2 Hypergeometric schemes

Let $f \colon U \to \operatorname {Spec} A_K$ be a family of affine schemes over K, whose general fibre $U_t=f^{-1}(t)$ is defined by

$$ \begin{align*}(1-x_0^N)\cdots (1-x_d^N)=t. \end{align*} $$

This is smooth over $A_K$ . We call U the hypergeometric scheme over $A_K$ . The group $\mu _N^{d+1}$ acts on U over $A_K$ by

$$ \begin{align*} &(\nu_0, \ldots, \nu_{d})\cdot (x_0, \ldots, x_d)=(\nu_0 x_0, \ldots, \nu_d x_d), \quad (\nu_0, \ldots, \nu_d) \in \mu_N^{d+1}. \end{align*} $$

This action induces an action on $H_{\mathrm {dR}}^d(U/A_K)$ and $H_{\mathrm {dR}}^d(U/A_K)$ becomes an $A_K[\mu _N^{d+1}]$ -module. Let $I_+=\{(i_0, \ldots , i_d) \in \mathbb {Z}^{d+1} \mid 0 < i_k < N\}$ . In this section, we suppose that $(i_0, \ldots , i_d) \in I_+$ . Put $\underline {a}=(1-i_0/N, \ldots , 1-i_d/N)$ and $\check {\underline {a}}=(i_0/N, \ldots , i_d/N)$ . Let $\mathscr {D}=K\langle t, (t-t^2)^{-1}, {d}/{dt} \rangle $ . For a character $\chi _{i_0, \ldots , i_d} \in \operatorname {Hom}(\mu _N^{d+1}, K^*)$ given by

$$ \begin{align*}(\nu_0, \ldots, \nu_d) \mapsto \nu_0^{i_0} \cdots \nu_d^{i_d}, \end{align*} $$

there is an isomorphism

$$ \begin{align*} \mathscr{D}/\mathscr{D} P_{\mathrm{HG}, \underline{a}} \xrightarrow{\simeq} H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}, \quad P \mapsto P(\omega_{i_0, \ldots, i_d}), \end{align*} $$

as $\mathscr {D}$ -modules (see [Reference Asakura2, Corollary 3.6]). Here, $\omega _{i_0, \ldots , i_d}$ is the holomorphic d-form defined by

$$ \begin{align*}\omega_{i_0, \ldots, i_d}=N^{-1} x_0^{i_0-N }x_1 ^{i_1-1} \cdots x_d^{i_d-1}\frac{dx_1 \wedge \cdots \wedge dx_d}{(1-x_1^N) \cdots (1-x_d^N)}. \end{align*} $$

We put $H^d_{\mathrm {dR}}(U/A_K)^{\,\chi _{i_0, \ldots , i_d}}_{K((t))}=K((t)) \otimes _{A_K} H^d_{\mathrm {dR}}(U/A_K)^{\,\chi _{i_0, \ldots , i_d}}$ .

Proposition 2.1 [Reference Asakura3, Proposition 4.1].

Put $D=t\,{d}/{dt}$ . There is an element $\widehat {\eta }_{i_0, \ldots , i_d} \in H^d_{\mathrm {dR}}(U/A_K)^{\,\chi _{i_0, \ldots , i_d}}_{K((t))}$ such that

$$ \begin{align*}\operatorname{Ker}\big[ D \colon H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}_{K((t))} \to H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}_{K((t))} \big] =K \,\widehat{\eta}_{i_0, \ldots, i_d}.\end{align*} $$

For the polynomial $h_{\underline {a}}(t)$ defined in (1.2), we put

$$ \begin{align*}h(t)=\prod_{(i_0, \ldots, i_d) \in I_+} h_{\underline{a}}(t). \end{align*} $$

Put $B=A[h(t)^{-1}]$ , $B_K=K \otimes _W B$ ,

$$ \begin{align*}\widehat{B}=A[h(t)^{-1}]^{\wedge}:=\varprojlim_{n} (W/p^nW[t, (t-t^2)^{-1}, h(t)^{-1}] ),\end{align*} $$

the p-adic completion of B, and $\widehat {B}_K= K \otimes _W \widehat {B}$ . By the congruence relation for Dwork’s p-adic hypergeometric function, $\mathscr {F}^{\mathrm {Dw}}_{\underline {a}}(t)$ becomes an element of $\widehat {B}_K$ , which is invertible (see [Reference Asakura, Bostan and Raschel1, Corollary 2.3]). Put

$$ \begin{align*}H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}_{\widehat{B}_K}= \widehat{B}_K \otimes_{A_K} H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}. \end{align*} $$

