# A Whipple ${}_{7}F_{6}$ formula revisited

Wen-Ching Winnie Li†, Ling Long‡ and Fang-Ting Tu‡ † Department of
Mathematics, Pennsylvania State University, University Park, PA 16802, USA
<EMAIL_ADDRESS>‡ Department of Mathematics, Louisiana State University,
Baton Rouge, LA 70803, USA<EMAIL_ADDRESS>http://www.math.lsu.edu/~llong/
<EMAIL_ADDRESS>https://sites.google.com/view/ft-tu/

###### Abstract.

A well-known formula of Whipple relates certain hypergeometric values
${}_{7}F_{6}(1)$ and ${}_{4}F_{3}(1)$. In this paper we revisit this relation
from the viewpoint of the underlying hypergeometric data $HD$, to which there
are also associated hypergeometric character sums and Galois representations.
We explain a special structure behind Whipple’s formula when the
hypergeometric data $HD$ are primitive and defined over $\mathbb{Q}$. In this
case, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible
families of $\ell$-adic representations of the absolute Galois group of
$\mathbb{Q}$ attached to $HD$. For specialized choices of $HD$, these Galois
representations are shown to be decomposable and automorphic. As a
consequence, the values of the corresponding hypergeometric character sums can
be explicitly expressed in terms of Fourier coefficients of certain modular
forms. We further relate the hypergeometric values ${}_{7}F_{6}(1)$ in
Whipple’s formula to the periods of modular forms occurred.

###### Key words and phrases:

Hypergeometric functions, Whipple’s ${}_{7}F_{6}$ formula, hypergeometric
character sums, Galois representations and modular forms

###### 2020 Mathematics Subject Classification:

11F11, 33C20, 11F80, 11F67, 11T24

##  1\. Introduction

###  1.1. Hypergeometric data

Hypergeometric functions have a long history in Mathematics, and they occur in
many different research branches. In recent years, rapid progress has been
made in hypergeometric character sums under the framework of hypergeometric
motive. It gives a unifying perspective to many hypergeometric structures and
identities, see [9, 10, 18, 17, 24, 27, 28, 31, 39, 43, 49, et al]. Classical
hypergeometric formulas such as Clausen’s formula have finite field analogues
by Evans and Greene [22]. McCarthy [43] obtained several finite field
analogues of well-posed hypergeometric values with argument being $\pm 1$
which was used in [44] by McCarthy and Papanikolas to relate hypergeometric
sums with Siegel eigenforms. Dembélé, Panchishkin, Voight, and Zudilin in [18]
provided numerical evidence relating special values of certain hypergeometric
L-functions and Asai L-functions of Hilbert modular forms. In [24], the
authors interpreted hypergeometric character sums parallel to the development
of classical hypergeometric functions.

Given a hypergeometric datum
$HD=\\{\alpha=\\{a_{1},a_{2},\cdots,a_{n}\\},\beta=\\{1,b_{2},\cdots,b_{n}\\};\lambda\\}$
with $a_{i},b_{j}\in\mathbb{Q}$ satisfying the primitive condition
$a_{i}-b_{j}\notin\mathbb{Z}$ for all $i,j$, and nonzero $\lambda$ in an
abelian extension of $\mathbb{Q}$, one can

  * (1)

Define a classical (generalized) hypergeometric series (HGS)
$F(\alpha,\beta;\lambda)$ by

$F(\alpha,\beta;\lambda):=\sum_{k=0}^{\infty}\frac{(a_{1})_{k}(a_{2})_{k}\cdots(a_{n})_{k}}{(1)_{k}(b_{2})_{k}\cdots(b_{n})_{k}}\lambda^{k},$

if it converges, where $(a)_{k}=\Gamma(a+k)/\Gamma(a)=a(a+1)\cdots(a+k-1)$ is
the Pochhammer symbol. It satisfies a Fuchsian differential equation of order
$n$ with singularities at $0$, $1$, and $\infty$. Solutions at nonsigular
points form a local system and give rise to a monodromy representation of the
fundamental group of $\mathbb{C}P^{1}\smallsetminus\\{0,1,\infty\\}$ with a
base point, see [8, 31, 60] for example. We use
$F(\alpha,\beta;\lambda)_{n}:=\sum_{k=0}^{n}\frac{(a_{1})_{k}(a_{2})_{k}\cdots(a_{n})_{k}}{(1)_{k}(b_{2})_{k}\cdots(b_{n})_{k}}\lambda^{k}$
to denote its truncation at $n$, which plays a vital role in Dwork’s unit root
theory for hypergeometric functions [21]; see [11] by Beukers and Vlasenko for
recent developments.

  * (2)

Compute hypergeometric character sums as hypergeometric functions over finite
fields $\mathbb{F}_{q}$. In this paper we focus on the finite period function
$\mathbb{P}(HD;q)$ in [24] and the finite hypergeometric function $H_{q}(HD)$
in [10].

  * (3)

Associate $\ell$-adic Galois representations, such as $\rho_{HD,\ell}$ in [31,
32] by Katz for the Galois group of a cyclotomic field, and its extension
$\rho_{HD,\ell}^{BCM}$ in [10] by Beukers, Cohen and Mellit to a
representation of the Galois group of $\mathbb{Q}$, when possible. The traces
of these representations are essentially given by $\mathbb{P}(HD;q)$ (see (17)
below) or $H_{p}(HD)$ (see (24) below), respectively. These are the
$\ell$-adic counterpart of (1).

These perspectives are interconnected and are compatible with each other. For
an overview of the overall package, known as hypergeometric motive, see [28]
by Hoffman and Tu. As we will see from the elaboration below, (2) and (3) only
depend on the multi-sets $\alpha$ and $\beta$ mod $\mathbb{Z}$, while
$\mathbb{P}(HD;q)$ and $\rho_{HD,\ell}$ also depend on the order of the
elements in the multi-sets.

There are different versions of hypergeometric character sums. One of the most
convenient choices for our purpose is the period function
${}_{n}\mathbb{P}_{n-1}$ (see (17) below) given in [24, §4]. For any finite
field $\mathbb{F}_{q}$ of characteristic $p>2$, let $\omega$ be a generator of
the group
$\widehat{\mathbb{F}_{q}^{\times}}:=\text{Hom}(\mathbb{F}_{q}^{\times},\mathbb{C}^{\times})$
of multiplicative characters of $\mathbb{F}_{q}$. Let
$\alpha=\\{a_{1},\cdots,a_{n}\\},\beta=\\{1,b_{2},\cdots,b_{n}\\}$ be two
multi-sets with entries in $\mathbb{Q}$. Denote by $M:=lcd(\alpha\cup\beta)$
the positive least common denominator of all $a_{i},b_{j}$. For a finite field
$\mathbb{F}_{q}$ containing a primitive $M$th root of $1$ and any
$\lambda\in\mathbb{F}_{q}$, we write

(1)
$\mathbb{P}(\alpha,\beta;\lambda;\mathbb{F}_{q};\omega):=\,_{n}{\mathbb{P}}_{n-1}\left[\begin{matrix}\omega^{(q-1)a_{1}}&\omega^{(q-1)a_{2}}&\cdots&\omega^{(q-1)a_{n}}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\omega^{(q-1)b_{2}}&\cdots&\omega^{(q-1)b_{n}}\end{matrix}\;;\;\lambda;q\right].$

The value depends on $a_{i}$ and $b_{j}$ modulo $\mathbb{Z}$, as well as their
order. All characters $\omega^{(q-1)a_{i}}$ and $\omega^{(q-1)b_{j}}$ have
order dividing $M$, so the above $\mathbb{P}$ function depends only on
$\omega^{(q-1)/M}$, which has order $M$. When the pair $\\{\alpha,\beta\\}$ is
primitive, that is, $a_{i}-b_{j}\notin\mathbb{Z}$ for all $i,j$, as shown in
[24], the above $\mathbb{P}$ function can be normalized to rid of the
dependence on the orders. For any number field $K$, denote by
$G_{K}=\text{Gal}(\overline{\mathbb{Q}}/K)$ the absolute Galois group of $K$.
When $K=\mathbb{Q}(\zeta_{M})$ is the cyclotomic field joining a primitive
$M$th root of unity $\zeta_{M}$, we further abbreviate
$G_{\mathbb{Q}(\zeta_{M})}$ by $G(M)$. Throughout this paper, a prime ideal
will always be nonzero. For a prime ideal $\wp$ of
$\mathbb{Z}[\zeta_{M},1/M]$, $\zeta_{M}\mod\wp$ in the residue field
$\kappa_{\wp}$ of $\wp$ has order $M$, and it generates the cyclic group
$(\kappa_{\wp}^{\times})^{(N(\wp)-1)/M}$. Put

$\mathbb{P}(\alpha,\beta;\lambda;\kappa_{\wp})=\mathbb{P}(\alpha,\beta;\lambda;\kappa_{\wp};\omega_{\wp})$

where $\omega_{\wp}$ is a generator of $\widehat{\kappa_{\wp}^{\times}}$ so
that

$\omega_{\wp}(\zeta_{M}\mod\wp)=\zeta_{M}^{-1}.$

Note that $\mathbb{P}(\alpha,\beta;\lambda;\kappa_{\wp})$ is independent of
the choice of $\omega_{\wp}$.

The connection between $\mathbb{P}(HD,\cdot)$ and Galois representation
$\rho_{HD,\ell}$ is the following result of Katz.

######  Theorem 1 (Katz [31, 32]).

Let $\ell$ be a prime. Given a primitive pair of multi-sets
$\alpha=\\{a_{1},\cdots,a_{n}\\}$, $\beta=\\{1,b_{2},\cdots,b_{n}\\}$ with
$M=lcd(\alpha\cup\beta)$, for any datum HD = $\\{\alpha,\beta;\lambda\\}$ with
$\lambda\in\mathbb{Z}[\zeta_{M},1/M]\smallsetminus\\{0\\}$ the following hold.

  * i).

There exists an $\ell$-adic Galois representation
$\rho_{HD,\ell}:G(M)\rightarrow GL(W_{\lambda})$ unramified almost everywhere
such that at each prime ideal $\wp$ of
$\mathbb{Z}[\zeta_{M},1/(M\ell\lambda)]$ with norm $N(\wp)=|\kappa_{\wp}|$,

(2)
$\text{Tr}\rho_{HD,\ell}(\text{Frob}_{\wp})=(-1)^{n-1}{\omega_{\wp}^{(N(\wp)-1)a_{1}}(-1)}\mathbb{P}(\alpha,\beta;1/\lambda;\kappa_{\wp}),$

where $\text{Frob}_{\wp}$ stands for the geometric Frobenius conjugacy class
of $G(M)$ at $\wp$.

  * iia).

When $\lambda\neq 0,1$, the dimension
$d:=dim_{\overline{\mathbb{Q}}_{\ell}}W_{\lambda}$ equals $n$ and all roots of
the characteristic polynomial of $\rho_{DH,\ell}(\text{Frob}_{\wp})$ are
algebraic numbers integers and have the same absolute value $N(\wp)^{(n-1)/2}$
under all archimedean embeddings.

  * iib).

When $\lambda\neq 0,1$ and $HD$ is self-dual (see Definition 1 below), then
$W_{\lambda}$ admits a symmetric (resp. alternating) bilinear pairing if $n$
is odd (resp. even).

  * iii).

When $\lambda=1$, dimension $d$ equals $n-1$. In this case if $HD$ is self-
dual, then $\rho_{HD,\ell}$ has a subrepresentation $\rho_{HD,\ell}^{prim}$ of
dimension $2\lfloor\frac{n-1}{2}\rfloor$ whose representation space admits a
symmetric (resp. alternating) bilinear pairing if $n$ is odd (resp. even). All
roots of the characteristic polynomial of
$\rho_{HD,\ell}^{prim}(\text{Frob}_{\wp})$ have absolute value
$N(\wp)^{(n-1)/2}$, the same as (iia).

Here and in what follows, when $\text{ord}_{\wp}\lambda\geq 0$, the $\lambda$
in $\mathbb{P}(\alpha,\beta;\lambda;\kappa_{\wp})$ is viewed as an element in
the residue field $\kappa_{\wp}$. The reader is referred to [41] by Long, Tu,
Yui and Zudilin for a derivation of the above theorem of Katz using [31] and
[32]. The relation (2) sometimes appears as equating certain hypergeometric
character sums with either coefficients of modular forms or Jacobi sums, see
[17, 23, 25, 26, 41, 45, 50, 56, etc.].

Remark 1.  The sign $\omega_{\wp}^{(N(\wp)-1)a_{1}}(-1)$ as $\wp$ varies
defines a character $\phi(M,a_{1})$ of $G(M)$, which is trivial unless
$\text{ord}_{2}~{}M=-\text{ord}_{2}~{}a_{1}=r\geq 1$. In the latter case it is
the quadratic character corresponding to the Hilbert’s quadratic norm residue
symbol $\left(\frac{\zeta_{2^{r}}}{\cdot}\right)_{2}$ on the field
$\mathbb{Q}(\zeta_{M})$.

Example 1.  Let
$\alpha=\\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\\},\beta=\\{1,1,1\\}$, which
form a primitive pair with $M=lcd(\alpha,\beta)=2$. For any
$\lambda\in\mathbb{Q}^{\times}$, $p\nmid lcd(\\{\frac{1}{2},\lambda\\})$, by
definition,

$\displaystyle\mathbb{P}(HD;p)=\mathbb{P}(\alpha,\beta;\lambda;\mathbb{F}_{p})$
$\displaystyle=$
$\,{}_{3}{\mathbb{P}}_{2}\left[\begin{matrix}\omega_{p}^{(p-1)/2}&\omega_{p}^{(p-1)/2}&\omega_{p}^{(p-1)/2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\varepsilon&\varepsilon\end{matrix}\;;\;\lambda;p\right]$ $\displaystyle=$
$\displaystyle\sum_{x,y\in\mathbb{F}_{p}}\phi_{p}(xy(y-1)(1-x)(x-\lambda y)),$

where $\varepsilon$ is the trivial character and
$\phi_{p}=\omega_{p}^{(p-1)/2}$ is the quadratic character of
$\mathbb{F}_{p}^{\times}$ extended to $\mathbb{F}_{p}$ by $\phi_{p}(0)=0$.

The K3 surface $\mathcal{A}_{\lambda}$ over $\mathbb{Q}$ with affine equation

$\mathcal{A}_{\lambda}:\quad s^{2}=xy(x-\lambda y)(y-1)(1-x)$

has good reduction at odd primes $p$ with the number of affine
$\mathbb{F}_{p}$-rational points being equal to

$\sum_{x,y\in\mathbb{F}_{p}}(1+\phi_{p}(xy(x-\lambda
y)(y-1)(1-x)))=p^{2}+\mathbb{P}(HD;p).$

For a generic $\lambda$, the Picard number of the K3 surface
$\mathcal{A}_{\lambda}$ is 19, as shown in [2] by Ahlgren, Ono, and Penniston
or [38] by Long. The associated representation $\rho_{DH,\ell}$ of
$G_{\mathbb{Q}}$ is 3-dimensional which is the symmetric square of a
2-dimensional Galois representation arising from an explicit elliptic curve
$E_{\lambda}$, see [2, (2)]. This essentially follows from the Clausen
formula, see (39) and Theorem 7 for the finite field version by Evans and
Greene. When $\lambda=1$, $\mathcal{A}_{1}$ has Picard number 20 and
$\rho_{HD,\ell}$ is 2-dimensional. As shown in [36, section 5] by the first
two authors and Liu, the action of the Galois group $G_{\mathbb{Q}}$ on the
$2$-dimensional subspace of the cohomological space
$H_{\acute{e}t}^{2}(\mathcal{A}_{1}\otimes_{\mathbb{Q}}\overline{\mathbb{Q}},\mathbb{Q}_{\ell})$
spanned by the transcendental cycles on $\mathcal{A}_{1}$ has the same trace
as $\rho_{DH,\ell}$ twisted by
$\phi(2,1/2)=\left(\frac{-1}{\cdot}\right){=\chi_{-1}}$. In this paper
$\chi_{d}$ for a square-free integer $d$ denotes the quadratic character of
$G_{\mathbb{Q}}$ with kernel $G_{\mathbb{Q}(\sqrt{d})}$; it is also identified
with the quadratic character of $\mathbb{Q}$ associated to the extension
$\mathbb{Q}(\sqrt{d})/\mathbb{Q}$. On the other hand, since
$\rho_{DH,\ell}(\text{Frob}_{p})$ has trace zero at $p\equiv 3\mod 4$ where
$\left(\frac{-1}{p}\right)=-1$, $\rho_{DH,\ell}$ is invariant under twist by
$\phi(2,1/2)$ introduced in Remark 1 and we have a model for $\rho_{DH,\ell}$.

It is explained in [54] by Stienstra and Beukers (see also [5, section 3] and
[36, section 5]) that $\mathcal{A}_{1}$ is birationally isomorphic to an
elliptic surface fibered over a genus $0$ curve $X$ over $\mathbb{Q}$. The
above $G_{\mathbb{Q}}$ action on the $2$-dimensional subspace of
$H_{\acute{e}t}^{2}(\mathcal{A}_{1}\otimes_{\mathbb{Q}}\overline{\mathbb{Q}},\mathbb{Q}_{\ell})$
can also be realized on $H^{1}$ of
$X\otimes_{\mathbb{Q}}\overline{\mathbb{Q}}$ with coefficients in an
$\ell$-adic sheaf on $X$. This is the prototype of the Katz representation
$\rho_{HD,\ell}$ in Theorem 1 for a general hypergeometric datum HD.

When $HD=\\{\alpha,\beta;\lambda\\}$ with $M=lcd(\alpha\cup\beta)$ is defined
over $\mathbb{Q}$ (see Definition 1), the representation $\rho_{HD,\ell}$ of
$G(M)$ can be extended to $G_{\mathbb{Q}}$. The extensions differ by twists by
characters of $G_{\mathbb{Q}}$ trivial on $G(M)$. The next result provides
such an extension expressed in terms of the hypergeometric sums $H_{q}(HD)$
defined in [10, Definition 1.1] by Beukers, Cohen and Mellit using Gauss sums
(see (24) for the case when $HD$ is defined over $\mathbb{Q}$). They depend
only on the elements in $\alpha$ and $\beta$ mod $\mathbb{Z}$ and are
independent of their order. For $\alpha=\\{a_{1},...,a_{n}\\}$,
$\beta=\\{1,b_{2},...,b_{n}\\}$ forming a primitive pair and $\lambda$ a
nonzero element in a finite field $\mathbb{F}_{q}$ containing $M$th roots of
$1$, $H_{q}$ and $\mathbb{P}$ are related by

(3)
$\mathbb{P}(\alpha,\beta;\lambda;\mathbb{F}_{q};\omega)=H_{q}(\alpha,\beta;\lambda;\omega)\cdot\prod_{i=2}^{n}\omega^{(q-1)a_{i}}(-1)J(\omega^{(q-1)a_{i}},\omega^{(q-1)(-b_{i})})\\\
=:H_{q}(\alpha,\beta;\lambda;\omega)\mathcal{J}(\alpha,\beta;\mathbb{F}_{q};\omega),$

where $J(A,B)=\sum_{x\in\mathbb{F}_{q}}A(x)B(1-x)$ is the Jacobi sum of
characters $A$ and $B$.

Given a hypergeometric datum
$HD^{\prime}=\\{\alpha^{\prime},\beta^{\prime};1\\}$, where
$\alpha^{\prime}=\\{c,a\\},\beta^{\prime}=\\{1,b\\}$ is a primitive pair with
$lcd(\alpha^{\prime}\cup\beta^{\prime})=M^{\prime}$, for a finite field
$\mathbb{F}_{q}$ containing $M^{\prime}$th roots of $1$, we have

(4)
$\mathbb{P}(HD^{\prime};\mathbb{F}_{q};\omega)=J(\omega^{(q-1)a},\omega^{(q-1)(b-c-a)})$

(from either the definition of the $\mathbb{P}$-function or (6.11) of [24]).
By Theorem 1, the representation $\rho_{HD^{\prime},\ell}$ is a character of
$G(M^{\prime})$. The relation (3) shows that when the elements in the multi-
sets $\alpha$ and $\beta$ are permuted, the associated representations in
Theorem 1 differ by a character of $G(M)$. On the other hand, the
representation remains the same if the elements in $\alpha$ and $\beta$ are
shuffled by the same permutation on $\\{2,...,n\\}$. For this reason, we shall
regard the pair $\alpha,\beta$ in HD as a multi-set of column vectors
$\left\\{\begin{pmatrix}a_{1}\\\ 1\end{pmatrix},\cdots,\begin{pmatrix}a_{n}\\\
b_{n}\end{pmatrix}\right\\}$ from now on.

Since the Jacobi sum $J(A,B)$ for any
$A,B\in\widehat{\mathbb{F}_{q}^{\times}}$ has absolute value $q^{1/2}$ as long
as $A$, $B$ and $AB$ are nontrivial, and is equal to $-1$ if one of $A,B$ is
trivial, for the primitive pair $\alpha,\beta$ above, we have

(5)
$\mathcal{J}(\alpha,\beta;\mathbb{F}_{q};\omega)=\prod_{i=2}^{n}{\omega^{(q-1)a_{i}}(-1)}J(\omega^{(q-1)a_{i}},\omega^{(q-1)(-b_{i})})=(-1)^{m-1}\chi(\alpha,\beta;\mathbb{F}_{q})q^{(n-m)/2},$

where $m$ is the number of elements in $\beta$ which are in $\mathbb{Z}$ and
$\chi(\alpha,\beta;\mathbb{F}_{q})$ is a root of unity. If, moreover, the pair
$\alpha,\beta$ is self-dual (see §2.2 for definition), then
$\chi(\alpha,\beta;\mathbb{F}_{q})=\prod_{i=2}^{n}\omega^{(q-1)a_{i}}(-1)=\pm
1$ and $n-m$ is even.

######  Theorem 2 (Katz, Beukers-Cohen-Mellit).

Let $\alpha=\\{a_{1},\cdots,a_{n}\\},\beta=\\{1,b_{2},\cdots,b_{n}\\}$ be a
primitive pair with $M=lcd(\alpha\cup\beta)$, and let
$\lambda\in\mathbb{Z}[1/M]\smallsetminus\\{0\\}$. Suppose the hypergeometric
datum $HD=\\{\alpha,\beta;\lambda\\}$ is defined over $\mathbb{Q}$ (see
Definition 1 below). Assume that exactly $m$ elements in $\beta$ are in
$\mathbb{Z}$. Then, for each prime $\ell$, there exists an $\ell$-adic
representation $\rho_{HD,\ell}^{BCM}$ of $G_{\mathbb{Q}}$ with the following
properties:

  * i).

