL}_{n}(\overline{\mathbb{Q}}_{\ell})$
be the Galois representation defined in [HLTT16]. Moreover, we will need the
notion of $\iota$-ordinarity of $\Pi$ at a prime $v$ of $F$, which is defined
in [Ger19, 5.3].

######  1.2.3 Theorem ( = Theorem 3.2.1).

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$ appearing in table 2.
There exists an integer $C>0$ with the following property. Given the following
objects:

  * •

a CM field $F$ containing $\mu_{d}$,

  * •

a prime $p$,

  * •

a prime $\ell\nmid pC$ such that $\ell\equiv 1\pmod{d}$,

  * •

a continuous semisimple representation
$\overline{\rho}:G_{F}\to\operatorname{GL}_{n}(\mathbb{F}_{\ell})$ such that
$\det\overline{\rho}=\overline{\chi}_{cyc}^{-n(n-1)/2}$,

  * •

a finite extension $F^{avoid}/F$,

there exists a finite totally real extension $F_{2}/\mathbb{Q}$, linearly
disjoint from $F^{avoid}$ over $\mathbb{Q}$ such that for $F^{\prime}=FF_{2}$,
there exists an isomorphism
$\iota:\overline{\mathbb{Q}}_{\ell}\cong\mathbb{C}$ and a cuspidal automorphic
representation $\Pi$ of $\operatorname{GL}_{n}(\mathbb{A}_{F^{\prime}})$ which
is $\iota$-ordinary of weight $0$ at all places above $\ell$ such that

$\overline{r_{\iota}(\Pi)}=\overline{\rho}$

and for all places $v\mid p$ of $F^{\prime}$

$WD(r_{\iota}(\Pi)\rvert_{G_{F_{v}^{\prime}}})^{F-ss}$

is generic with monodromy operator having Jordan blocks $J_{\alpha}$ of size
$\\#\\{i:\alpha_{i}=\alpha\\}$ for $\alpha\in\mathbb{Z}/d\mathbb{Z}$. If $d$
is odd, then there also exists another cuspidal automorphic representation
$\Pi_{2}$ of $\operatorname{GL}_{n}(\mathbb{A}_{F^{\prime}})$ with the same
properties as $\Pi$ except that the $v$-adic monodromy operator is nilpotent
of rank $1$.

######  1.2.4

This theorem can be seen as a refinement of the main theorem of [Qia22].
Namely, we prove potential automorphy together with a specified choice of
local monodromy. The proof of the theorem closely follows the methods of
[HSBT10]. However, one major new ingredient is a theorem of Scholze [Sch12],
which proves the weight-monodromy conjecture for hypersurfaces. We use it to
obtain the genericity in the conclusion of the theorem.

We require stronger assumptions on $\ell$ and the coefficient field of
$\overline{\rho}$ as in [Qia22], but these could be removed if we had more
examples of motives satsifying the hypotheses of proposition 2.5.9 with
sufficiently small $U$. Our assumption on $\ell$ also allows us to easily
prove the ordinarity of the Galois representations corresponding to our
families of motives. (The proof of ordinarity from [Qia22] does not apply in
our setting.)

######  1.2.5

As a consequence of theorem 3.2.1, we prove the existence of non conjugate
self-dual automorphic representations with local monodromy operator $N$ (see
corollary 3.2.3). We do not know how to construct such representations by
purely automorphic methods. For groups which have discrete series, there are
such methods [Shi12, Thm 5.7], but this does not apply to non conjugate self-
dual automorphic representations of $\operatorname{GL}_{n}$ for $n\geq 3$.

As another application, we generalise the method introduced by Allen–Newton
[AN20] to prove the following local-global compatibility theorem.

######  1.2.6 Theorem ( = Theorem 3.3.3).

Let $n\in\\{3,4,5\\}$ and $d_{3}=9$, $d_{4}=18$, $d_{5}=168$. There exists an
integer $C>0$ with the following property. Given the following objects:

  * •

a CM field $F$ containing $\mu_{d_{n}}$,

  * •

a prime $\ell$ such that $\ell\nmid C$ and $\ell\equiv 1\pmod{d_{n}}$,

  * •

a cuspidal cohomological automorphic representation $\Pi$ of
$\operatorname{GL}_{n}(\mathbb{A}_{F})$ which is $\iota$-ordinary at all
places above $\ell$,

such that

  * •

$\overline{r_{\iota}(\Pi)}(G_{F(\mu_{\ell})})$ is enormous [ACC+23, 6.2.28]
and $\overline{r_{\iota}(\Pi)}$ is absolutely irreducible and decomposed
generic [ACC+23, 4.3.1],

  * •

there exists $\sigma\in G_{F}\setminus G_{F(\mu_{\ell})}$ such that
$\overline{r_{\iota}(\Pi)}(\sigma)$ is a scalar,

  * •

there exists $\gamma\in\operatorname{GL}_{n}(\overline{\mathbb{F}}_{\ell})$
such that
$\gamma\overline{r_{\iota}(\Pi)}(G_{F})\gamma^{-1}\subset\operatorname{GL}_{n}(\mathbb{F}_{\ell})$,

  * •

$\det\overline{r_{\iota}(\Pi)}=\overline{\chi}_{cyc}^{-n(n-1)/2}$,

we have

$WD(r_{\iota}(\Pi)\rvert_{G_{F_{v}}})^{F-ss}\cong\operatorname{rec}_{F_{v}}(\Pi_{v}|\det|^{(1-n)/2}),$

for all places $v\nmid\ell$.

######  1.2.7

During the preparation of the present article, Matsumoto [Mat24] has proven a
very general local-global compatibility theorem which works for all $n$ and
includes our theorem 3.3.3 as a special case. Their proof is based on the
Harris tensor product trick and requires neither our potential automorphy
theorem nor any new families of motives.

Previously, Varma [Var15] has proven local-global compatibility up to
semisimplicity. For $n=2$, Allen–Newton [AN20] and Yang [Yan21] have proven
many cases of full local-global compatibility. Their work was the inspiration
for theorem 3.3.3.

When our theorem or Matsumoto’s theorem applies, then it implies that
condition (d) of [A’C23, Thm 5.3.5] is satisfied. This was the original
motivation for this paper.

### Acknowledgements

I am grateful to my advisor James Newton for his encouragement to write up
this paper and for many interesting discussions on the subject. I thank
Alexander Bertoloni Meli, Lie Qian, Peter Scholze, Richard Taylor and Pol van
Hoften for helpful conversations and correspondence.

##  2\. The Dwork Family

######  2.0.1

In this section we recall the properties of the (unweighted) Dwork family and
define the families of motives which are the subject of this paper. Many
authors have written about this subject, but our main references are [DMOS82],
[HSBT10], [Kat09], [Klo07].

Let $d>2$ be an integer, set $\mathcal{O}_{d}=\mathbb{Z}[\frac{1}{d},e^{2\pi
i/d}]\subset\mathbb{C}$ and let

$A_{d}=\mathcal{O}_{d}\left[t,\frac{1}{t^{d}-1}\right].$

######  2.0.2 Definition.

Let $d>2$ be an integer. We define the projective $A_{d}$-scheme

$\mathcal{Y}:=\operatorname{Proj}A_{d}[x_{0},\dots,x_{d-1}]/(F_{t}),$

where $F:=\sum_{i=0}^{d-1}x_{i}^{d}$ and $F_{t}=F-tdx_{0}\cdots x_{d-1}$.
Denote the structure morphism by

$f:\mathcal{Y}\to\operatorname{Spec}A_{d}.$

One can check that $f$ is smooth using the Jacobian criterion.

######  2.0.3

Let $\mu_{d}=\\{z\in\mathbb{C}:z^{d}=1\\}\subset\mathbb{C}$ denote the group
of $d$th roots of unity under multiplication. The group

$H_{0}:=\\{(\zeta_{0},\dots,\zeta_{d-1})\in\mu_{d}^{d}:\zeta_{0}\cdots\zeta_{d-1}=1\\}/\\{(\zeta,\dots,\zeta):\zeta\in\mu_{d}\\}$

acts on $\mathcal{Y}$ by

$(\zeta_{0},\dots,\zeta_{d-1})\cdot(x_{0}:\dots:x_{d-1})=(\zeta_{0}x_{0}:\dots:\zeta_{d-1}x_{d-1}),$

leaving $f$ invariant. On the fibre
$\mathcal{Y}_{0}:=\mathcal{Y}\times_{\operatorname{Spec}A_{d}}\operatorname{Spec}A_{d}/(t)$,
the same formula defines an action of the bigger group

$H:=\mu_{d}^{d}/\\{(\zeta,\dots,\zeta):\zeta\in\mu_{d}\\}.$

######  2.0.4

For $\underline{a}=(a_{0},\dots,a_{d-1})\in(\mathbb{Z}/d\mathbb{Z})^{d}$ such
that $\sum_{i}a_{i}=0$ we define the character

$\chi_{\underline{a}}:H\to\mathcal{O}_{d}^{\times}:(\zeta_{0},\dots,\zeta_{d-1})\mapsto\prod_{i=0}^{d-1}\zeta_{i}^{a_{i}}.$

Every character of $H$ is of this form for a unique $\underline{a}$ and
$\chi_{\underline{a}}\rvert_{H_{0}}=\chi_{\underline{a}^{\prime}}\rvert_{H_{0}}$
if and only if there exists $b\in\mathbb{Z}/d\mathbb{Z}$ such that

$\underline{a}^{\prime}=\underline{a}+b:=(a_{0}+b,a_{1}+b,\dots,a_{d-1}+b).$

######  2.0.5

If a group $G$ acts on the left on a topological space $X$, we define the
induced left group action on its cohomology $H^{\bullet}(X,\mathbb{Z})$ by

$g\cdot\omega:=(g^{-1})^{*}\omega$

for $g\in G$ and $\omega\in H^{\bullet}(X,\mathbb{Z})$. If $V$ is a
$\mathcal{O}_{d}[H_{0}]$-module we let $V_{\underline{a}}$ be the
$\chi_{\underline{a}}\rvert_{H_{0}}$-isotypic part of $V$. For example

$H^{d-2}(\mathcal{Y}_{t},\mathbb{C})_{\underline{a}}=\\{\omega\in
H^{d-2}(\mathcal{Y}_{t},\mathbb{C}):\forall h\in H_{0},\
(h^{-1})^{*}\omega=\chi_{\underline{a}}(h)\omega\\}.$

###  2.1. Statements

Here we define our families of motives and state their most important
properties. The proofs follow in the subsequent sections.

######  2.1.1 Definition.

Let $d>n>0$ be integers. A hypergeometric parameter of dimension $n$ modulo
$d$ is a tuple $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ of
elements of $\mathbb{Z}/d\mathbb{Z}$ such that

  * •

$\sum_{i}\alpha_{i}-\sum_{i}\beta_{i}=\binom{d}{2}$,

  * •

$\alpha_{i}\neq\beta_{j}$ for $i,j\in\\{1,\dots,n\\}$,

  * •

$\beta_{i}\neq\beta_{j}$ for $i\neq j$.

######  2.1.2 Definition.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$. Let

$\underline{a}=(-\alpha_{1},\dots,-\alpha_{n},s_{0},\dots,s_{d-n-1})\in(\mathbb{Z}/d\mathbb{Z})^{d},$

where
$\\{s_{1},\dots,s_{d-n-1}\\}=\\{0,\dots,d-1\\}\setminus\\{-\beta_{i}\\}$.
Since $(\alpha_{i};\beta_{j})$ is a hypergeometric parameter, we have
$\sum_{i}a_{i}=0$. Suppose $K$ is a field containing the $d$th roots of unity
and $t\in K\setminus\mu_{d}$. We define

  * •

$M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})_{dR,t}:=H^{d-2}_{dR}(\mathcal{Y}_{t}/K)_{\underline{a}}^{(prim)}$,

  * •

$M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})_{\lambda,t}=H^{d-2}_{et}(\mathcal{Y}_{t,\overline{K}},\mathcal{O}_{d,\lambda})_{\underline{a}}^{(prim)}$
for primes $\lambda<\mathcal{O}_{d}$,

  * •

$M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})_{B,t}:=H^{d-2}_{sing}(\mathcal{Y}_{t}(\mathbb{C}),\mathcal{O}_{d})_{\underline{a}}^{(prim)}$
if $K\subset\mathbb{C}$.

Here $(prim)$ refers to the primitive part of the cohomology and is only
relevant for the definition when $n$ is odd and $d=n+1$.

######  2.1.3 Proposition (= Proposition 2.3.2).

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$. For every
$t\in\mathbb{C}\setminus\mu_{d}$, we have:

  * •

$M_{*,t}$ is a free module of rank $n$ for $*\in\\{dR,B,\lambda\\}$.

  * •

The dimension of $\operatorname{gr}_{\mathrm{Hodge}}^{p}(M_{dR,t})$ is equal
to the number of indices $j\in\\{1,\dots,n\\}$ such that

$d(p+1)=\binom{d}{2}+\sum_{i=1}^{n}[\beta_{j}-\alpha_{i}]-\sum_{i=1}^{n}[\beta_{j}-\beta_{i}],$

where $[x]$ denotes the unique representative of $x$ such that $0\leq[x]\leq
d-1$.

######  2.1.4 Proposition (= Proposition 2.5.7).

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$ and

$M=M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n}).$

For $t\in\mathbb{C}\setminus\mu_{d}$ let
$\gamma_{\infty}\in\pi_{1}(\mathbb{C}\setminus\mu_{d},t)$ be a simple loop
based at $t$ going around $\infty$. Then $\gamma_{\infty}$ acts on $M_{B,t}$
as a unipotent operator with Jordan blocks $J_{\alpha}$ of size
$\\#\\{i:\alpha_{i}=\alpha\\}$ for $\alpha\in\\{\alpha_{i}\\}$. Similarly, let
$\gamma_{1}$ denote a simple loop around $1$. Then $\gamma_{1}$ acts as a
unipotent operator on $M_{B,t}$ and the rank of $\gamma_{1}-\operatorname{id}$
is $1$ if $d$ is odd and $0$ if $d$ is even.

######  2.1.5 Proposition (= Proposition 2.5.8).

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$ and
$t\in\mathbb{C}\setminus\mu_{d}$. If the following hypotheses are satisfied

  * •

$|\\{\alpha_{1},\dots,\alpha_{n}\\}|<n$,

  * •

$\\{\beta_{1},\dots,\beta_{n}\\}$ does not form an arithmetic progression,

  * •

there is no non-zero $s\in\mathbb{Z}/d\mathbb{Z}$ such that

$\\{\alpha_{i}+s\\}=\\{\alpha_{i}\\}\quad\wedge\quad\\{\beta_{i}+s\\}=\\{\beta_{i}\\},$

  * •

there is no $s\in\mathbb{Z}/d\mathbb{Z}$ such that

$\\{-\alpha_{i}-s\\}=\\{\alpha_{i}+s\\}\quad\wedge\quad\\{-\beta_{i}-s\\}=\\{\beta_{i}+s\\},$

then the identity component of the Zariski closure of the image of the
geometric monodromy representation

$\pi_{1}(\mathbb{C}\setminus\mu_{d},t)\to\operatorname{GL}(M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})_{B,t}\otimes_{\mathcal{O}_{d}}\mathbb{C})$

contains $\operatorname{SL}_{n}(\mathbb{C})$.

######  2.1.6 Proposition ( = Proposition 2.5.9).

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$ and
$t\in\mathbb{C}\setminus\mu_{d}$ satisfying the hypotheses of the previous
proposition. Moreover, let $U,V<(\mathbb{Z}/d\mathbb{Z})^{\times}$ be
subgroups such that $\\{1\\}=U\cap V$, $UV=(\mathbb{Z}/d\mathbb{Z})^{\times}$
and for $s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$ the equalities

$\displaystyle\\{s\alpha_{i}-s\alpha_{j}:1\leq i,j\leq n\\}$
$\displaystyle=\\{\alpha_{i}-\alpha_{j}:1\leq i,j\leq n\\}$
$\displaystyle\\{s\beta_{i}-s\beta_{j}:1\leq i,j\leq n\\}$
$\displaystyle=\\{\beta_{i}-\beta_{j}:1\leq i,j\leq n\\}$

imply that $s\in U$. Let $\lambda<\mathcal{O}_{d}$ be a sufficiently large
prime of residue characteristic $\ell$ such that $\ell\in V$. Then the image
of

$\pi_{1}(\operatorname{Spec}A_{d}(\mathbb{C}),t)\to\operatorname{GL}(M(\alpha_{i};\beta_{j})_{B,t}\otimes_{\mathcal{O}_{d}}\mathcal{O}_{d}/\lambda)$

contains $\operatorname{SL}_{n}(\mathcal{O}_{d}/\lambda)$.

######  2.1.7

For $a\in(\mathbb{Z}/d\mathbb{Z})\setminus\\{0\\}$ and a divisor $k\mid d$
such that $ka\neq 0$, we define the functions
$\delta_{a},\epsilon_{k,a}:(\mathbb{Z}/d\mathbb{Z})\setminus\\{0\\}\to\mathbb{Z}$
as follows

$\displaystyle\delta_{a}(x)=\begin{cases}1&x=a\\\ 0&\text{ otherwise
}\end{cases}$

and

$\epsilon_{k,a}(x)=\delta_{-ka}+\sum_{0\leq j\leq k-1}\delta_{a+jd/k}.$

We let $E(d)$ denote the set of functions consisting of $\epsilon_{1,a}$ and
$\epsilon_{p,a}$, where $p$ is a prime divisor of $d$.

######  2.1.8 Proposition (= Proposition 2.6.8).

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$ and $F$ a number field
containing the $d$th roots of unity. Let $t\in F\setminus\mu_{d}$ and
$\underline{c}\in(\mathbb{Z}/d\mathbb{Z})^{3}$ such that $c_{0}+c_{1}+c_{2}=0$
and either $c_{i}\neq 0$ for $i=0,1,2$ or $c_{0}=c_{1}=c_{2}=0$. Suppose the
following hypotheses are satisfied:

  * •

If $d$ is even, then $1-t^{d}$ is a square.