We define the unit root vector by

$$ \begin{align*}\eta_{i_0, \ldots, i_d}= F_{\check{\underline{a}}}(t)^{-1} \widehat{\eta}_{i_0, \ldots, i_d}, \end{align*} $$

which defines an element of $H^d_{\mathrm {dR}}(U/A_K)^{\,\chi _{i_0, \ldots , i_d}}_{\widehat {B}_K}$ (see [Reference Asakura3, Lemma 4.2]). Put

$$ \begin{align*}H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}_{\widehat{B}_K, \text{unit}} = \widehat{B}_K \eta_{i_0, \ldots, i_d}. \end{align*} $$

There is a perfect pairing

$$ \begin{align*} Q \colon H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}_{\widehat{B}_K} \otimes_{\widehat{B}_K} H_{\mathrm{dR}} (U / A_K)^{\,\chi_{-i_0, \ldots, -i_d}}_{\widehat{B}_K} \to \widehat{B}_K \end{align*} $$

(see [Reference Asakura3, Section 3.2, (3.5)]). Let

$$ \begin{align*}VH^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}_{\widehat{B}_K} \subset H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}_{\widehat{B}_K}\end{align*} $$

be a $\widehat {B}_K$ -submodule which is the exact annihilator of $H^d_{\mathrm {dR}}(U/A_K)^{\,\chi _{-i_0, \ldots , -i_d}}_{\widehat {B}_K, \text {unit}} $ . Then,

(2.1) $$ \begin{align} H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}_{\widehat{B}_K}/VH^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}_{\widehat{B}_K} \simeq \widehat{B}_K \omega_{i_0, \ldots, i_d} \end{align} $$

(see [Reference Asakura3, Lemma 4.3]).

Let $A_K^{\dagger }$ be the weak completion of $A_K$ (see [Reference Le Stum11, page 135]). Let $\sigma $ be the F-linear Frobenius on $A_K^{\dagger }$ given by $\sigma (t)=t^p$ , where F is the pth Frobenius on W. Then, there is a pth Frobenius $\Phi $ induced by $\sigma $ acting on

$$ \begin{align*}H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}_{A_K^{\dagger}}:=A_K^{\dagger}\otimes_{A_K} H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}. \end{align*} $$

The Frobenius $\sigma $ extends to $\widehat {B}_K$ , and hence we can extend $\Phi $ to the pth Frobenius on $H^d_{\mathrm {dR}}(U/A_K)^{\,\chi _{i_0, \ldots , i_d}}_{\widehat {B}_K}$ and denote it by the same notation.

Theorem 2.2 [Reference Asakura3, Theorem 4.6].

Let $(i_0, \ldots , i_d), (j_0, \ldots , j_d) \in I_+$ and suppose that $p j_k \equiv i_k \pmod {N}$ for $k=0, \ldots , d$ . Then,

$$ \begin{align*}\Phi(\omega_{j_0, \ldots, j_d}) \equiv p^d \mathscr{F}^{\mathrm{Dw}}_{\underline{a}} (t)^{-1} \omega_{i_0, \ldots, i_d} \,\mod{VH^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{i_0, \ldots, i_d}}_{\widehat{B}_K}}, \end{align*} $$

where $\underline {a}=(1-i_0/N, \ldots , 1-i_d/N)$ .

3 Proof of Theorem 1.2

Let $g \colon V^* \to \operatorname {Spec} A_K$ be a family of affine schemes over K, whose general fibre $V^*_t=g^{-1}(t)$ is defined by

$$ \begin{align*}w^N=z_1 \cdots z_d (1-z_1)^{N-1} \cdots (1-z_d)^{N-1}(z_1 \cdots z_d -t)^{N-1}, \quad w \neq 0, \end{align*} $$

which is smooth over $A_K$ . Let U be the hypergeometric scheme defined in Section 2 and let $U^*$ be the open subscheme of U defined by $x_0 \cdots x_d \neq 0$ . Then, there is a covering map

$$ \begin{align*} &\rho \colon U^* \to V^*, \quad \begin{cases} w=x_0^{N-1} \cdots x_d^{N-1} (1-x_1^N) \cdots (1-x_d^N), \\ z_i=1-x_i^N \quad (i=1, \ldots, d). \end{cases} \end{align*} $$