$\rho_{HD,\ell}^{BCM}|_{G(M)}\cong\rho_{HD,\ell}$.

  * ii).

For any prime $p\nmid\ell\cdot M$ such that $\text{ord}_{p}\lambda=0$,

(6)
$\text{Tr}\,\rho_{HD,\ell}^{BCM}(\text{Frob}_{p})=\phi(M,a_{1})(\text{Frob}_{p}){\chi(\alpha,\beta;\mathbb{F}_{p})}H_{p}(\alpha,\beta;1/\lambda)\cdot
p^{(n-m)/2}\in\mathbb{Z}.$

  * iii).

When $\lambda=1$, $\rho_{HD,\ell}^{BCM}$ is $(n-1)$-dimensional and it has a
subrepresentation, denoted by $\rho_{HD,\ell}^{BCM,prim}$, of dimension
$2\lfloor\frac{n-1}{2}\rfloor$ whose representation space admits a symmetric
(resp. alternating) bilinear pairing if $n$ is odd (resp. even). All roots of
the characteristic polynomial of $\rho_{HD,\ell}^{BCM,prim}(\text{Frob}_{p})$
have absolute value $p^{(n-1)/2}$.

The character $\phi(M,a_{1})$ in Remark 1 extends to a character of
$G_{\mathbb{Q}}$. The character $\phi(M,a_{1})$ in Theorem 2 refers to the
extension with minimal conductor. When it is nontrivial, that is, when
$\text{ord}_{2}~{}M=-\text{ord}_{2}~{}a_{1}=r\geq 1$, it has conductor
$2^{r+1}$. In particular, when $\text{ord}_{2}~{}M=-\text{ord}_{2}~{}a_{1}=1$,
$\phi(M,a_{1})(\text{Frob}_{p})=\left(\frac{-1}{p}\right)$ is given by the
Legendre symbol at odd primes $p$.

###  1.2. The Whipple ${}_{7}F_{6}(1)$ formula

The well-known Whipple ${}_{7}F_{6}(1)$ formula asserts that

(7) $\,_{7}F_{6}\left[\begin{matrix}a&1+\frac{a}{2}&c&d&e&f&g\vskip 3.0pt plus
1.0pt minus 1.0pt\\\
&\frac{a}{2}&1+a-c&1+a-d&1+a-e&1+a-f&1+a-g\end{matrix}\;;\;1\right]\\\
=\frac{\Gamma(1+a-e)\Gamma(1+a-f)\Gamma(1+a-g)\Gamma(1+a-e-
f-g)}{\Gamma(1+a)\Gamma(1+a-f-g)\Gamma(1+a-e-f)\Gamma(1+a-e-g)}\cdot\,_{4}F_{3}\left[\begin{matrix}a&e&f&g\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&e+f+g-a&1+a-c&1+a-d\end{matrix}\;;\;1\right],$

when both sides terminate; see Theorem 3.4.4 of the book [3] by Andrews, Askey
and Roy for backgrounds and details. The ${}_{7}F_{6}(1)$ on the left hand
side is well-posed, meaning the upper and lower parameters sum to the same
value (equal to $1+a$ in our case) in each column. By Bailey [6], the left
hand side can be also computed by a contour integral of Barnes type, which is
related to the irrationality of $\zeta(3)$, see [62] by Zudilin for the
related discussions. To hypergeometric systems Katz [31] attached $\ell$-adic
sheaves on the projective line with three points $0,1,\infty$ removed. In
order that the hypergeometric sheaves arising from the data in the Whipple
formula are self-dual, we impose additional assumptions:

$a=\frac{1}{2},\quad c+d=1,\quad f+g=1,\quad\text{and }e=\frac{1-p}{2}$

for an odd natural number $p$ so that both sides converge.111 Here $e$ is set
to be $\frac{1-p}{2}$ instead of the target $\frac{1}{2}$ so that the Gamma
quotient on the right hand side of (7) is a finite value.  Whipple’s formula
now becomes

(8)
$\,_{7}F_{6}\left[\begin{matrix}\frac{1}{2}&\frac{5}{4}&c&1-c&\frac{1-p}{2}&f&1-f\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{1}{4}&\frac{3}{2}-c&\frac{1}{2}+c&1+\frac{p}{2}&\frac{3}{2}-f&\frac{1}{2}+f\end{matrix}\;;\;1\right]=C\times\left(p\cdot\,_{4}F_{3}\left[\begin{matrix}\frac{1}{2}&\frac{1-p}{2}&f&1-f\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1-\frac{p}{2}&\frac{3}{2}-c&\frac{1}{2}+c\end{matrix}\;;\;1\right]\right),$

where
$C=\frac{\Gamma(\frac{p}{2})\Gamma(\frac{3}{2}-f)\Gamma(\frac{1}{2}+f)\Gamma(\frac{p}{2})}{\Gamma(\frac{1}{2})\Gamma(\frac{1}{2})\Gamma(1+\frac{p}{2}-f)\Gamma(\frac{p}{2}+f)}$.
Ignoring the deformation by $p$ and canceling $\frac{5}{4}$ and $\frac{1}{4}$
which correspond to the same character in the finite field setting, the left
hand side of (8) is associated to a hypergeometric datum $HD_{1}$ as below
depending on $c,f$

(9)
$HD_{1}(c,f)=\left\\{\alpha_{6}(c,f):=\left\\{\frac{1}{2},c,1-c,\frac{1}{2},f,1-f\right\\},\beta_{6}(c,f):=\left\\{1,\frac{3}{2}-c,\frac{1}{2}+c,1,\frac{3}{2}-f,\frac{1}{2}+f\right\\};1\right\\}$

and the right hand side ${}_{4}F_{3}(1)$ corresponds to another hypergeometric
datum

(10)
$HD_{2}=HD_{2}(c,f)=\left\\{\alpha_{4}(f):=\left\\{\frac{1}{2},\frac{1}{2},f,1-f\right\\},\beta_{4}(c):=\left\\{1,1,\frac{3}{2}-c,\frac{1}{2}+c\right\\};1\right\\}.$

Both $HD_{1}$ and $HD_{2}$ are self-dual, see §2.2 for definitions.

There are only 7 un-ordered pairs of $(c,f)$ such that $HD_{1}$ is defined
over $\mathbb{Q}$ (see next section for definition) and primitive. They are

(11)
$\left(\frac{1}{2},\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{3}\right),\left(\frac{1}{2},\frac{1}{6}\right),\left(\frac{1}{3},\frac{1}{3}\right),\left(\frac{1}{6},\frac{1}{6}\right),\left(\frac{1}{5},\frac{2}{5}\right),\left(\frac{1}{10},\frac{3}{10}\right).$

Among them, $HD_{2}$ is defined over $\mathbb{Q}$ for the first 5 pairs.

To each of the seven pairs $(c,f)$ as above, by Theorem 1 there are associated
representations $\rho_{HD_{1}(c,f),\ell}$ and $\rho_{HD_{2}(c,f),\ell}$ of the
Galois group $G(M(c,f))$, independent of the order of $(c,f)$, where
$M(c,f)=lcd(\alpha_{6}(c,f)\cup\beta_{6}(c,f))=lcd(\alpha_{4}(f),\beta_{4}(c))$,
of degrees $5$ and $3$, respectively. Moreover, $\rho_{HD_{1}(c,f),\ell}$ can
be extended to $\rho_{HD_{1}(c,f),\ell}^{BCM}$ of $G_{\mathbb{Q}}$ by Theorem
2. The same holds for $\rho_{HD_{2}(c,f),\ell}$ for the first 5 pairs; and as
shown in §3.2, this is also true for the remaining 2 pairs. For these $(c,f)$,
the characters $\chi$ and $\phi$ in Theorem 2 at unramified odd primes $p$ are
given by
$\phi(M(c,f),a_{1})(\text{Frob}_{p})=\left(\frac{-1}{p}\right)=\chi(\alpha_{6}(c,f),\beta_{6}(c,f);\mathbb{F}_{p})=\chi(\alpha_{4}(f),\beta_{4}(c);\mathbb{F}_{p})$
so that
$\text{Tr}\,\rho_{HD_{i}(c,f),\ell}^{BCM}(\text{Frob}_{p})=H_{p}(HD_{i}(c,f))\cdot
p^{(n-m)/2}$ for $i=1,2$.

###  1.3. This paper

In this paper, we revisit the Whipple ${}_{7}F_{6}(1)$ formula for the above
seven pairs from the perspective of the Galois representations
$\rho_{HD_{1},\ell}$ and $\rho_{HD_{2},\ell}$, and also address, for our
cases, three natural questions, explained below, connecting different aspects
of the hypergeometric data, extending the result in [39] by Long.

  * a)

Modularity. A Galois representation is called _modular_ or _automorphic_ if it
corresponds to an automorphic representation of a general linear group as
described in the Langlands program. Both $\rho_{DH,\ell}$ and
$\rho_{DH,\ell}^{BCM}$ are nice Galois representations arising from geometry
and unramified almost everywhere, they are expected to be automorphic. When
the representation involved has degree $1$ or $2$, the modularity can be
verified by showing its attached L-function agrees with the L-function of an
automorphic form. In view of the known modularity lifting theorems, we are
particularly interested in representations of $G_{\mathbb{Q}}$, such as
$\rho_{DH,\ell}^{BCM}$.

  * b)

Truncated HGS and $p$-adic periods. Assume that at $p$, the characteristic
polynomial of $\rho_{HD,\ell}^{BCM}(\text{Frob}_{p})$ has a unique unit root
$\gamma_{p}$. When $\beta=\\{1,1,\cdots,1\\}$, Dwork showed that $\gamma_{p}$
exists and can be computed from truncated HGS

$\gamma_{p}:=\gamma_{p}(HD)\equiv
F(\alpha,\beta;\lambda)_{p^{s}-1}/F(\alpha,\beta;\lambda^{p})_{p^{s-1}-1}\mod
p^{s}.$

If $\rho_{HD,\ell}^{BCM}$ is modular, is there a $2$-dimensional component
corresponding to a Hecke-eigenform $g$ such that at such $p$, the Hecke-
eigenvalue $a_{p}(g)$ of $g$ satisfies the congruence relation
$a_{p}(g)\equiv\gamma_{p}\mod p$?

  * c)

HGS and Complex periods. If the answers to both questions a) and b) are
positive, is the untruncated hypergeometric value $F(\alpha,\beta;\lambda)$
related to a period of the normalized Hecke-eigenform $g$ which affords the
unit roots in b)?

The example below illustrates the three questions raised above.

Example 2.  Let $HD$ denote the hypergeometric datum
$\left\\{\\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\\},\\{1,1,1\\};1\right\\}$ as
in Example 1. Ahlgren in [1] showed that for any prime $p>2$,

$\mathbb{P}(HD;p)=\mathbb{P}(\\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\\},\\{1,1,1\\};1;\mathbb{F}_{p})=a_{p}(\eta(4\tau)^{6}),$

the $p$th coefficient of $\eta(4\tau)^{6}$, which admits complex
multiplication (CM) by $\chi_{-1}$. Here $\eta(\tau)$ is the Dedekind eta
function. In other words, $\rho_{DH,\ell}$ is isomorphic to the Galois
representation attached to $\eta(4\tau)^{6}$, hence it is modular. This is a).
For b), Dwork’s theorem holds for the ordinary prime $p\equiv 1\mod 4$. In
particular, the unit root can be written as
$\gamma_{p}=-\Gamma_{p}(\frac{1}{4})^{4}$ where $\Gamma_{p}(\cdot)$ denotes
the $p$-adic Gamma function. In fact, Ahlgren showed that a stronger
congruence, called supercongruence, holds. Namely for each prime $p>2$,

(12)
$F\left(\left\\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\\},\\{1,1,1\\};1\right)_{p-1}\equiv
a_{p}(\eta(4\tau)^{6})\mod p^{2}.$

In [40] Long and Ramakrishna further showed that when $p>2$,

(13)
$F\left(\left\\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\\},\\{1,1,1\\};1\right)_{p-1}\equiv\begin{cases}-\Gamma_{p}(\frac{1}{4})^{4}\quad\text{
if }p\equiv 1\mod 4\\\ -\frac{p^{2}}{16}\Gamma_{p}(\frac{1}{4})^{4}\quad\text{
if }p\equiv 3\mod 4\end{cases}\mod p^{3}.$

An archimedean version for the untruncated hypergeometric value is proved in
our paper [37]:

(14)
$\,{}_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1&1\end{matrix}\;;\;1\right]=\frac{16}{\pi^{2}}L(\eta(4\tau)^{6},2)=\frac{\Gamma(\frac{1}{4})^{4}}{4\pi^{3}},$

which is c). Similar recent discussions relating $L$-values of hypergeometric
motives to $L$-functions of modular forms or Hilbert modular forms include
[46] by Osbrun, Straub, and [18] by Dembélé, Panchishkin, Voight, and Zudilin.

###  1.4. Main results

Our first main result is a representation-theoretic interpretation of the
Whipple ${}_{7}F_{6}(1)$ formula for the seven pairs of $(c,f)$ in (11).

######  Theorem 3.

For each pair $(c,f)$ in (11), let
$N(c,f)=lcd(\frac{1+2f-2c}{4},\frac{3-2f-2c}{4})$ and
$M(c,f)=lcd(\alpha_{6}(c,f)\cup\beta_{6}(c,f)\color[rgb]{0,0,0})$. Then
$N(c,f)=2M(c,f)$ except for $(c,f)=(\frac{1}{2},\frac{1}{3})$, in which case
$N(c,f)=M(c,f)$. Given any prime $\ell$, the $5$-dimensional representation
$\rho_{HD_{1}(c,f),\ell}$ of $G(M(c,f))$, when restricted to $G(N(c,f))$,
decomposes into the sum of a $2$\- and a $3$-dimensional $G(N(c,f))$-modules
such that the $3$-dimensional component is isomorphic to
$\epsilon_{\ell}\otimes\rho_{HD_{2}(c,f),\ell}$ restricted to $G(N(c,f))$.

Here and in what follows $\epsilon_{\ell}$ denotes the $\ell$-adic cyclotomic
character of $G_{\mathbb{Q}}$.

It follows from Theorem 2 that, for the 7 pairs (resp. the first 5 pairs) of
$(c,f)$ in (11), $\rho_{HD_{1}(c,f),\ell}^{BCM}$ (resp.
$\rho_{HD_{2}(c,f),\ell}^{BCM}$) has a $4$-dimensional (resp. $2$-dimensional)
sub-representation $\rho_{HD_{1}(c,f),\ell}^{BCM,prim}$ (resp.
$\rho_{HD_{2}(c,f),\ell}^{BCM,prim}$) and a $1$-dimensional complement
$\rho_{HD_{1}(c,f),\ell}^{BCM,1}$ (resp. $\rho_{HD_{2}(c,f),\ell}^{BCM,1}$).

Next we address the modularity question a). First consider
$\rho_{HD_{2}(c,f),\ell}^{BCM}$.

######  Theorem 4.

For each of the first 5 pairs of $(c,f)$ such that $HD_{2}(c,f)$ is defined
over $\mathbb{Q}$, both components of $\rho_{HD_{2}(c,f),\ell}^{BCM}$ are
modular, which are explicitly identified by the information at primes $p>7$ in
the table below. For each of the remaining two pairs, $M(c,f)=10$ and the
representation $\rho_{HD_{2}(c,f),\ell}$ of $G(10)$ can be extended to
$G_{\mathbb{Q}}$, with both components modular. For each pair we list
extensions with integral traces under “$\rho_{HD_{2}(c,f),\ell}^{BCM,prim}$”
and “$\rho_{HD_{2}(c,f),\ell}^{BCM,1}$”, resp.

$\begin{array}[]{|c|c|c|c|c|c|c|c|}\hline\cr(c,f)&\alpha&\beta&w(HD_{2})&\text{Tr}\,\rho_{HD_{2}(c,f),\ell}^{BCM,prim}(\text{Frob}_{p})&\rho_{HD_{2}(c,f),\ell}^{BCM,1}\\\
\hline\cr(\frac{1}{2},\frac{1}{2})&(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})&(1,1,1,1)&4&a_{p}(f_{8.4.a.a})&\epsilon_{\ell}\\\
\hline\cr(\frac{1}{2},\frac{1}{3})&(\frac{1}{2},\frac{1}{2},\frac{1}{3},\frac{2}{3})&(1,1,1,1)&4&a_{p}(f_{36.4.a.a})=\left(\frac{-3}{p}\right)a_{p}(f_{12.4.a.a})&\chi_{3}\cdot\epsilon_{\ell}\\\
\hline\cr(\frac{1}{3},\frac{1}{3})&(\frac{1}{2},\frac{1}{2},\frac{1}{3},\frac{2}{3})&(1,1,\frac{7}{6},\frac{5}{6})&4&a_{p}(f_{6.4.a.a})=\left(\frac{-3}{p}\right)a_{p}(f_{18.4.a.a})&\chi_{-3}\cdot\epsilon_{\ell}\\\
\hline\cr(\frac{1}{2},\frac{1}{6})&(\frac{1}{2},\frac{1}{2},\frac{1}{6},\frac{5}{6})&(1,1,1,1)&4&a_{p}(f_{72.4.a.b})=\left(\frac{-3}{p}\right)a_{p}(f_{24.4.a.a})&\epsilon_{\ell}\\\
\hline\cr(\frac{1}{6},\frac{1}{6})&(\frac{1}{2},\frac{1}{2},\frac{1}{6},\frac{5}{6})&(1,1,\frac{4}{3},\frac{2}{3})&2&a_{p}(f_{24.2.a.a})\cdot
p={\left(\frac{-3}{p}\right)}a_{p}(f_{72.2.a.a})\cdot
p&\chi_{-3}\cdot\epsilon_{\ell}\\\ \hline\cr(\frac{1}{5},\frac{2}{5})&\
(\frac{1}{2},\frac{1}{2},\frac{2}{5},\frac{3}{5})&(1,1,\frac{13}{10},\frac{7}{10})&4&a_{p}(f_{50.4.a.b})=\left(\frac{5}{p}\right)a_{p}(f_{50.4.a.d})&\epsilon_{\ell}\text{
or }\chi_{5}\cdot\epsilon_{\ell}\\\
\hline\cr(\frac{1}{10},\frac{3}{10})&(\frac{1}{2},\frac{1}{2},\frac{3}{10},\frac{7}{10})&(1,1,\frac{7}{5},\frac{3}{5})&2&a_{p}(f_{200.2.a.b})\cdot
p=\left(\frac{5}{p}\right)a_{p}(f_{200.2.a.d})\cdot p&\epsilon_{\ell}\text{ or
}\chi_{5}\cdot\epsilon_{\ell}\\\ \hline\cr\end{array}$

The modular forms above are denoted by their LMFDB labels; for instance
$f_{8.4.a.a}$ means a normalized cuspidal newform of weight 4 and level 8. See
(25) for the definition of $w(HD_{2})$.

Our next theorem records the modularity of $\rho_{HD_{1}(c,f),\ell}^{BCM}$.

######  Theorem 5.

For each pair $(c,f)$ in (11), $\rho_{HD_{1}(c,f),\ell}^{BCM,prim}$ decomposes
as a sum of two $2$-dimensional $G_{\mathbb{Q}}$-modules, both are modular.
There are two normalized cuspidal newforms $\mathfrak{f}_{c,f}$ and
$\mathfrak{g}_{c,f}$ (expressed in their LMFDB labels) such that for each
prime $p\geq 7$, $\text{Tr}\,\rho_{HD_{1}(c,f),\ell}^{BCM}(\text{Frob}_{p})$
is explicitly described by the Fourier coefficients
$a_{p}(\mathfrak{f}_{c,f})$ and $a_{p}(\mathfrak{g}_{c,f})$ in the following
table, where $\mathfrak{f}_{c,f}$ (resp. $\mathfrak{g}_{c,f}$ ) is the first
(resp. second) modular form listed in the last column.

$\begin{array}[]{|c|c|c|c|c|c|c|c|}\hline\cr(c,f)&\mathcal{J}(HD_{1}(c,f);\mathbb{F}_{p})&w(HD_{1})&w(HD_{2})&\text{Tr}\,\rho_{HD_{1}(c,f),\ell}^{BCM}(\text{Frob}_{p})\\\
\hline\cr(\frac{1}{2},\frac{1}{2})&-\left(\frac{-1}{p}\right)&6&4&a_{p}(f_{8.6.a.a})+p\cdot
a_{p}(f_{8.4.a.a})+\left(\frac{-1}{p}\right)p^{2}\\\
\hline\cr\left(\frac{1}{2},\frac{1}{3}\right)&-\left(\frac{-1}{p}\right)p&6&4&a_{p}(f_{4.6.a.a})+p\cdot
a_{p}(f_{12.4.a.a})+\left(\frac{3}{p}\right)p^{2}\\\
\hline\cr(\frac{1}{3},\frac{1}{3})&-\left(\frac{-1}{p}\right)p^{2}&6&4&a_{p}(f_{6.6.a.a})+p\cdot
a_{p}(f_{18.4.a.a})+\left(\frac{-1}{p}\right)p^{2}\\\
\hline\cr(\frac{1}{2},\frac{1}{6})&-\left(\frac{-1}{p}\right)p&4&4&p\cdot
a_{p}(f_{8.4.a.a})+p\cdot a_{p}(f_{24.4.a.a})+\left(\frac{3}{p}\right)p^{2}\\\
\hline\cr(\frac{1}{6},\frac{1}{6})&-\left(\frac{-1}{p}\right)p^{2}&2&2&p^{2}\cdot
a_{p}(f_{24.2.a.a})+p^{2}\cdot
a_{p}(f_{72.2.a.a})+\left(\frac{-1}{p}\right)p^{2}\\\
\hline\cr(\frac{1}{5},\frac{2}{5})&-\left(\frac{-1}{p}\right)p^{2}&4&4&p\cdot
a_{p}(f_{10.4.a.a})+p\cdot
a_{p}(f_{50.4.a.d})+\left(\frac{-5}{p}\right)p^{2}\\\
\hline\cr(\frac{1}{10},\frac{3}{10})&-\left(\frac{-1}{p}\right)p^{2}&2&2&p^{2}\cdot
a_{p}(f_{40.2.a.a})+p^{2}\cdot
a_{p}(f_{200.2.a.b})+\left(\frac{-5}{p}\right)p^{2}\\\ \hline\cr\end{array}$

In the above Table, $\mathcal{J}(HD_{1}(c,f);\mathbb{F}_{p})$ is defined by
(5), and $w(HD_{1})$ and $w(HD_{2})$ are defined in (25).