  * •

For all $s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$, the following $n$ integers
are pairwise distinct

$\sum_{i}[s\beta_{j}-s\alpha_{i}]-\sum_{i}[s\beta_{j}-s\beta_{i}]\qquad
j=1,\dots,n.$

  * •

The following quantity is independent of
$s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$

$\sum_{i,j}[s\beta_{j}-s\alpha_{i}]-\sum_{i,j}[s\beta_{j}-s\beta_{i}]+n\sum_{i=0}^{2}[sc_{i}].$

  * •

For each $s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$ we have

$\sum_{i,j}[s\beta_{j}-s\alpha_{i}]=n\sum_{i}[s\beta_{i}-s\alpha_{i}].$

  * •

There are integers $x_{\epsilon}$ for $\epsilon\in E(d)$ such that

$n+\sum_{i,j}\delta_{\beta_{j}-\alpha_{i}}-\sum_{i\neq
j}\delta_{\beta_{j}-\beta_{i}}+n\sum_{i=0}^{2}\delta_{c_{i}}=\sum_{\epsilon\in
E(d)}x_{\epsilon}\epsilon.$

Let $b_{1}$ and $b_{p}$ be the integers defined in corollary 2.6.7. The
integer $\varphi(\operatorname{lcm}(2b_{1},d))/\varphi(d)$ is coprime to $n$
and $b_{p}$ is coprime to $n$.

Then there exists a character
$\psi:G_{\mathbb{Q}(\mu_{d})}\to\mathcal{O}_{d,\lambda}^{\times}$, such that

$\det(M(\alpha_{i};\beta_{j})_{\lambda,t}\otimes\psi)=\chi_{cyc}^{-n(n-1)/2}.$

######  2.1.9 Proposition.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$ appearing with
$U<(\mathbb{Z}/d\mathbb{Z})^{\times}$ in table 2 at the end of the paper. Let
$V<(\mathbb{Z}/d\mathbb{Z})^{\times}$ be a subgroup such that $\\{1\\}=U\cap
V$, $UV=(\mathbb{Z}/d\mathbb{Z})^{\times}$. Then the family of motives
$M=M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ enjoys the
following properties

  1. (BM)

For $t\in\mathbb{C}\setminus\mu_{d}$, the Zariski closure of the image of

$\pi_{1}(\mathbb{C}\setminus\mu_{d},t)\to\operatorname{GL}(M_{B,t}\otimes_{\mathcal{O}_{d}}\mathbb{C})$

contains $\operatorname{SL}_{n}(\mathbb{C})$. Moreover, there exists a
constant $C=C(d,\alpha_{i};\beta_{i})$ such that if
$\lambda\neq\lambda^{\prime}<\mathcal{O}_{d}$ are primes above
$\ell,\ell^{\prime}$ not dividing $C$ and $\ell,\ell^{\prime}\in V$, then the
image of

$\pi_{1}(\mathbb{C}\setminus\mu_{d},t)\to\operatorname{GL}(M_{B,t}\otimes_{\mathcal{O}_{d}}\mathcal{O}_{d}/\lambda\lambda^{\prime})$

contains $\operatorname{SL}_{n}(\mathcal{O}_{d}/\lambda\lambda^{\prime})$.

  2. (R)

For $t\in\mathbb{C}\setminus\mu_{d}$, the Hodge numbers of
$M_{dR,t}\otimes_{\iota}\mathbb{C}$ are $\leq 1$ for every embedding
$\iota:\mathcal{O}_{d}\to\mathbb{C}$.

  3. (UM)

For $t\in\mathbb{C}\setminus\mu_{d}$, let $\gamma_{\infty}$ denote a simple
loop around $\infty$ based at $t$. Then $\gamma_{\infty,*}$ acts on $M_{B,t}$
as a unipotent operator with Jordan blocks $J_{\alpha}$ of size
$\\#\\{i:\alpha_{i}=\alpha\\}$ for $\alpha\in\\{\alpha_{i}\\}$.

  4. (D)

Let $F$ be a number field containing the $d$th roots of unity and $t\in
F\setminus\mu_{d}$ such that if $d$ is even, then $1-t^{d}$ is a square. The
Galois group $G_{F}$ acts on $\bigwedge^{n}M_{\lambda,t}$ as
$\chi_{cyc}^{-n(n-1)/2}\psi^{-n}$, for some continuous character
$\psi:G_{F}\to\mathcal{O}_{d,\lambda}^{\times}$ which is independent of $t$.

###### Proof.

With the previous propositions this proof becomes a purely computational task
which has been completed by the computer program described in §4. ∎

###  2.2. The Method of Griffiths

The work of Griffiths [Gri69] conveniently describes the Hodge theory and
periods of projective hypersurfaces such as $\mathcal{Y}_{t}$. Thus, we recall
a few aspects of Griffiths’ theory and then apply it to the local system
$M(\alpha_{i};\beta_{j})_{B,t}$ and the variation of Hodge structures given by
$M(\alpha_{i};\beta_{j})_{dR,t}$.

######  2.2.1

For $t\in\operatorname{Spec}A_{d}(\mathbb{C})=\mathbb{C}\setminus\mu_{d}$,
there is the tube map

$\tau:H_{d-2}(\mathcal{Y}_{t}(\mathbb{C}),\mathcal{O}_{d})^{(prim)}\to
H_{d-1}(\mathbb{P}^{d-1}(\mathbb{C})\setminus\mathcal{Y}_{t}(\mathbb{C}),\mathcal{O}_{d})$

defined in [Gri69, §3]. It is an isomorphism by [Gri69, Prop 3.5].

######  2.2.2

We describe the cohomology of the smooth projective hypersurface
$\mathcal{Y}_{t}\subset\mathbb{P}^{d-1}$ in terms of the de Rham cohomology of
the affine variety $\mathbb{P}^{d-1}\setminus\mathcal{Y}_{t}$ as follows.
Define the vector spaces

$A_{l}^{q}(\mathcal{Y}_{t}):=\\{\omega\in
H^{0}(\mathbb{P}^{d-1}_{\mathbb{C}}\setminus\mathcal{Y}_{t,\mathbb{C}},\Omega^{q}_{\mathbb{P}^{d-1}}):F_{t}^{l}\omega\text{
is regular on }\mathbb{P}^{d-1}_{\mathbb{C}}\\}$

and let $F^{\bullet}H^{d-2}(\mathcal{Y}_{t},\mathbb{C})$ denote the Hodge
filtration. Then [Gri69, Thm 8.1] implies that we have linear maps

$A^{d-1}_{l}(\mathcal{Y}_{t})/dA^{d-1}_{l-1}(\mathcal{Y}_{t})\xrightarrow{R}F^{d-1-l}H^{d-2}(\mathcal{Y}_{t}(\mathbb{C}),\mathbb{C})$

for all $l\geq 1$ which satisfy the following characterising property. If
$\gamma\in H_{d-2}(\mathcal{Y}_{t}(\mathbb{C}),\mathcal{O}_{d})$, then

$\frac{1}{2\pi i}\int_{\tau(\gamma)}\omega=\int_{\gamma}R(\omega).$

In particular, for all $h\in H_{0}$ we have $R(h^{*}\omega)=h^{*}R(\omega)$.
We moreover know that for $l=1,2,\dots,d-1$, the map

$A^{d-1}_{l}(\mathcal{Y}_{t})/dA^{d-1}_{l-1}(\mathcal{Y}_{t})\xrightarrow{R}F^{d-1-l}H^{d-2}(\mathcal{Y}_{t}(\mathbb{C}),\mathbb{C})$

is injective and the image is the primitive part of the cohomology [Gri69, Thm
8.3].

######  2.2.3

Let $\mathbb{C}[x_{0},\dots,x_{d-1}]_{k}$ denote the vector space of
homogeneous degree $k$ polynomials. By [Gri69, Cor 2.11], we have an
isomorphism

$\mathbb{C}[x_{0},\dots,x_{d-1}]_{dl}\to
A_{l+1}^{d-1}(\mathcal{Y}_{t}):P\mapsto\frac{P}{F_{t}^{l+1}}\Omega$

for $l=0,\dots,d-2$, where

$\Omega=\sum_{i=0}^{d-1}(-1)^{i}x_{i}dx_{0}\wedge\dots\wedge\widehat{dx_{i}}\wedge\dots
dx_{d-1}.$

Let $J_{t}=(\frac{\partial}{\partial
x_{0}}F_{t},\dots,\frac{\partial}{\partial
x_{d-1}}F_{t})<\mathbb{C}[x_{0},\dots,x_{d-1}]$ be the Jacobian ideal of
$F_{t}$. Then [Gri69, Prop 4.6] implies that the map

$\displaystyle\left(\mathbb{C}[x_{0},\dots,x_{d-1}]/J_{t}\right)_{dl}$
$\displaystyle\to H^{d-2-l,l}(\mathcal{Y}_{t}(\mathbb{C}))$ $\displaystyle P$
$\displaystyle\mapsto R\left(\frac{P}{F_{t}^{l+1}}\Omega\right)$

is a well-defined injection whose image is the primitive part of the
cohomology.

###  2.3. The Cohomology at $t=0$

The cohomology of $\mathcal{Y}_{0}$ is described in detail in [DMOS82, I. §7].
We recall parts of this description here, since we will need it for our study
of the periods of $\mathcal{Y}_{t}$ for $t\neq 0$.

######  2.3.1

For $t=0$, we have the particularly simple Jacobian ideal

$J_{0}=(x_{0}^{d-1},\dots,x_{d-1}^{d-1}).$

Thus, $\mathbb{C}[x_{0},\dots,x_{d-1}]/J_{0}$ has the monomial basis

$\\{x^{\underline{e}}=x_{0}^{e_{0}}x_{1}^{e_{1}}\dots
x_{d-1}^{e_{d-1}}:\forall i=0,\dots,d-1,\ e_{i}<d-1\\}.$

Moreover, if $h=(\zeta_{0},\dots,\zeta_{d-1})\in H$, then

$(h^{-1})^{*}\frac{x^{\underline{e}}\Omega}{F^{l+1}}=\prod_{i=0}^{d-1}\zeta_{i}^{-e_{i}-1}\frac{x^{\underline{e}}\Omega}{F^{l+1}}.$

Consequently, paragraph 2.2.3 implies that for
$\underline{a}\in(\mathbb{Z}/d\mathbb{Z})^{d}$ such that
$\sum_{i=0}^{d-1}a_{i}=0$, we have the decomposition

$H^{d-2-l,l}(\mathcal{Y}_{0}(\mathbb{C}))_{\underline{a}}^{(prim)}=\bigoplus_{\underline{e}\in
B_{\underline{a}}^{l}}R\left(\frac{x^{\underline{e}}\Omega}{F^{l+1}}\right)\mathbb{C},$

where

$B_{\underline{a}}^{l}=\left\\{\underline{e}\in\\{0,\dots,d-2\\}^{d}:\exists
b\in\mathbb{Z}/d\mathbb{Z},\begin{array}[]{l}-e_{i}-1\equiv a_{i}+b\pmod{d}\\\
\sum_{i=0}^{d-1}e_{i}=dl\end{array}\right\\}.$

Equivalently, the dimension of
$H^{d-2-l,l}(\mathcal{Y}_{0}(\mathbb{C}))_{\underline{a}}^{(prim)}$ is equal
to the number of solutions to the equation

$dl=\sum_{i=0}^{d-1}[b-a_{i}]$

with $b\in\mathbb{Z}/d\mathbb{Z}\setminus\\{-a_{0},\dots,-a_{d-1}\\}$, where
$[x]$ denotes the unique representative of $x\in\mathbb{Z}/d\mathbb{Z}$ in
$\\{0,1,\dots,d-1\\}$. Summing over $l=0,\dots,d-2$ we find that

$\displaystyle\dim
H^{d-2}(\mathcal{Y}_{0}(\mathbb{C}),\mathbb{C})_{\underline{a}}^{(prim)}$
$\displaystyle=|\mathbb{Z}/d\mathbb{Z}\setminus\\{-a_{0},\dots,-a_{d-1}\\}|.$

As a consequence, we obtain

######  2.3.2 Proposition.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$. For every
$t\in\mathbb{C}\setminus\mu_{d}$, we have:

  * •

$M_{*,t}$ is a free module of rank $n$ for $*\in\\{dR,B,\lambda\\}$.

  * •

The dimension of $\operatorname{gr}_{\mathrm{Hodge}}^{p}(M_{dR,t})$ is equal
to the number of indices $j\in\\{1,\dots,n\\}$ such that

$d(p+1)=\binom{d}{2}+\sum_{i=1}^{n}[\beta_{j}-\alpha_{i}]-\sum_{i=1}^{n}[\beta_{j}-\beta_{i}],$

where $[x]$ denotes the unique representative of $x$ such that $0\leq[x]\leq
d-1$.

###### Proof.

The singular cohomology of projective hypersurfaces is torsion-free, thus by
the usual comparison theorems it suffices to prove the first part for
$M_{B,t}\otimes_{\mathcal{O}_{d}}\mathbb{C}$. Moreover, the dimension of
$M_{B,t}\otimes_{\mathcal{O}_{d}}\mathbb{C}$ is locally constant by the
theorem of Ehresmann so it suffices to prove the claim for $t=0$. By
definition,
$M_{B,0}=H^{d-2}(\mathcal{Y}_{0}(\mathbb{C}),\mathcal{O}_{d})_{\underline{a}}$,
where

$\underline{a}=(-\alpha_{1},\dots,-\alpha_{n},s_{0},\dots,s_{d-n-1})\in(\mathbb{Z}/d\mathbb{Z})^{d}$

and $\\{s_{1},\dots,s_{d-n-1}\\}=\\{0,\dots,d-1\\}\setminus\\{-\beta_{i}\\}$.
Hence,

$\mathbb{Z}/d\mathbb{Z}\setminus\\{-a_{0},\dots,-a_{d-1}\\}=\\{\beta_{1},\dots,\beta_{n}\\}$

has cardinality $n$ and paragraph 2.3.1 implies that

$\dim M_{B,0}\otimes\mathbb{C}=\dim
H^{d-2}(\mathcal{Y}_{0}(\mathbb{C}),\mathbb{C})^{(prim)}_{\underline{a}}=n.$

For the second part we observe that

$\sum_{i=0}^{d-1}[\beta_{j}-a_{i}]=\binom{d}{2}+\sum_{i=0}^{d-1}[\beta_{j}-\alpha_{i}]-\sum_{i=0}^{d-1}[\beta_{j}-\beta_{i}].$

This proves the claim for $t=0$ by paragraph 2.3.1. Since the Hodge filtration
is given by vector bundles (which have locally constant dimension), this also
proves the claim for general $t$. ∎

######  2.3.3 Example.

For $d=3$, we have the isomorphisms

$H^{1,0}(\mathcal{Y}_{0}(\mathbb{C}))=R\left(\frac{\Omega}{F}\right)\mathbb{C}\qquad
H^{0,1}(\mathcal{Y}_{0}(\mathbb{C}))=R\left(\frac{x_{0}x_{1}x_{2}\Omega}{F^{2}}\right)\mathbb{C}$

######  2.3.4 Example.

For $d=5$ the dimension of $H^{3}(\mathcal{Y}_{0}(\mathbb{C}))$ is 204 and we
have the isomorphisms

$H^{3-l,l}(\mathcal{Y}_{0}(\mathbb{C}))^{H_{0}}=R\left(\frac{(x_{0}x_{1}x_{2}x_{3}x_{4})^{l}\Omega}{F^{l+1}}\right)\mathbb{C}.$

Let $\underline{a}=(0,-1,-1,-1,-2)$. Then there are isomorphisms

$\displaystyle H^{2,1}(\mathcal{Y}_{0}(\mathbb{C}))_{\underline{a}}$
$\displaystyle=R\left(\frac{x_{1}x_{2}x_{3}x_{4}^{2}\Omega}{F^{2}}\right)\mathbb{C}$
$\displaystyle H^{1,2}(\mathcal{Y}_{0}(\mathbb{C}))_{\underline{a}}$
$\displaystyle=R\left(\frac{x_{0}x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{3}\Omega}{F^{3}}\right)\mathbb{C}.$

######  2.3.5 Lemma.

Suppose $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ is a
hypergeometric parameter of dimension $n$ modulo $d$ such that for every
$s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$, the following $n$ integers are
pairwise distinct

$\sum_{i=1}^{n}[-s\alpha_{i}+s\beta_{j}]-\sum_{i\neq
j}[s\beta_{j}-s\beta_{i}]\qquad j=1,\dots,n.$

Let $\lambda<\mathcal{O}_{d}$ be a prime above $\ell$ and let
$\iota:\mathcal{O}_{d,\lambda}\to\overline{\mathbb{Q}}_{\ell}$ be an
embedding. If $F\subset\overline{\mathbb{Q}}_{\ell}$ is a finite extension of
$\mathbb{Q}_{\ell}(\mu_{d})$ and $t\in F\setminus\mu_{d}$, then for every
embedding $\tau:F\hookrightarrow\overline{\mathbb{Q}}_{\ell}$, the
$\tau$-Hodge–Tate weights of

$M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})_{\lambda,t}\otimes_{\iota}\overline{\mathbb{Q}}_{\ell}$

are pairwise distinct.