The group $\mu _N$ acts on $V^*$ over $A_K$ by

$$ \begin{align*}\nu \cdot (z_1, \ldots, z_d, w)=(z_1, \ldots, z_d, \nu^{-1} w), \quad \nu \in \mu_N. \end{align*} $$

The action induces an action on $H^d_{\mathrm {dR}}(V^*/A_K)$ , and hence $H^d_{\mathrm {dR}}(V^*/A_K)$ becomes an $A_K[\mu _N]$ -module. For $n \in \mathbb {Z}$ , let

$$ \begin{align*}\chi_n \colon \mu_N \to K^*, \quad \nu \mapsto \nu^n\end{align*} $$

be a character of $\mu _N$ . This leads to the following proposition.

Proposition 3.1. If $n \not \equiv 0 \pmod {N}$ , then there is an isomorphism of $\mathscr {D}$ -modules,

$$ \begin{align*}H^d_{\mathrm{dR}}(V^*/A_K)^{\,\chi_{n}} \simeq H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{n, \ldots, n}}.\end{align*} $$

Proof. Since $\rho \colon U^* \to V^*$ is a finite étale morphism of degree $N^{d}$ with $U^*$ connected, and the automorphism group of $U^*$ over $V^*$ is given by

$$ \begin{align*}G:=\{(\nu_0, \ldots, \nu_d) \in \mu_N^{d+1} \mid \nu_0 \cdots \nu_d=1\} \simeq \mu_N^d, \end{align*} $$

$U^*$ is Galois over $V^*$ with Galois group G. Therefore,

$$ \begin{align*} H^d_{\mathrm{dR}}(U^*/A_K)^{\,\chi_{n, \ldots, n}} \simeq H^d_{\mathrm{dR}}(V^*/A_K)^{\,\chi_n}. \end{align*} $$

Put $D=U \backslash U^*$ . We have a localisation exact sequence

$$ \begin{align*} \cdots &\to \mathscr{H}^{d-2}_{\mathrm{dR}}(D/A_K) \to \mathscr{H}^d_{\mathrm{dR}}(U/A_K) \to \mathscr{H}^d_{\mathrm{dR}}(U^*/A_K) \to \mathscr{H}^{d-1}_{\mathrm{dR}}(D/A_K) \to \cdots \end{align*} $$

(see [Reference Hartshorne8, Theorem 3.3]). Since each irreducible component of D is given by $\{x_i=0\}$ and the ith component of $\mu _N^{d+1}$ acts trivially on $\{x_i=0\}$ , we have

$$ \begin{align*}\mathscr{H}^j_{\mathrm{dR}}(D/A_K)^{\,\chi_{n, \ldots, n}}=0\end{align*} $$

by the assumption $n \not \equiv 0 \pmod {N}$ . Therefore, we have an isomorphism

$$ \begin{align*}\mathscr{H}^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{n, \ldots, n}} \simeq \mathscr{H}^d_{\mathrm{dR}}(U^*/A_K)^{\,\chi_{n, \ldots, n}}, \end{align*} $$

which induces an isomorphism

$$ \begin{align*} H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{n, \ldots, n}} \simeq H^d_{\mathrm{dR}}(U^*/A_K)^{\,\chi_{n, \ldots, n}}.\\[-34pt] \end{align*} $$

Let $\omega _n$ be the holomorphic d-form on $V^*$ defined by

$$ \begin{align*}\omega_n=(1-z_1)^{n-1} \cdots (1-z_n)^{n-1}(z_1 \cdots z_d -t)^{n-1}\frac{dz_1 \wedge \cdots \wedge dz_d}{w^n}, \end{align*} $$

which defines an element of $H^d_{\mathrm {dR}}(V^*/A_K)^{\,\chi _n}$ . Then,

(3.1) $$ \begin{align} \rho^*(\omega_n) =(-1)^d N^d x_0^{n-N }x_1 ^{n-1} \cdots x_d^{n-1}\frac{dx_1 \wedge \cdots \wedge dx_d}{(1-x_1^N) \cdots (1-x_d^N)} =(-1)^d N^{d+1} \omega_{n, \ldots, n}. \end{align} $$

Hence, $H^d_{\mathrm {dR}}(V^*/A_K)^{\,\chi _n}$ is a $\mathscr {D}$ -module generated by $\omega _n$ .