When $(c,f)=(\frac{1}{2},\frac{1}{2})$, the conclusion above is equivalent to
a conjecture of Koike [34], proved in [23] by Frechette, Ono and Papanikolas.
Comparing Theorems 4 and 5, we see that the extension to $G_{\mathbb{Q}}$ of
$\epsilon_{\ell}|_{G(M(c,f))}\otimes\rho_{HD_{2}(c,f),\ell}^{prim}$ in
$\rho_{HD_{1}(c,f),\ell}^{BCM}$ differs from
$\epsilon_{\ell}\otimes\rho_{HD_{2}(c,f),\ell}^{BCM,prim}$ by a character of
$G_{\mathbb{Q}}/G(N(c,f))$. The same phenomenon occurs for the $1$-dimensional
component.

For the question b), we see from the above table that only when
$(c,f)=(\frac{1}{2},\frac{1}{2}),(\frac{1}{2},\frac{1}{3}),(\frac{1}{3},\frac{1}{3})$,
$\rho_{HD_{1}(c,f),\ell}^{BCM}$ could possibly have a unique unit root
$\gamma_{p}$ at an unramified prime $p$. In each of these cases, there is a
congruence relation modulo $p$ between a truncated hypergeometric series and
the coefficients of $\mathfrak{f}_{c,f}$, see Theorem 4 of [39]. Numerical
computations suggest the following supercongruences for primes $p>5$:

$\displaystyle(c,f)=(\frac{1}{2},\frac{1}{2}):$
$\,{}_{6}F_{5}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1&1&1&1&1\end{matrix}\;;\;1\right]_{p-1}$
$\displaystyle\overset{?}{\equiv}a_{p}(f_{8.6.a.a})\mod p^{5}.$
$\displaystyle(c,f)=(\frac{1}{2},\frac{1}{3}):$ $\displaystyle
p\cdot\,_{6}F_{5}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{3}&\frac{2}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{5}{6}&\frac{7}{6}&1&1&1\end{matrix}\;;\;1\right]_{p-1}$
$\displaystyle\overset{?}{\equiv}a_{p}(f_{4.6.a.a})\mod p^{5}.$
$\displaystyle(c,f)=(\frac{1}{3},\frac{1}{3}):$ $\displaystyle
p^{2}\cdot\,_{6}F_{5}\left[\begin{matrix}\frac{1}{2}&\frac{1}{3}&\frac{2}{3}&\frac{1}{2}&\frac{1}{3}&\frac{2}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{5}{6}&\frac{7}{6}&1&\frac{5}{6}&\frac{7}{6}\end{matrix}\;;\;1\right]_{p-1}$
$\displaystyle\overset{?}{\equiv}a_{p}(f_{6.6.a.a})\mod p^{4}.$

In the above formulas, each ${}_{6}F_{5}$ is multiplied by a suitable $p$
power to ensure the resulting product is $p$-adically integral. Of these, the
first congruence is shown to hold mod $p^{3}$ by Osburn, Straub and Zudilin in
[47]. It would be interesting to know the truth. The reader is referred to
[33, 39, 41, 42, 48] for related discussions on hypergeometric
supercongruences.

Finally we provide an archimedean version of these congruences for
$(c,f)=(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},\frac{1}{3})$. Namely we
give precise descriptions of the untruncated hypergeometric values as periods
of $\mathfrak{f}_{c,f}$, which addresses the question c). Interestingly, for
$(c,f)=(\frac{1}{2},\frac{1}{6})$, we find an expression of $F(HD_{1}(c,f))$
in terms of a period of $f_{8.4.a.a}$, see (44).

Similar to the identification of $\mathfrak{f}_{c,f}$ at the archimedean
place, we will also explain in the $(\frac{1}{2},\frac{1}{2})$ case the
natural appearance of $\mathfrak{g}_{c,f}$ from the perspective of modular
forms. Meanwhile, numerical data suggest for any prime $p>3$

(15)
$\,_{7}F_{6}\left[\begin{matrix}\frac{1}{2}&\frac{5}{4}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{1}{4}&1&1&1&1&1\end{matrix}\;;\;1\right]_{p-1}\overset{?}{\equiv}p\cdot
a_{p}(f_{8.4.a.a})\mod p^{4}.$

The next result expresses the untruncated hypergeometric ${}_{6}F_{5}(1)$ and
${}_{7}F_{6}(1)$ values in terms of periods of modular forms
$\mathfrak{f}_{\frac{1}{2},\frac{1}{2}}$ and
$\mathfrak{g}_{\frac{1}{2},\frac{1}{2}}$ respectively. For a general frame
work for periods in terms of integrals, see [35] by Kontsevich and Zagier, in
which the authors pointed out that “Periods appear surprisingly often in
various formulas and conjectures in mathematics, and often provide a bridge
between problems coming from different disciplines.”

######  Theorem 6.

The following two equalities hold:

$\,{}_{6}F_{5}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1&1&1&1&1\end{matrix}\;;\;1\right]=16\oint_{|t_{2}(\tau)|=1}\tau^{2}f_{8.6.a.a}\left(\frac{\tau}{2}\right)d\tau=16\int_{1/2+i/2}^{-1/2+i/2}\tau^{2}f_{8.6.a.a}\left(\frac{\tau}{2}\right)d\tau$

and

(16)
$\,_{7}F_{6}\left[\begin{matrix}\frac{1}{2}&\frac{5}{4}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{1}{4}&1&1&1&1&1\end{matrix}\;;\;1\right]=\frac{32i}{\pi}\oint_{|t_{2}(\tau)|=1}\tau
f_{8.4.a.a}(\frac{\tau}{2})d\tau=\frac{32i}{\pi}\int_{1/2+i/2}^{-1/2+i/2}\tau
f_{8.4.a.a}(\frac{\tau}{2})d\tau,$

where $t_{2}(\tau)=-64\frac{\eta(2\tau)^{24}}{\eta(\tau)^{24}}$, and the path
in $\tau$ such that $|t_{2}(\tau)|=1$ is the hyperbolic geodesic from
$\frac{1+i}{2}$ to $\frac{-1+i}{2}$ going counter clockwise.

We believe the appearance of $\mathfrak{g}_{c,f}$ in the
$(\frac{1}{2},\frac{1}{3})$ case can be explained in a similar way. This is
left to any interested reader.

###  1.5. Geometric perspective

Given $k\in\mathbb{C}\smallsetminus\\{0,\pm 1\\}$, the 1st de Rham cohomology
$H^{1}_{DR}(L(k)/\mathbb{C})$ of the elliptic curve $L(k):\quad
y^{2}=(1-x^{2})(1-k^{2}x^{2})$ is isomorphic to the differentials on $L(k)$
with at most a double pole at infinity (see [30, Appendix 1] by Katz). It is a
2-dimensional vector space over $\mathbb{C}$ generated by
$\mathfrak{w}_{1}(k)=\frac{dx}{y}$ and
$\mathfrak{w}_{2}(k)=(1-k^{2}x^{2})\frac{dx}{y}$. Here $\mathfrak{w}_{1}(k)$
is holomorphic, which is unique up to scalar. Typically there is no a priori
‘functorial’ choice for the second generator of $H_{DR}^{1}(L(k)/\mathbb{C})$
unless $L(k)$ admits complex multiplication (CM), see [30]. The two periods

$K(k)=\int_{0}^{1}\mathfrak{w}_{1}(k)=\frac{\pi}{2}\,_{2}F_{1}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1\end{matrix}\;;\;k^{2}\right]$

and

$E(k)=\int_{0}^{1}\mathfrak{w}_{2}(k)=\frac{\pi}{2}\,_{2}F_{1}\left[\begin{matrix}-\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1\end{matrix}\;;\;k^{2}\right]$

are complete elliptic integrals of first and second kind respectively, see
[12, Chapter 1] by Borwein-Borwein. As $k$ varies, these two period functions
are related by

$E(k)=k(1-k^{2})\frac{d}{dk}K(k)+(1-k^{2})K(k).$

In our situation, the hypergeometric functions

$F(\lambda):=F(\alpha_{6}(c,f),\beta_{6}(c,f);\lambda)$

and

${}_{7}F_{6}(\lambda):=\,_{7}F_{6}\left[\begin{matrix}\frac{1}{2}&\frac{5}{4}&c&1-c&\frac{1}{2}&f&1-f\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{1}{4}&\frac{3}{2}-c&\frac{1}{2}+c&1&\frac{3}{2}-f&\frac{1}{2}+f\end{matrix}\;;\;\lambda\right]=4\frac{d}{d\lambda}F(\lambda)+F(\lambda)$

play roles parallel to $K(k)$ and $E(k)$, respectively. Geometrically, for any
given primitive pair $\alpha,\beta$ of length $n$ and
$\lambda\in\overline{\mathbb{Q}}^{\times}$, one can write down (using the
inductive formula via Euler integral (see [53, §2.1]) for ${}_{n}F_{n-1}$) an
affine equation, which defines a variety
$V(\lambda):=V_{\alpha,\beta}(\lambda)$. The surface $\mathcal{A}_{\lambda}$
in Example 1 is such an example. The variation of the de Rham cohomology
$H_{DR}^{n-1}(V(\lambda)/\mathbb{C})$ can be described by the hypergeometric
differential equation satisfied by $F(\alpha,\beta;\lambda)$. By the Hodge
filtration, we have
$H_{DR}^{n-1}(V(\lambda))\cong\bigoplus_{i=0}^{n-1}H^{i}(V(\lambda),\Omega_{V(\lambda)}^{n-1-i})$.
For example, $H^{0}(V(\lambda),\Omega_{V(\lambda)}^{n-1})$ consists of
holomorphic differential $n-1$st forms on $V(\lambda)$. For
$HD_{1}(c,f)=\\{\alpha_{6}(c,f),\beta_{6}(c,f);1\\}$ with $(c,f)$ in (11),
$\dim H_{DR}^{0}(V_{\alpha_{6}(c,f),\beta_{6}(c,f)}(1),\Omega^{5})=1$ only
occurs for the first 3 cases of Theorem 5.

From this perspective, Whipple’s formula (8) suggests that when $\lambda=1$,
up to a Tate twist, there is a correspondence between
$H_{DR}^{5}(V_{\alpha_{6}(c,f),\beta_{6}(c,f)}(1))$ and
$H_{DR}^{3}(V_{\alpha_{4}(c,f),\beta_{4}(c,f)}(1))$. Firstly, such a
correspondence only exists for special fibers like $\lambda=1$, similar to
$H_{DR}^{1}(L(k))$ admitting a canonical splitting only when $L(k)$ has CM.
Secondly, such a correspondence, when exists, can be recognized simultaneously
from different aspects in compatible ways, including the statement of
Whipple’s formula (8), the statements of Theorems 3 and 5, the conjectural
supercongruence (15), as well as (16). In particular, the factor
$\frac{1}{\pi}$ on the right hand side of (16) and the $p$ on the right hand
side of (15) both reflect a Tate twist by 1. 222Legendre relation for $L(k)$
is
$E(k)K(\sqrt{1-k^{2}})+K(k)E(\sqrt{1-k^{2}})-K(k)K(\sqrt{1-k^{2}})=\frac{\pi}{2}$,
which is a coupling between $H^{0}(L(k),\Omega_{V(1)})$ and
$H^{1}(L(k),\mathcal{O}_{V(1)})$

Finally we comment on the role of Theorem 6. From transcendence theory, given
a normalized weight-$k$ cusp form $\mathfrak{f}(\tau)$, the values
$\oint_{\gamma}\mathfrak{f}(\tau)p(\tau)d\tau$ for all polynomials
$p(\tau)\in\overline{\mathbb{Q}}[\tau]$ of degree at most $k-1$ and all closed
paths $\gamma$ on the underlying modular curve generate a finite dimensional
vector space over $\overline{\mathbb{Q}}(\pi)$. Thus the equalities in Theorem
6 are not coincidental by nature. On the other hand, from the perspective of
de Rham cohomology, one expects that untruncated hypergeometric values arise
from line integrals of some differentials in $H_{DR}^{n-1}(V(\lambda))$. The
challenge becomes how to realize hypergeometric values as periods.

This paper is organized in the following way. In §2, we recall basic notation,
relevant results for hypergeometric data, character sums, and modular forms.
Section 3 is devoted to the proofs of Theorems 3, 4 and 5. In §4, Theorem 6
and other computations for untruncated hypergeometric functions are obtained
using properties of modular forms and hypergeometric functions. It is worth
pointing out that in the proofs of Theorems 3 and 6, the classical Clausen
formula (19) plays a prominent role. The finite field version of the well-
posedness of the ${}_{6}F_{5}(HD_{1})$ under consideration gives rise to Lemma
1, using which we decompose the representation $\rho_{HD_{1},\ell}$.

##  2\. Preliminaries

###  2.1. Hypergeometric functions over finite fields

We recall the period functions for finite fields with odd characteristic. All
multiplicative characters $A$ are extended to $0$ by letting $A(0)=0$. In what
follows, denote by $\varepsilon$ the trivial character, by $\phi_{p}$ or
simply $\phi$ the quadratic character, and by $\overline{A}$ the inverse of
the character $A$. For any characters $A_{i}$, $B_{i}$, $i=1,\cdots,n+1$ with
$B_{1}=\varepsilon$ in $\widehat{\mathbb{F}_{q}^{\times}}$, the
${}_{n+1}\mathbb{P}_{n}$-function is defined as follows: set

$\,{}_{1}{\mathbb{P}}_{0}\left[\begin{matrix}A_{1}\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\
\,\end{matrix}\;;\;\lambda;q\right]:=\overline{A}_{1}(1-\lambda)$

and construct ${}_{n+1}\mathbb{P}_{n}$ inductively by

(17)
$\,_{n+1}{\mathbb{P}}_{n}\left[\begin{matrix}A_{1}&A_{2}&\cdots&A_{n+1}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&B_{2}&\cdots&B_{n+1}\end{matrix}\;;\;\lambda;q\right]:=\sum_{x\in\mathbb{F}_{q}}A_{n+1}(x)\overline{A}_{n+1}B_{n+1}(1-x)\cdot\,_{n}{\mathbb{P}}_{n-1}\left[\begin{matrix}A_{1}&A_{2}&\cdots&A_{n}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &B_{2}&\cdots&B_{n}\end{matrix}\;;\;\lambda
x;q\right].$

For simplicity, we may leave out the $q$ in the notation when there is no
danger of ambiguity in the underlying finite field $\mathbb{F}_{q}$.

This $\mathbb{P}$-function is defined as an analog of the integral expression
of the classical hypergeometric series. Note that for any character $A$, we
have

$\overline{A}(1-\lambda)=\delta(\lambda)+\frac{1}{q-1}\sum_{\chi\in\widehat{\mathbb{F}_{q}^{\times}}}J(A\chi,\overline{\chi})\chi(-\lambda)=\delta(\lambda)+\frac{-1}{q-1}\sum_{\chi\in\widehat{\mathbb{F}_{q}^{\times}}}\binom{A\chi}{\chi}\chi(\lambda),$

where

$\delta(t):=\begin{cases}1,&\mbox{ if }t=0,\\\ 0,&\mbox{ otherwise,
}\end{cases}\quad\quad
J(A,B):=\sum_{t\in\mathbb{F}_{q}}A(t)B(1-t),\quad\text{and
}\displaystyle\binom{A}{B}:=-B(-1)J(A,\overline{B}).$

Thus, we can rewrite the $\mathbb{P}$-function in terms of finite field
version of binomial coefficients $\binom{A}{B}$ defined above:

(18)
$\,_{n+1}{\mathbb{P}}_{n}\left[\begin{matrix}A_{1}&A_{2}&\cdots&A_{n+1}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&B_{2}&\cdots&B_{n+1}\end{matrix}\;;\;\lambda;q\right]\\\
=\frac{(-1)^{n+1}}{q-1}\cdot\left(\prod_{i=2}^{n+1}A_{i}B_{i}(-1)\right)\cdot\sum_{\chi\in\widehat{\mathbb{F}_{q}^{\times}}}\binom{A_{1}\chi}{\chi}\binom{A_{2}\chi}{B_{2}\chi}\cdots\binom{A_{n+1}\chi}{B_{n+1}\chi}\chi(\lambda)+\delta(\lambda)\prod_{i=2}^{n+1}J(A_{i},\overline{A_{i}}B_{i}).$

Here are some properties of these character sums that are useful for our later
discussion.

######  Proposition 1 (Theorem 4.2, [27]).

For any fixed $\mathbb{F}_{q}$ of characteristic $p>2$, $A_{i}$,
$B_{j}\in\widehat{\mathbb{F}_{q}^{\times}}$, and $t\neq 0$,

$\,{}_{n}{\mathbb{P}}_{n-1}\left[\begin{matrix}A_{1}&A_{2}&\cdots&A_{n}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&B_{2}&\cdots&B_{n}\end{matrix}\;;\;\frac{1}{t}\right]=A_{1}(-t)\left(\prod_{i=2}^{n}A_{i}B_{i}(-1)\right)\cdot\,_{n}{\mathbb{P}}_{n-1}\left[\begin{matrix}A_{1}&A_{1}\overline{B_{2}}&\cdots&A_{1}\overline{B_{n}}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&A_{1}\overline{A_{2}}&\cdots&A_{1}\overline{A_{n}}\end{matrix}\;;\;t\right]\\\
=A_{2}(t)\left(\prod_{i=3}^{n}A_{i}B_{i}(-1)\right)\cdot\,_{n}{\mathbb{P}}_{n-1}\left[\begin{matrix}A_{2}\overline{B}_{2}&A_{2}&A_{2}\overline{B_{3}}&\cdots&A_{2}\overline{B_{n}}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&A_{2}\overline{A_{1}}&A_{2}\overline{A_{3}}&\cdots&A_{2}\overline{A_{n}}\end{matrix}\;;\;t\right].$

It corresponds to the following classical result.

######  Proposition 2.

Given $\alpha=\\{a_{1},a_{2},\cdots,a_{n}\\}$,
$\beta=\\{1,b_{2},b_{3},\cdots,b_{n}\\}$, if we denote
$1+a_{j}-\beta:=\\{a_{j},1+a_{j}-b_{2},\cdots,1+a_{j}-b_{n}\\}$ and
$1+a_{j}-\alpha:=\\{1+a_{j}-a_{1},\cdots,1+a_{j}-a_{n}\\}$ then the functions
$F(\alpha,\beta;z)$ and $(-z)^{-a_{j}}F(1+a_{j}-\beta,1+a_{j}-\alpha;1/z)$ for
any $j=1,\ldots,n$, satisfy the same differential equation

$\left[\theta\left(\theta+b_{2}-1\right)\cdots\left(\theta+b_{n}-1\right)-z\left(\theta+a_{1}\right)\cdots\left(\theta+a_{n}\right)\right]F=0,\quad\text{where
}\theta=z\frac{d}{dz}.$

Next we recall a version of the classical Clausen formula ([3, pp 184]):

(19) $\,_{2}F_{1}\left[\begin{matrix}a&b\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&a+b+\frac{1}{2}\end{matrix}\;;\;x\right]\,_{2}F_{1}\left[\begin{matrix}\frac{1}{2}-a&\frac{1}{2}-b\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{3}{2}-a-b\end{matrix}\;;\;x\right]=\,_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&a-b+\frac{1}{2}&b-a+\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&a+b+\frac{1}{2}&\frac{3}{2}-a-b\end{matrix}\;;\;x\right].$

Below is the Clausen formula over finite fields obtained by Evans and Greene.