###### Proof.

Let $\underline{a}\in(\mathbb{Z}/d\mathbb{Z})^{d}$ be the vector from
definition 2.1.2. Let $h\geq 0$ be an integer and
$\tau:F\hookrightarrow\overline{\mathbb{Q}}_{\ell}$ a field embedding. By the
Hodge–Tate comparison theorem [Fal88], $h$ is a $\tau$-Hodge–Tate weight of

$M(\alpha_{i};\beta_{j})_{\lambda,t}\otimes_{\iota}\overline{\mathbb{Q}}_{\ell}=H^{d-2}(\mathcal{Y}_{t,\overline{F}},\mathcal{O}_{d,\lambda}\otimes_{\iota}\overline{\mathbb{Q}}_{\ell})_{\underline{a}}$

of multiplicity

$\dim
H^{d-2-h}(\mathcal{Y}_{\tau(t),\overline{\mathbb{Q}}_{\ell}},\Omega^{h})_{\underline{a}}.$

Let $s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$ be the unique element such that
$\iota^{-1}(\tau(\iota(e^{2\pi i/d})))=e^{2\pi is/d}$. We have a commutative
diagram

${\mathcal{Y}_{\tau(t),\overline{\mathbb{Q}}_{\ell}}}$${\mathcal{Y}_{t,\overline{\mathbb{Q}}_{\ell}}}$${\operatorname{Spec}\mathcal{O}_{d}}$${\operatorname{Spec}\mathcal{O}_{d}}$$\scriptstyle{\tau^{*}}$$\scriptstyle{\sigma_{s}}$

where $\sigma_{s}$ is the automorphism of $\mathcal{O}_{d}$ such that
$\sigma_{s}(e^{2\pi i/d})=e^{2\pi is/d}$. Thus, pullback by $\tau$ induces an
isomorphism

$H^{d-2-h}(\mathcal{Y}_{t,\overline{\mathbb{Q}}_{\ell}},\Omega^{h})_{s^{-1}\underline{a}}\cong
H^{d-2-h}(\mathcal{Y}_{\tau(t)},\Omega^{h})_{\underline{a}}$

and the claim follows from lemma 2.3.2. ∎

###  2.4. The Picard–Fuchs Equation

In the previous section we studied the de Rham cohomology of $\mathcal{Y}_{0}$
and computed its Hodge filtration. We are now ready to extend this to a study
of the whole local system $R^{d-2}f_{*}(\underline{\mathbb{C}})$ on
$\operatorname{Spec}A_{d}(\mathbb{C})=\mathbb{C}\setminus\mu_{d}$ via the
differential equations satisfied by the periods of $\mathcal{Y}_{t}$.

######  2.4.1

Let $t_{0}\in\mathbb{C}\setminus\mu_{d}$ and
$U\subset\mathbb{C}\setminus\mu_{d}$ a connected and simply connected open
neighbourhood of $t_{0}$. For $\gamma_{0}\in
H_{d-2}(\mathcal{Y}_{t_{0}}(\mathbb{C}),\mathcal{O}_{d})$ and $t\in U$, there
exists a well-defined homology class $\gamma_{t}\in
H_{d-2}(\mathcal{Y}_{t}(\mathbb{C}),\mathcal{O}_{d})$ such that $\gamma_{0}$
and $\gamma_{t}$ pull back to the same class in
$H_{d-2}(f^{-1}(U),\mathcal{O}_{d})$.

Moreover, $\tau(\gamma_{0})$ and $\tau(\gamma_{t})$ are the pull-back of a
single class lying in $H_{d-1}(\mathbb{P}^{d-1}(\mathbb{C})\setminus
f^{-1}(U),\mathcal{O}_{d})$. In particular, one can integrate a $(d-1)$-form
$\omega$ on $\mathbb{P}^{d-1}(\mathbb{C})\setminus\mathcal{Y}_{t}(\mathbb{C})$
over $\tau(\gamma_{0})$ and if $d\omega=0$, we have

$\int_{\tau(\gamma_{0})}\omega=\int_{\tau(\gamma_{t})}\omega.$

######  2.4.2 Definition.

For a complex number $z$ and a non-negative integer $j$, we define the
Pochhammer symbol

$(z)_{j}=\begin{cases}1&j=0\\\ (z+j-1)(z)_{j-1}&j>0\end{cases}.$

######  2.4.3 Lemma (Kloosterman).

Let $e_{i}$ be non-negative integers such that $\sum_{i}e_{i}=dm$. Let
$q_{i},r_{i}$ be non-negative integers such that $e_{i}=dq_{i}+r_{i}$. If
there exists $i\in\\{0,\dots,d-1\\}$ such that $r_{i}=d-1$, then

$R\left(\frac{x^{\underline{e}}\Omega}{F^{m+1}}\right)=0.$

In general, we have

$R\left(\frac{x^{\underline{e}}\Omega}{F^{m+1}}\right)=R\left(\frac{\prod_{i=0}^{{d-1}}((r_{i}+1)/d)_{q_{i}}}{(l+1)_{m-l}}\frac{x^{\underline{r}}\Omega}{F^{l+1}}\right),$

where $dl=\sum_{i}r_{i}$.

###### Proof.

When $r_{i}<d$ this is precisely the case of [Klo07, Lemma 5.1] with $w_{i}=1$
and $w=d$. The proof also works for general $r_{i}$ without change. ∎

######  2.4.4 Lemma.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter modulo $d$ and define
$\underline{a}\in(\mathbb{Z}/d\mathbb{Z})^{d}$ as in definition 2.1.2. Let
$r_{i}=[-1-\beta_{1}-a_{i}]$, $m=\frac{1}{d}\sum_{i=0}^{d-1}r_{i}$ and

$\omega_{t}:=\frac{x^{\underline{r}}\Omega}{F_{t}^{m+1}}.$

For $\gamma_{0}\in H_{d-2}(\mathcal{Y}_{0}(\mathbb{C}),\mathcal{O}_{d})$ there
exists $\varepsilon>0$ such that for all $|t|<\varepsilon$, we have

$\int_{\tau(\gamma_{t})}\omega_{t}=\sum_{j=1}^{n}\binom{m+[\beta_{1}-\beta_{j}]}{m}(td)^{[\beta_{1}-\beta_{j}]}G_{j}(t^{d})\int_{\tau(\gamma_{0})}\frac{x^{\underline{r}+[\beta_{1}-\beta_{j}]}\Omega}{F^{m+[\beta_{1}-\beta_{j}]+1}},$

where

$G_{j}(z)=\sum_{k\geq
0}\prod_{i=1}^{n}\frac{((d+[\alpha_{i}-\beta_{1}]-[\beta_{j}-\beta_{1}])/d)_{k}}{((d+[\beta_{i}-\beta_{1}]-[\beta_{j}-\beta_{1}])/d)_{k}}z^{k}$

###### Proof.

Choose a $(d-1)$-cycle $\sigma$ on $\mathbb{P}^{d-1}(\mathbb{C})\setminus
f^{-1}(U)$ representing $\tau(\gamma_{0})$. If $\varepsilon<1/M$, where $M$ is
the minimum of $|F|$ on $\sigma$, then the following power series expansion
converges on $\sigma$ for $|t|<\varepsilon$

$\omega_{t}=\sum_{k\geq
0}\binom{k+m}{m}d^{k}t^{k}\frac{x^{\underline{r}+k}\Omega}{F^{m+k+1}}.$

Lemma 2.4.3 implies that

$R\left(\frac{x^{\underline{r}+k}\Omega}{F^{m+k+1}}\right)=0$

unless $k\equiv\beta_{1}-\beta_{j}\pmod{d}$ for some $j\in\\{1,\dots,n\\}$.
Thus, the formula in lemma 2.4.3 and the fact that $\tau(\gamma_{t})$ and
$\tau(\gamma_{0})$ are homologous imply that

$\int_{\tau(\gamma_{t})}\omega_{t}=\sum_{j=1}^{n}\binom{m+[\beta_{1}-\beta_{j}]}{m}(td)^{[\beta_{1}-\beta_{j}]}G_{j}(t^{d})\int_{\tau(\gamma_{0})}\frac{x^{\underline{r}+[\beta_{1}-\beta_{j}]}\Omega}{F^{m+[\beta_{1}-\beta_{j}]+1}},$

where

$\displaystyle G_{j}(z)$ $\displaystyle=\sum_{k\geq
0}\frac{([\beta_{1}-\beta_{j}]+1+dk)_{m}}{([\beta_{1}-\beta_{j}]+1)_{m}}d^{dk}\frac{\prod_{i=0}^{d-1}((r_{i}+[\beta_{1}-\beta_{j}]+1)/d)_{k}}{(m+[\beta_{1}-\beta_{j}]+1)_{dk}}z^{k}$
$\displaystyle=\sum_{k\geq
0}\frac{\prod_{i=0}^{d-1}((r_{i}+[\beta_{1}-\beta_{j}]+1)/d)_{k}}{d^{-dk}([\beta_{1}-\beta_{j}]+1)_{dk}}z^{k}$
$\displaystyle=\sum_{k\geq
0}\prod_{i=1}^{d-1}\frac{((r_{i}+[\beta_{1}-\beta_{j}]+1)/d)_{k}}{((i+[\beta_{1}-\beta_{j}]+1)/d)_{k}}z^{k}.$

Cancelling factors in numerator and denominator we obtain the equality

$\displaystyle\prod_{i=1}^{d-1}\frac{((r_{i}+[\beta_{1}-\beta_{j}]+1)/d)_{k}}{((i+[\beta_{1}-\beta_{j}]+1)/d)_{k}}$
$\displaystyle=\prod_{i=1}^{n}\frac{(([-1-\beta_{1}+\alpha_{i}]+[\beta_{1}-\beta_{j}]+1)/d)_{k}}{(([-1-\beta_{1}+\beta_{i}]+[\beta_{1}-\beta_{j}]+1)/d)_{k}}$
$\displaystyle=\prod_{i=1}^{n}\frac{((d+[-\beta_{1}+\alpha_{i}]-[\beta_{j}-\beta_{1}])/d)_{k}}{((d+[-\beta_{1}+\beta_{i}]-[\beta_{j}-\beta_{1}])/d)_{k}}$

and hence the claimed formula for $G_{j}$. ∎

######  2.4.5 Lemma.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter modulo $d$ and define $\underline{a}$ as in
definition 2.1.2. Let $r_{i}=[-1-\beta_{1}-a_{i}]$,
$m=\frac{1}{d}\sum_{i=0}^{d-1}r_{i}$ and

$\omega_{t}:=\frac{x^{\underline{r}}\Omega}{F_{t}^{m+1}}.$

Let $t_{0}\in\mathbb{C}\setminus\mu_{d}$ and $\gamma_{0}\in
H_{d-2}(\mathcal{Y}_{t_{0}}(\mathbb{C}),\mathbb{Z})$. For $|t-t_{0}|$
sufficiently small, the function

$u(t)=\int_{\gamma_{t}}R(\omega_{t})$

is holomorphic and is annihilated by the differential operator

$D:=\prod_{i=1}^{n}(\theta+[\beta_{i}-\beta_{1}]-d)-t^{d}\prod_{i=1}^{n}(\theta+[\alpha_{i}-\beta_{1}]),$

where $\theta=t\frac{d}{dt}$.

###### Proof.

We can differentiate under the integral sign in the formula

$u(t)=\int_{\gamma_{t}}R(\omega_{t})=\int_{\tau(\gamma_{t})}\omega_{t}=\int_{\tau(\gamma_{0})}\omega_{t}$

to prove that $u(t)$ is holomorphic.

If $t_{0}=0$, the differential equation is satisfied on a neighbourhood of
$t_{0}$ by [BH89, (2.9)] since

$G_{j}(z)={}_{n}F_{n-1}\left(\begin{array}[]{c c
c}1+\alpha^{\prime}_{1}/d-\beta^{\prime}_{j}/d,&\dots,&1+\alpha^{\prime}_{n}/d-\beta^{\prime}_{j}/d\\\
1+\beta^{\prime}_{1}/d-\beta^{\prime}_{j}/d,&\widehat{\dots},&1+\beta^{\prime}_{n}/d-\beta^{\prime}_{j}/d\end{array};z\right),$

where $\beta_{i}^{\prime}=[\beta_{i}-\beta_{1}]$ and
$\alpha_{i}^{\prime}=[\alpha_{i}-\beta_{1}]$. For arbitrary $t_{0}$ we can
find a simply-connected neighbourhood $U$ of $t_{0}$ which contains $0$ and
extend $\gamma_{t}$ to $U$. Then $Du$ is a holomorphic function on $U$ which
vanishes on a non-empty open set. Consequently, $Du=0$ on $U$. ∎

######  2.4.6 Corollary.

In the notation of the previous lemma, the following map is an isomorphism of
local systems

$(R^{d-2}f_{*}(\underline{\mathcal{\mathbb{C}}}))_{\underline{a}}\to\operatorname{Sol}(D):\sigma_{t}\mapsto\int_{\mathcal{Y}_{t}(\mathbb{C})}R(\omega_{t})\wedge\sigma_{t}$

on $\mathbb{C}\setminus\mu_{d}$, where $\operatorname{Sol}(D)$ denotes the
local system of solutions of $Du=0$.

###### Proof.

By lemma 2.4.5, the map is well-defined. Both source and target have rank $n$.
For the source it follows from the computation at $t=0$ in paragraph 2.3.1 and
for the target we refer to [BH89].

Lemma 2.4.4 implies that over a small neighbourhood of $t=0$, the image of the
map in question contains the $n$ independent hypergeometric series $G_{j}$.
Thus, the map of local systems is surjective, hence an isomorphism. ∎

###  2.5. The Geometric Monodromy

Now that we have determined the Picard–Fuchs equation we can describe the
image of the geometric monodromy representation with the results of [BH89].

######  2.5.1

Let $d>2$ be an integer. We identify
$\pi_{1}(\mathbb{P}^{1}(\mathbb{C})\setminus\\{0,1,\infty\\},1/2^{d})$ with
the group

$\Gamma_{0,1,\infty}:=\langle\gamma_{0},\gamma_{1},\gamma_{\infty}\mid\gamma_{0}\gamma_{1}\gamma_{\infty}\rangle.$

Let $[d]:\mathbb{P}^{1}\to\mathbb{P}^{1}$ denote the map $z\mapsto z^{d}$. It
induces an injective group homomorphism

$[d]_{*}:\pi_{1}(\mathbb{P}^{1}(\mathbb{C})\setminus(\mu_{d}\cup\\{0,\infty\\}),1/2)\to\pi_{1}(\mathbb{P}^{1}(\mathbb{C})\setminus\\{0,1,\infty\\},1/2^{d}).$

The inclusion
$\mathbb{P}^{1}(\mathbb{C})\setminus(\mu_{d}\cup\\{0,\infty\\})\to\operatorname{Spec}A_{d}(\mathbb{C})$
induces a surjective group homomorphism

$\eta:\pi_{1}(\mathbb{P}^{1}(\mathbb{C})\setminus(\mu_{d}\cup\\{0,\infty\\}),1/2)\twoheadrightarrow\pi_{1}(\operatorname{Spec}A_{d}(\mathbb{C}),1/2).$

######  2.5.2 Definition.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$. Define coefficients
$A_{i}\in\mathcal{O}_{d}$ and $B_{i}\in\mathcal{O}_{d}$ by the equations

$\prod_{j=1}^{R}(X-e^{2\pi
i\alpha_{j}/d})=\sum_{j=0}^{R}A_{j}X^{j}\qquad\prod_{j=1}^{R}(X-e^{2\pi
i\beta_{j}/d})=\sum_{j=0}^{R}B_{j}X^{j}.$

We define the hypergeometric monodromy representation

$HG(\alpha_{i};\beta_{i}):\Gamma_{0,1,\infty}\to\operatorname{GL}_{n}(\mathcal{O}_{d})$

by $\gamma_{\infty}\mapsto A$, $\gamma_{0}\mapsto B^{-1}$ and
$\gamma_{1}\mapsto A^{-1}B$, where

$A=\begin{bmatrix}0&0&\dots&0&-A_{0}\\\ 1&0&\dots&0&-A_{1}\\\
0&1&\dots&0&-A_{2}\\\ &&\ddots&&\\\ 0&0&\dots&1&-A_{n-1}\end{bmatrix},\qquad
B=\begin{bmatrix}0&0&\dots&0&-B_{0}\\\ 1&0&\dots&0&-B_{1}\\\
0&1&\dots&0&-B_{2}\\\ &&\ddots&&\\\ 0&0&\dots&1&-B_{n-1}\end{bmatrix}.$

######  2.5.3 Lemma.

The matrix $u=HG(\alpha_{i};\beta_{i})(\gamma_{1})$ is a pseudoreflection,
i.e. $u-\operatorname{id}$ has rank one. If $d$ is odd, then $\det u=1$ and if
$d$ is even, then $\det u=-1$.

###### Proof.

By [BH89, Prop 2.10], we know that $u-\operatorname{id}$ has rank one and
$\det u=e^{2\pi i\gamma/d}$, where

$\gamma=\sum_{i=1}^{n}(\beta_{i}-\beta_{1})-\sum_{i=1}^{n}(\alpha_{i}-\beta_{1})=\sum_{i=1}^{n}\beta_{i}-\sum_{i=1}^{n}\alpha_{i}.$

Hence, the definition of hypergeometric parameters implies
$\gamma=\binom{d}{2}$ and the claim follows. ∎

######  2.5.4 Lemma.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$. There exists a basis of
$M(\alpha_{i};\beta_{j})_{B,1/2}\otimes_{\mathcal{O}}\mathbb{C}$ in which the
action of $\pi_{1}(\operatorname{Spec}A_{d}(\mathbb{C}),1/2)$ on
$M(\alpha_{i};\beta_{j})_{B,1/2}\otimes_{\mathcal{O}}\mathbb{C}$ is given by
the unique group homomorphism

$\rho_{1/2}:\pi_{1}(\operatorname{Spec}A_{d}(\mathbb{C}),1/2)\to\operatorname{GL}_{n}(\mathbb{C})$

making the following diagram commute.

${\pi_{1}(\mathbb{P}^{1}(\mathbb{C})\setminus(\mu_{d}\cup\\{0,\infty\\}),1/2)}$${\pi_{1}(\mathbb{P}^{1}(\mathbb{C})\setminus\\{0,1,\infty\\},1/2^{d})}$${\pi_{1}(\operatorname{Spec}A_{d}(\mathbb{C}),1/2)}$${\operatorname{GL}_{n}(\mathbb{C})}$$\scriptstyle{[d]_{*}}$$\scriptstyle{\eta}$$\scriptstyle{HG(\alpha_{i};\beta_{i})}$$\scriptstyle{\rho_{1/2}}$

###### Proof.

Let $D$ be the differential operator from lemma 2.4.5. Observe that $d^{-n}D$
is the pullback under $[d]$ of the hypergeometric differential operator
studied in [BH89]. Hence, the claim follows from lemma 2.4.6 and the Theorem
of Levelt [BH89, Thm 3.5]. ∎

######  2.5.5 Remark.

The homomorphism $\rho_{1/2}$ actually maps into
$\\{g\in\operatorname{GL}_{n}(\mathcal{O}_{d}):\det(g)^{2}=1\\}\subset\operatorname{GL}_{n}(\mathbb{C})$
since $\det(A^{d})=\det(B^{d})=1$ and $\det(A^{-1}B)\in\\{\pm 1\\}$. If $d$ is
odd, then $\rho_{1/2}$ maps into $\operatorname{SL}_{n}(\mathcal{O}_{d})$.