Let $\widehat {f} \colon \widehat {U} \to \operatorname {Spec} A_K$ (respectively $\widehat {g} \colon \widehat {V}^* \to \operatorname {Spec} A_K$ ) be a family of affine schemes over K, whose general fibre is $\widehat {f}^{-1}(t)=U_{t^{-1}}$ (respectively $\widehat {g}^{-1}(t)=V^*_{t^{-1}}$ ).

Put $t=t_0^N$ . From now on, we change the notation, and let $A=W[t_0, t_0^{-1}, (1-t_0^N)^{-1}]$ , $A_K=K \otimes _W A$ and $\widehat {B}_K=K \otimes _W A[h(t_0^N)^{-1}]^{\wedge }$ , where $h(t)$ is the polynomial defined in Section 2. Then, there is an isomorphism over $A_K$ ,

(3.2) $$ \begin{align} &\iota \colon V^* \to \widehat{V}^*, \quad (z_1, \ldots, z_d, w) \mapsto (u_1, \ldots, u_d, v), \end{align} $$

which is given by

$$ \begin{align*} u_i=z_i^{-1} (i=1, \ldots, d), \quad v= \begin{cases} \displaystyle\frac{\xi w}{z_1^2 \cdots z_d^2 t_0^{N-1}} & \text{if } N \text{ and } d \text{ are even}, \\[2ex] \displaystyle\frac{w}{z_1^2 \cdots z_d^2 t_0^{N-1}}& \text{otherwise}, \end{cases} \end{align*} $$

where $\xi \in K^*$ such that $\xi ^N=-1$ . For $n \in \{1, \ldots , N-1\}$ , define

$$ \begin{align*}\varphi_{N, d}(n)= \begin{cases} {n} & \text{if } N \text{ and } d \text{ are even}, \\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

Since

(3.3) $$ \begin{align} \iota^*(\omega_{n})=(-1)^{(d+1)n-1}\xi^{-\varphi_{N, d}(n)}t_0^{N-n} \omega_{n}, \end{align} $$

the map (3.2) induces an isomorphism

$$ \begin{align*}\iota^* \colon H_{\mathrm{dR}}^d(\widehat{V}^* /A_K)^{\,\chi_n} \xrightarrow{\simeq} H_{\mathrm{dR}}^d(V^*/A_K)^{\,\chi_n}. \end{align*} $$

We define a morphism

$$ \begin{align*}\widetilde{\iota^*} \colon H^d_{\mathrm{dR}}(\widehat{U}/A_K)^{\,\chi_{n, \ldots, n}} \to H^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{n, \ldots, n}}\end{align*} $$

so that the following diagram commutes:

Then, by (3.1) and (3.3), we compute explicitly

(3.4) $$ \begin{align} \widetilde{\iota^*}(\omega_{n, \ldots, n})=(-1)^{(d+1)n-1}\xi^{-\varphi_{N, d}(n)}t_0^{N-n} \omega_{n, \ldots, n}, \end{align} $$

and hence by (2.1),

(3.5) $$ \begin{align} \widetilde{\iota^*}(VH^d_{\mathrm{dR}}(\widehat{U}/A_K)^{\,\chi_{n, \ldots, n}}_{\widehat{B}_K}) \simeq VH^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{n, \ldots, n}} _{\widehat{B}_K}. \end{align} $$

Proof of Theorem 1.2.