######  Theorem 7 (Evans-Greene [22]).

Assume $\eta,K\in\widehat{\mathbb{F}_{q}^{\times}}$ such that none of
$\eta,K\phi,\eta K,\eta\overline{K}$ is trivial. Suppose $\eta K=S^{2}$ is a
square. When $t\neq 0,1$, we have

$\,{}_{3}{\mathbb{P}}_{2}\left[\begin{matrix}\phi&\eta&\overline{\eta}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &K&\overline{K}\end{matrix}\;;\;t\right]$
$\displaystyle=\phi(1-t)\left(\phi(t-1)K(t)\frac{J(\phi S,\phi
K\overline{S})}{J(S,K\overline{S})}\,_{2}{\mathbb{P}}_{1}\left[\begin{matrix}\phi
K\overline{S}&S\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&K\end{matrix}\;;\;t\right]^{2}-q\right)$
$\displaystyle=\phi(1-t)\left(\,{}_{2}{\mathbb{P}}_{1}\left[\begin{matrix}\phi
K\overline{S}&S\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&K\end{matrix}\;;\;t\right]\,_{2}{\mathbb{P}}_{1}\left[\begin{matrix}\phi\overline{K}S&\overline{S}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\overline{K}\end{matrix}\;;\;t\right]-q\right),$

where

(20) $\,_{2}{\mathbb{P}}_{1}\left[\begin{matrix}\phi K\overline{S}&S\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&K\end{matrix}\;;\;t\right]=\overline{K}(t)\phi(t-1)\frac{J(S,K\overline{S})}{J(\phi
K\overline{S},\phi
S)}\,_{2}{\mathbb{P}}_{1}\left[\begin{matrix}\phi\overline{K}S&\overline{S}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &\overline{K}\end{matrix}\;;\;t\right].$

When $t=1$, we have

(21)
$\,_{3}{\mathbb{P}}_{2}\left[\begin{matrix}\phi&\eta&\overline{\eta}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&K&\overline{K}\end{matrix}\;;\;1\right]=\phi\eta(-1)q\frac{J(\eta
K,\overline{\eta}K)J(\phi,\phi K)}{J(\phi
S,\overline{K})J(S,\overline{K})}\left(\frac{J(S,\phi\overline{S})}{J(S\overline{\eta},\phi\overline{S}\eta)}+\frac{J(\phi
S,\overline{S})}{J(\eta\overline{S},\phi S\overline{\eta})}\right)\\\
=\frac{J(\eta
K,\overline{\eta}K)}{J(\phi,\overline{K})}\left(J(S\overline{K},\phi\overline{S})^{2}+J(\phi
S\overline{K},\overline{S})^{2}\right).$

In particular, when $K=\varepsilon$, this gives

(22)
$\,_{3}{\mathbb{P}}_{2}\left[\begin{matrix}\phi&\eta&\overline{\eta}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\varepsilon&\varepsilon\end{matrix}\;;\;1\right]=J(S,\phi\overline{S})^{2}+J(\overline{S},\phi
S)^{2}.$

When $\eta K$ is not a square, we have

(23)
$\,_{3}{\mathbb{P}}_{2}\left[\begin{matrix}\phi&\eta&\overline{\eta}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &K&\overline{K}\end{matrix}\;;\;1\right]=0.$

###### Proof.

When $t\neq 0$,

$\,{}_{3}{\mathbb{P}}_{2}\left[\begin{matrix}\phi&\eta&\overline{\eta}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&K&\overline{K}\end{matrix}\;;\;t\right]=\frac{-1}{q-1}\sum_{\chi}\binom{\phi\chi}{\chi}\binom{\eta\chi}{K\chi}\binom{\overline{\eta}\chi}{\overline{K}\chi}\chi(t)\\\
\overset{\chi\mapsto K\chi}{=}K(t)\frac{-1}{q-1}\sum_{\chi}\binom{\phi
K\chi}{K\chi}\binom{\eta
K\chi}{K^{2}\chi}\binom{\overline{\eta}K\chi}{\chi}\chi(t)=\phi(-1)K(t)\,_{3}{\mathbb{P}}_{2}\left[\begin{matrix}\overline{\eta}K&\eta
K&\phi K\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&K^{2}&K\end{matrix}\;;\;t\right].$

Apply the Clausen’s formula over $\mathbb{F}_{q}$ in [22, Theorem 1.5] with
$C=K$ and $S^{2}=\eta K$, when $t\neq 1$, one has

$\,{}_{3}{\mathbb{P}}_{2}\left[\begin{matrix}\overline{\eta}K&\eta K&\phi
K\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ &K^{2}&K\end{matrix}\;;\;t\right]$
$\displaystyle=\frac{J(\phi S,\phi
K\overline{S})}{J(S,K\overline{S})}\left(\,{}_{2}{\mathbb{P}}_{1}\left[\begin{matrix}\phi
K\overline{S}&S\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&K\end{matrix}\;;\;t\right]^{2}-\phi(1-t)\overline{K}(t)\cdot\phi(-1)q\cdot\frac{J(S,K\overline{S})}{J(\phi
S,\phi K\overline{S})}\right).$

On the other hand, the claim (20) follows from [24, Proposition 8.7 (2)].
Therefore, when $t\neq 0$, $1$,

$\,{}_{3}{\mathbb{P}}_{2}\left[\begin{matrix}\phi&\eta&\overline{\eta}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &K&\overline{K}\end{matrix}\;;\;t\right]$
$\displaystyle=\phi(-1)K(t)\frac{J(\phi S,\phi
K\overline{S})}{J(S,K\overline{S})}\,_{2}{\mathbb{P}}_{1}\left[\begin{matrix}\phi
K\overline{S}&S\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&K\end{matrix}\;;\;t\right]^{2}-\phi(1-t)\cdot q$
$\displaystyle=\phi(1-t)\,_{2}{\mathbb{P}}_{1}\left[\begin{matrix}\phi
K\overline{S}&S\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&K\end{matrix}\;;\;t\right]\,_{2}{\mathbb{P}}_{1}\left[\begin{matrix}\phi\overline{K}S&\overline{S}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\overline{K}\end{matrix}\;;\;t\right]-\phi(1-t)\cdot q$

When $t=1$, according to [27, Theorem 4.38] (or [24, Equation (7.2)]), the
evaluation is

$\,{}_{3}{\mathbb{P}}_{2}\left[\begin{matrix}\phi&\eta&\overline{\eta}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &K&\overline{K}\end{matrix}\;;\;1\right]$
$\displaystyle=\phi(-1)\,_{3}{\mathbb{P}}_{2}\left[\begin{matrix}\overline{\eta}K&\eta
K&\phi K\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&K^{2}&K\end{matrix}\;;\;1\right]$ $\displaystyle=\phi(-1)J(\eta
K,\overline{\eta}K)J(\phi,\phi K)\left(\sum_{D=S,\phi
S}\frac{J(D,K\overline{S}^{2})J(\phi\overline{D},\overline{K}S^{2})}{J(\phi
S,\overline{K})J(S,\overline{K})}\right)$

which can be converted to the right hand side of (21) using properties of
Jacobi sums. The identity (22) can be verified directly, and (23) follows from
Theorem 4.38(ii) of [27] by Greene. ∎

######  Lemma 1.

For a finite field $\mathbb{F}_{q}$ of odd characteristic and $A$, $B$, $C$,
$D$, $E\in\widehat{\mathbb{F}_{q}^{\times}}$, we have

$\,{}_{6}{\mathbb{P}}_{5}\left[\begin{matrix}A&B&C&A&D&E\vskip 3.0pt plus
1.0pt minus 1.0pt\\\
&A\overline{D}&A\overline{E}&\varepsilon&A\overline{B}&A\overline{C}\end{matrix}\;;\;1\right]=BCDE(-1)\sum_{t\in\mathbb{F}_{q}}A(t)\,_{3}{\mathbb{P}}_{2}\left[\begin{matrix}A&B&C\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&A\overline{D}&A\overline{E}\end{matrix}\;;\;t\right]^{2}\\\
=BCDE(-1)\sum_{t\in\mathbb{F}_{q}}A(t)\,_{3}{\mathbb{P}}_{2}\left[\begin{matrix}A&D&E\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&A\overline{B}&A\overline{C}\end{matrix}\;;\;t\right]^{2}.$

###### Proof.

$\displaystyle
BCDE(-1)\sum_{t\in\mathbb{F}_{q}}A(t)\,_{3}{\mathbb{P}}_{2}\left[\begin{matrix}A&B&C\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&A\overline{D}&A\overline{E}\end{matrix}\;;\;t\right]^{2}$
$\displaystyle\overset{Prop.\ref{prop:
Kummer}}{=}A(-1)\sum_{t\in\mathbb{F}_{q}^{\times}}\,{}_{3}{\mathbb{P}}_{2}\left[\begin{matrix}A&B&C\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&A\overline{D}&A\overline{E}\end{matrix}\;;\;t\right]\,_{3}{\mathbb{P}}_{2}\left[\begin{matrix}A&D&E\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&A\overline{B}&A\overline{C}\end{matrix}\;;\;1/t\right]$
$\displaystyle=A(-1)\sum_{t\in\mathbb{F}_{q}^{\times}}\frac{-1}{q-1}\sum_{\chi\in\widehat{\mathbb{F}_{q}^{\times}}}\binom{A\chi}{\chi}\binom{B\chi}{A\overline{D}\chi}\binom{C\chi}{A\overline{E}\chi}\chi(t)\frac{-1}{q-1}\sum_{\psi\in\widehat{\mathbb{F}_{q}^{\times}}}\binom{A\psi}{\psi}\binom{D\psi}{A\overline{B}\psi}\binom{E\psi}{A\overline{C}\psi}\psi(1/t)$
$\displaystyle=A(-1)\frac{1}{(q-1)^{2}}\sum_{\chi\in\widehat{\mathbb{F}_{q}^{\times}}}\binom{A\chi}{\chi}\binom{B\chi}{A\overline{D}\chi}\binom{C\chi}{A\overline{E}\chi}\sum_{\psi\in\widehat{\mathbb{F}_{q}^{\times}}}\binom{A\psi}{\psi}\binom{D\psi}{A\overline{B}\psi}\binom{E\psi}{A\overline{C}\psi}\sum_{t\in\mathbb{F}_{q}^{\times}}\chi\psi^{-1}(t)$
$\displaystyle=A(-1)\frac{1}{q-1}\sum_{\chi\in\widehat{\mathbb{F}_{q}^{\times}}}\binom{A\chi}{\chi}\binom{B\chi}{A\overline{D}\chi}\binom{C\chi}{A\overline{E}\chi}\binom{A\chi}{\chi}\binom{D\chi}{A\overline{B}\chi}\binom{E\chi}{A\overline{C}\chi}$
$\displaystyle=\,_{6}{\mathbb{P}}_{5}\left[\begin{matrix}A&B&C&A&D&E\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&A\overline{D}&A\overline{E}&\varepsilon&A\overline{B}&A\overline{C}\end{matrix}\;;\;1\right].$

∎

###  2.2. Hypergeometric character sums and Galois representations

To a hypergeometric datum $HD=\\{\alpha,\beta;\lambda\\}$ consisting of a
primitive pair of multi-sets
$\alpha=\\{a_{1},...,a_{n}\\},\beta=\\{1,b_{2},...,b_{n}\\}$ with
$M=lcd(\alpha\cup\beta)$ and
$\lambda\in\mathbb{Z}[\zeta_{M},1/M]\setminus\\{0\\}$, and a prime $\ell$, we
associate the $\ell$-adic representation $\rho_{HD,\ell}$ of
$G(M)=G_{\mathbb{Q}(\zeta_{M})}$ as in Theorem 1. We are interested in when it
can be extended to a representation of $G_{K}$ for a subfield $K$ of
$\mathbb{Q}(\zeta_{M})$, in particular $K=\mathbb{Q}$. This happens if and
only if $\rho_{HD,\ell}$ is isomorphic to its conjugation by each element in
Gal$(\mathbb{Q}(\zeta_{M})/K)$. This is a standard result by the Clifford
theory [14] and Frobenius reciprocity. For related discussion, see [36, §2.6]
by Li, Liu and Long. If this happens we say $\\{\alpha,\beta;\lambda\\}$ can
be _extended_ to $K$.

We now explain the Galois action. An element $\tau\in$
Gal$(\mathbb{Q}(\zeta_{M})/\mathbb{Q})$ sends $\zeta_{M}$ to $\zeta_{M}^{c}$
for some $c=c(\tau)\in(\mathbb{Z}/M\mathbb{Z})^{\times}$. The conjugation of
$\rho_{HD,\ell}$ by $\tau$, denoted by $\rho_{HD,\ell}^{\tau}$, satisfies

$\text{Tr}\rho_{HD,\ell}^{\tau}(\text{Frob}_{\wp})=\text{Tr}\rho_{HD,\ell}(\text{Frob}_{\tau(\wp)})=\text{Tr}\rho_{c\cdot
HD,\ell}(\text{Frob}_{\wp})$

for almost all prime ideals $\wp$ of $\mathbb{Z}[\zeta_{M}]$. Here $c\cdot HD$
denotes the datum

$\\{c\cdot\alpha,c\cdot\beta;\lambda\\}=\left\\{\begin{pmatrix}ca_{1}\\\
c\end{pmatrix},...,\begin{pmatrix}ca_{n}\\\
cb_{n}\end{pmatrix};\lambda\right\\}.$

This motivates the following definitions.

######  Definition 1.

For a primitive pair $\alpha,\beta$ with $M=lcd(\alpha\cup\beta)$ as above and
$c\in(\mathbb{Z}/M\mathbb{Z})^{\times}$, the pairs $\alpha,\beta$ and
$c\cdot\alpha,c\cdot\beta$ are _congruent mod $\mathbb{Z}$_ if the two multi-
sets $\left\\{\begin{pmatrix}ca_{1}\\\ c\end{pmatrix},\begin{pmatrix}ca_{2}\\\
cb_{2}\end{pmatrix},...,\begin{pmatrix}ca_{n}\\\ cb_{n}\end{pmatrix}\right\\}$
and $\left\\{\begin{pmatrix}a_{1}\\\ 1\end{pmatrix},\begin{pmatrix}a_{2}\\\
b_{2}\end{pmatrix},...,\begin{pmatrix}a_{n}\\\ b_{n}\end{pmatrix}\right\\}$
are congruent mod $\mathbb{Z}$. The pair $\alpha,\beta$ is said to be _self-
dual_ if it is congruent to the pair $-\alpha,-\beta$ mod $\mathbb{Z}$, and it
is _defined over a subfield $K$ of $\mathbb{Q}(\zeta_{M})$_ if it is congruent
to $c\cdot\alpha,c\cdot\beta$ for all $c=c(\tau)$, where $\tau\in$
Gal$(\mathbb{Q}(\zeta_{M})/K)$. A hypergeometric datum
$HD=\\{\alpha,\beta;\lambda\\}$ is said to be _self-dual_ if the pair
$\alpha,\beta$ is self-dual; it is _defined over $\mathbb{Q}$_ if the pair
$\alpha,\beta$ is defined over $\mathbb{Q}$ and
$\lambda\in\mathbb{Q}^{\times}$.

If $HD=\\{\alpha,\beta;\lambda\\}$ is defined over $\mathbb{Q}$, writing
$\displaystyle\prod_{j=1}^{n}\frac{X-e^{2\pi ia_{j}}}{X-e^{2\pi
ib_{j}}}=\frac{\prod_{j=1}^{r}(X^{p_{j}}-1)}{\prod_{k=1}^{s}(X^{q_{k}}-1)}$
where $p_{j},q_{k}\in\mathbb{Z}_{>0}$ and $p_{j}\neq q_{k}$ for all $j,k$, by
Theorem 1.3 of [10], the character sum $H_{q}(\alpha,\beta;\lambda)$ mentioned
in the introduction can be defined in the following way for any finite field
$\mathbb{F}_{q}$ as long as $q$ is coprime to $lcd(\\{1/M,\lambda\\})$

(24)
$H_{q}(\alpha,\beta;\lambda):=\frac{(-1)^{r+s}}{1-q}\sum_{m=0}^{q-2}q^{-s(0)+s(m)}\prod_{j=1}^{r}\mathfrak{g}(\omega^{mp_{j}})\prod_{k=1}^{s}\mathfrak{g}(\omega^{-mq_{k}})\omega(N^{-1}\lambda),$

where $\mathfrak{g}(\chi)$ denotes the Gauss sum of the character $\chi$,
$N=(-1)^{q_{1}+\cdots+q_{s}}\frac{p_{1}^{p_{1}}\cdots
p_{r}^{p_{r}}}{q_{1}^{q_{1}}\cdots q_{s}^{q_{s}}}$ and $s(m)$ is the
multiplicity of $X-e^{2\pi im/(q-1)}$ in the factorization of $\displaystyle
gcd(\prod_{j=1}^{r}(X^{p_{j}}-1),\prod_{k=1}^{s}(X^{q_{k}}-1)).$ Further,
since $H_{q}$ is $\mathbb{Q}$-valued, it is independent of the choice of
$\omega$.

Clearly, if the pair $\alpha,\beta$ in $HD$ above is defined over $K$ and
$\lambda\in K^{\times}$, then the representation $\rho_{HD,\ell}$ can be
extended to $G_{K}$.

Remark 2.  Let $K$ be a number field with the ring of integers
$\mathbb{Z}_{K}$. Since the prime ideals of $\mathbb{Z}_{K}$ of degree 1 have
density one among all prime ideals of $\mathbb{Z}_{K}$, by Chebotarev density
theorem, if two $\ell$-adic representations of $G_{K}$ ramified only at
finitely many prime ideals have the same traces at $\text{Frob}_{\wp}$ for
almost all prime ideals $\wp$ of $\mathbb{Z}_{K}$ of degree 1, then they are
isomorphic as $G_{K}$ representations provided that they are semi-simple. In
particular, if $K=\mathbb{Q}(\zeta_{M})$ is a cyclotomic field, the semi-
simplification of an $\ell$-adic representation of $G_{K}$ ramified only at
finitely many prime ideals, such as those in Theorems 1 and 2, is determined
by its traces at $\text{Frob}_{\wp}$ for almost all prime ideals $\wp$ of
$\mathbb{Z}_{K}$ above primes $p\equiv 1\mod M$.

To obtain modularity result for degree-2 Galois representations of
$G_{\mathbb{Q}}$, we shall use the following version of modularity theorem;
see Theorem 2.1.4 of [4].

######  Theorem 8.

Given a prime $\ell$ and a 2-dimensional absolutely irreducible representation
$\rho$ of $G_{\mathbb{Q}}$ over $\overline{\mathbb{Q}}_{\ell}$ that is odd,
unramified at almost all primes, and its restriction to a decomposition
subgroup $D_{\ell}$ at $\ell$ is crystalline with Hodge-Tate weight
$\\{0,r\\}$ where $1\leq r\leq\ell-2$ and $\ell+1\nmid 2r$, then $\rho$ is
modular and corresponds to a weight $r+1$ holomorphic Hecke eigenform.

The actual identification of the target modular form is carried out by
computing the trace of the representation at various Frobenius conjugacy
classes $\text{Frob}_{p}$. In the next subsection we discuss key issues
involved in computation. Since representations from Theorem 2 have integral
traces, we use the following theorem of Serre (cf. [51] by Serre or [19,
Theorem 2.2] by Dieulefait) to narrow the search for the corresponding modular
forms.

######  Theorem 9 (Serre).

Let $f$ be an integral weight holomorphic Hecke eigenform with coefficients in
$\mathbb{Z}$. Then the $p$-exponents of the level of $f$ are bounded by $8$
for $p=2$, by $5$ for $p=3$, and by $2$ for all other bad primes.

###  2.3. Some computational issues

Given $\alpha=\\{a_{1},...,a_{n}\\}$ and $\beta=\\{b_{1},...,b_{n}\\}$ with
$a_{i},b_{j}\in\mathbb{Q}\cap[0,1)$ and $M=lcd(\alpha\cup\beta)$, the
following step function on the interval $[0,1)$ is introduced in [39] by the
second author:

$e_{\alpha,\beta}(x):=\sum_{i=1}^{n}-\left\lfloor
a_{i}-x\right\rfloor-\left\lfloor x+b_{i}\right\rfloor.$

The value of $e_{\alpha,\beta}(x)$ jumps up (resp. down) only at $a_{i}$
(resp. $1-b_{j}$). When $p$ is an odd prime not dividing $M$, $0\leq k<p-1$ is
an integer and the pair $\alpha,\beta$ is defined over $\mathbb{Q}$,
$e_{\alpha,\beta}(\frac{k}{p-1})$ gives the collective exponent of $p$ in the
$k$th summand of $H_{p}(\alpha,\beta;\lambda)$ recalled in (24). Moreover when
the pair $\alpha,\beta$ is defined over $\mathbb{Q}$, the graph of
$e_{c\alpha,c\beta}(x)$ is independent of any integer $c$ prime to $M$.

The weight $w(HD)$ of a datum $HD=\\{\alpha,\beta;\lambda\\}$ is defined as

(25) $w(HD)=w(\alpha,\beta):=\max e_{\alpha,\beta}(x)-\min
e_{\alpha,\beta}(x).$

When $\rho_{HD,\ell}$ can be extended to $G_{\mathbb{Q}}$, $w(c\alpha,c\beta)$
is independent of $c$, although the graph of $e_{c\alpha,c\beta}(x)$ may.

For $HD$ defined over $\mathbb{Q}$, there is an efficient Magma program called
“Hypergeometric Motives over $\mathbb{Q}$” implemented by Watkins which
computes the characteristic polynomial of
$\rho_{\\{\alpha,\beta;\lambda\\},\ell}^{BCM}[t]$ (resp.
$\rho_{\\{\alpha,\beta;\lambda\\},\ell}^{BCM,prim}[t]$) at $\text{Fr}_{p}$,
the inverse of $\text{Frob}_{p}$, for $p\nmid M\ell$ efficiently when
$\lambda\neq 0,1$ (resp. $\lambda=1$), where

(26) $t=-\min\\{e_{\alpha,\beta}(x)\mid 0\leq x<1\\}-\frac{n-m}{2}.$

Here $\rho[t]$ denotes the weight-$t$ Tate twist of a representation $\rho$ of
$G_{\mathbb{Q}}$.

For example, for
$\alpha=\\{\frac{1}{2},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}\\}$,
$\beta=\\{1,\frac{1}{10},\frac{3}{10},\frac{7}{10},\frac{9}{10}\\}$, the graph
of the function is as follows.

The graph of $e_{\alpha,\beta}(x)$ with
$\alpha=\\{\frac{1}{2},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}\\}$,
$\beta=\\{1,\frac{1}{10},\frac{3}{10},\frac{7}{10},\frac{9}{10}\\}$

In this case, $\min e_{\alpha,\beta}=-1,$ $\max e_{\alpha,\beta}=2$,
$w(HD)=2-(-1)=3$, $n=5,m=1$, so $t=-\min
e_{\alpha,\beta}-\frac{n-m}{2}=1-\frac{5-1}{2}=-1$. When $\lambda=1$,
$\dim\rho_{HD,\ell}^{BCM,prim}=2\lfloor\frac{5-1}{2}\rfloor=4$. We exhibit
below the commands used in the Magma program to compute the characteristic
polynomial of $\rho_{HD,\ell}^{BCM,prim}[-1]$ at $\text{Fr}_{7}$:

H:=HypergeometricData([1/2,1/5,2/5,3/5,4/5],[1,1/10,3/10,7/10,9/10]);

Factorization(EulerFactor(H,1,7));

The out put is

$<2401*\$.1^{4}-88*\$.1^{2}+1,1>$

It confirms that that $\dim\rho_{HD,\ell}^{BCM,prim}=4$ and the roots of the
characteristic polynomial of $\rho_{HD,\ell}^{BCM,prim}[-1]$ at
$\text{Fr}_{7}$ all have absolute value $1/7$ under any embedding, which is
$1/p^{(n-1)/2+t}$ at $p=7$.