######  2.5.6 Lemma.

If $d$ is even, let $\widetilde{A}_{d}=A_{d}[\sqrt{1-t^{d}}]$ and if $d$ is
odd, let $\widetilde{A}_{d}=A_{d}$. Let
$(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a hypergeometric
parameter of dimension $n$ modulo $d$. Denote by $\mathcal{M}_{\lambda}$ the
$\mathcal{O}_{d,\lambda}$-etale local system on $\operatorname{Spec}A_{d}$
corresponding to
$M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})_{\lambda}$. For any
map $\iota:\mathcal{O}_{d}\to\mathbb{C}$, the local system

$\det(\mathcal{M}_{\lambda})\rvert_{\operatorname{Spec}(\widetilde{A}_{d}\otimes_{\iota}\mathbb{C})}$

is constant.

###### Proof.

After replacing $(\alpha_{i};\beta_{j})$ by $(s\alpha_{i};s\beta_{j})$, we may
assume that $\iota$ is the inclusion. Thus, the geometric monodromy action on
$\mathcal{M}_{\lambda}$ is given by $\rho_{1/2}$. When $d$ is odd, the claim
follows from the previous remark. So we assume that $d$ is even. The image of
$\pi_{1}(\operatorname{Spec}\widetilde{A}_{d}(\mathbb{C}))$ in
$\pi_{1}(\operatorname{Spec}A_{d}(\mathbb{C}),1/2)$ is the kernel of the
homomorphism $\pi_{1}(\operatorname{Spec}A_{d}(\mathbb{C}),1/2)\to\\{\pm 1\\}$
which sends any simple loop around a $d$th root of unity to $-1$. By lemma
2.5.3, this homomorphism is equal to $\det\rho_{1/2}$. Thus, the restriction
of $\det(\mathcal{M}_{\lambda})$ to
$\operatorname{Spec}(\widetilde{A}_{d}\otimes_{\iota}\mathbb{C})$ is constant.
∎

######  2.5.7 Proposition.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$ and

$M=M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n}).$

For $t\in\mathbb{C}\setminus\mu_{d}$ let
$\gamma_{\infty}\in\pi_{1}(\mathbb{C}\setminus\mu_{d},t)$ be a simple loop
based at $t$ going around $\infty$. Then $\gamma_{\infty}$ acts on $M_{B,t}$
as a unipotent operator with Jordan blocks $J_{\alpha}$ of size
$\\#\\{i:\alpha_{i}=\alpha\\}$ for $\alpha\in\\{\alpha_{i}\\}$. Similarly, let
$\gamma_{1}$ denote a simple loop around $1$. Then $\gamma_{1}$ acts as a
unipotent operator on $M_{B,t}$ and the rank of $\gamma_{1}-\operatorname{id}$
is $1$ if $d$ is odd and $0$ if $d$ is even.

###### Proof.

The first part follows directly from lemma 2.5.4 and the definition of the
matrix $A$. The second part is implied by 2.5.3. ∎

######  2.5.8 Proposition.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$ and
$t\in\mathbb{C}\setminus\mu_{d}$. If the following hypotheses are satisfied

  * •

$|\\{\alpha_{1},\dots,\alpha_{n}\\}|<n$,

  * •

$\\{\beta_{1},\dots,\beta_{n}\\}$ does not form an arithmetic progression,

  * •

there is no non-zero $s\in\mathbb{Z}/d\mathbb{Z}$ such that

$\\{\alpha_{i}+s\\}=\\{\alpha_{i}\\}\quad\wedge\quad\\{\beta_{i}+s\\}=\\{\beta_{i}\\},$

  * •

there is no $s\in\mathbb{Z}/d\mathbb{Z}$ such that

$\\{-\alpha_{i}-s\\}=\\{\alpha_{i}+s\\}\quad\wedge\quad\\{-\beta_{i}-s\\}=\\{\beta_{i}+s\\},$

then the identity component of the Zariski closure of the image of the
geometric monodromy representation

$\pi_{1}(\mathbb{C}\setminus\mu_{d},t)\to\operatorname{GL}(M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})_{B,t}\otimes_{\mathcal{O}_{d}}\mathbb{C})$

contains $\operatorname{SL}_{n}(\mathbb{C})$.

###### Proof.

It follows from [BH89, Theorem 6.5] that the Zariski closure of the image of
$HG(\alpha_{i},\beta_{j})$ contains $\operatorname{SL}_{n}(\mathbb{C})$.
Namely, the conditions are satisfied by [BH89, Theorem 5.3 and Theorem 5.8]
and the fact that any scalar shift of the hypergeometric group with parameters
$(e^{2\pi i\alpha_{i}/d};e^{2\pi i\beta_{j}/d})$ contains an element of
infinite order since $\\{\alpha_{i}\\}$ contains a repeated element.

Using lemma 2.5.4 we see that the identity component of the Zariski closure of
the image of

$\pi_{1}(\mathbb{C}\setminus\mu_{d},t)\to\operatorname{GL}(M(\alpha_{i};\beta_{i})_{B,t}\otimes_{\mathcal{O}_{d}}\mathbb{C})$

is a Zariski-closed finite index subgroup of
$\operatorname{SL}_{n}(\mathbb{C})$. But $\operatorname{SL}_{n}(\mathbb{C})$
is connected, hence any such group is equal to
$\operatorname{SL}_{n}(\mathbb{C})$ itself. ∎

######  2.5.9 Proposition.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$ and
$t\in\mathbb{C}\setminus\mu_{d}$ satisfying the hypotheses of the previous
proposition. Moreover, let $U,V<(\mathbb{Z}/d\mathbb{Z})^{\times}$ be
subgroups such that $\\{1\\}=U\cap V$, $UV=(\mathbb{Z}/d\mathbb{Z})^{\times}$
and for $s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$ the equalities

$\displaystyle\\{s\alpha_{i}-s\alpha_{j}:1\leq i,j\leq n\\}$
$\displaystyle=\\{\alpha_{i}-\alpha_{j}:1\leq i,j\leq n\\}$
$\displaystyle\\{s\beta_{i}-s\beta_{j}:1\leq i,j\leq n\\}$
$\displaystyle=\\{\beta_{i}-\beta_{j}:1\leq i,j\leq n\\}$

imply that $s\in U$. Let $\lambda<\mathcal{O}_{d}$ be a sufficiently large
prime of residue characteristic $\ell$ such that $\ell\in V$. Then the image
of

$\pi_{1}(\operatorname{Spec}A_{d}(\mathbb{C}),t)\to\operatorname{GL}(M(\alpha_{i};\beta_{j})_{B,t}\otimes_{\mathcal{O}_{d}}\mathcal{O}_{d}/\lambda)$

contains $\operatorname{SL}_{n}(\mathcal{O}_{d}/\lambda)$.

###### Proof.

Let $F\subset\mathbb{Q}(\mu_{d})$ be the fixed field of
$U<(\mathbb{Z}/d\mathbb{Z})^{\times}=\operatorname{Gal}(\mathbb{Q}(\mu_{d})/\mathbb{Q})$
and $K\subset\mathbb{Q}(\mu_{d})$ the fixed field of $V$. Let
$X=\operatorname{Spec}\mathcal{O}_{K}[1/(n|V|-1)!]$ and let $G$ be the group
scheme $(\operatorname{Res}_{F/\mathbb{Q}}\operatorname{SL}_{n})_{X}$ and
consider the homomorphism

$\displaystyle\Gamma^{\prime}$
$\displaystyle\xrightarrow{\Phi}G(\mathcal{O}_{K}[1/(n|V|-1)!])=\operatorname{SL}_{n}(\mathcal{O}_{d}[1/(n|V|-1)!])$
$\displaystyle\gamma$ $\displaystyle\mapsto HG(\alpha_{i};\beta_{j})(\gamma),$

where $\Gamma^{\prime}<\Gamma_{0,1,\infty}$ is the kernel of $\det
HG(\alpha_{i};\beta_{j})$. Choose an embdedding
$G\hookrightarrow\operatorname{GL}_{n|V|,X}$.

Let $W\subset\mathfrak{g}:=\operatorname{Lie}(G)$ be the
$\mathcal{O}_{K}[1/(n|V|-1)!]$-module generated by elements $\log\gamma$,
where $\gamma$ is a unipotent element in the image of $\Phi\circ[d]_{*}$.
Equip $\mathfrak{g}$ with the conjugation action of $\Gamma_{0,1,\infty}$ via
$\Phi$. Then $W$ is a $\Gamma_{0,1,\infty}$-equivariant subspace.

We have $\mathfrak{g}\otimes_{\mathcal{O}_{K}}\mathbb{C}=\prod_{v\in
V}\mathfrak{sl}_{n}$. Since the image of $HG(\alpha_{i};\beta_{j})$ is
Zariski-dense and $\mathfrak{sl}_{n}$ is an irreducible
$\operatorname{SL}_{n}(\mathbb{C})$-representation, we see that for each $v\in
V$, the $\mathfrak{sl}_{n}$-summand of $\mathfrak{g}$ is an irreducible
$\Gamma_{0,1,\infty}$-representation. Now the assumption on the sets
$\\{\alpha_{i}-\alpha_{j}\\}$ and $\\{\beta_{i}-\beta_{j}\\}$ imply that
$\mathfrak{g}\otimes\mathbb{C}$ is a multiplicity-free semisimple
$\Gamma_{0,1,\infty}$-module.

The element $\log\gamma_{\infty}^{d}\in W$ has a non-zero projection in each
irreducible summand of $\mathfrak{g}\otimes\mathbb{C}$. Thus,
$W\otimes\mathbb{C}=\mathfrak{g}\otimes\mathbb{C}$ and there exists a constant
$C$ such that $W[1/C]=\mathfrak{g}[1/C]$. Let
$\lambda^{\prime}<\mathcal{O}_{K}$ be the prime below $\lambda$. Since
$\ell\in V$, we have $\mathcal{O}_{K}/\lambda^{\prime}\cong\mathbb{F}_{\ell}$.
We apply [Kat88, Theorem 12.4.1] with $R=\mathcal{O}_{K}/\lambda^{\prime}$ to
see that
$G(R)=\operatorname{SL}_{n}(\mathcal{O}_{d}/\lambda^{\prime}\mathcal{O}_{d})$
is generated by the unipotent elements in the image of
$\pi_{1}(\mathbb{C}\setminus\mu_{d},t)\to\operatorname{GL}_{n}(\mathcal{O}_{d}/\lambda^{\prime}\mathcal{O}_{d})$.
In particular, the image of
$\pi_{1}(\mathbb{C}\setminus\mu_{d},t)\to\operatorname{GL}_{n}(\mathcal{O}_{d}/\lambda)$
contains $\operatorname{SL}_{n}(\mathcal{O}_{d}/\lambda)$. ∎

######  2.5.10 Remark.

If we take $V=\\{1\\}$, $U=(\mathbb{Z}/d\mathbb{Z})^{\times}$ and $\ell\equiv
1\pmod{d}$, then the conditions in the proposition are satisfied and
$\mathcal{O}/\lambda\cong\mathbb{F}_{\ell}$.

######  2.5.11 Corollary.

If $\lambda_{1},\lambda_{2}<\mathcal{O}_{d}$ are sufficiently large primes
above $\ell_{1}\neq\ell_{2}$ such that $\ell_{1},\ell_{2}\in V$, then the
image of

$\pi_{1}(\operatorname{Spec}A_{d}(\mathbb{C}),t)\to\operatorname{GL}(M(\alpha_{i};\beta_{j})_{B,t}\otimes_{\mathcal{O}_{d}}\mathcal{O}_{d}/\lambda_{1}\lambda_{2})$

contains
$\operatorname{SL}(M(\alpha_{i};\beta_{j})_{B,t}\otimes_{\mathcal{O}_{d}}\mathcal{O}_{d}/\lambda_{1}\lambda_{2})$.

###### Proof.

By the proposition we already know that each of the projections
$\ker\rho_{1/2}\to\operatorname{SL}_{n}(\mathcal{O}_{d}/\lambda_{i})$ is
surjective. But for different primes $\ell_{1},\ell_{2}>3$, there are no
proper normal subgroups
$N_{i}<\operatorname{SL}_{n}(\mathcal{O}_{d}/\lambda_{i})$ such that there
exists an isomorphism
$\operatorname{SL}_{n}(\mathcal{O}_{d}/\lambda_{1})/N_{1}\cong\operatorname{SL}_{n}(\mathcal{O}_{d}/\lambda_{2})/N_{2}$
because $\operatorname{PSL}_{n}(\mathcal{O}_{d}/\lambda_{1})$ and
$\operatorname{PSL}_{n}(\mathcal{O}_{d}/\lambda_{2})$ are distinct simple
groups. Thus, the claim follows from Goursat’s lemma. ∎

###  2.6. The Arithmetic Monodromy

Now that we have studied the action of the fundamental group on
$M(\alpha_{i},\beta_{j})_{B,t}$ we turn to the Galois action on
$M(\alpha_{i},\beta_{j})_{\lambda,t}$. In particular, we will describe its
determinant.

######  2.6.1

Let $d$ be a positive integer and $k$ a finite field of cardinality $q$
coprime to $d$ and let $h:\mathcal{O}_{d}\to k$ be a ring homomorphism. Let

$t:\\{y\in k:y^{d}=1\\}\to\mathcal{O}_{d}^{\times}$

be the unique map such that $h(t(y))=y$. We define $\tau(y):=t(y^{(q-1)/d})$.
Choose a non-trivial additive character $e:k\to\mathbb{C}^{\times}$ and define
the Gauss sums

$g(k,a)=-\sum_{x\in k^{\times}}\tau(x)^{-a}e(x)\in\mathbb{C}$

for $a\in\mathbb{Z}/d\mathbb{Z}$. For
$\underline{a}\in(\mathbb{Z}/d\mathbb{Z})^{m+1}$ we also define the Jacobi sum

$J_{\underline{a}}(k)=(-1)^{m}\sum_{\begin{subarray}{c}x_{1},\dots,x_{m}\in
k\\\
x_{1}+\dots+x_{m}=-1\end{subarray}}\prod_{i=1}^{m}\tau(x_{i})^{-a_{i}}\in\mathcal{O}_{d}.$

When $\underline{a}\neq(0,\dots,0)$, the relation

$J_{\underline{a}}(k)=q^{-1}g(k,a_{0})\cdots g(k,a_{m})$

holds [DMOS82, I, Lemma 7.9]. It is proved in [Wei52], that

$\psi_{\underline{a}}:\mathfrak{p}\mapsto J_{\underline{a}}(k(\mathfrak{p}))$

defines an algebraic Hecke character of $\mathbb{Q}(\mu_{d})$, unramified
outside $d$. By abuse of notation we will also denote the corresponding
character $G_{\mathbb{Q}(\mu_{d})}\to\mathcal{O}_{d,\lambda}^{\times}$ by
$\psi_{\underline{a}}$.

######  2.6.2 Lemma.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$. Then for all primes
$\lambda<\mathcal{O}_{d}$ with residue characteristic different from the
characteristic of $k(\lambda)$, the geometric Frobenius $F_{\lambda}$ acts on

$\bigwedge^{n}M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})_{\lambda,0}$

as multiplication by

$\prod_{i=1}^{n}J_{\underline{a}+\beta_{i}}(k(\lambda)),$

where $\underline{a}$ is the vector defined in 2.1.2.

###### Proof.

This follows from [DMOS82, I, Proposition 7.10]. ∎

######  2.6.3

For $a\in(\mathbb{Z}/d\mathbb{Z})\setminus\\{0\\}$ and a divisor $m\mid d$
such that $ma\neq 0$, we define the functions
$\delta_{a},\epsilon_{m,a}:(\mathbb{Z}/d\mathbb{Z})\setminus\\{0\\}\to\mathbb{Z}$
as follows

$\displaystyle\delta_{a}(x)=\begin{cases}1&x=a\\\ 0&\text{ otherwise
}\end{cases}$

and

$\epsilon_{m,a}(x)=\delta_{-ma}+\sum_{0\leq j\leq m-1}\delta_{a+jd/m}.$

We let $E(d)$ denote the set of functions consisting of $\epsilon_{1,a}$ and
$\epsilon_{p,a}$, where $p$ is a prime divisor of $d$. It is proved in [Del79,
Appendix] that any $f:(\mathbb{Z}/d\mathbb{Z})\setminus\\{0\\}\to\mathbb{Q}$,
such that

$\langle f\rangle(s):=\frac{1}{d}\sum_{a=1}^{d-1}f(a)[sa]$

is independent of $s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$, is a linear
combination of elements of $E(d)$. For such a function $f$, we define the
complex number

$\Gamma(f):=\frac{1}{(2\pi i)^{\langle
f\rangle(1)}}\prod_{a=1}^{d-1}\Gamma(a/d)^{f(a)}.$

######  2.6.4 Lemma.

Let $f:(\mathbb{Z}/d\mathbb{Z})\setminus\\{0\\}\to\mathbb{Z}$ be a function
such that $w=\langle f\rangle(s)$ is an integer independent of
$s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$. Then $\Gamma(f)$ is an algebraic
number and generates an abelian extension of $\mathbb{Q}(\mu_{d})$ unnramified
outside $2d$. For $\mathfrak{p}<\mathcal{O}_{d}$ a prime not dividing $2d$ we
have

$F_{\mathfrak{p}}(\Gamma(f))=\frac{1}{(N\mathfrak{p})^{w}}\prod_{a=1}^{d-1}g(k(\mathfrak{p}),a)^{f(a)}\Gamma(f),$

where
$F_{\mathfrak{p}}\in\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}(\mu_{d}))^{ab}$
is the geometric Frobenius element attached to $\mathfrak{p}$.