Let $\sigma $ be the F-linear pth Frobenius on $\widehat {B}_K$ given by $\sigma (t_0)=t_0^p$ and let $\Phi $ be the pth Frobenius induced by $\sigma $ acting on the de Rham cohomology groups. Let $m, n \in \{1, \ldots , N-1\}$ be integers such that $pm \equiv n \pmod {N}$ . By Theorem 2.2 and [Reference Wang16, Proposition 4.11(3)],

(3.6) $$ \begin{align} \Phi(\omega_{m, \ldots, m}) \equiv p^d\mathscr{F}^{\mathrm{Dw}}_{\underline{a}}(t_0^{-N})^{-1} \omega_{n \ldots, n} \,\mod{VH^d_{\mathrm{dR}}(\widehat{U}/A_K)^{\,\chi_{n, \ldots, n}} _{\widehat{B}_K}}, \end{align} $$

where $\underline {a}=(1-n/N, \ldots , 1-n/N)$ . Apply $\widetilde {\iota ^*}$ on both sides of (3.6). By (3.4) and (3.5),

$$ \begin{align*} &\Phi((-1)^{(d+1)m-1}\xi^{-\varphi_{N, d}(m)}t_0^{N-m}\omega_{m, \ldots, m}) \\ & \equiv p^d\mathscr{F}^{\mathrm{Dw}}_{\underline{a}}(t_0^{-N})^{-1} ((-1)^{(d+1)n-1}\xi^{-\varphi_{N, d}(n)}t_0^{N-n} \omega_{n, \ldots, n}) \,\mod{VH^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{n, \ldots, n}} _{\widehat{B}_K}}. \end{align*} $$

By Theorem 2.2, the left-hand side is

$$ \begin{align*} & (-1)^{(d+1)m-1}\xi^{-p\varphi_{N, d}(m)}t_0^{p(N-m)} p^d \mathscr{F}^{\mathrm{Dw}}_{\underline{a}}(t_0^N)^{-1}\omega_{n, \ldots, n} \,\mod{VH^d_{\mathrm{dR}}(U/A_K)^{\,\chi_{n, \ldots, n}} _{\widehat{B}_K}}. \end{align*} $$

Hence,

$$ \begin{align*} \mathscr{F}^{\mathrm{Dw}}_{\underline{a}}(t)&=(-1)^{(d+1)(m-n)}\xi^{-(p\varphi_{N, d}(m) - \varphi_{N, d}(n))} t^{pa_m-a_n}\mathscr{F}^{\mathrm{Dw}}_{\underline{a}}(t^{-1}) \\ &=(-1)^{(d+1)(m-n)-{(p\varphi_{N, d}(m) - \varphi_{N, d}(n))}/{N}} t^{pa_m-a_n}\mathscr{F}^{\mathrm{Dw}}_{\underline{a}}(t^{-1}), \end{align*} $$

where we put $a_i=1-i/N$ .

First, we prove the case $p \neq 2$ . Since

$$ \begin{align*}pa_m -a_n=p-1-\frac{pm-n}{N}=l, \end{align*} $$

it suffices to show that

(3.7) $$ \begin{align} (d+1)(m-n) - \dfrac{p\varphi_{N, d}(m) - \varphi_{N, d}(n)}{N} \equiv (d+1)l \pmod{2}. \end{align} $$

Suppose that N and d are even. Then, we compute

$$ \begin{align*} (d+1)(m-n)-\dfrac{p\varphi_{N, d}(m) - \varphi_{N, d}(n)}{N} & = (d+1)(m-n)-\frac{pm-n}{N} \\ & \equiv N \dfrac{pm-n}{N} -\frac{pm-n}{N} \\ & \equiv -\frac{pm-n}{N} \equiv l \equiv (d+1)l \pmod{2}, \end{align*} $$

and hence (3.7) holds.

Next, suppose that N is odd or d is odd. Then, the left-hand side of (3.7) is

$$ \begin{align*} (d+1)(m-n)-\dfrac{p\varphi_{N, d}(m) - \varphi_{N, d}(n)}{N} = (d+1)(m-n). \end{align*} $$

If d is odd, then (3.7) holds. If N is odd and d is even, then

$$ \begin{align*} l=p-1-\frac{pm-n}{N} \equiv m-n \pmod{2}, \end{align*} $$

and hence (3.7) holds.