##  3\. Galois representations associated to hypergeometric data

Recall the hypergeometric data $HD_{1}$ and $HD_{2}$ given by

$HD_{1}=HD_{1}(c,f)=\left\\{\alpha_{6}(c,f):=\left\\{\frac{1}{2},c,1-c,\frac{1}{2},f,1-f\right\\},\beta_{6}(c,f):=\left\\{1,\frac{3}{2}-c,\frac{1}{2}+c,1,\frac{3}{2}-f,\frac{1}{2}+f\right\\};1\right\\}$

$HD_{2}=HD_{2}(c,f)=\left\\{\alpha_{4}(f):=\left\\{\frac{1}{2},\frac{1}{2},f,1-f\right\\},\beta_{4}(c):=\left\\{1,1,\frac{3}{2}-c,\frac{1}{2}+c\right\\};1\right\\}$

with $c,f\in\mathbb{Q}\cap(0,1)$ as in §1.2. They are both self-dual from
construction. Let

$M=M(c,f)=lcd(\alpha_{6}(c,f)\cup\beta_{6}(c,f))=lcd(\alpha_{4}(c,f)\cup\beta_{4}(c,f)).$

Then both $\alpha_{6}(c,f)$ and $\beta_{6}(c,f)=1+\frac{1}{2}-\alpha_{6}(c,f)$
in $HD_{1}$ are defined over $\mathbb{Q}(\zeta_{M}+\zeta_{M}^{-1})$. In
particular, only 14 un-ordered couples $(c,f)$ give rise to $HD_{1}$ defined
over $\mathbb{Q}$:

$\text{either
}c,f\in\left\\{\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{6}\right\\},\quad\text{
or}\quad(c,f)=\left(\frac{1}{5},\frac{2}{5}\right),\left(\frac{1}{8},\frac{3}{8}\right),\left(\frac{1}{10},\frac{3}{10}\right)$

as enumerated in [41]. Among them, only 7 pairs of $(c,f)$ give rise to
primitive $HD_{1}$, as listed in (11).

###  3.1. A proof of Theorem 3

Let $c,f$ in $\mathbb{Q}^{\times}$ be such that

(27)
$\alpha_{3}(f)=\\{\frac{1}{2},f,1-f\\},\quad\beta_{3}(c)=\\{1,\frac{3}{2}-c,\frac{1}{2}+c\\}$

form a primitive pair with $M(c,f)=lcd(\alpha_{3}(f)\cup\beta_{3}(c))$.
Motivated by the finite field Clausen formula (Theorem 7), we define the
multi-sets

$\alpha_{2}(c,f):=\left\\{\frac{1+2f-2c}{4},\frac{3-2f-2c}{4},\right\\},\quad\beta_{2}(c):=\left\\{1,\frac{3}{2}-c\right\\}$

with $N=N(c,f)=lcd(\alpha_{2}(c,f)\cup\beta_{2}(c))$. Then either
$N(c,f)=2M(c,f)$ or $N(c,f)=M(c,f)$. Among the 7 pairs listed in (11),
$N(c,f)=M(c,f)$ only happens when $(c,f)=(\frac{1}{2},\frac{1}{3}).$

Fix a prime $\ell$. By Theorem 1, for any nonzero
$\lambda\in\mathbb{Z}[\zeta_{M(c,f)},1/M(c,f)]$ there exists an $\ell$-adic
representation

(28)
$\rho_{\\{\alpha_{3}(f),\beta_{3}(c);\lambda\\},\ell}:G(M(c,f))\rightarrow
GL_{\overline{\mathbb{Q}}_{\ell}}(W_{\lambda}(c,f))$

such that at any prime ideal $\wp$ of
$\mathbb{Z}[\zeta_{M(c,f)},1/(2N(c,f)\ell)]$ and $\text{ord}_{\wp}\lambda=0$

(29)
$\text{Tr}\rho_{\\{\alpha_{3}(f),\beta_{3}(c),\lambda\\};\ell}(\text{Frob}_{\wp})=\phi_{\wp}(-1)\
\mathbb{P}\left(\alpha_{3}(f),\beta_{3}(c);{1/\lambda};\kappa_{\wp}\right).$

Moreover, $\dim W_{1}(c,f)=2$ and $\dim W_{\lambda}(c,f)=3$ when $\lambda\neq
0,1$. We first deal with the $\lambda=1$ case.

######  Proposition 3.

Let $\ell$ be a prime. For each pair $(c,f)$ in (11), let
$HD_{3}(c,f)=\\{\alpha_{3}(f),\beta_{3}(c);1\\}$, $\rho_{HD_{3}(c,f),\ell}$ be
the $2$-dimensional $\ell$-adic representation of $G(M(c,f))$ as in Theorem 1.
Then $\rho_{HD_{3}(c,f),\ell}$ is invariant under the twist by the quadratic
character $\chi_{-1}|_{G(M(c,f))}$ when $(c,f)\neq(\frac{1}{2},\frac{1}{3})$.
Further, each $\rho_{HD_{3}(c,f),\ell}$ can be extended to a representation of
$G_{\mathbb{Q}}$, which is modular and invariant under a quadratic twist. Each
extension has determinant $\psi_{c,f}\epsilon_{\ell}^{2}$, where $\psi_{c,f}$
is a quadratic character of $G_{\mathbb{Q}}$ shown in the table below. Listed
below are the modular forms corresponding to $\rho_{HD_{3}(c,f),\ell}^{BCM}$
when $HD_{3}(c,f)$ is defined over $\mathbb{Q}$ and one of the extensions for
the last two cases.

$\begin{array}[]{|c|c|c|c|c|}\hline\cr(c,f)&w(HD_{3})&\text{Tr}\rho_{\\{\alpha_{3}(f),\beta_{3}(c);1\\},\ell}(\text{Frob}_{p})&\det\rho_{\\{\alpha_{3}(f),\beta_{3}(c);1\\},\ell}=\psi_{c,f}\epsilon_{\ell}^{2}&CM\\\
\hline\cr(\frac{1}{2},\frac{1}{2})&3&a_{p}(\eta(4\tau)^{6})&\chi_{-1}\epsilon_{\ell}^{2}&\chi_{-1}\\\
\hline\cr\left(\frac{1}{2},\frac{1}{3}\right)&3&{a_{p}(\eta(2\tau)^{3}\eta(6\tau)^{3})}&\chi_{-3}\epsilon_{\ell}^{2}&\chi_{-3}\\\
\hline\cr(\frac{1}{3},\frac{1}{3})&3&a_{p}(f_{36.3.d.b})&\chi_{-1}\epsilon_{\ell}^{2}&\chi_{-1}\\\
\hline\cr(\frac{1}{2},\frac{1}{6})&3&\left(\frac{-3}{p}\right)a_{p}(\eta(4\tau)^{6})&\chi_{-1}\epsilon_{\ell}^{2}&\chi_{-1}\\\
\hline\cr(\frac{1}{6},\frac{1}{6})&1&p\cdot
a_{p}(\eta(12\tau)^{2})&\chi_{-1}\epsilon_{\ell}^{2}&\chi_{-1},\chi_{-3}\\\
\hline\cr(\frac{1}{5},\frac{2}{5})&3&a_{p}(f_{20.3.d.a})&\chi_{-5}\epsilon_{\ell}^{2}&\chi_{-5}\\\
\hline\cr(\frac{1}{10},\frac{3}{10})&1&p\cdot
a_{p}(\eta(4\tau)\eta(20\tau))&\chi_{-5}\epsilon_{\ell}^{2}&\chi_{-1},\chi_{-5}\\\
\hline\cr\end{array}$

###### Proof.

It follows from Theorem 7 and (4) that, for each of the $7$ pairs,
$\rho_{HD_{3},\ell}|_{G(N)}$ is a direct sum of two explicit characters
$\xi_{1}$ and $\xi_{2}$ of $G(N)$ associated to Jacobi sums, see [58]. When
$(c,f)\neq(\frac{1}{6},\frac{1}{6})$ nor $(\frac{1}{10},\frac{3}{10})$,
$\xi_{1}$ is not equal to $\xi_{2}$. Furthermore, by (23), when $N=2M$, that
is, $(c,f)\neq(\frac{1}{2},\frac{1}{3})$, $\mathbb{P}(HD_{3},q)=0$ when
$q\equiv 3\mod 4$. This implies that $\rho_{HD_{3},\ell}$ is invariant under
twist by $\chi_{-1}|_{G(M)}$.

For the first 5 pairs, $HD_{3}$ is defined over $\mathbb{Q}$. The determinant
of $\rho_{HD_{3},\ell}^{BCM}$, by Theorem 2, can be determined explicitly as
it is $\epsilon_{\ell}^{2}$ times a character of order at most 2 and conductor
dividing $24$. They turn out to be either $\chi_{-1}\epsilon_{\ell}^{2}$ or
$\chi_{-3}\epsilon_{\ell}^{2}$, as listed in the table. For example, when
$(c,f)=(\frac{1}{2},\frac{1}{3})$, the determinant at $\text{Frob}_{p}$ equals
$-25,49,-121,169$ for $p=5,7,11,13$, respectively. One sees that the
determinant is $\chi_{-3}\epsilon_{\ell}^{2}$. This shows that
$\rho_{HD_{3},\ell}^{BCM}$ is odd. When $(c,f)\neq(\frac{1}{6},\frac{1}{6})$,
from $\xi_{1}\neq\xi_{2}$ we know $\rho_{HD_{3},\ell}^{BCM}$ is irreducible.
When $(c,f)=(\frac{1}{6},\frac{1}{6})$, by (24) and reflection formula for
Gauss sum, we have for any prime $p$ not dividing $6$

$H_{p}(HD_{3})=\frac{1}{p-1}\sum_{\chi\in\widehat{\mathbb{F}_{q}^{\times}}}\frac{\mathfrak{g}(\chi^{6})\mathfrak{g}(\overline{\chi}^{3})}{\mathfrak{g}(\chi^{3})}\chi(-2^{-6}).$

When $p\equiv 2\mod 3$, the map $\chi\mapsto\chi^{3}$ gives an automorphism of
$\widehat{\mathbb{F}_{p}^{\times}}$. It follows that

$\displaystyle H_{p}(HD_{3})=$
$\displaystyle\frac{1}{p-1}\sum_{\chi}\frac{\mathfrak{g}(\chi^{2})\mathfrak{g}(\overline{\chi})}{\mathfrak{g}(\chi)}\chi(-2^{-2})=\frac{1}{p-1}\sum_{\chi}\frac{\mathfrak{g}(\chi)\mathfrak{g}(\phi_{p}\chi)\mathfrak{g}(\overline{\chi})}{\mathfrak{g}(\phi_{p})\mathfrak{g}(\chi)}\chi(-1)$
$\displaystyle=$
$\displaystyle\frac{1}{p-1}\sum_{\chi}J(\phi_{p}\chi,\overline{\chi})\chi(-1)=\frac{1}{p-1}\sum_{x}\phi_{p}(x)\sum_{\chi}\chi(-x)\overline{\chi}(1-x)=0.$

If $\rho_{HD_{3},\ell}^{BCM}$ is reducible, then it is isomorphic to
$\xi\oplus\xi\chi_{-3}$ for some 1-dimensional representation $\xi$ of
$G_{\mathbb{Q}}$. Meanwhile, as $\rho_{HD_{3},\ell}$ is invariant when twisted
by $\chi_{-1}$, we have
$(\xi\oplus\xi\chi_{-3})|_{G(6)}\cong(\xi\oplus\xi)|_{G(6)}\cong(\xi\chi_{-1}\oplus\xi\chi_{-1})|_{G(6)}$,
which is impossible.

For the remaining 2 pairs, we have
$\rho_{HD_{3},\ell}\cong(\chi_{-1})|_{G(M)}\otimes\rho_{HD_{3},\ell}$ as noted
before. In these two cases $M=10$ and the field $\mathbb{Q}(\zeta_{M})$ has
degree $4$ over $\mathbb{Q}$. We show that $\rho_{HD_{3},\ell}$ can be
extended to $G_{\mathbb{Q}}$ for the case $(c,f)=(\frac{1}{5},\frac{2}{5})$;
the same conclusion holds for $(c,f)=(1/10,3/10)$ by a similar argument. The
group Gal$(\mathbb{Q}(\zeta_{5})/\mathbb{Q})$ is generated by the automorphism
$\tau:\zeta_{5}\mapsto\zeta_{5}^{3}$. It maps
$\alpha_{3}(\frac{2}{5})=\\{\frac{1}{2},\frac{2}{5},\frac{3}{5}\\}$ to
$\\{\frac{1}{2},\frac{1}{5},\frac{4}{5}\\}$ and
$\beta_{3}(\frac{1}{5})=\\{1,\frac{13}{10},\frac{7}{10}\\}$ to
$\\{1,\frac{9}{10},\frac{11}{10}\\}$. By Proposition 1, for prime power $q$
coprime to $M$,

$\mathbb{P}\left(\alpha_{3}({f}),\beta_{3}({c});1;\mathbb{F}_{q}\right)={\phi_{q}(-1)}\mathbb{P}\left({\left\\{\frac{1}{2},\frac{1}{5},\frac{4}{5}\right\\},\left\\{1,\frac{9}{10},\frac{1}{10}\right\\}};1;\mathbb{F}_{q}\right).$

In terms of representations, this means that
$\rho_{HD_{3},\ell}\cong\chi_{-1}|_{G(M)}\otimes\rho_{HD_{3},\ell}^{\tau}$,
where $\rho_{HD_{3},\ell}^{\tau}$ is the conjugate of $\rho_{HD_{3},\ell}$ by
$\tau$. As shown above,
$\rho_{HD_{3},\ell}\otimes\chi_{-1}|_{G(M)}\cong\rho_{HD_{3},\ell}$, therefore
$\rho_{HD_{3},\ell}\cong\rho_{HD_{3},\ell}^{\tau}$. This proves that
$\rho_{HD_{3},\ell}$ can be extended to $G_{\mathbb{Q}}$, denoted by
$\rho_{HD_{3},\ell}^{BCM}$ by abuse of notation.

For $(\frac{1}{5},\frac{2}{5})$, $\xi_{1}\neq\xi_{2}$, which implies that
$\rho_{HD_{3},\ell}^{BCM}$ is irreducible. By Clifford theory, $\xi_{1}$
extends to an index-2 subgroup $G_{K}$ of $G_{\mathbb{Q}}$ so that
$\rho_{HD_{3},\ell}^{BCM}=\text{Ind}_{G_{K}}^{G_{\mathbb{Q}}}\xi_{1}$. Since
the map $\iota:\zeta_{4}\mapsto-\zeta_{4}$ in
Gal$(\mathbb{Q}(\zeta_{20})/\mathbb{Q}(\zeta_{10}))$ swaps $\xi_{1}$ and
$\xi_{2}$, $K$ is not contained in $\mathbb{Q}(\zeta_{10})$. On the other
hand, $\mathbb{Q}(\zeta_{20})$ contains three quadratic subfields, so $K$ is
one of the two imaginary extensions outside $\mathbb{Q}(\zeta_{10})$. This
shows that $\rho_{HD_{3},\ell}^{BCM}$ is odd.

For $(\frac{1}{10},\frac{3}{10})$, by formula (21) and simplification using
the reflection and multiplication formulas of Gauss sums, we get
$\xi_{1}=\xi_{2}=\left(\frac{\sqrt{5}}{\cdot}\right)_{2}\epsilon_{\ell}|_{G(20)}$,
where $\left(\frac{\sqrt{5}}{\cdot}\right)_{2}$ is a Hilbert quadratic norm
residue symbol on $\mathbb{Q}(\zeta_{20})$. Clearly $\xi_{1}$ extends to
$G_{\mathbb{Q}(\sqrt{5})}$ and not $G_{\mathbb{Q}}$. Hence
$\rho_{HD_{3},\ell}^{BCM}$ is irreducible and it equals
$\text{Ind}_{G_{\mathbb{Q}(\sqrt{5})}}^{G_{\mathbb{Q}}}\xi_{1}$. Moreover,
viewed as an idele class character of $\mathbb{Q}(\sqrt{5})$ by class field
theory, $\left(\frac{\sqrt{5}}{\cdot}\right)_{2}$ is trivial at one real place
of $\mathbb{Q}(\sqrt{5})$, and is the sign character at the other real place,
resulting from the two real embeddings of $\sqrt{5}$. This shows that
$\rho_{HD_{3},\ell}^{BCM}$ is odd.

In conclusion, for each of the 7 cases, $\rho_{HD_{3},\ell}^{BCM}$ is odd,
irreducible and, by Clifford theory, induced from a character of an index-2
subgroup of $G_{\mathbb{Q}}$. Hence it is modular by the work of Hecke. When
$HD_{3}$ is defined over $\mathbb{Q}$, we compute the characteristic
polynomials of each representation at suitable Frobenius conjugacy classes and
identify the corresponding modular forms using the method described in §2.3.
For other two cases, from (21) we identify the corresponding modular forms.
For example, for primes $p<200$ and $p\equiv 1\mod 10$,
$\mathbb{P}(HD_{3}(\frac{1}{5},\frac{2}{5});1;\mathbb{F}_{p})=a_{p}(f_{20.3.d.a})=a_{p}(f_{100.3.b.c})$.
Here $f_{20.3.d.a}$ is a cusp form of character $\chi_{-5}$ with coefficients
in $\mathbb{Z}$ while the coefficients of $f_{100.3.b.c}$ are no longer in
$\mathbb{Z}$. ∎

######  Corollary 1.

In Proposition 3 above, the character $\psi_{c,f}$ is trivial on $G(N(c,f))$.

As shown in the proof of Proposition 3, $\rho_{HD_{3},\ell}|_{G(N(c,f))}$ is a
direct sum of two 1-dimensional representations and $\rho_{HD_{3},\ell}$ can
be extended to $G_{\mathbb{Q}}$, and any extended representation admits CM by
an imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$ with $d\geq 1$. Also
Proposition 3 says
$\det\rho_{HD_{3},\ell}(\text{Frob}_{\wp})=\psi_{c,f}(\text{Frob}_{\wp})N(\wp)^{2}$
at prime ideals $\wp$ of $\mathbb{Z}[\zeta_{M(c,f)},1/M(c,f)\ell]$.
Consequently,

(30)
$\text{Tr}\,\text{Alt}^{2}\rho_{HD_{3},\ell}(\text{Frob}_{\wp})=\det\rho_{HD_{3},\ell}(\text{Frob}_{\wp})=\psi_{c,f}(\text{Frob}_{\wp})N(\wp)^{2};$
(31)
$\text{Tr}\,\text{Sym}^{2}\rho_{HD_{3},\ell}(\text{Frob}_{\wp})+\det\rho_{{HD_{3}},\ell}(\text{Frob}_{\wp})=\mathbb{P}(HD_{3};\kappa_{\wp})^{2},$

where Alt${}^{2}\rho$ and Sym${}^{2}\rho$ stand for the alternating square and
symmetric square of a representation $\rho$ respectively.

Next we consider the case when $\lambda\neq 1$.

######  Proposition 4.

Let $\ell$ be a prime and $c,f\in\mathbb{Q}^{\times}$ be such that
$\alpha_{3}(f)=\\{\frac{1}{2},f,1-f\\},\beta_{3}(c)=\\{1,\frac{3}{2}-c,\frac{1}{2}+c\\}$
form a primitive pair. Let $N(c,f)=lcd(\alpha_{2}(c,f)\cup\beta_{2}(c))$ as
before. Then for any
$\lambda\in\mathbb{Z}[\zeta_{M(c,f)}]\smallsetminus\\{0,1\\}$ the
3-dimensional $\ell$-adic Galois representation
$\rho_{{\\{\alpha_{3}(f),\beta_{3}(c);\lambda\\}},\ell}|_{G(N(c,f))}$ as in
(28) possesses the following properties at a prime ideal $\wp$ of
$\mathbb{Z}[\zeta_{N(c,f)},1/N(c,f)\ell]$:

(32)
$\text{Tr}\,\text{Alt}^{2}\rho_{\\{\alpha_{3}(f),\beta_{3}(c);\lambda\\},\ell}|_{G(N(c,f))}(\text{Frob}_{\wp})=\phi_{\wp}(1-1/\lambda)N(\wp)\cdot\mathbb{P}(\alpha_{3}(f),\beta_{3}(c);1/\lambda;\kappa_{\wp})$

and

(33)
$\text{Tr}\,\text{Sym}^{2}\rho_{HD_{3},\ell}|_{G(N(c,f))}(\text{Frob}_{\wp})=\mathbb{P}(\alpha_{3}(f),\beta_{3}(c);1/\lambda;\kappa_{\wp})^{2}-\phi_{\wp}(1-1/\lambda)N(\wp)\cdot\mathbb{P}(\alpha_{3}(f),\beta_{3}(c);1/\lambda;\kappa_{\wp}),$

where $\phi_{\wp}(\cdot)$ is the quadratic character of $\kappa_{\wp}$.
Further,
$\text{Sym}^{2}\rho_{\\{\alpha_{3}(f),\beta_{3}(c);\lambda\\},\ell}|_{G(N(c,f))}$
contains a $1$-dimensional sub-representation isomorphic to
$\epsilon_{\ell}^{2}|_{G(N(c,f))}$.