###### Proof.

See [DMOS82, I, Theorem 7.15]. ∎

######  2.6.5 Lemma.

For an integer $1\leq a\leq d/2$, we have

$\Gamma(\epsilon_{1,a})=\frac{1}{e^{\pi ia/d}-e^{-\pi ia/d}}\in e^{\pi
ia/d}\cdot\mathbb{Q}(\mu_{d})$

and for a prime divisor $p\mid d$ and $1\leq a<d/p$, we have

$\Gamma(\epsilon_{p,a})=\frac{p^{1/2-a/d}}{i^{(p-1)/2}(e^{\pi iap/d}-e^{-\pi
iap/d})}\in e^{\pi i(ap/d+(p-1)/4)}p^{1/2-a/d}\cdot\mathbb{Q}(\mu_{d}).$

###### Proof.

This follows from the two classical identities

$\Gamma(x)\Gamma(1-x)=\pi\sin(\pi x)^{-1}$

and

$\prod_{m=0}^{p-1}\Gamma(x+m/p)=p^{1/2-x}(2\pi)^{(p-1)/2}\Gamma(px).\qed$

######  2.6.6 Lemma.

Suppose $f=\sum_{1\leq a\leq d/2}x_{1,a}\epsilon_{1,a}+\sum_{p\mid
d}\sum_{1\leq a<d/p}x_{p,a}\epsilon_{p,a}$ for some integer coefficients
$x_{m,a}$. Then

$\Gamma(f)\in e^{\pi iy_{1}}\prod_{p\mid d}p^{y_{p}}\cdot\mathbb{Q}(\mu_{d}),$

where

$\displaystyle y_{p}$ $\displaystyle=\sum_{1\leq a<d/p}x_{p,a}(1/2-a/d)$
$\displaystyle y_{1}$ $\displaystyle=\sum_{1\leq a\leq
d/2}x_{1,a}a/d+\sum_{p\mid d}\sum_{1\leq a<d/p}x_{p,a}(ap/d+(p-1)/4)$

###### Proof.

Note that given two functions
$f_{1},f_{2}:(\mathbb{Z}/d\mathbb{Z})\setminus\\{0\\}\to\mathbb{Q}$ such that
$\langle f_{i}\rangle$ are constant, we have
$\Gamma(f_{1}+f_{2})=\Gamma(f_{1})\Gamma(f_{2})$. The lemma follows from
expanding $\Gamma(f)$ as a product of factors of the form
$\Gamma(\epsilon_{m,a})$ and collecting terms using lemma 2.6.5. ∎

######  2.6.7 Corollary.

Let $n>1$ be an integer. Suppose $f=\sum_{1\leq a\leq
d/2}x_{1,a}\epsilon_{1,a}+\sum_{p\mid d}\sum_{1\leq
a<d/p}x_{p,a}\epsilon_{p,a}$ for some integer coefficients $x_{m,a}$. Let
$y_{p}$ and $y_{1}$ be defined as in the lemma and let $b_{1}$ be the
denominator of $y_{1}$ and

$b_{p}=\begin{cases}\text{ denominator of }2y_{p}&4\mid d\text{ or }p\equiv
1\pmod{4}\\\ \text{ denominator of }y_{p}&\text{ otherwise.}\end{cases}$

If the $b_{p}$ are coprime to $n$ and
$\varphi(\operatorname{lcm}(2b_{1},d))/\varphi(d)$ is coprime to $n$, then

$[\mathbb{Q}(\mu_{d},\Gamma(f)):\mathbb{Q}(\mu_{d})]$

is coprime to $n$.

###### Proof.

By the lemma, we know that $\mathbb{Q}(\mu_{d},\Gamma(f))$ is contained in the
composite field $E=\mathbb{Q}(\mu_{d},e^{\pi
iy_{1}})\prod_{p}\mathbb{Q}(\mu_{d},p^{y_{p}})$. Note that if $4\mid d$ or
$p\equiv 1\pmod{4}$, then $p^{1/2}\in\mathbb{Q}(\mu_{d})$. Hence, the
assumptions imply that

$[E:\mathbb{Q}(\mu_{d})]\mid\varphi(\operatorname{lcm}(2b_{1},d))/\varphi(d)\prod_{p}b_{p}$

is coprime to $n$ and the claim follows from multiplicativity of degrees in
field extensions. ∎

######  2.6.8 Proposition.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$ and $F$ a number field
containing the $d$th roots of unity. Let $t\in F\setminus\mu_{d}$ and
$\underline{c}\in(\mathbb{Z}/d\mathbb{Z})^{3}$ such that $c_{0}+c_{1}+c_{2}=0$
and either $c_{i}\neq 0$ for $i=0,1,2$ or $c_{0}=c_{1}=c_{2}=0$. Suppose the
following hypotheses are satisfied:

  * •

If $d$ is even, then $1-t^{d}$ is a square.

  * •

For all $s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$, the following $n$ integers
are pairwise distinct

$\sum_{i}[s\beta_{j}-s\alpha_{i}]-\sum_{i}[s\beta_{j}-s\beta_{i}]\qquad
j=1,\dots,n.$

  * •

The following quantity is independent of
$s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$

$\sum_{i,j}[s\beta_{j}-s\alpha_{i}]-\sum_{i,j}[s\beta_{j}-s\beta_{i}]+n\sum_{i=0}^{2}[sc_{i}].$

  * •

For each $s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$ we have

$\sum_{i,j}[s\beta_{j}-s\alpha_{i}]=n\sum_{i}[s\beta_{i}-s\alpha_{i}].$

  * •

There are integers $x_{\epsilon}$ for $\epsilon\in E(d)$ such that

$n+\sum_{i,j}\delta_{\beta_{j}-\alpha_{i}}-\sum_{i\neq
j}\delta_{\beta_{j}-\beta_{i}}+n\sum_{i=0}^{2}\delta_{c_{i}}=\sum_{\epsilon\in
E(d)}x_{\epsilon}\epsilon.$

Let $b_{1}$ and $b_{p}$ be the integers defined in corollary 2.6.7. The
integer $\varphi(\operatorname{lcm}(2b_{1},d))/\varphi(d)$ is coprime to $n$
and $b_{p}$ is coprime to $n$.

Then there exists a character
$\psi:G_{\mathbb{Q}(\mu_{d})}\to\mathcal{O}_{d,\lambda}^{\times}$, such that

$\det(M(\alpha_{i};\beta_{j})_{\lambda,t}\otimes\psi)=\chi_{cyc}^{-n(n-1)/2}.$

###### Proof.

Let $f:(\mathbb{Z}/d\mathbb{Z})\setminus\\{0\\}\to\mathbb{Z}$ be defined by

$f(x)=n+\sum_{i,j}\delta_{\beta_{j}-\alpha_{i}}(x)-\sum_{i\neq
j}\delta_{\beta_{j}-\beta_{i}}(x)+n\sum_{i=0}^{2}\delta_{c_{i}}(x).$

Then by assumption, $\langle f\rangle(s)$ is an integer $w$ independent of
$s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$. Thus, $\Gamma(f)$ is an algebraic
number by lemma 2.6.4 and if $g\in G_{\mathbb{Q}(\mu_{d})}$, we have

$g(\Gamma(f))=\chi(g)\Gamma(f),$

where $\chi:G_{F}\to\mathbb{Q}(\mu_{d})^{\times}$ is the continuous character
such that

$\chi(F_{\mathfrak{p}})=\frac{1}{(N\mathfrak{p})^{w}}\prod_{a=1}^{d-1}g(k(\mathfrak{p}),a)^{f(a)}$

for geometric Frobenius elements $F_{\mathfrak{p}}\in G_{F}^{ab}$. On the
other hand, $t$ defines an $F$-point of $\widetilde{A}_{d}$, hence lemmas
2.5.6 and 2.6.2 show that $F_{\mathfrak{p}}$ acts on
$\bigwedge^{n}(M_{\lambda,t})$ as multiplication by

$\prod_{i=1}^{n}J_{\underline{a}+\beta_{i}}(k(\mathfrak{p}))=(N\mathfrak{p})^{w-2n}J_{\underline{c}}(k(\mathfrak{p}))^{-n}\chi(F_{\mathfrak{p}}).$

Moreover, the third assumption of the proposition implies that

$w\equiv-\sum_{i\neq j}[\beta_{j}-\beta_{i}]=-n(n-1)/2\pmod{n}.$

Corollary 2.6.7 shows that $\Gamma(f)$ generates an extension of
$\mathbb{Q}(\mu_{d})$ of degree coprime to $n$. Equivalently, $\chi=\eta^{n}$
is an $n$th power. Thus, we may take
$\psi=\eta^{-1}\psi_{\underline{c}}\chi_{cyc}^{2-m}$, where $m$ is an integer
such that $w=mn-n(n-1)/2$. ∎

######  2.6.9 Lemma.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$ such that for all
$s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$, the following $n$ integers are
pairwise distinct

$\sum_{i}[s\beta_{j}-s\alpha_{i}]-\sum_{i}[s\beta_{j}-s\beta_{i}]\qquad
j=1,\dots,n.$

Let $\ell\equiv 1\pmod{d}$ and $\lambda<\mathcal{O}_{d}$ be a prime above
$\ell$. Suppose $K/\mathbb{Q}_{\ell}$ is a finite Galois extension and
$\eta:\mathcal{O}_{d}\to K$ a morphism of rings inducing $\lambda$. If $t\in
K$ satisfies $|t|<1$, then the Galois representation

$\rho:G_{K}\to\operatorname{GL}(M(\alpha_{i};\beta_{j})_{\lambda,t}\otimes_{\eta}K)$

is ordinary in the sense of [Ger19, §5.2].

###### Proof.

Note that $\mathcal{Y}_{t}$ is smooth and proper over $\mathcal{O}_{K}$.
Hence, $\rho$ is crystalline and by [Ger19, Lemma 2.32] it remains to compute
the valuations of the eigenvalues of the geometric Frobenius acting on

$H^{d-2}_{cris}(\mathcal{Y}_{t}/\mathcal{O}_{K})_{\underline{a}}\cong
H^{d-2}_{cris}(\mathcal{Y}_{0}/\mathcal{O}_{d,\lambda})_{\underline{a}}\otimes_{\eta}\mathcal{O}_{K}.$

One can use [KM74, Theorem 2(2)] to show that these eigenvalues coincide with
the Frobenius eigenvalues on $M(\alpha_{i};\beta_{j})_{\mathfrak{p},0}$ for
some auxiliary prime $\mathfrak{p}\nmid\ell$ of $\mathcal{O}_{d}$. It follows
from [DMOS82, I, Proposition 7.10] that the set of eigenvalues is

$\\{\eta(J_{\underline{a}+\beta_{i}}(k(\lambda))):i=1,2,\dots,n\\}.$

The formula [Wei52, (9)] with the congruence $\ell\equiv 1\pmod{d}$ implies
that there exists an element $s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$ depending
on $\eta$ such that

$v_{\ell}(\eta(J_{\underline{a}+\beta_{i}}(k(\lambda))))=\sum_{j=0}^{d-1}[sa_{j}+s\beta_{i}]=\binom{d}{2}+\sum_{j=0}^{d-1}[s\beta_{i}-s\alpha_{j}]-\sum_{j=0}^{d-1}[s\beta_{i}-s\beta_{j}].$

By assumption, these are $n$ distinct integers and [Ger19, Lemma 2.32]
applies. ∎

##  3\. Applications to Galois Representations

###  3.1. Preliminaries on Weil–Deligne Representations

Let $K/\mathbb{Q}_{p}$ be a finite extension and
$W_{K}\subset\operatorname{Gal}(\overline{K}/K)$ the Weil group of $K$. In
this section we prove some lemmas about Weil–Deligne representations, which we
use in the applications below. For a general introduction to Weil–Deligne
representations see [Tat79, §4].

######  3.1.1 Definition.

A Weil–Deligne representation of $W_{K}$ is a triple $(\rho,V,N)$ where $V$ is
a finite dimensional complex vector space, $\rho:W_{K}\to\operatorname{GL}(V)$
is a continuous representation and $N\in\operatorname{End}(V)$ is a nilpotent
operator such that $\rho(g)N=\|g\|N\rho(g)$ for all $g\in W_{K}$.

######  3.1.2 Definition.

We define the Frobenius semisimplification of a Weil–Deligne representation
$(\rho,V,N)$ as

$(\rho,V,N)^{F-ss}:=(\rho^{ss},V,N),$

where $\rho^{ss}:W_{K}\to\operatorname{GL}(V)$ denotes the semisimplification
of $\rho$. The semisimplification of $(\rho,V,N)$ is defined as

$(\rho,V,N)^{ss}:=(\rho^{ss},V,0).$

######  3.1.3 Definition.

A Weil–Deligne representation $(\rho,V,N)$ is generic if

$\operatorname{Hom}((\rho,V,N),(\rho(1),V,N))=0.$

######  3.1.4 Example.

For any integer $m\geq 1$, the indecomposable Weil–Deligne representation
$\operatorname{Sp}(m)$ of rank $m$ defined in [Tat79, 4.1.4] is generic.

######  3.1.5 Lemma.

Given a nilpotent operator $N$ acting on a finite dimensional vector space
$V$, there exists a unique exhaustive filtration $\dots\subset M_{i}\subset
M_{i+1}\subset\dots$ of $V$ such that

  * •

$NM_{i}\subset M_{i-2}$

  * •

$N^{i}$ induces an isomorphism
$\operatorname{gr}_{i}^{M}V\to\operatorname{gr}_{-i}^{M}V$

$M_{\bullet}$ is called the monodromy filtration.

###### Proof.

Apply [Del80, Prop 1.6.1] to the category of vector spaces. ∎

######  3.1.6 Corollary.

If $f:V\to V$ is a linear automorphism commuting with $N$, then
$fM_{i}=M_{i}$.

###### Proof.

Consider the filtration $fM_{i}$. We have $NfM_{i}=fNM_{i}\subset fM_{i-2}$.
Since $f$ is an automorphism, we also know that

$fM_{i}/fM_{i-1}\xrightarrow{N^{i}}fM_{-i}/fM_{-i-1}$

is an isomorphism. Thus, the uniqueness claim in the lemma implies that
$fM_{i}=M_{i}$. ∎

######  3.1.7 Definition.

A Weil–Deligne representation $(\rho,V,N)$ is pure of weight $w$ if the
eigenvalues of any lift of geometric Frobenius acting on
$\operatorname{gr}_{i}^{M}V$ are Weil numbers of weight $w+i$.

######  3.1.8 Lemma.

If a Weil–Deligne representation $(\rho,V,N)$ is pure, then
$(\rho,V,N)^{F-ss}$ is generic.

###### Proof.

Suppose $f:V\to V(1)$ is a morphism of Weil–Deligne representations. By
corollary 3.1.6, $f$ preserves the filtration $M_{i}$ and induces morphisms of
$W_{K}$-representations $f_{i}:gr_{i}^{M}V\to gr_{i}^{M}V(1)$. By the purity,
the eigenvalues of $\rho^{ss}(\operatorname{Frob}_{K})$, which are the same as
the eigenvalues of $\rho(\operatorname{Frob}_{K})$, on these two vector spaces
have different weights. Hence, $f_{i}=0$ for all $i$ and $f=0$. Consequently,
$(\rho,V,N)^{F-ss}$ is generic. ∎

######  3.1.9 Lemma.

Let $(\rho,V,0)$ be a Weil–Deligne representation such that $\rho$ is
semisimple. There exists a unique generic Weil–Deligne representation of the
form $(\rho,V,N)$.

###### Proof.

Decompose $(\rho,V)=\bigoplus_{i=1}^{t}(\rho_{i},V_{i})$ into irreducible
summands and assume that they are ordered such that for every
$i\in\\{1,\dots,t\\}$, we have $\rho_{i}\not\cong\rho_{j}(k)$ for $j>i$ and
$k\geq 1$. We will prove the claim by induction on $t$. If $t=0$, we can take
$N=0$. Suppose $t>0$. We treat uniqueness first.

Our ordering implies that $\rho_{1}$ cannot lie in the image of $N$. Thus,
there is a direct sum decomposition of Weil–Deligne representations $V=U\oplus
S$, where $U=\bigoplus_{k\geq 0}N^{k}\rho_{1}$. Since $(\rho,V,N)$ is generic,
we have $\operatorname{Hom}(S,U(1))=0$. This is implies that
$\rho_{1}(a)\otimes\operatorname{Sp}(m^{\prime})$ for $1\leq a\leq m$ and
$m^{\prime}>m-a$ is not a direct summand of $S$. In particular, $\rho_{1}(m)$
is not a $W_{K}$-direct summand of $S$ and $m$ is the maximal integer such
that $\rho_{1}(k)\in\\{\rho_{i}\\}$ for $k=0,\dots,m-1$. Hence, the
isomorphism class of $U\cong\rho_{1}\otimes\operatorname{Sp}(m)$ is
independent of $N$. By the inductive hypothesis, the isomorphism class of $S$
is independent of $N$, too. We conclude that $(\rho,V,N)$ is uniquely
determined.

For the existence we let $m$ be maximal such that
$\rho_{1}(k)\in\\{\rho_{i}\\}$ for all $k=0,\dots,m-1$. There are indices
$i_{0},i_{1},\dots,i_{m-1}\in\\{1,\dots,t\\}$ such that $V_{i_{j}}\cong
V_{i_{j-1}}(1)$ and $i_{0}=1$. We define a nilpotent operator

$N:\bigoplus_{j=0}^{m-1}V_{i_{j}}\to\bigoplus_{j=0}^{m-1}V_{i_{j}}(-1)$

such that the restrictions induce isomorphisms $N:V_{i_{j-1}}\to
V_{i_{j}}(-1)$ for $j>0$. This operator defines a Weil–Deligne representation
on $U:=\bigoplus_{j=0}^{m-1}V_{i_{j}}$ which is isomorphic to the generic
representation $\rho_{1}\otimes\operatorname{Sp}(m)$. Using the inductive
hypothesis on $S:=\bigoplus_{i\neq i_{j}}V_{i}$, we can extend $N$ to all of
$V$. We have $\operatorname{Hom}(U,S(1))=0$ since $U$ is generated by
$\rho_{1}$ and $\rho_{1}(1)\not\cong\rho_{i}$ for all $i>1$. We have
$\operatorname{Hom}(S,U(1))=0$ since otherwise there would be an index $i>1$
such that $\rho_{i}\cong\rho_{1}(m)$, contradicting the maximality of $m$.
Hence $(\rho,V,N)$ is generic. ∎

######  3.1.10 Remark.

The previous lemma has the following geometric interpretation: There is an
affine variety $X/\mathbb{C}$ (the Vogan variety introduced in [Vog93, §4])
which parametrises all the possible nilpotent matrices $N$ such that
$(\rho,V,N)$ is a Weil–Deligne representation. The centraliser $G$ of
$(\rho,V)$ acts on $X$ by conjugation. The space of (framed) deformations of
$(\rho,V,N)$ has minimal dimension if and only if $(\rho,V,N)$ lies in an open
$G$-orbit.