Second, we prove the case $p=2$ . Since N is odd, it suffices to show that

(3.8) $$ \begin{align} (d+1)(m-n)\equiv \begin{cases} 1+(d+1)l & \text{if }d \text{ is even and } (a_n)'\equiv 1 \pmod{2},\\ (d+1)l & \text{otherwise}, \end{cases} \end{align} $$

modulo $2$ . If d is odd, then (3.8) holds. Suppose that d is even. Then,

$$ \begin{align*}(d+1)(m-n) \equiv m-n \pmod{2}\end{align*} $$

and

$$ \begin{align*}(d+1)l \equiv l \equiv 1-\dfrac{2m-n}{N} \equiv 1-n \pmod{2}. \end{align*} $$

Therefore,

$$ \begin{align*}(d+1)(m-n) \equiv 1+(d+1)l \pmod{2}\end{align*} $$

holds if and only if $m\hspace{-1pt} \equiv\hspace{-1pt} 0 \pmod {2}$ , which means that ${(a_n)'\hspace{-1pt}=\hspace{-1pt}a_m\hspace{-1pt}=\hspace{-1pt}1-m/N\hspace{-1pt} \equiv\hspace{-1pt} 1} \pmod {2}$ . Hence, (3.8) holds.

For a positive integer f, put $q=p^f$ and define

$$ \begin{align*}\mathscr{F}^{\mathrm{Dw}, f}_{\underline{a}}(t)= \prod_{i=0}^f \mathscr{F}^{\mathrm{Dw}}_{\underline{a}^{(i)}}(t^{p^i})= \dfrac{F_{\underline{a}}(t)}{F_{\underline{a}^{(f)}}(t^q)} \in \mathbb{Z}_p\langle t, h_{\underline{a}}(t)^{-1}\rangle. \end{align*} $$

Repeatedly using Theorem 1.2 gives the following corollary.

Corollary 3.2. Suppose conditions (i) and (ii) in Theorem 1.2 hold. Let $l \in \{1, \ldots , q-1\}$ be the unique integer such that $a + l \equiv 0 \pmod {q}$ . If $p \neq 2$ , then

$$ \begin{align*}\mathscr{F}^{\mathrm{Dw}, f}_{a, \ldots, a}(t)= ((-1)^{d+1}t)^l \mathscr{F}^{\mathrm{Dw}, f}_{a, \ldots, a}(t^{-1}) \end{align*} $$

in $\mathbb {Z}_p\langle t, t^{-1}, h_{a, \ldots , a}(t)^{-1}\rangle $ . If $p=2$ , then this transformation formula holds up to sign.

A Transformation formula of hypergeometric functions over finite fields

Here, we introduce a transformation formula of a hypergeometric function over a finite field, which is a finite analogue of Theorem 1.2. Over finite fields, there are several definitions of hypergeometric functions due to Koblitz [Reference Koblitz10], Greene [Reference Greene7], McCarthy [Reference McCarthy12], Fuselier et al. [Reference Fuselier, Long, Ramakrishna, Swisher and Tu6], Katz [Reference Katz9] and Otsubo [Reference Otsubo13]. We use Otsubo’s definition.

For a positive integer n, let $\mu _n \subset \mathbb {C}^*$ denote the group of nth roots of unity. Let $\mathbb {F}_q$ be a finite field of characteristic p with $q=p^f$ elements. Put $\widehat {\mathbb {F}_q^{*}}=\operatorname {Hom}(\mathbb {F}_q^*, \mathbb {C}^*)$ and for any $\varphi \in \widehat {\mathbb {F}_q^{*}}$ , set $\varphi (0)=0$ and write $\overline {\varphi }=\varphi ^{-1}$ . Let $\varepsilon \in \widehat {\mathbb {F}_q^{*}}$ be the trivial character, that is, for any $x \in \mathbb {F}_q$ ,

$$ \begin{align*}\varepsilon(x)= \begin{cases} 1 & \text{if } x \neq 0, \\ 0 & \text{if } x=0. \end{cases} \end{align*} $$

Fix a nontrivial additive character $\psi \in \operatorname {Hom}(\mathbb {F}_q, \mathbb {C}^*)$ . For a multiplicative character $\varphi \in \widehat {\mathbb {F}_q^*}$ , the Gauss sum, which is a finite analogue of the gamma function, and its variant are defined by

$$ \begin{align*}g(\varphi)=-\sum_{x \in \mathbb{F}_q}\varphi(x) \psi(x), \quad g^{\circ}(\varphi)= \begin{cases} g(\varphi) & \text{if } \varphi \neq \varepsilon, \\ q & \text{if } \varphi = \varepsilon, \end{cases} \quad \text{in } \mathbb{Q}(\mu_{p(q-1)}). \end{align*} $$