###### Proof.

Write $\rho$ for the $2$-dimensional $\ell$-adic representation
$\rho_{\\{\alpha_{2}(c,f),\beta_{2}(c);\lambda\\},\ell}$ of $G(N(c,f))$ and by
$\overline{\rho}$ the representation
$\rho_{\\{\overline{\alpha_{2}}(c,f),\overline{\beta_{2}}(c);\lambda\\},\ell}$
of the same group, where $\overline{\alpha_{2}}(c,f)=1-\alpha_{2}(c,f)$ and
$\overline{\beta_{2}}(c)=\\{1,\frac{1}{2}+c\\}$. Then at a prime ideal $\wp$
of $\mathbb{Z}[\zeta_{N(c,f)},1/N(c,f)\ell]$ and such that
$\text{ord}_{\wp}\lambda=0$, we have

$\text{Tr}\rho(\text{Frob}_{\wp})=-\omega^{(q-1){\frac{1+2f-2c}{4}}}(-1)\mathbb{P}(\alpha_{2}(c,f),\beta_{2}(c);1/\lambda;\kappa_{\wp})$

and similarly for $\overline{\rho}$. The eigenvalues $u_{\wp},v_{\wp}$ of
$\rho(\text{Frob}_{\wp})$ have absolute value $N(\wp)^{1/2}$, and those of
$\overline{\rho}(\text{Frob}_{\wp})$ are $N(\wp)/u_{\wp}$ and
$N(\wp)/v_{\wp}$. Thus the $4$ eigenvalues of
$\rho\otimes\overline{\rho}\,(\text{Frob}_{\wp})$ are
$N(\wp)u_{\wp}/v_{\wp},N(\wp)v_{\wp}/u_{\wp},N(\wp),N(\wp)$. Write
$\det\rho=\chi_{\rho}\epsilon_{\ell}$ for a finite order character
$\chi_{\rho}$ of $G(N(c,f))$. Then
$u_{\wp}v_{\wp}=\chi_{\rho}(\text{Frob}_{\wp})N(\wp)$ implies
$N(\wp)/u_{\wp}=\chi_{\rho}(\text{Frob}_{p})^{-1}v_{\wp}$ and
$N(\wp)/v_{\wp}=\chi_{\rho}(\text{Frob}_{p})^{-1}u_{\wp}$. Since the
eigenvalues of Sym${}^{2}\rho(\text{Frob}_{\wp})$ are
$u_{\wp}^{2},v_{\wp}^{2},u_{\wp}v_{\wp}$, we get

$\chi_{\rho}^{-1}\,(\text{Frob}_{\wp})\text{Tr}\,\text{Sym}^{2}\rho(\text{Frob}_{\wp})=\text{Tr}\,\rho\otimes\overline{\rho}(\text{Frob}_{\wp})-N(\wp).$

Theorem 7 then implies that

$\text{Tr}\rho_{\\{\alpha_{3}(f),\beta_{3}(c);\lambda\\},\ell}|_{G(N(c,f))}(\text{Frob}_{\wp})=\phi_{\wp}(1-1/\lambda)\chi_{\rho}^{-1}(\text{Frob}_{\wp})\text{Tr}\,\text{Sym}^{2}\rho(\text{Frob}_{\wp}).$

In other words, as representations of $G(N(c,f))$,
$\rho_{\\{\alpha_{3}(f),\beta_{3}(c);\lambda\\},\ell}|_{G(N(c,f))}$ is
isomorphic to $\chi_{\rho}^{-1}\otimes\text{Sym}^{2}\rho$ twisted by the
Hilbert quadratic norm residue symbol
$\left(\frac{1-1/\lambda}{\cdot}\right)_{2}$ on $\mathbb{Q}(\zeta_{N(c,f)})$.
Hence the eigenvalues of
$\rho_{\\{\alpha_{3}(f),\beta_{3}(c),\lambda\\},\ell}|_{G(N(c,f))}(\text{Frob}_{\wp})$
are

$c\cdot N(\wp)u_{\wp}/v_{\wp},\quad c\cdot N(\wp)v_{\wp}/u_{\wp},\quad c\cdot
N(\wp),\quad\text{where }\,c=\phi_{\wp}(1-1/\lambda).$

Therefore,

$\text{Sym}^{2}\rho_{\\{\alpha_{3}(f),\beta_{3}(c);\lambda\\},\ell}|_{G(N(c,f))}\simeq\text{Sym}^{2}(\chi_{\rho}^{-1}\otimes\text{Sym}^{2}\rho)\simeq\chi_{\rho}^{-2}\otimes\text{Sym}^{2}(\text{Sym}^{2}\rho)$

and

$\text{Alt}^{2}\rho_{\\{\alpha_{3}(f),\beta_{3}(c);\lambda\\},\ell}|_{G(N(c,f)}\simeq\text{Alt}^{2}(\chi_{\rho}^{-1}\otimes\text{Sym}^{2}\rho).$

The last relation implies the eigenvalues of
$\text{Alt}^{2}\rho_{\\{\alpha_{3}(c),\beta_{3}(f);\lambda\\},\ell}|_{G(N(c,f)}(\text{Frob}_{\wp})$
are $N(\wp)^{2}$, $N(\wp)^{2}u_{\wp}/v_{\wp}$ and $N(\wp)^{2}v_{\wp}/u_{\wp}$.
This proves (32). The equality (33) follows from the general identity for a
representation $\rho^{\prime}$ (see [16, (67.8)]):

$\text{Tr}\text{Alt}^{2}\rho^{\prime}+\text{Tr}\text{Sym}^{2}\rho^{\prime}=\text{Tr}\rho^{\prime}\otimes\rho^{\prime}=(\text{Tr}\rho^{\prime})^{2}.$

For the last assertion, note the following relation for 2-dimensional
semisimple representation $\rho$ which can be proved by comparing the
characters of both sides :

$\text{Sym}^{2}(\text{Sym}^{2}\rho)=\text{Sym}^{4}\rho\oplus(\det\rho)^{2},$

from which it follows that the degree-6 representation
$\text{Sym}^{2}\rho_{\\{\alpha_{3}(f),\beta_{3}(c);\lambda\\},\ell}|_{G(N(c,f))}$
decomposes into the sum of a degree-5 and a degree-1 sub-representations:

(34)
$\displaystyle\text{Sym}^{2}\rho_{\\{\alpha_{3}(f),\beta_{3}(c);\lambda\\},\ell}|_{G(N(c,f))}\simeq(\chi_{\rho}^{-2}\otimes\text{Sym}^{4}\rho)\oplus\chi_{\rho}^{-2}(\det\rho)^{2}.$

Since $\det\rho=\chi_{\rho}\epsilon_{\ell}|_{G(N(c,f))}$, the degree-1 sub-
representation of
$\text{Sym}^{2}\rho_{\\{\alpha_{3}(c),\beta_{3}(f);\lambda\\},\ell}|_{G(N(c,f))}$
is $\epsilon_{\ell}^{2}|_{G(N(c,f))}$. ∎

###### Proof of Theorem 3.

Below we fix a pair $(c,f)$ in (11) and drop the reference to $c,f$ when there
is no danger of confusion, and fix a prime $\ell$. Using Lemma 1, we have, at
a prime ideal $\wp$ of $\mathbb{Z}[\zeta_{M},1/M\ell]$,

(35)
$\mathbb{P}(HD_{1};\kappa_{\wp})=\sum_{t\in\kappa_{\wp}}\phi_{\wp}(t)\cdot\,\mathbb{P}\left(\alpha_{3}(f),\beta_{3}(c);t;\kappa_{\wp}\right)^{2}.$

Recall that finite dimensional semisimple representations of a group of any
size are determined by their traces (see [16, (27.13)] by Curtis and Reiner).
We will use (35) to obtain an isomorphism between two Galois representations
of $G(N)$ with difference origins. For the left hand side, recall from Theorem
1 that
$\text{Tr}\rho_{HD_{1},\ell}(\text{Frob}_{\wp})=-\phi_{\wp}(-1)\mathbb{P}(HD_{1};\kappa_{\wp})$.

For the right hand side, by Katz [31, 32] there exists an $\ell$-adic sheaf
$\mathcal{H}$ of $\mathbb{A}$ over $R=\mathbb{Z}[\zeta_{N},1/\ell N]$ such
that the action of $G(N)$ on the stalk of $\mathcal{H}_{\lambda}$ at each
nonzero $\lambda\in R$ is a generically 3-dimensional representation,
characterized by the following trace formula: At any prime ideal $\wp$ of $R$,
and $\lambda\in R$ when $\text{ord}_{\wp}\lambda=0$

$\text{Tr}\,\text{Frob}_{\wp}(\mathcal{H}_{\lambda})=\phi_{\wp}(-1)\mathbb{P}\left(\alpha_{3}(f),\beta_{3}(c);1/\lambda;\kappa_{\wp}\right).$

Here and thereafter, to ease our notation, we sometimes identify a
representation $\rho^{\prime}$ with the underlying space $V$ on which it acts,
and use $\text{Frob}_{\wp}(V)$ to denote $\rho^{\prime}(\text{Frob}_{\wp})$.
By [16, (30.9)],
$\text{Tr}\,\text{Frob}_{\wp}(\mathcal{H}_{\lambda}\otimes\mathcal{H}_{\lambda})=\mathbb{P}\left(\alpha_{3}(f),\beta_{3}(c);1/\lambda;\kappa_{\wp}\right)^{2}.$
Now consider the representation $\tilde{\rho}$ obtained from $G(N)$ acting on
$V:=H^{1}_{\acute{e}t}(\mathbb{A}\otimes_{R}\overline{\mathbb{Q}},\mathcal{L}_{\phi}\otimes\mathcal{H}^{\otimes
2})$ where $\mathcal{L}_{\phi}$ is a rank-1 sheaf on $\mathbb{A}$
corresponding to the quadratic character $\phi$. At a prime ideal $\wp$ where
the representation is unramified, the trace of
$\tilde{\rho}(\text{Frob}_{\wp})$ on $V$ can be computed from the action of
the Frobenius element in Gal$(\overline{\kappa_{\wp}}/\kappa_{\wp})$ on
$H^{1}_{\acute{e}t}(\mathbb{A}\otimes_{\kappa_{\wp}}\overline{\kappa_{\wp}},\mathcal{L}_{\phi}\otimes\mathcal{H}^{\otimes
2})$ since
$H^{i}_{\acute{e}t}(\mathbb{A}\otimes_{\kappa_{\wp}}\overline{\kappa_{\wp}},\mathcal{L}_{\phi}\otimes\mathcal{H}^{\otimes
2})=0$ for $i\neq 1$. Expressed in terms of the action of the Frobenius
$F_{x}$ at each point $x\in\kappa_{\wp}$, the Leftschetz formula gives
$\text{Tr}\tilde{\rho}(\text{Frob}_{\wp})=-\sum_{x\in\kappa_{\wp}^{\times}}\text{Tr}(x)$
where

$\text{Tr}(x)=\text{Tr}F_{x}((\mathcal{L}_{\phi}\otimes\mathcal{H}^{\otimes
2})_{x})=\phi_{\wp}(x)\mathbb{P}(\alpha_{3}(f),\beta_{3}(c);1/x;\kappa_{\wp})^{2}.$

Putting together we get from (35) that, up to semisimplification,
$(\chi_{-1}\otimes\rho_{HD_{1},\ell})|_{G(N)}\cong\tilde{\rho}.$

Next we will use this isomorphism to obtain a decomposition of
$\rho_{HD_{1},\ell}|_{G(N)}$. Note that $\mathcal{H}^{\otimes
2}=\mathcal{H}\otimes\mathcal{H}$ decomposes naturally into
$\text{Sym}^{2}\mathcal{H}\oplus\text{Alt}^{2}\mathcal{H}$ as $G(N)$-invariant
sheaves. From the discussion of Proposition 4, we further know that
$\text{Sym}^{2}\mathcal{H}=\text{Sym}^{4}\rho\oplus\det(\rho)^{2}$, where
$\rho$ stands for the $\ell$-adic sheaf on $\mathbb{A}$ such that at each
$t\in\mathbb{A}$, $t\neq 0$, the stalk at $t$ gives rise to the representation
in Theorem 1 associated to the pair $\alpha_{2}(c,f),\beta_{2}(c)$.
Accordingly
$H^{1}(\mathbb{A}\otimes_{R}\overline{\mathbb{Q}},\mathcal{L}_{\phi}\otimes\mathcal{H}^{\otimes
2})$ decomposes into
$H^{1}(\mathbb{A}\otimes_{R}\overline{\mathbb{Q}},\mathcal{L}_{\phi}\otimes\text{Alt}^{2}\mathcal{H})\oplus
H^{1}(\mathbb{A}\otimes_{R}\overline{\mathbb{Q}},\mathcal{L}_{\phi}\otimes(\det\rho)^{2})\oplus
H^{1}(\mathbb{A}\otimes_{R}\overline{\mathbb{Q}},\mathcal{L}_{\phi}\otimes\text{Sym}^{4}\rho)$
as $G(N)$-modules, and we write the corresponding decomposition of
representations of $G(N)$ on these cohomology spaces in order as

$\tilde{\rho}=\tilde{\rho}_{AS}\oplus\tilde{\rho}_{ES}\oplus\tilde{\rho}_{SS}.$

This naturally yields the decomposition of the trace at each prime ideal $\wp$
of $R$:
$\text{Tr}\tilde{\rho}(\text{Frob}_{\wp})=-\mathbb{P}(HD_{1},\wp)=AS_{c,f}(\wp)+ES_{c,f}(\wp)+SS_{c,f}(\wp)$
where

$\displaystyle AS_{c,f}(\wp)$ $\displaystyle:=$
$\displaystyle\text{Tr}\,{\tilde{\rho}_{AS}}(\text{Frob}_{\wp})$
$\displaystyle=$
$\displaystyle-\text{Tr}\,F_{1}((\text{Alt}^{2}\mathcal{H})_{1})-\sum_{t\in\kappa_{\wp}^{\times},t\neq
1}\phi_{\wp}(t)\text{Tr}\,F_{t}((\text{Alt}^{2}\,\mathcal{H})_{t})$
$\displaystyle\overset{\eqref{eq:tr-Alt-1},\eqref{eq:tr-Alt}}{=}$
$\displaystyle-\psi_{c,f}(\text{Frob}_{\wp})N(\wp)^{2}-\sum_{t}\phi_{\wp}(t)\phi_{\wp}(1-1/t)N(\wp)\cdot\mathbb{P}(\alpha_{3}(f),\beta_{3}(c);1/t;\kappa_{\wp})$
$\displaystyle\overset{\eqref{eq:P-by-induction}}{=}-$
$\displaystyle\psi_{c,f}(\text{Frob}_{\wp})N(\wp)^{2}-N(\wp)\cdot\mathbb{P}\left(\\{\frac{1}{2},\frac{1}{2},f,1-f\\},\\{1,1,\frac{3}{2}-c,\frac{1}{2}+c\\};1;\kappa_{\wp}\right)$
$\displaystyle\overset{Cor.\ref{cor:psi}}{=}$
$\displaystyle-N(\wp)^{2}-N(\wp)\cdot\mathbb{P}(HD_{2};\kappa_{\wp})$
$\displaystyle=$
$\displaystyle-N(\wp)^{2}+\phi_{\wp}(-1)N(\wp)\text{Tr}\,\rho_{HD_{2},\ell}(\text{Frob}_{\wp});$
$\displaystyle ES_{c,f}(\wp)$ $\displaystyle:=$
$\displaystyle\text{Tr}\,{\tilde{\rho}_{ES}}(\text{Frob}_{\wp})=-\sum_{t\neq
0,1}\phi_{\wp}(t)N(\wp)^{2}=N(\wp)^{2};$ $\displaystyle SS_{c,f}(\wp)$
$\displaystyle:=$
$\displaystyle\text{Tr}\,{\tilde{\rho}_{SS}}(\text{Frob}_{\wp}).$

From the above expressions, we see that the representation of $G(N)$ on
$H^{1}(\mathbb{A}\otimes_{R}\overline{\mathbb{Q}},\mathcal{L}_{\phi}\otimes\text{Alt}^{2}\mathcal{H})\oplus
H^{1}(\mathbb{A}\otimes_{R}\overline{\mathbb{Q}},\mathcal{L}_{\phi}\otimes(\det\rho)^{2})$
is isomorphic to
$\epsilon_{\ell}\otimes\chi_{-1}\otimes\rho_{HD_{2},\ell}|_{G(N)}$. Further,
explicit computations as explained in §2.3 show that the
$\rho_{HD_{1},\ell}|_{G(N)}$ contains no irreducible subrepresentations of
multiplicity larger than 1. Consequently, we obtain the following isomorphism:

(36)
$\displaystyle\rho_{HD_{1},\ell}|_{G(N)}\simeq(\epsilon_{\ell}\otimes\rho_{HD_{2},\ell})|_{G(N)}\oplus(\tilde{\rho}_{SS}\otimes\chi_{-1})|_{G(N)},$

showing the decomposition of $\rho_{HD_{1},\ell}|_{G(N)}$ into a sum of a
degree-3 and a degree-2 representations. ∎

By Theorem 1, iii), we can decompose the degree-$3$ representation
$\rho_{HD_{2},\ell}$ as
$\rho_{HD_{2},\ell}^{prim}\oplus\rho_{HD_{2},\ell}^{1}$, and similarly,
$\rho_{HD_{1},\ell}=\rho_{HD_{1},\ell}^{prim}\oplus\rho_{HD_{1},\ell}^{1}$.
Contained in the proof above is the information on $\rho_{HD_{2},\ell}^{1}$
and $\rho_{HD_{1},\ell}^{1}$.

######  Corollary 2.

For any pair $(c,f)$ in (11) and any prime $\ell$,

$\rho_{HD_{2}(c,f),\ell}^{1}|_{G(N(c,f))}=\chi_{-1}\epsilon_{\ell}|_{G(N(c,f))}\quad
and\quad\rho_{HD_{1}(c,f),\ell}^{1}|_{G(N(c,f))}=\chi_{-1}\epsilon_{\ell}^{2}|_{G(N(c,f))}.$

As a corollary, we can write down the following finite field analogue of (8).

######  Corollary 3.

For any pair $(c,f)$ in (11) and any prime ideal $\wp$ of
$\mathbb{Z}[\zeta_{N(c,f)},1/N(c,f)]$ we have

(37)
$-\mathbb{P}(HD_{1};\kappa_{\wp})=-N(\wp)\cdot\mathbb{P}(HD_{2};\kappa_{\wp})+SS_{c,f}(\wp).$

###  3.2. A proof of Theorem 4

For any $(c,f)$ as in (11), $HD_{2}(c,f)$ and $HD_{2}(f,c)$ may not agree by
definition. Nonetheless the representations $\rho_{HD_{2}(c,f),\ell}$ and
$\rho_{HD_{2}(f,c),\ell}$ are isomorphic because of the following Lemma.

######  Lemma 2.

Given any $(c,f)$ in (11), for any prime power $q\equiv 1\mod M(c,f)$,

$\mathbb{P}(\alpha_{4}(f),\beta_{4}(c);1;\mathbb{F}_{q})=\mathbb{P}(\alpha_{4}(c),\beta_{4}(f);1;\mathbb{F}_{q}).$

###### Proof.

$\mathbb{P}(\alpha_{4}(f),\beta_{4}(c);1;\mathbb{F}_{q})=\,_{4}{\mathbb{P}}_{3}\left[\begin{matrix}\phi&\phi&\omega^{(q-1)f}&\omega^{(q-1)(1-f)}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\varepsilon&\omega^{(q-1)(\frac{1}{2}-c)}&\omega^{(q-1)(\frac{1}{2}+c)}\end{matrix}\;;\;1;q\right]$

By Proposition 1, the right hand side coincides with
$\,{}_{4}{\mathbb{P}}_{3}\left[\begin{matrix}\phi&\phi&\omega^{(q-1)c}&\omega^{(q-1)(1-c)}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\varepsilon&\omega^{(q-1)(\frac{1}{2}-f)}&\omega^{(q-1)(\frac{1}{2}+f)}\end{matrix}\;;\;1;q\right]$
which is $\mathbb{P}(\alpha_{4}(c),\beta_{4}(f);1;\mathbb{F}_{q})$. ∎

It is clear that when $(c,f)$ is one of the first five pair, $HD_{2}$ is also
defined over $\mathbb{Q}$. When $(c,f)$ is either
$\left(\frac{1}{5},\frac{2}{5}\right)$ or
$\left(\frac{1}{10},\frac{3}{10}\right)$, one has

######  Lemma 3.

Let $(c,f)$ be either $\left(\frac{1}{5},\frac{2}{5}\right)$ or
$\left(\frac{1}{10},\frac{3}{10}\right)$. Then the hypergeometric data
$HD_{2}=\\{\alpha_{4}(f),\beta_{4}(c);1\\}$ can be extended to $\mathbb{Q}$.

###### Proof.

When $(c,f)=(\frac{1}{5},\frac{2}{5})$, $M=10$ and $\rho_{HD,\ell}$ is a
representation of $G(10)=G(5)$. Since $HD_{2}$ is self-dual and defined over
$K=\mathbb{Q}(\zeta_{5}+\zeta_{5}^{-1})=\mathbb{Q}(\sqrt{5})$,
$\rho_{HD_{2},\ell}$ is invariant under the Galois conjugation sending
$\zeta_{5}$ to $\zeta_{5}^{-1}$ so that it can be extended to $G_{K}$, the
absolute Galois group of $K$. The Galois group $\text{Gal}(K/\mathbb{Q})$ is
generated by $\tau:\zeta_{5}\mapsto\zeta_{5}^{3}$, which maps
$\alpha_{4}(f)=\\{\frac{1}{2},\frac{1}{2},\frac{2}{5},\frac{3}{5}\\}$ to
$\\{\frac{1}{2},\frac{1}{2},\frac{1}{5},\frac{4}{5}\\}$ and
$\beta_{4}(c)=\\{1,1,\frac{3}{10},\frac{7}{10}\\}$ to
$\\{1,1,\frac{9}{10},\frac{1}{10}\\}$. For all primes $p\equiv 1\mod 5$, we
have $p\equiv 1\mod 10$ and by Lemma 2,

$\mathbb{P}(\alpha_{4}(f),\beta_{4}(c);1;\mathbb{F}_{p})=\mathbb{P}(\alpha_{4}(c),\beta_{4}(f);1;\mathbb{F}_{p})=\mathbb{P}\left(\left\\{\frac{1}{2},\frac{1}{2},\frac{1}{5},\frac{4}{5}\right\\},\left\\{1,1,\frac{9}{10},\frac{1}{10}\right\\};1;\mathbb{F}_{p}\right).$

This shows that $\rho_{HD_{2},\ell}$ of $G_{K}$ is invariant under conjugation
by $\tau$, and hence it can be extended to $\mathbb{Q}$. The case of
$(c,f)=\left(\frac{1}{10},\frac{3}{10}\right)$ can be handled in a parallel
manner. ∎

Now that we know each $\rho_{HD_{2},\ell}$ has an extension to
$G_{\mathbb{Q}}$, we proceed to prove the assertions about
$\rho_{HD_{2},\ell}^{BCM}$ in Theorem 4. For
$(c,f)=(\frac{1}{2},\frac{1}{2}),(\frac{1}{2},\frac{1}{3}),(\frac{1}{2},\frac{1}{6})$,
this has been established in [41]. We prove the remaining cases. If $HD_{2}$
is defined over $\mathbb{Q}$, from Theorems 1 and 2, we know each
$\rho_{HD_{2},\ell}^{BCM}\cong\rho_{HD_{2},\ell}^{BCM,prim}\oplus\rho_{HD_{2},\ell}^{BCM,1}$
where $\rho_{HD_{2},\ell}^{BCM,1}$ is one-dimensional. When $HD_{2}$ is not
defined over $\mathbb{Q}$, we continue to use $\rho_{HD_{2},\ell}^{BCM,prim}$
and $\rho_{HD_{2},\ell}^{BCM,1}$ to denote an extension of the degree-2 and
degree-1 pieces of $\rho_{HD_{2},\ell}$ to $G_{\mathbb{Q}}$ with integral
traces.

We first deal with $\rho_{HD_{2},\ell}^{BCM,prim}$ by using Theorem 8 to prove
the modularity of $\rho_{HD_{2},\ell}^{BCM,prim}[t]$ where $t$ is computed by
(26) and identify the corresponding cuspidal Hecke eigenform. Then we
determine $\rho_{HD_{2},\ell}^{BCM,1}$.