A Galois cohomology argument shows that the (framed) deformation space of
$(\rho,V,N)$ has dimension

$(\dim V)^{2}+\dim\operatorname{Hom}((\rho,V,N),(\rho(1),V,N)),$

which is minimal if and only if $(\rho,V,N)$ is generic. Thus the lemma could
be restated as saying that $X$ has a unique open $G$-orbit, which follows from
[Vog93, Prop 4.5 d)].

######  3.1.11

Let $K/\mathbb{Q}_{p}$ be a finite extension and $\ell$ a prime number. Let
$\operatorname{Frob}_{K}\in G_{K}$ be a Frobenius element and
$t_{\ell}:I_{K}\to\mathbb{Z}_{\ell}$ a choice of tame character, where
$I_{K}<G_{K}$ is the inertia group. Given an isomorphism
$\iota:\overline{\mathbb{Q}}_{\ell}\cong\mathbb{C}$ and a continuous Galois
representation
$\rho:G_{K}\to\operatorname{GL}_{n}(\overline{\mathbb{Q}}_{\ell})$ there
exists a unique Weil–Deligne representation
$WD_{\iota}(\rho)=(r,\mathbb{C}^{n},N)$ such that for all $a\in\mathbb{Z}$ and
$x\in I_{K}$, we have
$\rho^{\iota}(\operatorname{Frob}_{K}^{a}x)=r(\operatorname{Frob}_{K}^{a}x)\exp(t_{\ell}(x)N)$.
Moreover, the isomorphism class of $WD_{\iota}(\rho)$ does not depend on
$\operatorname{Frob}_{K}$ and $t_{\ell}$. [Tat79, 4.2.1]

######  3.1.12 Lemma.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter modulo $d$. Let $F$ be a CM field containing the
$d$th roots of unity and $t\in F\setminus\mu_{d}$. Let
$\lambda<\mathcal{O}_{d}$ be a prime of residue characteristic $\ell$ and
$v\nmid\ell$ a finite place of $F$. For any isomorphism
$\iota:\overline{\mathbb{Q}}_{\ell}\cong\mathbb{C}$, the Weil–Deligne
representation

$WD_{\iota}(M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})_{\lambda,t}\rvert_{G_{F_{v}}}\otimes_{\mathcal{O}_{d,\lambda}}\overline{\mathbb{Q}}_{\ell})$

is pure.

###### Proof.

The representation
$M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})_{\lambda,t}\rvert_{G_{F_{v}}}$
is a direct summand of the cohomology of a smooth projective hypersurface.
Thus, it is pure by [Sch12, Thm 1.14]. ∎

######  3.1.13

For a smooth irreducible representation $\pi$ of $\operatorname{GL}_{n}(K)$,
we denote by $\operatorname{rec}_{K}(\pi)$ the Weil–Deligne representation
attached to $\pi$ via the local Langlands correspondence of Harris–Taylor
[HT01].

######  3.1.14 Theorem (Varma).

Let $F$ be a CM or totally real field and $\ell$ a prime number. Let $\pi$ be
a cuspidal cohomological automorphic representation of
$\operatorname{GL}_{n}(\mathbb{A}_{F})$. For a place $v\nmid\ell$ of $F$ we
have

$WD_{\iota}(r_{\iota}(\pi)\rvert_{G_{F_{v}}})^{ss}\cong\operatorname{rec}_{F_{v}}(\pi_{v}|\det|^{(1-n)/2})$

and

$WD_{\iota}(r_{\iota}(\pi)\rvert_{G_{F_{v}}})^{F-ss}\cong\operatorname{rec}_{F_{v}}(\pi_{v}|\det|^{(1-n)/2})$

holds if and only if $WD_{\iota}(r_{\iota}(\pi)\rvert_{G_{F_{v}}})^{F-ss}$ is
generic.

###### Proof.

The first part follows from the main theorem of [Var15]. Since $\pi$ is a
cuspidal representation of $\operatorname{GL}_{n}(\mathbb{A}_{F})$, it is
generic, and its local factors $\pi_{w}$ are generic for all finite places
$w$. By [All16, Lemma 1.1.3], the Weil–Deligne representation
$\operatorname{rec}_{F_{v}}(\pi_{v}|\det|^{(1-n)/2})$ is generic. The second
part of the claim follows from lemma 3.1.9. ∎

###  3.2. The Potential Automorphy Theorem

Now we have collected all the necessary prerequisites to prove our potential
automorphy theorem.

######  3.2.1 Theorem.

Let $(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ be a
hypergeometric parameter of dimension $n$ modulo $d$ appearing in table 2.
There exists an integer $C>0$ with the following property. Given the following
objects:

  * •

a CM field $F$ containing $\mu_{d}$,

  * •

a prime $p$,

  * •

a prime $\ell\nmid pC$ such that $\ell\equiv 1\pmod{d}$,

  * •

a continuous semisimple representation
$\overline{\rho}:G_{F}\to\operatorname{GL}_{n}(\mathbb{F}_{\ell})$ such that
$\det\overline{\rho}=\overline{\chi}_{cyc}^{-n(n-1)/2}$,

  * •

a finite extension $F^{avoid}/F$,

there exists a finite totally real extension $F_{2}/\mathbb{Q}$, linearly
disjoint from $F^{avoid}$ over $\mathbb{Q}$ such that for $F^{\prime}=FF_{2}$,
there exists an isomorphism
$\iota:\overline{\mathbb{Q}}_{\ell}\cong\mathbb{C}$ and a cuspidal automorphic
representation $\Pi$ of $\operatorname{GL}_{n}(\mathbb{A}_{F^{\prime}})$ which
is $\iota$-ordinary of weight $0$ at all places above $\ell$ such that

$\overline{r_{\iota}(\Pi)}=\overline{\rho}$

and for all places $v\mid p$ of $F^{\prime}$

$WD(r_{\iota}(\Pi)\rvert_{G_{F_{v}^{\prime}}})^{F-ss}$

is generic with monodromy operator having Jordan blocks $J_{\alpha}$ of size
$\\#\\{i:\alpha_{i}=\alpha\\}$ for $\alpha\in\mathbb{Z}/d\mathbb{Z}$. If $d$
is odd, then there also exists another cuspidal automorphic representation
$\Pi_{2}$ of $\operatorname{GL}_{n}(\mathbb{A}_{F^{\prime}})$ with the same
properties as $\Pi$ except that the $v$-adic monodromy operator is nilpotent
of rank $1$.

###### Proof.

Pick a prime $\lambda<\mathcal{O}_{d}$ above $\ell$ and choose an isomorphism
$\iota:\overline{\mathbb{Q}}_{\ell}\cong\mathbb{C}$. Let
$M=M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$. Property (D) of
proposition 2.1.9 shows that there is an isomorphism

$\alpha_{\lambda}:\det(M_{\lambda,0}\otimes\psi_{\lambda})\cong\chi_{cyc}^{-n(n-1)/2}$

for some character
$\psi_{\lambda}:G_{\mathbb{Q}(\mu_{d})}\to\mathcal{O}_{d,\lambda}^{\times}$.
Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$. Denote by
$r_{E,\ell^{\prime}}:G_{\mathbb{Q}}\to\operatorname{GL}_{2}(\mathbb{Q}_{\ell^{\prime}})$
the Galois representation acting on the dual of the $\ell^{\prime}$-adic Tate
module of $E$. As in the first part of the proof of [Qia22, Proposition 4.1],
there exists $\ell^{\prime}\equiv 1\pmod{d}$ a large enough prime such that

  * •

$\ell^{\prime}$ is unramified in $F$,

  * •

$\overline{r}_{E,\ell^{\prime}}(G_{\tilde{F}})=\operatorname{GL}_{2}(\mathbb{F}_{\ell^{\prime}})$,
where $\tilde{F}$ denotes the normal closure of $F/\mathbb{Q}$,

  * •

$\exists\sigma\in G_{F}\setminus G_{F(\mu_{\ell^{\prime}})}$ such that
$\overline{r}_{E,\ell^{\prime}}(\sigma)$ is a scalar,

  * •

$E$ has good ordinary reduction at $\ell^{\prime}$.

Let $F^{av}$ be the normal closure of
$F^{avoid}\overline{\mathbb{Q}}^{\ker\overline{r}_{E,\ell^{\prime}}}$ over
$\mathbb{Q}$. By [ACC+23, Corollary 7.2.4], there exists a finite totally real
Galois extension $F^{suff}/\mathbb{Q}$ and a finite Galois extension
$F^{av}_{2}/\mathbb{Q}$ such that

  * •

$F_{2}^{av}/\mathbb{Q}$ is linearly disjoint from $F^{av}/\mathbb{Q}$.

  * •

$F^{suff}/\mathbb{Q}$ is linearly disjoint from $F^{av}F_{2}^{av}/\mathbb{Q}$.

  * •

For any totally real extension $F^{\prime}/F^{suff}$ which is linearly
disjoint from $F_{2}^{av}$ over $\mathbb{Q}$, the Galois representation
$\operatorname{Sym}^{n-1}r_{E,\ell^{\prime}}\rvert_{G_{F^{\prime}}}$ is
automorphic.

Let $\lambda^{\prime}<\mathcal{O}_{d}$ be a prime above $\ell^{\prime}$ and
$K/F$ a number field. Let $B$ be the etale
$\mathcal{O}_{d}/\lambda\lambda^{\prime}$-local system on
$\operatorname{Spec}A_{d}\otimes_{\mathcal{O}_{d}}K$ corresponding to
$M_{\lambda}/\lambda(\psi_{\lambda})\times
M_{\lambda^{\prime}}/\lambda^{\prime}(\psi_{\lambda^{\prime}})$. Let
$\widetilde{A}_{d}$ be defined as in lemma 2.5.6. Consider the following
moduli problem.

$S/\operatorname{Spec}\widetilde{A}_{d}\otimes_{\mathcal{O}_{d}}K\mapsto\left\\{\eta:B\rvert_{S}\cong\overline{\rho}\times\operatorname{Sym}^{n-1}\overline{r}_{E,\ell^{\prime}}\mid\det(\eta)=\alpha_{\lambda}\times\alpha_{\lambda^{\prime}}\right\\}$

It is represented by a finite etale scheme

$X\to\operatorname{Spec}\widetilde{A}_{d}\otimes_{\mathcal{O}_{d}}K.$

If $K\to\mathbb{C}$ is a field homomorphism, then $X(\mathbb{C})$ is the
covering space of $\operatorname{Spec}\widetilde{A}_{d}(\mathbb{C})$
corresponding to the $\pi_{1}(\mathbb{C}\setminus\mu_{d},t)$-set

$\operatorname{SL}(M(s\alpha_{i};s\beta_{j})_{\lambda,t}/\lambda)\times\operatorname{SL}(M(s\alpha_{i};s\beta_{j})_{\lambda^{\prime},t}/\lambda^{\prime})$

for some $s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$. It follows from property
(BM) of proposition 2.1.9 that $\pi_{1}(\mathbb{C}\setminus\mu_{d},t)$ acts
transitively on this set, hence $X(\mathbb{C})$ is connected. Since
$K\to\mathbb{C}$ was arbitrary, $X$ is geometrically connected.

Let $K=FF^{suff}$ and let $q$ be the composition

$X\to\operatorname{Spec}\widetilde{A}_{d}\otimes_{\mathcal{O}_{d}}K\to\operatorname{Spec}A_{d}\otimes_{\mathcal{O}_{d}}K.$

Let $T=\operatorname{Res}_{K/\mathbb{Q}}X$, $S_{1}=\\{\infty\\}$,
$S_{2}=\emptyset$ and $S_{3}=\\{\ell,\ell^{\prime},p\\}$. We apply the theorem
of Moret-Bailly [HSBT10, Proposition 2.1] to $T/\mathbb{Q}$, $S_{1}$, $S_{2}$,
$S_{3}$, $L=F^{suff}F^{av}F_{2}^{av}$ and

  * •

if $u\in S_{1}$, then
$\Omega_{u}=X(\mathbb{R}\otimes_{\mathbb{Q}}K)=X(\prod_{v\mid\infty}\mathbb{C})$,

  * •

if $u\in\\{\ell,\ell^{\prime}\\}$, then

$\Omega_{u}=q^{-1}\\{(x_{\tau})_{\tau:K\hookrightarrow\overline{\mathbb{Q}}_{u}}\in\operatorname{Spec}A_{d}(\prod_{K\hookrightarrow\overline{\mathbb{Q}}_{u}}\overline{\mathbb{Q}}_{u}):\forall\,\tau,\,|x_{\tau}|_{u}<1\\}$

  * •

if $u=p$, then

$\Omega_{u}=q^{-1}\\{(x_{\tau})_{\tau:K\hookrightarrow\overline{\mathbb{Q}}_{u}}\in\operatorname{Spec}A_{d}(\prod_{K\hookrightarrow\overline{\mathbb{Q}}_{u}}\overline{\mathbb{Q}}_{u}):\forall\,\tau,\,|x_{\tau}|_{u}>1\\}.$

Thus, there exists a finite totally real Galois extension $F_{1}/\mathbb{Q}$
such that for $F^{\prime}:=KF_{1}=FF_{1}F^{suff}$, there exists
$t\in\operatorname{Spec}\widetilde{A}_{d}(F^{\prime})$ such that

  * •

$F_{1}$ is linearly disjoint from $F^{suff}F^{av}F_{2}^{av}$ over
$\mathbb{Q}$,

  * •

$M_{t,\lambda}/\lambda\otimes\psi_{\lambda}\cong\overline{\rho}\rvert_{G_{F^{\prime}}}$
and
$M_{t,\lambda^{\prime}}/\lambda^{\prime}\otimes\psi_{\lambda^{\prime}}\cong\operatorname{Sym}^{n-1}\overline{r}_{E,\ell^{\prime}}\rvert_{G_{F^{\prime}}}$,

  * •

$|t|_{v}>1$ for all $v\mid p$,

  * •

$M_{t,\lambda}\otimes\psi_{\lambda}$ and
$M_{t,\lambda^{\prime}}\otimes\psi_{\lambda^{\prime}}$ are ordinary of weight
0 by lemma 2.6.9.

Note that this also implies the following inclusions

$F^{\prime}\cap F_{2}^{av}\subset K\cap F_{2}^{av}\subset F^{suff}F^{av}\cap
F_{2}^{av}=\mathbb{Q}$

and

$F_{1}F^{suff}\cap F^{av}=\mathbb{Q}.$

Hence, $F^{\prime}$ is linearly disjoint from $F_{2}^{av}$ over $\mathbb{Q}$
and $F_{2}:=F_{1}F^{suff}$ is linearly disjoint from $F^{av}$ over $F$. In
particular $F_{2}$ is linearly disjoint from $F^{avoid}\subset F^{av}$ as
claimed in the theorem.

Since $(F^{\prime})^{+}$ contains $F^{suff}$, and is linearly disjoint from
$F_{2}^{av}$, we know that
$\operatorname{Sym}^{n-1}r_{E,\ell^{\prime}}\rvert_{G_{(F^{\prime})^{+}}}$ is
automorphic. By quadratic base change, the same holds for
$\operatorname{Sym}^{n-1}r_{E,\ell^{\prime}}\rvert_{G_{F^{\prime}}}$. Since
$\overline{\mathbb{Q}}^{\ker\overline{r}_{E,\ell^{\prime}}}\subset F^{av}$, we
have

  * •

$\overline{r}_{E,\ell^{\prime}}(G_{F^{\prime}})=\operatorname{GL}_{2}(\mathbb{F}_{\ell^{\prime}})$,

  * •

$\exists\sigma\in G_{F^{\prime}}\setminus G_{F^{\prime}(\mu_{\ell^{\prime}})}$
such that $\overline{r}_{E,\ell^{\prime}}(\sigma)$ is a scalar.

By [ACC+23, Lemma 7.1.6 (2)], we know that
$\operatorname{Sym}^{n-1}\overline{r}_{E,\ell^{\prime}}(G_{F^{\prime}(\mu_{\ell^{\prime}})})$
is enormous and [Qia22, Lemma 4.3(4)] implies that
$\operatorname{Sym}^{n-1}\overline{r}_{E,\ell^{\prime}}\rvert_{G_{F^{\prime}}}$
is decomposed generic.

The automorphy lifting theorem [ACC+23, Theorem 6.1.2] implies that
$M_{\lambda^{\prime},t}\otimes\psi_{\lambda^{\prime}}$ is automorphic. This
yields a cuspidal cohomological automorphic representation $\Pi$ of
$\operatorname{GL}_{n}(\mathbb{A}_{F^{\prime}})$ such that
$r_{\iota^{\prime}}(\Pi)=M_{\lambda^{\prime},t}\otimes\psi_{\lambda^{\prime}}$
for some isomorphism
$\iota^{\prime}:\overline{\mathbb{Q}}_{\ell^{\prime}}\cong\mathbb{C}$ such
that $\iota^{\prime}\circ\iota^{-1}$ fixes $\mathbb{Q}(\mu_{d})$. Since
$M_{\lambda^{\prime},t}\otimes\psi_{\lambda^{\prime}}$ and
$M_{\lambda,t}\otimes\psi_{\lambda}$ have the same Frobenius eigenvalues, this
also implies that $r_{\iota}(\Pi)=M_{\lambda,t}\otimes\psi_{\lambda}$.
Moreover, since $|t|_{v}>1$ we can prove as in [HSBT10, Lemma 1.15], that the
$v$-adic monodromy operator acting on $M_{\lambda,t}$ coincides with the
monodromy around $\infty$ acting on $M_{B,t}$. This has the correct Jordan
blocks by property (UM) of proposition 2.1.9. By lemma 3.1.12, we know that
$WD(M_{\lambda,t}\rvert_{G_{F_{w}}})^{F-ss}$ is pure and hence generic by
lemma 3.1.8.