Using the Magma program called “Hypergeometric Motives over $\mathbb{Q}$”
implemented by Watkins [57], the characteristic polynomial of
$\rho_{HD_{2},\ell}^{BCM,prim}{[t]}(\text{Fr}_{p})$ for $p\geq 5$ can be
computed efficiently by choosing $\ell\neq p$ as mentioned in §2.3. Thus for
each $p>3$, the trace and determinant of
$\rho_{HD_{2},\ell}^{BCM,prim}{[t]}(\text{Fr}_{p})$ can be read off. As
$HD_{2}$ is defined over $\mathbb{Q}$, the characteristic polynomials of
$\rho_{HD_{2},\ell}^{BCM,prim}{[t]}(\text{Fr}_{p})$ all have coefficients in
$\mathbb{Z}$. The representation $\rho_{HD_{2},\ell}^{BCM,prim}$ is unramified
outside $\\{2,3,\ell\\}$ for $(c,f)=(\frac{1}{3},\frac{1}{3})$ and
$(\frac{1}{6},\frac{1}{6})$, and these $\ell$-adic representations are
compatible as $\ell$ varies. Hence the determinant of this system can ramify
only at $2$ and $3$. To compute it, we may choose $\ell=2$ or 3. As $n=4$ and
dimension of $\rho_{HD_{2},\ell}^{BCM,prim}$ is 2, by iii) of Theorem 2, and
the absolute value of the roots of
$\rho_{HD_{2},\ell}^{BCM,prim}(\text{Frob}_{p})$ are $p^{3/2}$ for $p>3$, one
sees that $\det\rho_{HD_{2},\ell}^{BCM,prim}=\chi_{c,f}\epsilon_{\ell}^{3}$
where $\epsilon_{\ell}$ denotes the unramified $\ell$-adic cyclotomic
character with $\epsilon_{\ell}(\text{Frob}_{p})=p$ and $\chi_{c,f}$ has order
at most 2. Since $\chi_{c,f}$ is unramified outside $2$ and $3$, its conductor
divides 24. From computing the first few characteristic polynomials of
$\rho_{HD_{2},\ell}^{BCM,prim}{[t]}(\text{Fr}_{p})$, we confirm that
$\chi_{c,f}$ is the trivial character for each case. So
$\det\rho_{HD_{2},\ell}^{BCM,prim}=\epsilon_{\ell}^{3}$ which implies
$\rho_{HD_{2},\ell}^{BCM,prim}$ and hence each
$\rho_{HD_{2},\ell}^{BCM,prim}[t]$ (which is $2$-dimensional) is odd for one
and hence all $\ell$. The irreducibility of $\rho_{HD_{2},\ell}^{BCM,prim}$
follows from the lemma below since it has degree $2$ and determinant
$\epsilon_{\ell}^{3}$.

######  Lemma 4.

Suppose the hypergeometric datum $HD=\\{\alpha,\beta;1\\}$ in Theorem 1 is
such that, for all primes $\ell$, the representation $\rho_{HD,\ell}^{prim}$
of $G(M)$ has a degree-$2$ subrepresentation which can be extended to a
representation $\rho_{\ell}$ of $G_{\mathbb{Q}}$ and the $\rho_{\ell}$ form a
compatible family. If $\det\rho_{\ell}$, up to a finite twist, is
$\epsilon_{\ell}^{m}$ for an odd natural number $m$, then $\rho_{\ell}$ is
irreducible.

###### Proof.

Suppose otherwise. Then $\rho_{\ell}$ is of the form
$\chi_{1}\epsilon_{\ell}^{a}\oplus\chi_{2}\epsilon_{\ell}^{b}$, where $a,b$
are non-negative integers with $a+b=m$ and $\chi_{1},\chi_{2}$ are characters
of $G_{\mathbb{Q}}$ of finite order. Thus the eigenvalues of
$\rho_{HD,\ell}^{prim}(\text{Frob}_{\wp})$ would not have the same absolute
values at almost all prime ideals $\wp$, contradicting Theorem 1, iii). ∎

To apply Theorem 8 to conclude the modularity of
$\rho_{HD_{2},\ell}^{BCM,prim}[t]$, choose $\ell>5$ to be a prime which splits
completely in $\mathbb{Q}(\zeta_{M})$. Then the action of
$\rho_{HD_{2},\ell}^{BCM,prim}[t]$ restricted to a decomposition group
$D_{\ell}$ at $\ell$ is isomorphic to the action of
$\rho_{HD_{2},\ell}^{prim}[t]$ restricted to a decomposition group $D_{\wp}$
at a prime $\wp$ of $\mathbb{Q}(\zeta_{M})$ above $\ell$, which is known to be
crystalline by Katz [31].

For $(c,f)=(\frac{1}{3},\frac{1}{3})$, the graph of $e_{\alpha,\beta}$ with
$\alpha=\\{\frac{1}{2},\frac{1}{2},\frac{1}{3},\frac{2}{3}\\},\beta=\\{1,1,\frac{7}{6},\frac{5}{6}\\}$
is as follows.

In this case $\min e_{\alpha,\beta}=-1$, $t=1-\frac{4-2}{2}=0$, and
$w(HD_{2})=3-(-1)=4.$ This means $\rho_{HD_{2},\ell}^{BCM,prim}$ has Hodge-
Tate weights $\\{0,3\\}$. By Theorem 8, $\rho_{HD_{2},\ell}^{BCM,prim}$ is
modular and the corresponding modular form is of weight-4 with integer
coefficients and by Theorem 9 its level divides $2^{8}\cdot 3^{5}$. From the
list of cuspidal Hecke eigenforms of weight-4 and level dividing $2^{8}\cdot
3^{5}$ listed in the database. We see that only $f_{6.4.a.a}$ has the matching
$p$-coefficients for all primes within the range $5\leq p<100$.

When $(c,f)=(\frac{1}{6},\frac{1}{6})$, $\min=0,\,\max=2$, $w(HD_{2})=2$ and
$t=-1$. By a similar analysis we know $\rho_{HD_{2},\ell}^{BCM,prim}[-1]$ is
modular and is isomorphic to the $\ell$-adic Galois representation attached to
$f_{24.2.a.a}$.

Next we determine $\rho_{HD_{2},\ell}^{BCM,1}$. We implemented a SageMath
program to compute $H_{p}(\alpha,\beta;1)$. From Theorem 2, we know
$\rho_{HD_{2},\ell}^{BCM,1}$ is 1-dimensional and its value at
$\text{Frob}_{p}$ can be computed by

$\text{Tr}\rho_{HD_{2},\ell}^{BCM,1}(\text{Frob}_{p})=p^{{(n-m)/2}}H_{p}(\alpha,\beta;1)-\text{Tr}\rho_{HD_{2},\ell}^{BCM,prim}(\text{Frob}_{p}).$

So $\rho_{HD_{2},\ell}^{BCM,1}(\text{Frob}_{p})=\varphi_{c,f}(p)p$ for some
character $\varphi_{c,f}$ of order at most $2$ and conductor dividing 24. By
computing the first few primes $p$, $\varphi_{c,f}$ is identified quickly. For
example, when $\alpha=\\{\frac{1}{2},\frac{1}{2},\frac{1}{3},\frac{2}{3}\\}$,
$\beta=\\{1,1,\frac{7}{6},\frac{5}{6}\\}$, for $p\in\\{7,11,13,17,19\\}$ the
values of $\rho_{HD_{2},\ell}^{BCM,1}(\text{Frob}_{p})$ computed from
$pH_{p}(\alpha,\beta;1)-\text{Tr}\rho_{HD_{2},\ell}^{BCM,prim}(\text{Frob}_{p})$
are $7,-11,13,-17,19$, respectively, which agree with
$\left(\frac{-3}{p}\right)p.$

For the last two cases $(c,f)=\left(\frac{1}{5},\frac{2}{5}\right)$ and
$\left(\frac{1}{10},\frac{3}{10}\right)$, each
$\rho_{HD_{2},\ell}=\rho_{HD_{2},\ell}^{prim}+\rho_{HD_{2},\ell}^{1}$ is a
representation of $G(10)$. Since $\rho_{HD_{2},\ell}$ and
$\rho_{HD_{2},\ell}^{1}$ can be extended to $G_{\mathbb{Q}}$, so can
$\rho_{HD_{2},\ell}^{prim}$, and each extension is unique up to twists by the
$4$ characters of $G_{\mathbb{Q}}$ trivial on $G(10)$. Unlike the previous $5$
cases, there is no preferred choice; we shall choose extensions with integral
traces. From iii) of Theorem 1 and numeric computation we know
$\det\rho_{HD_{2},\ell}^{prim}=(\epsilon_{\ell}^{3})|_{G(10)}$. Thus any
extension of $\rho_{HD_{2},\ell}^{prim}$ from $G(10)$ to $G_{\mathbb{Q}}$ has
determinant either $\epsilon_{\ell}^{3}$ or
$\chi_{5}\otimes\epsilon_{\ell}^{3}$, and hence is odd. It is also irreducible
by Lemma 4. By Corollary 2,
$\rho_{HD_{2},\ell}^{1}|_{G(20)}=\epsilon_{\ell}|_{G(20)}$. Thus we can obtain
the trace of $\rho_{HD_{2},\ell}^{BCM,prim}$ at any $\text{Frob}_{p}$ with
$p\equiv 1\mod 20$ from computing
$\text{Tr}\rho_{HD_{2},\ell}(\text{Frob}_{\wp})$ at any prime ideal $\wp$ of
$\mathbb{Z}[\zeta_{20}]$ above such $p$ and coprime to $\ell$ which, by Remark
1, allows us to identify the corresponding modular form up to quartic twists,
parallel to the first 5 cases. The degree-1 piece can be determined
accordingly.

###  3.3. Proof of Theorem 5

For each pair $(c,f)$ listed in (11), we will proceed in a way similar to the
proof of Theorem 4 by dealing with $\rho_{HD_{1},\ell}^{BCM,prim}$ and then
its $1$-dimensional complement $\rho_{HD_{1},\ell}^{BCM,1}$ which can be
determined the same way as the end of the proof of Theorem 4.

######  Proposition 5.

For each pair $(c,f)$ in (11), the 4-dimensional representation
$\rho_{HD_{1},\ell}^{BCM,prim}$ is the sum of two odd degree-2 irreducible
automorphic representations of $G_{\mathbb{Q}}$.

###### Proof.

Fix a pair $(c,f)$ in (11). Since $HD_{1}$ is defined over $\mathbb{Q}$ and
$HD_{2}$ can be extended to $\mathbb{Q}$, the representations
$\rho_{HD_{1},\ell}|_{G(N)}$ and
$(\epsilon_{\ell}\otimes\rho_{HD_{2},\ell})|_{G(N)}$ can be extended to
$G_{\mathbb{Q}}$ as $\rho_{HD_{1},\ell}^{BCM}$ and
$\epsilon_{\ell}\otimes\rho_{HD_{2},\ell}^{BCM}$ respectively. Hence by (36),
$(\tilde{\rho}_{SS}\otimes\chi_{-1})|_{G(N)}$ can be extended to
$\tilde{\rho}_{SS}^{BCM}\otimes\chi_{-1}$ of $G_{\mathbb{Q}}$. Therefore
$\rho_{HD_{1},\ell}^{BCM}$ decomposes as the sum of a finite twist of
$\epsilon_{\ell}\otimes\rho_{HD_{2},\ell}^{BCM}$ and a finite twist of
$\tilde{\rho}_{SS}^{BCM}\otimes\chi_{-1}$, written as

(38)
$\rho_{HD_{1},\ell}^{BCM,prim}\cong\sigma_{HD_{1},alt,\ell}\oplus\sigma_{HD_{1},sym,\ell}$

where $\sigma_{HD_{1},sym,\ell}$ and $\sigma_{HD_{1},alt,\ell}$ are both
2-dimensional. Here we use $sym$ and $alt$ in the subscripts in order to keep
track of their origin. It is useful to keep in mind that they will play
asymmetric roles in general. From Theorem 4,
$\det\sigma_{HD_{1},alt,\ell}=\det(\epsilon_{\ell}\otimes\rho_{HD_{2},\ell}^{prim})=\epsilon_{\ell}^{5}$
or $\chi_{5}\epsilon_{\ell}^{5}$. We also determine
$\det\rho_{HD_{1},\ell}^{BCM,prim}=\epsilon_{\ell}^{10}$ the same way as the
proof of Theorem 4. Hence $\sigma_{HD_{1},sym,\ell}$ and
$\sigma_{HD_{1},alt,\ell}$ have the same determinant and are both odd. They
are irreducible by Lemma 4. Since $\rho_{HD_{1},\ell}^{BCM,prim}$ and
$\sigma_{HD_{1},alt,\ell}$ are both crystalline for a large prime $\ell\equiv
1\mod N$ by Katz [31], so is $\sigma_{HD_{1},sym,\ell}$. By Theorem 8, both
$\sigma_{HD_{1},alt,\ell}$ and $\sigma_{HD_{1},sym,\ell}$ are modular. ∎

Our next task is to determine the modular forms $\mathfrak{f}_{c,f}$ and
$\mathfrak{g}_{c,f}$ corresponding to $\sigma_{HD_{1}(c,f),sym,\ell}$ and
$\sigma_{HD_{1}(c,f),alt,\ell}$ respectively in the decomposition (38) of
$\rho_{HD_{1},\ell}^{BCM,prim}$.

As each $HD_{1}$ is defined over $\mathbb{Q}$, using Magma, one can compute
the factorization of $\rho_{HD_{1},\ell}^{BCM,prim}[t](\text{Fr}_{p})$ over
$\mathbb{Z}$ for any unramified prime $p$. It turns out that the
characteristic polynomial of $\rho_{HD_{1},\ell}^{BCM,prim}[t](\text{Fr}_{p})$
always factorizes into 2 degree-2 factors over $\mathbb{Z}$. For example,

H1:=HypergeometricData([1/2,1/2,1/2,1/2,1/3,2/3],[1,1,1,1,1/6,5/6]);

Factorization(EulerFactor(H1,1,7));

The out put is

$<16807*\$.1^{2}-56*\$.1+1,1>,$

$<16807*\$.1^{2}+88*\$.1+1,1>$

Our next task is to assign the degree-2 factors to $\sigma_{HD_{1},sym,\ell}$
and $\sigma_{HD_{1},alt,\ell}$ respectively. By Theorem 3,
$\sigma_{HD_{1},alt,\ell}|_{G(N)}$ and
$\left(\epsilon_{\ell}\otimes\rho_{HD_{2},\ell}^{BCM,prim}\right)|_{G(N)}$
agree. This implies that at each unramified prime $p$,
$\text{Tr}\sigma_{HD_{1},alt,\ell}(\text{Frob}_{p})$ and
$p\cdot\text{Tr}\rho_{HD_{2},\ell}^{BCM,prim}(\text{Frob}_{p})$ differ by a
root of unity. Since they both have integer coefficients, they differ at most
by a quadratic character. This allows us to recognize which quadratic factor
is from $\sigma_{HD_{1},alt,\ell}|_{G(N)}$. For instance,

H2:=HypergeometricData([1/2,1/2,1/3,2/3],[1,1,1,1]);

Factorization(EulerFactor(H2,1,7));

The out put is

$<343*\$.1^{2}-8*\$.1+1,1>$

From which we know the first quadratic factor computed from $H1$ is from
$\sigma_{HD_{1},alt,\ell}.$ The separation of the quadratic factors allows us
to compute the trace and determinant of $\sigma_{HD_{1},sym,\ell}$ and
$\sigma_{HD_{1},alt,\ell}$ respectively. Similar to the proof of Theorem 4,
the determination of $\mathfrak{f}_{c,f}$ and $\mathfrak{g}_{c,f}$ boils down
to the application of Theorem 9 as outlined in §2.3.

##  4\. Hypergeometric values and periods of modular forms

In this section we study untruncated hypergeometric series. Our main purpose
is to relate hypergeometric values and periods of modular forms.

###  4.1. Modular forms and differential equations

In [61], Zagier used

$\,{}_{4}F_{3}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1&1&1\end{matrix}\;;\;1\right]=\frac{1}{2\pi
i}\oint_{|t|=1}\,{}_{2}F_{1}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1\end{matrix}\;;\;t\right]\,_{2}F_{1}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1\end{matrix}\;;\;1/t\right]\frac{dt}{t}$

with $t$ to be the modular lambda function to obtain

$\,{}_{4}F_{3}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1&1&1\end{matrix}\;;\;1\right]=\frac{16}{\pi^{2}}L(f_{8.4.a.a},2).$

In [55] Stiller studied periods of modular forms and their relation to the
Eichler-Shimura parabolic cohomology theory. It is known in the literature
(see [55] by Stiller and [59, Theorem 1] by Yang) that

######  Theorem 10.

Given a weight-$k$ meromorphic modular form $f(\tau)$ of a Fuschian group
$\Gamma$ in $SL_{2}(\mathbb{R})$ which is commensurable with
$SL_{2}(\mathbb{Z})$ and a non-constant modular function $t$ of $\Gamma$, then
$f(t),\tau f(t),$ $\cdots,\tau^{k}f(t)$ satisfy an order $k+1$ differential
equation in variable $t$ which has only regular singularities.

Because of this reason the period formulas below often involve polynomials in
$\tau$.

To prove Theorem 6, we first recall some relevant properties of hypergeometric
functions [20, Chapters 15, 16] (For more detail, please see [3, 53] for
example.):

  1. (P.1)

[20, (16.8.7)] The functions
$(-z)^{-a_{1}}\,_{3}F_{2}\left[\begin{matrix}a_{1}&1+a_{1}-b_{2}&1+a_{1}-b_{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1+a_{1}-a_{2}&1+a_{1}-a_{3}\end{matrix}\;;\;\frac{1}{z}\right]$ and
$\,{}_{3}F_{2}\left[\begin{matrix}a_{1}&a_{2}&a_{3}\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\ &b_{2}&b_{3}\end{matrix}\;;\;z\right]$ satisfy the same 3rd
order differential equation.

  2. (P.2)

[20, §15.10, §16.8] The following four functions

$\,{}_{2}F_{1}\left[\begin{matrix}a&b\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&a+b+\frac{1}{2}\end{matrix}\;;\;z\right]^{2},\quad\,_{3}F_{2}\left[\begin{matrix}2a&2b&a+b\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&2a+2b&a+b+\frac{1}{2}\end{matrix}\;;\;z\right],$
$z^{\frac{1}{2}-a-b}\,_{2}F_{1}\left[\begin{matrix}a&b\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\
&a+b+\frac{1}{2}\end{matrix}\;;\;z\right]\,_{2}F_{1}\left[\begin{matrix}\frac{1}{2}-a&\frac{1}{2}-b\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{3}{2}-a-b\end{matrix}\;;\;z\right],\quad
z^{\frac{1}{2}-a-b}\,_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&a-b+\frac{1}{2}&b-a+\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&a+b+\frac{1}{2}&\frac{3}{2}-a-b\end{matrix}\;;\;z\right]$

satisfy the same 3rd order differential equation (ODE) and hence are linearly
dependent.

  3. (P.3)

[20, (16.12.2)] The Clausen’s formulas give two identities related to the
functions in (P.2):

(39) $\,_{2}F_{1}\left[\begin{matrix}a&b\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&a+b+\frac{1}{2}\end{matrix}\;;\;z\right]^{2}=\,_{3}F_{2}\left[\begin{matrix}2a&2b&a+b\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&2a+2b&a+b+\frac{1}{2}\end{matrix}\;;\;z\right]$
$\,{}_{2}F_{1}\left[\begin{matrix}a&b\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&a+b+\frac{1}{2}\end{matrix}\;;\;z\right]\,_{2}F_{1}\left[\begin{matrix}\frac{1}{2}-a&\frac{1}{2}-b\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{3}{2}-a-b\end{matrix}\;;\;z\right]=\,_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&a-b+\frac{1}{2}&b-a+\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&a+b+\frac{1}{2}&\frac{3}{2}-a-b\end{matrix}\;;\;z\right].$

Moreover, $\,{}_{2}F_{1}\left[\begin{matrix}a&b\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ &a+b+\frac{1}{2}\end{matrix}\;;\;z\right]$ and
$z^{\frac{1}{2}-a-b}\,_{2}F_{1}\left[\begin{matrix}\frac{1}{2}-a&\frac{1}{2}-b\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &\frac{3}{2}-a-b\end{matrix}\;;\;z\right]$
satisfy the same ordinary differential equation.

Put together, we see that
$z^{-1}\,_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&a-b+\frac{1}{2}&b-a+\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&a+b+\frac{1}{2}&\frac{3}{2}-a-b\end{matrix}\;;\;1/z\right]\,_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&a-b+\frac{1}{2}&b-a+\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&a+b+\frac{1}{2}&\frac{3}{2}-a-b\end{matrix}\;;\;z\right]$ and
$z^{-1/2}\,_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&a-b+\frac{1}{2}&b-a+\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&a+b+\frac{1}{2}&\frac{3}{2}-a-b\end{matrix}\;;\;z\right]^{2}$ satisfy the
same ODE. Letting $z=u^{2}$, we get the same conclusion for
$\,{}_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&a-b+\frac{1}{2}&b-a+\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&a+b+\frac{1}{2}&\frac{3}{2}-a-b\end{matrix}\;;\;u^{2}\right]^{2}$ and
$\,{}_{2}F_{1}\left[\begin{matrix}a&b\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&a+b+\frac{1}{2}\end{matrix}\;;\;u^{2}\right]^{4}$.

###  4.2. The occurrence of $\mathfrak{f}_{c,f}$ in $w(HD_{1})=6$ cases

We show how $\mathfrak{f}_{c,f}$ in Theorem 5 for the three cases where
$w(HD_{1})=6$ arises naturally in connection with a ${}_{2}F_{1}$ function.

######  Theorem 11.

When
$(c,f)=(\frac{1}{2},\frac{1}{2}),(\frac{1}{2},\frac{1}{3}),(\frac{1}{3},\frac{1}{3})$,
the monodromy group $\Gamma(c,f)$ of the order-2 ODE satisfied by
$\,{}_{2}F_{1}\left[\begin{matrix}\frac{1+2f-2c}{4}&\frac{3-2c-2f}{4}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &\frac{3}{2}-c\end{matrix}\;;\;z^{2}\right]$
is a genus 0 arithmetic Fuchsian group. The space of weight-6 holomorphic
modular forms for $\Gamma(c,f)$, is $1$-dimensional, generated by
$\mathfrak{h}_{c,f}$, which is a Hecke eigenform with eigenvalue
$a_{p}(\mathfrak{f}_{c,f})$ for almost all primes $p>7$.