If $d$ is odd, then we simply repeat the same proof but for $u=p$ replace
$\Omega_{u}$ by

$\Omega_{u}=q^{-1}\\{(x_{\tau})_{\tau:K\hookrightarrow\overline{\mathbb{Q}}_{u}}\in\operatorname{Spec}A_{d}(\prod_{K\hookrightarrow\overline{\mathbb{Q}}_{u}}\overline{\mathbb{Q}}_{u}):\forall\,\tau,\,|x_{\tau}-1|_{u}<1\\},$

to obtain $\Pi_{2}$. ∎

######  3.2.2 Corollary.

Let $n\in\\{3,4,5\\}$ and $d_{3}=9$, $d_{4}=18$,
$d_{5}=168=\operatorname{lcm}(21,24)$. There exists an integer $C>0$ with the
following property. Given the following objects:

  * •

a nilpotent $n\times n$ matrix $N$,

  * •

a CM field $F$ containing $\mu_{d_{n}}$,

  * •

a prime $p$,

  * •

a prime $\ell\neq p$ such that $\ell\nmid C$ and $\ell\equiv 1\pmod{d_{n}}$,

  * •

a continuous semisimple representation
$\overline{\rho}:G_{F}\to\operatorname{GL}_{n}(\mathbb{F}_{\ell})$ such that
$\det\overline{\rho}=\overline{\chi}_{cyc}^{-n(n-1)/2}$,

  * •

a finite Galois extension $F^{avoid}/F$,

there exists a finite totally real extension $F_{2}/\mathbb{Q}$, linearly
disjoint from $F^{avoid}$ over $\mathbb{Q}$. For $F^{\prime}=FF_{2}$, there
exists a cuspidal automorphic representation $\Pi$ of
$\operatorname{GL}_{n}(\mathbb{A}_{F^{\prime}})$ which is ordinary of weight
$0$ at all places above $\ell$ such that

$\overline{r_{\iota}(\Pi)}=\overline{\rho}\rvert_{G_{F^{\prime}}}$

and for all places $v\mid p$ of $F^{\prime}$

$WD(r_{\iota}(\Pi)\rvert_{G_{F_{v}^{\prime}}})^{F-ss}$

is generic with monodromy operator conjugate to $N$.

###### Proof.

If $N=0$, then this follows from the main theorem of [Qia22]. If $N\neq 0$,
use proposition 2.5.7 to select the hypergeometric parameter corresponding to
$N$ from table 1, which is a subset of table 2. Then apply the theorem with
this parameter. ∎

$n$ | $d$ | $\alpha_{i}$ | $\beta_{j}$ | $\underline{c}$
---|---|---|---|---
$3$ | $9$ | $0,0,0$ | $1,2,6$ | $3,7,8$
$4$ | $9$ | $0,0,0,0$ | $1,2,7,8$ | $0,0,0$
$4$ | $9$ | $0,0,1,1$ | $2,4,6,8$ | $0,0,0$
$4$ | $18$ | $0,0,0,3$ | $4,11,16,17$ | $1,7,10$
$5$ | $21$ | $0,0,0,0,0$ | $1,2,4,15,20$ | $6,17,19$
$5$ | $21$ | $0,0,0,0,1$ | $4,10,12,18,20$ | $1,1,19$
$5$ | $24$ | $0,0,0,1,1$ | $2,6,12,19,23$ | $0,0,0$
$5$ | $24$ | $0,0,1,1,7$ | $8,12,13,17,19$ | $0,0,0$
$5$ | $24$ | $0,0,0,2,6$ | $7,8,12,19,22$ | $0,0,0$
Figure 1. Hypergeometric parameters sufficient for Corollary 3.2.2

######  3.2.3 Corollary.

Let $n\in\\{3,4,5\\}$ and $d_{3}=9$, $d_{4}=18$, $d_{5}=168$ and $p$ a prime.
For any nilpotent $n\times n$ matrix $N$, there exists a CM field $F^{\prime}$
and a cuspidal cohomological automorphic representation $\Pi$ of
$\operatorname{GL}_{n}(\mathbb{A}_{F^{\prime}})$ of weight $0$, which is not
essentially conjugate self-dual, such that the monodromy operator of
$\operatorname{rec}_{F_{v}^{\prime}}(\Pi_{v}|\det|^{(1-n)/2})$ coincides with
$N$ for all $v\mid p$.

###### Proof.

Let $\ell\equiv 1\pmod{504}$ be a sufficiently large prime and
$G=\operatorname{PSL}_{2}(\mathbb{F}_{7})$. There is an irreducible non-self-
dual representation $r:G\to\operatorname{GL}_{3}(\mathbb{F}_{\ell})$. Let $L$
be the splitting field of $x^{7}-154x+99$. We have
$\operatorname{Gal}(L/\mathbb{Q})\cong G$ by [EFM79]. Consider
$F=\mathbb{Q}(\mu_{504\ell})$ and
$\overline{\rho}:G_{F}\to\operatorname{GL}_{3}(\mathbb{F}_{\ell})$ the
restriction of $r$ and
$F^{avoid}=\overline{\mathbb{Q}}^{\ker\overline{\rho}}$. Now apply the
previous corollary to $\overline{\rho},\overline{\rho}\oplus\mathbf{1}$ and
$\overline{\rho}\oplus\mathbf{1}\oplus\mathbf{1}$. The resulting
representation $\Pi$ cannot be essentially conjugate self-dual since that
would imply the existence of a character
$\overline{\chi}:G_{F^{\prime}}\to\overline{\mathbb{F}}_{\ell}^{\times}$ such
that $\overline{\rho}^{\vee}\cong\overline{\rho}\otimes\overline{\chi}$. In
particular, $\overline{\chi}$ must factor through the simple group $G$ and is
trivial. Absurd. Finally, the claim follows from theorem 3.1.14. ∎

###  3.3. Local-Global Compatibility

In this section we explain the consequences of the potential automorphy
theorem for the compatibility of local and global Langlands correspondences.
Fix an isomorphism $\iota:\overline{\mathbb{Q}}_{\ell}\cong\mathbb{C}$. First
we need a variant of [ACC+23, Thm 6.1.2].

######  3.3.1 Lemma.

Let $F$ be an imaginary CM or totally real field, let
$c\in\operatorname{Aut}(F)$ be complex conjugation, and let $p$ be a prime.
Let $w\nmid p$ be a place of $F$ and
$K_{w}\subset\operatorname{GL}_{n}(F_{w})$ an open compact subgroup containing
the Iwahori subgroup. If
$\rho:G_{F}\to\operatorname{GL}_{n}(\overline{\mathbb{Q}}_{\ell})$ is a
continuous representation satisfying the following conditions:

  1. (1)

$\rho$ is unramified almost everywhere.

  2. (2)

For each place $v\mid\ell$ of $F$, the representation $\rho\rvert_{G_{F_{v}}}$
is potentially semistable, ordinary with regular Hodge–Tate weights.

  3. (3)

$\overline{\rho}$ is absolutely irreducible and decomposed generic [ACC+23,
4.3.1]. The image of $\overline{\rho}\rvert_{F(\zeta_{\ell})}$ is enormous
[ACC+23, 6.2.28].

  4. (4)

There exist $\sigma\in G_{F}\setminus G_{F(\zeta_{\ell})}$ such that
$\overline{\rho}(\sigma)$ is a scalar. We have $\ell>n$.

  5. (5)

There exists a regular algebraic cuspidal automorphic representation $\pi$ of
$\operatorname{GL}_{n}(\mathbb{A}_{F})$ and an isomorphism
$\iota:\overline{\mathbb{Q}}_{\ell}\to\mathbb{C}$ such that $\pi$ is
$\iota$-ordinary and $\overline{r_{\iota}(\pi)}\cong\overline{\rho}$.

  6. (6)

The restriction $\overline{\rho}\rvert_{G_{F_{w}}}$ is trivial and
$q_{w}\equiv 1\pmod{\ell}$.

  7. (7)

The space of invariants $\pi_{w}^{K_{w}}$ contains a non-zero vector and the
monodromy operators of $r_{\iota}(\pi)\rvert_{G_{F_{w}}}$ and
$\rho\rvert_{G_{F_{w}}}$ are conjugate to each other.

Then $\rho$ is ordinarily automorphic of weight $\iota\lambda$: there exists
an $\iota$-ordinary cuspidal automorphic representation $\Pi$ of
$\operatorname{GL}_{n}(\mathbb{A}_{F})$ of weight $\iota\lambda$ such that
$\rho\cong r_{\iota}(\Pi)$ and $\Pi_{w}^{K_{w}}\neq 0$. Moreover, if $v\nmid
p$ is a finite place of $F$ and both $\rho$ and $\pi$ are unramified at $v$,
then $\Pi_{v}$ is unramified.

###### Proof.

Given [ACC+23, Thm 6.1.2], the only new thing we need to prove is that
$\Pi_{w}^{K_{w}}\neq 0$. As in [ACC+23, §6.6], after a solvable base change
which is totally split at $w$, it suffices to prove this under the additional
conditions listed in [ACC+23, §6.6.1].

Consider the patched homology complex $\mathcal{C}_{\infty}$ of
$S_{\infty}$-modules used in the proof of theorem [ACC+23, Thm 6.6.2] and the
$S_{\infty}$-algebras $R_{\infty}$ and
$T_{\infty}\subset\operatorname{End}_{\mathbf{D}(S_{\infty})}(\mathcal{C}_{\infty})$.
By replacing the Iwahori subgroup in the level subgroups defining
$\mathcal{C}_{\infty}$ with the group $K_{w}$ we obtain a similar bounded
complex of $S_{\infty}$-modules $\mathcal{C}_{\infty}(K_{w})$ and an
$S_{\infty}$-algebra
$T_{\infty}(K_{w})\subset\operatorname{End}_{\mathbf{D}(S_{\infty})}(\mathcal{C}_{\infty}(K_{w}))$.
Moreover, there is a natural morphism of complexes
$\mathcal{C}_{\infty}\to\mathcal{C}_{\infty}(K_{w})$ and a compatible map of
$S_{\infty}$-algebras $T_{\infty}\to T_{\infty}(K_{w})$.

The Galois representation $\rho$ defines a point $y$ of
$\operatorname{Spec}R_{\infty}$. Denote its image in
$\operatorname{Spec}S_{\infty}$ by $\mathfrak{p}$. We wish to show that $y$
lies in the support of
$H^{\bullet}(\mathcal{C}_{\infty}(K_{w})\otimes_{S_{\infty}}^{\mathbb{L}}S_{\infty}/\mathfrak{p})[1/p]$,
since then the lemma follows as in the proof of [ACC+23, Thm 6.6.2]. Arguing
as in the proof of [ACC+23, Corollary 6.3.9], we see that it suffices to show
that $y$ lies in the support of $H^{\bullet}(\mathcal{C}_{\infty}(K_{w}))$ or
equivalently that
$y\in\operatorname{Spec}T_{\infty}(K_{w})\subset\operatorname{Spec}R_{\infty}$.

Let $x\in\operatorname{Spec}R_{\infty}$ be the point corresponding to the
Galois representation $r_{\iota}(\pi)$. As in the beginning of the proof of
[ACC+23, prop 6.3.8], the key lemma [CG18, Lemma 6.2] implies that
$\operatorname{Supp}_{S_{\infty,x}}H^{\bullet}(\mathcal{C}_{\infty}(K_{w})_{x})$
contains a point of codimension at most $l_{0}$. Let $x_{1}$ denote a preimage
of such a point in $\operatorname{Spec}T_{\infty}(K_{w})$. If we view
$\operatorname{Spec}T_{\infty}(K_{w})$ as a closed subset of
$\operatorname{Spec}R_{\infty}$, then $x_{1}$ must be a generic point of
$\operatorname{Spec}R_{\infty}$ of maximal dimension.

Let $R$ be the set of places used in [ACC+23, §6.6.1]. In particular, $w\in
R$. The proofs of [Tay08, Prop 3.1] and [ACC+23, Lemma 6.2.26] show that the
irreducible components of maximal dimension of $\operatorname{Spec}R_{\infty}$
are in bijection with tuples $(N_{v})_{v\in R}$ where $N_{v}$ is a conjugacy
class of a $n\times n$ nilpotent matrix. Moreover, a point of
$\operatorname{Spec}R_{\infty}$ corresponding to a Galois representation
$\rho^{\prime}:G_{F}\to\operatorname{GL}_{n}(\overline{\mathbb{Q}}_{\ell})$
lies on the irreducible component of maximal dimension indexed by
$N_{v}=N(\rho^{\prime}\rvert_{G_{F_{v}}})$.

If $R=\\{w\\}$, then we use assumption (7) to directly conclude that $y$ lies
in the closure of $x_{1}$, hence $y\in\operatorname{Spec}T_{\infty}(K_{w})$.
If $\\#R>1$, then we consider the deformation datum
$\mathcal{S}_{\chi}^{ord}$, where $\chi_{w,i}=1$ and for $v\in
R\setminus\\{w\\}$ we choose
$\chi_{v,1},\dots,\chi_{v,1}:\mathcal{O}_{F_{v}}^{\times}\to\mathcal{O}^{\times}$
pairwise distinct characters which are trivial mod $\varpi$. Similarly to the
proof of [ACC+23, Thm 6.6.2], this leads to a setup
$R_{\infty}^{{}^{\prime}},\mathcal{C}_{\infty}^{{}^{\prime}}(K_{w}),T_{\infty}^{{}^{\prime}}(K_{w})$
with an isomorphism $R_{\infty}^{\prime}/\varpi\cong R_{\infty}/\varpi$ which
is compatible with an isomorphism
$\mathcal{C}_{\infty}^{\prime}(K_{w})\otimes^{\mathbb{L}}_{S_{\infty}}S_{\infty}/\varpi\cong\mathcal{C}_{\infty}(K_{w})\otimes_{S_{\infty}}^{\mathbb{L}}S_{\infty}/\varpi$.
Moreover, [ACC+23, Lemma 6.3.7] applies to
$R_{\infty}^{{}^{\prime}},\mathcal{C}_{\infty}^{{}^{\prime}}(K_{w}),T_{\infty}^{{}^{\prime}}(K_{w})$.
The irreducible components of $\operatorname{Spec}R^{\prime}_{\infty}$ of
maximal dimension are indexed by a single nilpotent conjugacy class
corresponding to the monodromy at $w$.

Let $y_{1}$ be the maximal dimension generic point of
$\operatorname{Spec}R_{\infty}$ containing $y$. Choose generic points
$\overline{x}_{1}$ and $\overline{y}_{1}$ of
$\operatorname{Spec}R_{\infty}/(x_{1},\varpi)$ and
$\operatorname{Spec}R_{\infty}(y_{1},\varpi)$. By assumption on the monodromy
at $w$, we see that both these points lie on the irreducible component of
$R_{\infty}^{\prime}$ indexed by $N_{w}=N(\rho\rvert_{G_{F_{w}}})$, i.e. both
$\overline{x}_{1}$ and $\overline{y}_{1}$ generalise to the same generic point
$x^{\prime}\in\operatorname{Spec}R_{\infty}^{\prime}$. Finally, we deduce in
exactly the same way as at the end of the proof of [ACC+23, Prop 6.3.8] that
$y_{1}\in\operatorname{Spec}T_{\infty}(K_{w})$, hence also
$y\in\operatorname{Spec}T_{\infty}(K_{w})$, as desired. ∎

######  3.3.2 Lemma.

Let $K/\mathbb{Q}_{p}$ be a finite extension with residue field $k$ and $\pi$
a generic (i.e. admits a Whittaker model) smooth irreducible representation of
$\operatorname{GL}_{n}(K)$ such that $\pi^{\operatorname{Iw}}\neq 0$, where
$\operatorname{Iw}=\operatorname{Iw}_{n}\subset\operatorname{GL}_{n}(K)$ is
the Iwahori subgroup. Let $m_{1}+\dots+m_{s}=n$ be a partition. Let
$P<\operatorname{GL}_{n}$ be the standard parabolic subgroup corresponding to
the conjugate partition $n_{j}=\\#\\{i:m_{i}\geq j\\}$ and
$U=\\{g\in\operatorname{GL}_{n}(\mathcal{O}_{K}):\overline{g}\in
P(k)\\}<\operatorname{GL}_{n}(K)$.

  1. (a)

If the monodromy operator of $\operatorname{rec}_{K}(\pi)$ has Jordan block
sizes given by $m_{1},\dots,m_{s}$, then $\pi^{U}\neq 0$.

  2. (b)

If $\pi^{U}\neq 0$, then the rank of the monodromy operator of
$\operatorname{rec}_{K}(\pi)$ is at most $n-s$.

###### Proof.

As $\pi^{\operatorname{Iw}}\neq 0$, it follows from [Cas80, Proposition 2.6]
that the supercuspidal support of $\pi$ consists of unramified characters of
$K^{\times}$.

Since $\pi$ is generic, [Zel80, Thm 9.7] implies that there are segments
$\Delta_{1},\dots,\Delta_{t}$ of lengths $r_{i}$ such that
$\pi=\langle\Delta_{1}\rangle\times\dots\times\langle\Delta_{t}\rangle$ and
$\Delta_{i},\Delta_{j}$ are not linked for $i\neq j$. See [Zel80] for the
definitions of these terms. As explained in [Rod82, §4.4], it follows from
[HT01, Thm VII.2.20] that

$\operatorname{rec}_{K}(\pi)=\bigoplus_{i=1}^{t}\chi_{i}\otimes\operatorname{Sp}_{r_{i}}$

for some unramified characters $\chi_{i}$.