###### Proof.

We first compute the monodromy group $G(c,f)$ of the 2nd order ODE satisfied
by

(40)
${}_{2}F_{1}(x):=\,_{2}F_{1}\left[\begin{matrix}\frac{1+2f-2c}{4}&\frac{3-2c-2f}{4}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{3}{2}-c\end{matrix}\;;\;x\right]=\,_{2}F_{1}\left[\begin{matrix}a&b\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &a+b+\frac{1}{2}\end{matrix}\;;\;x\right],$

(namely letting $a=\frac{1+2f-2c}{4},b=\frac{3-2c-2f}{4}$) which is an
arithmetic triangle group with angles
$\pi|a+b-\frac{1}{2}|,\frac{\pi}{2},\pi|a-b|$. For related background, see
[24, §3.2.4]. Then under the change of variable $x=z^{2}$, the singularities
of the ODE in variable $z$ satisfied by ${}_{2}F_{1}(z^{2})$ are
$\\{\infty,0,1,-1\\}$ and the monodromy group is an index-2 subgroup
$\Gamma(c,f)$ of $G(c,f)$. We will give a description for $\Gamma(c,f)$ as a
matrix group for each case, and then describe the space $S_{6}(\Gamma(c,f))$
of weight-6 holomorphic modular forms on $\Gamma(c,f)$ which vanish at each
cusp if there is any.

In what follows, we use the signature $(e_{1},e_{2},\cdots,e_{r})$ to denote a
genus-zero group $\Gamma$ which has $r$ in-equivalent elliptic points and
cusps together of respective order $e_{1}$, $e_{2}$, $\cdots$, $e_{r}$, where
$e_{i}\in\mathbb{Z}_{>1}\cup\\{\infty\\}$. According to [52, Theorem 2.23] by
Shimura, when $k\geq 4$ is even, the dimension of the space of weight-$k$
holomorphic cusp forms on $\Gamma$ is

$\dim
S_{k}(\Gamma)=-k+1+\sum_{i=1}^{r}\lfloor\frac{k}{2}(1-1/e_{i})\rfloor-\\#\\{cusps\\}.$

It follows that $\dim S_{6}(\Gamma(c,f))=1$ when $w(HD_{1})=6$.

For $(c,f)=(\frac{1}{2},\frac{1}{2})$, by (40),
${}_{2}F_{1}(x)=\,_{2}F_{1}\left[\begin{matrix}\frac{1}{4}&\frac{1}{4}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1\end{matrix}\;;\;x\right]$. In this case the
signature of $G(\frac{1}{2},\frac{1}{2})$ is $(\infty,\infty,2)$, and
$G(\frac{1}{2},\frac{1}{2})$ is isomorphic to $\Gamma_{0}(2)$. The group
$\Gamma(\frac{1}{2},\frac{1}{2})$ has signature $(\infty,\infty,2,2)$ and is
isomorphic to the level 4 index-6 subgroup of $SL_{2}(\mathbb{Z})$ labelled by
$4C^{0}$ in [15]. The group $4C^{0}$ is a supergroup of $\Gamma_{0}(8)$. By
the above dimension formula, $\dim
S_{6}(\Gamma(\frac{1}{2},\frac{1}{2}))=-5+2\lfloor\frac{6}{4}\rfloor+2(3-1)=1$.
The space is generated by $f_{8.6.a.a}(\tau)$, which is
$\mathfrak{f}_{\frac{1}{2},\frac{1}{2}}(\tau)$ in Theorem 5.

For $(c,f)=(\frac{1}{2},\frac{1}{3})$,
${}_{2}F_{1}(x)=\,_{2}F_{1}\left[\begin{matrix}\frac{1}{6}&\frac{1}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1\end{matrix}\;;\;x\right]$. In this case the
signature of $G(\frac{1}{2},\frac{1}{3})$ is $(\infty,6,2)$. The group
$G(\frac{1}{2},\frac{1}{3})$ can be realized as

$\Gamma_{0}^{+}(3)=\left\langle
w_{3}:=\frac{1}{\sqrt{3}}\begin{pmatrix}0&-1\\\
3&0\end{pmatrix},T:=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}\right\rangle,$

and its index-2 subgroup $\Gamma(\frac{1}{2},\frac{1}{3})$ has signature
$(2,2,3,\infty)$. This group and $\Gamma_{0}(3)$ are generated by

$\left\\{w_{3},\,(w_{3}T^{-1})^{2}=\begin{pmatrix}-1&1\\\
-3&2\end{pmatrix},T^{2}\right\\},\quad\mbox{and}\quad\left\\{T,\,\begin{pmatrix}-1&1\\\
-3&2\end{pmatrix},\,\begin{pmatrix}-1&0\\\ 3&-1\end{pmatrix}\right\\},$

respectively. Their intersection has signature $(3,3,\infty,\infty)$,
generated by

$\left\\{T^{2},\,\begin{pmatrix}-1&1\\\
-3&2\end{pmatrix},\,\begin{pmatrix}1&0\\\
-6&1\end{pmatrix},\,\begin{pmatrix}-1&-1\\\
3&2\end{pmatrix}=T^{-1}\begin{pmatrix}-1&1\\\ -3&2\end{pmatrix}T\right\\}.$

This group has a supergroup, which is the index-2 subgroup of
$SL_{2}(\mathbb{Z})$ with signature $(3,3,\infty)$. It has a weight-6 cuspform
$f_{4.6.a.a}(\tau)=\eta(2\tau)^{12}$, which is
$\mathfrak{f}_{\frac{1}{2},\frac{1}{3}}(\tau)$ in Theorem 5. So
$S_{6}(\Gamma(\frac{1}{2},\frac{1}{3}))=\langle f_{4.6.a.a}(\tau)\rangle.$

For $(c,f)=(\frac{1}{3},\frac{1}{3})$,
${}_{2}F_{1}(x)=\,_{2}F_{1}\left[\begin{matrix}\frac{1}{4}&\frac{5}{12}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &\frac{7}{6}\end{matrix}\;;\;x\right]$. Its
monodromy group has signature $(6,6,2)$, and the index-2 subgroup
$\Gamma(\frac{1}{3},\frac{1}{3})$ has signature $(3,3,2,2)$, which can be
realized as the subgroup $\Gamma_{0}^{6}(1)$ of $PSL_{2}(\mathbb{R})$ arising
from the norm $1$ group of the maximal order of the quaternion algebra over
$\mathbb{Q}$ with discriminant $6$. By the Jacquet-Langlands correspondence
[29], the generator of $S_{6}(\Gamma(\frac{1}{3},\frac{1}{3}))$ and the Hecke
eigenform $f_{6.6.a.a}(\tau)$, which is
$\mathfrak{f}_{\frac{1}{3},\frac{1}{3}}(\tau)$ in Theorem 5, are both Hecke
eigenforms with the same eigenvalue at each prime $p>3$. ∎

###  4.3. The case $(c,f)=(\frac{1}{2},\frac{1}{2})$ \- a proof of Theorem 6

The goal of this subsection is to prove Theorem 6.

Let
$F(t)=\,_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1&1\end{matrix}\;;\;t\right]$. Then

$\,{}_{6}F_{5}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1&1&1&1&1\end{matrix}\;;\;1\right]=\frac{1}{2\pi
i}\oint_{|t|=1}F(t)F(1/t)\frac{dt}{t}.$

Using Clausen formula (39) with $a=b=\frac{1}{4}$ and a Kummer quadratic
formula

$\,{}_{2}F_{1}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}\vskip 3.0pt plus
1.0pt minus 1.0pt\\\
&1\end{matrix}\;;\;z\right]=(1-z)^{-1/2}\,_{2}F_{1}\left[\begin{matrix}\frac{1}{4}&\frac{1}{4}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1\end{matrix}\;;\;\frac{-4z}{(1-z)^{2}}\right],$

we have the following identity

$\,{}_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1&1\end{matrix}\;;\;\frac{-4z}{(1-z)^{2}}\right]=(1-z)\,_{2}F_{1}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1\end{matrix}\;;\;z\right]^{2}.$

Letting $z=\lambda(2\tau)$ and

(41)
$t_{2}(\tau)=\frac{-4\lambda(2\tau)}{(1-\lambda(2\tau))^{2}}=-64\frac{\eta(2\tau)^{24}}{\eta(\tau)^{24}},$

we have

$\displaystyle t_{2}\left(-\frac{1}{2\tau}\right)=\frac{1}{t_{2}(\tau)},$
$\displaystyle\frac{dt_{2}(\tau)}{d\tau}=\frac{4(\lambda(2\tau)+1)}{(\lambda(2\tau)-1)^{3}}\frac{d\lambda(2\tau)}{d\tau}=\frac{8\pi
i\lambda(2\tau)(\lambda(2\tau)+1)}{(1-\lambda(2\tau))^{3}}\theta_{4}^{4}(2\tau).$

A classical result says

$\theta_{3}(\tau)^{4}=\,_{2}F_{1}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1\end{matrix}\;;\;\lambda(\tau)\right]^{2},\quad\theta_{4}(\tau)^{4}=(1-\lambda(\tau))\,_{2}F_{1}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1\end{matrix}\;;\;\lambda(\tau)\right]^{2},$

where

$\displaystyle\theta_{2}(\tau)$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{k\in\mathbb{Z}}q^{(2k+1)^{2}/8}=2\frac{\eta(2\tau)^{2}}{\eta(\tau)},$
$\displaystyle\theta_{3}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z}}q^{k^{2}/2}=\frac{\eta(\tau)^{5}}{\eta^{2}(\tau/2)\eta^{2}(2\tau)},$
$\displaystyle\theta_{4}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z}}(-1)^{k}q^{k^{2}/2}=\frac{\eta^{2}(\tau/2)}{\eta(\tau)},$

are Jacobi theta functions. It follows

$\,{}_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1&1\end{matrix}\;;\;t_{2}(\tau)\right]=\theta_{4}^{4}(2\tau)=\frac{\eta(\tau)^{8}}{\eta(2\tau)^{4}},\quad\,_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1&1\end{matrix}\;;\;1/t_{2}(\tau)\right]={16}\tau^{2}\frac{\eta(2\tau)^{8}}{\eta(\tau)^{4}}.$

Also, we have
$\displaystyle\frac{1+\lambda(2\tau)}{1-\lambda(2\tau)}=1+32\left(\frac{\eta(4\tau)}{\eta(\tau)}\right)^{8},$
and thus

$\,{}_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1&1\end{matrix}\;;\;t_{2}(\tau)\right]\,_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1&1\end{matrix}\;;\;1/t_{2}(\tau)\right]\frac{dt_{2}}{t_{2}}~{}={16}\cdot
2\pi i\tau^{2}\cdot f_{8.6.a.a}(\tau/2){d\tau},$

where
$f_{8.6.a.a}(\tau/2)=\eta^{12}(\tau)+32\eta^{4}(\tau)\eta(4\tau)^{8}=-\left(\eta^{12}(\tau)-2\frac{\eta(2\tau)^{24}}{\eta(\tau)^{4}\eta(4\tau)^{8}}\right).$

Next we demonstrate the appearance of
$\mathfrak{g}_{c,f}=f_{8.4.a.a}(\tau)=\eta(2\tau)^{4}\eta(4\tau)^{4}$ by
showing

$\,_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1&1\end{matrix}\;;\;t_{2}(\tau)\right]\,_{4}F_{3}\left[\begin{matrix}\frac{1}{2}&\frac{5}{4}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{1}{4}&1&1\end{matrix}\;;\;1/t_{2}(\tau)\right]\frac{dt_{2}}{t_{2}}\sim_{e}-64\tau
f_{8.4.a.a}(\tau)d\tau,$

where the notation $\sim_{e}$ means the two differentials differ by an exact
differential form.

Observe that

$\,{}_{4}F_{3}\left[\begin{matrix}\frac{5}{4}&a&b&c\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\
&\frac{1}{4}&d&e\end{matrix}\;;\;1/z\right]=-4z\frac{d}{dz}\,_{3}F_{2}\left[\begin{matrix}a&b&c\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&d&e\end{matrix}\;;\;1/z\right]+\,_{3}F_{2}\left[\begin{matrix}a&b&c\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &d&e\end{matrix}\;;\;1/z\right].$

provided that the series converge. Therefore,

$\,{}_{7}F_{6}\left[\begin{matrix}\frac{1}{2}&\frac{5}{4}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{1}{4}&1&1&1&1&1\end{matrix}\;;\;1\right]=\frac{1}{2\pi
i}\oint_{|t_{2}|=1}\left(-4t\frac{d}{dt_{2}}F(1/t_{2})+F(1/t_{2})\right)F(t_{2})\frac{dt_{2}}{t_{2}}.$

In the computation below, we use the following notations:
$\theta_{i}:=\theta_{i}(2\tau)$, $k:=\frac{\theta_{2}^{2}}{\theta_{3}^{2}}$,
$\lambda=k^{2}$. We rewrite $F$-functions as

$F(t_{2}(\tau))=\theta_{4}^{4},\quad
F\left(\frac{1}{t_{2}(\tau)}\right)=4{\tau^{2}}\theta_{3}^{4}k.$

Firstly,

$-4t_{2}F(t_{2})\frac{d}{dt_{2}}F(1/t_{2})\frac{dt_{2}}{t_{2}}=-16\theta_{4}^{4}\frac{d(\tau^{2}\theta_{3}^{4}k)}{d\tau}d\tau,$

and

$F(1/t_{2})F(t_{2})\frac{dt_{2}}{t_{2}}=8\pi
i\tau^{2}\theta_{3}^{4}\theta_{4}^{2}k\cdot\theta_{4}^{4}\frac{1+\lambda}{1-\lambda}d\tau=8\pi
i\tau^{2}\theta_{3}^{4}\theta_{4}^{2}k\cdot(\theta_{4}^{4}+2\theta_{2}^{4})d\tau=8\pi
i\tau^{2}\theta_{3}^{4}\theta_{4}^{2}k\cdot(\theta_{3}^{4}+\theta_{2}^{4})d\tau.$

Note that

${d(\tau^{2}k\theta_{3}^{4}\theta_{4}^{4})}=\theta_{4}^{4}d(\tau^{2}k\theta_{3}^{4})+4\tau^{2}k\theta_{3}^{4}\theta_{4}^{4}\frac{d\theta_{4}}{\theta_{4}}\sim_{e}0,$

one has

$-4t_{2}F(t_{2})\frac{d}{dt_{2}}F(1/t_{2})\frac{dt_{2}}{t_{2}}\sim_{e}-16\left(-4\tau^{2}k\theta_{3}^{4}\theta_{4}^{4}\frac{d\theta_{4}}{\theta_{4}}\right)=64\tau^{2}k\theta_{3}^{4}\theta_{4}^{4}\left(\frac{d\theta_{4}}{\theta_{4}}\right)\\\
=64\tau^{2}k\theta_{3}^{4}\theta_{4}^{4}\frac{2\pi i}{24}\cdot
2\left(E_{2}(\tau)-E_{2}(2\tau)\right)d\tau=8\pi
i\tau^{2}k\theta_{3}^{4}\theta_{4}^{4}\cdot\frac{4}{3}\left(E_{2}(\tau)-E_{2}(2\tau)\right)d\tau,$

where $\displaystyle
E_{2}(\tau)=1-24\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{n}}$ is the normalized
weight-2 Eisenstein series. Hence,

$\displaystyle
F(1/t_{2})F(t_{2})\frac{dt_{2}}{t_{2}}-4t_{2}F(t_{2})\frac{d}{dt_{2}}F(1/t_{2})\frac{dt_{2}}{t_{2}}$
$\displaystyle\sim_{e}$ $\displaystyle 8\pi
i\tau^{2}\theta_{3}^{4}\theta_{4}^{2}k\cdot(\theta_{3}^{4}+\theta_{2}^{4}+\frac{4}{3}\left(E_{2}(\tau)-E_{2}(2\tau)\right))d\tau$
$\displaystyle=$ $\displaystyle 8\pi
i\tau^{2}\theta_{3}^{4}\theta_{4}^{2}k\cdot(2E_{2}(2\tau)-E_{2}(\tau)+\frac{4}{3}\left(E_{2}(\tau)-E_{2}(2\tau)\right))d\tau$
$\displaystyle=$ $\displaystyle 8\pi
i\tau^{2}\theta_{3}^{4}\theta_{4}^{2}k\cdot\frac{1}{12}(8E_{2}(2\tau)+4E_{2}(\tau))d\tau$
$\displaystyle=$ $\displaystyle 8\pi
i\tau^{2}\theta_{3}^{4}\theta_{4}^{2}k\cdot\frac{1}{12}\frac{24}{2\pi
i}\frac{d(\eta(2\tau)^{4}\eta(\tau)^{4})}{\eta(2\tau)^{4}\eta(\tau)^{4}}=8\tau^{2}\cdot
4d(\eta(2\tau)^{4}\eta(\tau)^{4}).$

Thus

$\,{}_{7}F_{6}\left[\begin{matrix}\frac{1}{2}&\frac{5}{4}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{1}{4}&1&1&1&1&1\end{matrix}\;;\;1\right]=\frac{1}{2\pi
i}\oint_{|t_{2}|=1}32\tau^{2}d\,\eta(2\tau)^{4}=\frac{32}{2\pi
i}\left(-2\oint_{|t_{2}|=1}\tau\eta(2\tau)^{4}\eta(\tau)^{4}d\tau\right).$

We now consider the path $|t_{2}|=1$. As $\lambda(2\tau)$ is a Hauptmodul on
$\Gamma_{0}(4)$ and $t_{2}(\tau)$ is a Hauptmodul on $\Gamma_{0}(2)$
satisfying $t_{2}\left(\frac{i}{\sqrt{2}}\right)=-1$,
$t_{2}\left(\frac{1+i}{2}\right)=1$, where $i/\sqrt{2}$ is fixed by $w_{2}$,
we interpret $|t_{2}|=1$ as the hyperbolic geodesic from $\frac{1+i}{2}$ to
$\frac{-1+i}{2}$, since $t_{2}$ has a simple pole at $0$. Wrapping up the
above discussion, we conclude the two identities stated in Theorem 6. 333If we
denote $g(\tau)=f_{8.6..a.a}(\tau/2)$, then  $\displaystyle g(\tau\pm 1/2)=$
$\displaystyle
i^{\pm}\left(\left(\frac{\eta(2\tau)^{3}}{\eta(\tau)\eta(4\tau)}\right)^{12}-32\eta(2\tau)^{12}\left(\frac{\eta(4\tau)}{\eta(\tau)}\right)^{4}\right)=i^{\pm}f_{16.6.a.a}\left(\frac{\tau}{2}\right),$
where $f_{16.6.a.a}$ is the quadratic twist of $f_{8.6.a.a}$ by $\chi_{-1}$.
It follows that  (42)
${}_{6}F_{5}(HD_{1}(\frac{1}{2},\frac{1}{2}))=-4i\left(\int_{i/2}^{i\infty}f_{16.6.a.a}\left(\frac{\tau}{2}\right)(4\tau^{2}+1)d\tau\right).$

###  4.4. The case $(c,f)=(\frac{1}{2},\frac{1}{3})$

In this subsection, we will prove the following identity

$\frac{1}{\pi}\,_{6}F_{5}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{3}&\frac{2}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{5}{6}&\frac{7}{6}&1&1&1\end{matrix}\;;\;1\right]=6i\oint_{|t_{3}|=1}\left(\frac{1}{3}+\tau+\tau^{2}\right)\cdot(f_{4.6.a.a}(\tau/2)-27f_{4.6.a.a}(3\tau/2))d\tau,$

where

(43) $\displaystyle
t_{3}(\tau)=4\left(\frac{1}{3\sqrt{3}}\frac{\eta^{6}(\tau)}{\eta^{6}(3\tau)}+3\sqrt{3}\frac{\eta^{6}(3\tau)}{\eta^{6}(\tau)}\right)^{-2}$

is a Hauptmodul of $\Gamma_{0}^{+}(3)$ taking values $0$, $1$, $\infty$ at the
elliptic points of order $6$, $2$, and at the cusp (of order $\infty$),
respectively. To prove it, we use

$\,{}_{6}F_{5}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{3}&\frac{2}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{5}{6}&\frac{7}{6}&1&1&1\end{matrix}\;;\;1\right]=\frac{1}{2\pi
i}\oint_{|t|=1}\,{}_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{5}{6}&\frac{7}{6}\end{matrix}\;;\;1/t\right]\,_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{3}&\frac{2}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1&1\end{matrix}\;;\;t\right]\frac{dt}{t}$

The second ${}_{3}F_{2}$ term can be handled by the Clausen formula (39)

$\,{}_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{3}&\frac{2}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1&1\end{matrix}\;;\;4z(1-z)\right]=\,_{2}F_{1}\left[\begin{matrix}\frac{1}{3}&\frac{2}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1\end{matrix}\;;\;z\right]^{2}.$

To deal with the first one, we use Property (P.2). By a straightforward
verification, one has

$\displaystyle\frac{\sqrt{3}}{\pi}$
$\displaystyle(-4z(1-z))^{-1/2}\,_{3}F_{2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&\frac{7}{6}&\frac{5}{6}\end{matrix}\;;\;\frac{1}{4z(1-z)}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{3}\,_{2}F_{1}\left[\begin{matrix}\frac{1}{3}&\frac{2}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1\end{matrix}\;;\;z\right]^{2}+\frac{i}{\sqrt{3}}\,_{2}F_{1}\left[\begin{matrix}\frac{1}{3}&\frac{2}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1\end{matrix}\;;\;z\right]\,_{2}F_{1}\left[\begin{matrix}\frac{1}{3}&\frac{2}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&1\end{matrix}\;;\;1-z\right]-\frac{1}{3}\,_{2}F_{1}\left[\begin{matrix}\frac{1}{3}&\frac{2}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ &1\end{matrix}\;;\;1-z\right]^{2}.$

When $z=\left(1+\eta(\tau)^{12}/27\eta(3\tau)^{12}\right)^{-1}$, we can
express a point on the upper-half plane by

$\tau=\frac{i}{\sqrt{3}}\,_{2}F_{1}\left[\begin{matrix}\frac{1}{3}&\frac{2}{3}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\