Let $Q<\operatorname{GL}_{n}$ be the standard parabolic subgroup corresponding
to the partition $r_{1}+\dots+r_{t}=n$. Consider the Bruhat decomposition

$\operatorname{GL}_{n}(K)=\bigcup_{w\in W_{Q}\backslash W/W_{P}}Q(K)wU,$

where $W$ is the Weyl group of $\operatorname{GL}_{n}$ and $W_{Q}$ and $W_{P}$
are defined as the Weyl groups of the Levi subgroups of $Q$ and $P$. By
definition of normalised induction, we have

$(\langle\Delta_{1}\rangle\times\dots\times\langle\Delta_{t}\rangle)^{\operatorname{U}}\cong\bigoplus_{w\in
W_{Q}\backslash
W/W_{P}}(\langle\Delta_{1}\rangle\otimes\dots\otimes\langle\Delta_{t}\rangle)^{Q(K)\cap
wUw^{-1}}.$

Now we can prove part (a). The assumption on the monodromy of
$\operatorname{rec}_{K}(\pi)$ implies that the partition $r_{1}+\dots+r_{t}=n$
is equivalent to $m_{1}+\dots+m_{s}=n$.

Hence, the choice of $P$ implies that there exists an element $w\in W$ such
that $M\cap wPw^{-1}=M\cap B$, where $M$ is the Levi subgroup of $Q$ and $B$
is the group of upper triangular matrices. Namely, if we assume $m_{1}\geq
m_{2}\geq\dots\geq m_{s}$, then we can take $w(x)=j+\sum_{i<y}m_{i}$, where
$x=y+\sum_{i<j}n_{i}$ and $y\leq n_{j}$. With this choice of $w$ we have
$M(K)\cap
wUw^{-1}=\operatorname{Iw}_{m_{1}}\times\dots\times\operatorname{Iw}_{m_{s}}$
and in particular, $\pi^{U}\neq 0$.

For part (b) it suffices to show that $t\geq s$. The first proof of [Cas80,
Proposition 2.3] implies that
$\langle\Delta_{i}\rangle^{\operatorname{Iw}_{r_{i}}}\cong\phi_{w_{0}}\mathbb{C}$,
where $w_{0}$ is the longest Weyl group element of $\operatorname{GL}_{r_{i}}$
and $\phi_{w_{0}}$ is the function defined in [Cas80, §2]. Any simple
reflection acts non-trivially on $\phi_{w_{0}}$. In particular, if
$U<\operatorname{GL}_{r_{i}}(K)$ is a parahoric subgroup such that
$\langle\Delta_{i}\rangle^{U}\neq 0$, then $U$ is conjugate to
$\operatorname{Iw}_{r_{i}}$.

Hence,
$(\langle\Delta_{1}\rangle\times\dots\times\langle\Delta_{t}\rangle)^{\operatorname{U}}\neq
0$ implies that $M(K)\cap wUw^{-1}$ equals
$\operatorname{Iw}_{r_{1}}\times\dots\times\operatorname{Iw}_{r_{t}}$ for some
choice of $w$. If $t<s$, then the first block of $P$, which is of size $s$, is
larger than the number of factors of $M$. Thus, for any $w$, the group $P\cap
wMw^{-1}$ has a quotient isomorphic to $\operatorname{GL}_{j}$ for some $j>1$.
In particular we cannot have $M\cap B=M\cap wPw^{-1}$ and $M(K)\cap
wUw^{-1}=\operatorname{Iw}_{r_{1}}\times\dots\times\operatorname{Iw}_{r_{s}}$.
∎

######  3.3.3 Theorem.

Let $n\in\\{3,4,5\\}$ and $d_{3}=9$, $d_{4}=18$, $d_{5}=168$. There exists an
integer $C>0$ with the following property. Given the following objects:

  * •

a CM field $F$ containing $\mu_{d_{n}}$,

  * •

a prime $\ell$ such that $\ell\nmid C$ and $\ell\equiv 1\pmod{d_{n}}$,

  * •

a cuspidal cohomological automorphic representation $\Pi$ of
$\operatorname{GL}_{n}(\mathbb{A}_{F})$ which is $\iota$-ordinary at all
places above $\ell$,

such that

  * •

$\overline{r_{\iota}(\Pi)}(G_{F(\mu_{\ell})})$ is enormous [ACC+23, 6.2.28]
and $\overline{r_{\iota}(\Pi)}$ is absolutely irreducible and decomposed
generic [ACC+23, 4.3.1],

  * •

there exists $\sigma\in G_{F}\setminus G_{F(\mu_{\ell})}$ such that
$\overline{r_{\iota}(\Pi)}(\sigma)$ is a scalar,

  * •

there exists $\gamma\in\operatorname{GL}_{n}(\overline{\mathbb{F}}_{\ell})$
such that
$\gamma\overline{r_{\iota}(\Pi)}(G_{F})\gamma^{-1}\subset\operatorname{GL}_{n}(\mathbb{F}_{\ell})$,

  * •

$\det\overline{r_{\iota}(\Pi)}=\overline{\chi}_{cyc}^{-n(n-1)/2}$,

we have

$WD(r_{\iota}(\Pi)\rvert_{G_{F_{v}}})^{F-ss}\cong\operatorname{rec}_{F_{v}}(\Pi_{v}|\det|^{(1-n)/2}),$

for all places $v\nmid\ell$.

###### Proof.

Let $v\nmid\ell$ be a place of $F$ dividing $p$. With the theorem of Varma
(theorem 3.1.14), it suffices to prove that
$WD(r_{\iota}(\Pi)\rvert_{G_{F_{v}}})^{F-ss}$ is generic. Let $N$ denote the
monodromy operator of $WD(r_{\iota}(\Pi)\rvert_{G_{F_{v}}})^{F-ss}$.

Let $F^{avoid}$ be the normal closure of
$\overline{F}^{\ker\overline{r_{\iota}(\Pi)}}(\mu_{\ell})$ over $\mathbb{Q}$.
We apply Corollary 3.2.2 to find a finite totally real extension
$F_{2}/\mathbb{Q}$ linearly disjoint from $F^{avoid}$ over $\mathbb{Q}$ such
that for $F^{\prime}=FF_{2}$ we have a cuspidal automorphic representation
$\Pi^{\prime}$ of $\operatorname{GL}_{n}(\mathbb{A}_{F^{\prime}})$ such that

$\overline{r_{\iota}(\Pi)}\rvert_{G_{F^{\prime}}}=\overline{r_{\iota}(\Pi^{\prime})}$

and the Weil–Deligne representation

$WD(r_{\iota}(\Pi^{\prime})\rvert_{G_{F^{\prime}_{v}}})^{F-ss}$

is generic with monodromy operator conjugate to $N$ for all places $v\mid p$
of $F^{\prime}$.

By [ACC+23, Lemma 7.1.7], $\overline{r_{\iota}(\Pi)}\rvert_{G_{F^{\prime}}}$
is decomposed generic and absolutely irreducible. Moreover, since
$F^{\prime}(\mu_{\ell})$ is linearly disjoint from $F^{avoid}$ over
$F(\mu_{\ell})$, we find that
$\overline{r_{\iota}(\Pi)}(G_{F^{\prime}(\mu_{\ell})})=\overline{r_{\iota}(\Pi)}(G_{F(\mu_{\ell})})$
is enormous. After a cyclic base change we may moreover assume that
$\Pi^{\prime}_{v}$ has Iwahori fixed vectors by [A’C23, 5.2.3 and 5.2.4], that
$q_{v}\equiv 1\pmod{\ell}$ and that
$\overline{r_{\iota}(\Pi)}\rvert_{G_{F_{v}}}$ is trivial.

Theorem 3.1.14 implies that

$\operatorname{rec}_{F^{\prime}_{v}}(\Pi^{\prime}_{v}|\det|^{(1-n)/2})=WD(r_{\iota}(\Pi^{\prime})\rvert_{G_{F^{\prime}_{v}}})^{F-ss}.$

Hence, part (a) of lemma 3.3.2 implies that $(\Pi^{\prime}_{v})^{K_{v}}\neq
0$, where
$K_{v}=\\{k\in\operatorname{GL}_{n}(\mathcal{O}_{F_{v}^{\prime}}):\overline{k}\in
P(k(v))\\}$ and $P\subset\operatorname{GL}_{n}$ is the standard parabolic
subgroup corresponding to the conjugate partition of the Jordan block sizes of
$N$.

By lemma 3.3.1, there exists another automorphic representation
$\Pi^{\prime\prime}$ of $\operatorname{GL}_{n}(\mathbb{A}_{F^{\prime}})$ such
that

$r_{\iota}(\Pi^{\prime\prime})=r_{\iota}(\Pi)\rvert_{G_{F^{\prime}}}$

and $(\Pi^{\prime\prime}_{v})^{K_{v}}\neq 0$. Part (b) of lemma 3.3.2 together
with the main theorem of [Var15] implies that the rank of the monodromy
operator of the Weil–Deligne representation
$WD(r_{\iota}(\Pi^{\prime\prime})\rvert_{G_{F_{v}}})$ is as big as possible
and that local-global compatibility holds for $\Pi^{\prime\prime}$. It follows
from theorem 3.1.14 that
$WD(r_{\iota}(\Pi^{\prime\prime}))=WD(r_{\iota}(\Pi)\rvert_{G_{F^{\prime}}})$
is generic. Thus, $WD(r_{\iota}(\Pi))$ is generic, too. ∎

##  4\. Searching Hypergeometric Parameters

###  4.1. The Computational Check

In this section we describe a few explicit numerical criteria which follow
from the results stated in §2. We have implemented these in a straightforward
python program (available at https://github.com/LAC1213/dworkmotives) to
produce the hypergeometric parameters listed in table 2.

######  4.1.1

Fix the following data.

  * •

Positive integers $d>n$.

  * •

A hypergeometric parameter
$(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$ modulo $d$.

  * •

A vector $\underline{c}\in(\mathbb{Z}/d\mathbb{Z})^{3}$ such that
$c_{0}+c_{1}+c_{2}=0$.

  * •

A subgroup $U<(\mathbb{Z}/d\mathbb{Z})^{\times}$.

Let $M=M(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n})$.

######  4.1.2

It follows from Proposition 2.3.2 that $M$ satisfies (R) if and only if for
every $s\in(\mathbb{Z}/d\mathbb{Z})^{\times}$, the following $n$ integers are
pairwise distinct

$\sum_{i}[s\beta_{j}-s\alpha_{i}]-\sum_{i}[s\beta_{j}-s\beta_{i}]\qquad
j=1,\dots,n.$

Equivalently, they form a set of cardinality $n$. If this condition is
satisfied we set $R(\alpha_{i};\beta_{j})=\operatorname{True}$, otherwise
$R(\alpha_{i};\beta_{j})=\operatorname{False}$. If $n$ is small and $d$ grows,
then computing $R(\alpha_{i};\beta_{j})$ takes $O(d)$ operations.

######  4.1.3

We define $BM(\alpha_{i};\beta_{j})=\operatorname{True}$ if the conditions of
proposition 2.5.8 are satisfied. For small $n$ this requires $O(d)$ operations
to compute.

######  4.1.4

We define $BM_{fin}(\alpha_{i};\beta_{j};U)=\operatorname{True}$ if
additionally the conditions of proposition 2.5.9 are satisfied. Note that this
requires $O(d)$ operations to compute.

######  4.1.5

We define $D(\alpha_{i};\beta_{j};\underline{c})=\operatorname{True}$ if the
conditions of proposition 2.6.8 are satisfied. To check for the existence of
the $x_{\epsilon}\in\mathbb{Z}$, we do linear algebra in the vector space
spanned by the functions in $E(d)$. Namely we have to invert the matrix given
by the functions of a set of basis elements in $E(d)$. This costs $O(d^{3})$
operations.

######  4.1.6

Let $p_{1}\geq p_{2}\geq\dots\geq p_{r}\geq 1$ be a partition of $n$. Here is
a simple algorithm which finds hypergeometric parameters
$(\alpha_{i};\beta_{j})$ modulo $d$ such that the properties (BM), (R), (D)
stated in proposition 2.1.9 are satisfied and the monodromy operator at
$t=\infty$ has Jordan blocks sizes $p_{1},\dots,p_{r}$.

  1. (1)

Enumerate all possible hypergeometric parameters $(\alpha_{i};\beta_{j})$
modulo $d$ of the form

$(\underbrace{0,\dots,0}_{p_{1}},\underbrace{\gamma_{2},\dots,\gamma_{2}}_{p_{2}},\dots,\underbrace{\gamma_{r},\dots,\gamma_{r}}_{p_{r}};\beta_{1},\dots,\beta_{n}).$

There are $O(d^{n+r-1})$ of these.

  2. (2)

For each such $(\alpha_{i};\beta_{j})$ compute

$x=R(\alpha_{i};\beta_{j})\wedge BM(\alpha_{i};\beta_{j}).$

  3. (3)

If $x=\operatorname{True}$, then enumerate all possibilities for
$\underline{c}\in(\mathbb{Z}/d\mathbb{Z})^{3}$. If
$D(\alpha_{i};\beta_{j};\underline{c})$ is satisfied for any of them, then
output $\alpha_{i};\beta_{j};\underline{c}$.

###  4.2. Results of the Program

We have run this algorithm for $d\leq 30$ and partitions of $n\in\\{4,5,6\\}$
which are not of the form $(2,1,\dots,1)$ or $(1,1,\dots,1)$. (For the
excluded partitions one can simply use a family of motives with $d$ odd and
partition $(n)$.) We have listed some of the resulting parameters in table 2.

$n$ | $d$ | $\alpha_{i}$ | $\beta_{j}$ | $\underline{c}$ | $U<(\mathbb{Z}/d\mathbb{Z})^{\times}$
---|---|---|---|---|---
$3$ | $9$ | $0,0,0$ | $1,2,6$ | $3,7,8$ | $\\{\pm 1\\}$
$4$ | $9$ | $0,0,0,0$ | $1,2,7,8$ | $0,0,0$ | $\\{\pm 1\\}$
$5$ | $9$ | $0,0,0,0,0$ | $1,2,3,4,8$ | $5,5,8$ | $\\{\pm 1\\}$
$6$ | $9$ | $0,0,0,0,0,0$ | $1,2,3,6,7,8$ | $0,0,0$ | $\\{\pm 1\\}$
$4$ | $9$ | $0,0,1,1$ | $2,4,6,8$ | $0,0,0$ | $\\{\pm 1\\}$
$5$ | $14$ | $0,0,0,1,1$ | $2,4,7,11,13$ | $0,0,0$ | $\\{\pm 1\\}$
$4$ | $15$ | $0,0,1,1$ | $2,6,10,14$ | $0,0,0$ | $\\{\pm 1\\}$
$6$ | $15$ | $0,0,0,1,1,1$ | $3,5,7,9,11,13$ | $0,0,0$ | $\\{\pm 1\\}$
$4$ | $18$ | $0,0,0,3$ | $4,11,16,17$ | $1,7,10$ | $\\{\pm 1\\}$
$4$ | $20$ | $0,0,0,2$ | $3,4,9,16$ | $2,19,19$ | $\\{\pm 1\\}$
$6$ | $20$ | $0,0,0,0,1,1$ | $2,4,8,10,11,17$ | $4,18,18$ | $\\{\pm 1\\}$
$6$ | $20$ | $0,0,0,0,0,2$ | $3,10,12,13,16,18$ | $1,8,11$ | $(\mathbb{Z}/20\mathbb{Z})^{\times}$
$6$ | $20$ | $0,0,0,0,1,3$ | $4,10,11,13,17,19$ | $1,3,16$ | $\\{\pm 1\\}$
$5$ | $21$ | $0,0,0,0,0$ | $1,2,4,15,20$ | $6,17,19$ | $\\{\pm 1\\}$
$5$ | $21$ | $0,0,0,0,1$ | $4,10,12,18,20$ | $1,1,19$ | $\\{\pm 1\\}$
$5$ | $22$ | $0,0,0,1,1$ | $2,6,11,17,21$ | $0,0,0$ | $\\{\pm 1\\}$
$5$ | $24$ | $0,0,0,1,1$ | $2,6,12,19,23$ | $0,0,0$ | $\\{\pm 1\\}$
$5$ | $24$ | $0,0,1,1,7$ | $8,12,13,17,19$ | $0,0,0$ | $\\{\pm 1\\}$
$5$ | $24$ | $0,0,0,2,6$ | $7,8,12,19,22$ | $0,0,0$ | $\\{\pm 1,\pm 11\\}$
Figure 2. some special hypergeometric parameters

######  4.2.1

Given a partition of $n\in\\{4,5,6\\}$ our program computed those values of
$d\leq 30$ for which there exists a hypergeometric parameter satisfying (BM),
(R), and (D). We display this data in table 3. We omitted partitions of the
form $(n),(2,1,\dots,1)$ or $(1,1,\dots,1)$ since for these one can use any
odd value of $d$ as shown in [Qia22]. For the partitions
$(2,2,2),(3,2,1),(2,2,1,1),(3,1,1,1)$ our program did not find any
hypergeometric parameters satisfying (BM), (R), and (D).

$n$ | $\\{p_{i}\\}$ | $d$
---|---|---
$4$ | $2,2$ | $9,12,15,20,21,24,27$
$4$ | $3,1$ | $18,20,24,28,30$
$5$ | $3,2$ | $12,14,16,18,22,24,26,28$
$5$ | $4,1$ | $18,21,24$
$5$ | $2,2,1$ | $18,24$
$5$ | $3,1,1$ | $24$
$6$ | $3,3$ | $15,20,21,24,30$
$6$ | $4,2$ | $20,24,28$
$6$ | $5,1$ | $20,24,30$
$6$ | $4,1,1$ | $20,24$
Figure 3. values of $d\leq 30$ for which the program found a hypergeometric
parameter satisfying (BM), (R), (D)

######  4.2.2

Unfortunately, our algorithm above is very slow, hence we only ran it for
$d\leq 30$. Moreover, for any fixed partition we have no theoretical
understanding for which $d$ we can expect a suitable hypergeometric parameter
to exist. Thus, further investigations are necessary. It is not clear if one
has to use varieties beyond Dwork hypersurfaces to find all the families of
motives satisfying (BM), (R), (UM).

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