We define the Pochhammer symbol and its variant by

$$ \begin{align*}(\varphi)_{\nu}=\frac{g(\varphi \nu)}{g(\varphi)}, \quad (\varphi)^{\circ}_{\nu}=\frac{g^{\circ}(\varphi \nu)}{g^{\circ}(\varphi)}, \end{align*} $$

where $\varphi , \nu \in \widehat {\mathbb {F}_q^*}$ . For any $\varphi \in \widehat {\mathbb {F}_q^*}$ ,

(A.1) $$ \begin{align} g(\varphi) g^{\circ}(\overline{\varphi})=\varphi(-1) q. \end{align} $$

With the notation as above, for $\alpha _i, \beta _j, \in \widehat {\mathbb {F}_q^*}$ , we define the hypergeometric function over the finite field $\mathbb {F}_q$ by

$$ \begin{align*} {_{d+1}F_{d}}\bigg( \begin{matrix} \alpha_0, \ldots, \alpha_{d} \\ \beta_1, \ldots, \beta_d \end{matrix}; t \bigg) = \dfrac1{1-q}\sum_{\nu \in \widehat{\mathbb{F}_v}} \dfrac{(\alpha_0)_{\nu} \cdots (\alpha_{d})_{\nu} }{(\varepsilon)^{\circ}_{\nu} (\beta_1)^{\circ}_{\nu} \cdots (\beta_d)^{\circ}_{\nu} } \nu(t) \in \mathbb{Q}(\mu_{q-1}). \end{align*} $$

The transformation formula of the hypergeometric function over the finite field between t and $t^{-1}$ is as follows.

Proposition A.1. Let $\alpha _i$ , $\beta _j \in \widehat {\mathbb {F}_q^*} (1 \leq i, j \leq d)$ and suppose that $\alpha _i \neq \beta _i$ for all i. For any $\alpha _0 \in \widehat {\mathbb {F}_q^*}$ , $\alpha _0 \neq \varepsilon $ and $t \in \mathbb {F}_q^*$ ,

$$ \begin{align*}{_{d+1}F_{d}}\bigg( \begin{matrix} \alpha_0, \ldots, \alpha_d \\ \beta_1, \ldots \beta_d \end{matrix}; t \bigg)= \overline{\alpha_0}(-t) \prod_{i=1}^d \dfrac{g(\alpha_0 \overline{\beta_i}) g^{\circ} (\overline{\alpha_i})}{g^{\circ}(\alpha_0 \overline{\alpha_i}) g(\overline{\beta_i})} {_{d+1}F_{d}}\bigg( \begin{matrix} \alpha_0, \alpha_0 \overline{\beta_1}, \ldots, \alpha_0 \overline{\beta_d} \\ \alpha_0 \overline{\alpha_1}, \alpha_0 \overline{\alpha_2}, \ldots \alpha_0 \overline{\alpha_d} \end{matrix}; t^{-1} \bigg). \end{align*} $$

In particular, if $\alpha _0 = \cdots =\alpha _d =\alpha \neq \varepsilon $ and $\beta _1 = \cdots =\beta _d = \varepsilon $ , then

$$ \begin{align*}{_{d+1}F_{d}}\bigg( \begin{matrix} \alpha, \ldots, \alpha \\ \varepsilon, \ldots \varepsilon \end{matrix}; t \bigg) = \overline{\alpha}((-1)^{d+1}t) {_{d+1}F_{d}}\bigg( \begin{matrix} \alpha, \ldots, \alpha \\ \varepsilon, \ldots \varepsilon \end{matrix}; t^{-1} \bigg), \end{align*} $$

which is a finite analogue of Theorem 1.2.

Remark A.2. For $t \in \mathbb {F}_q \setminus \{0, 1\}$ and a positive integer $N \geq 2$ such that $N \mid q-1$ , let $X_t$ be the smooth projective curve over $\mathbb {F}_q$ birational to the affine curve

$$ \begin{align*}(1-x_0^N)(1-x_1^N)=t. \end{align*} $$

It is known that the Frobenius trace of $X_t$ , that is, the unit root of the Artin L-function of $X_t$ , can be written in terms of hypergeometric functions over $\mathbb {F}_q$ (see [Reference Asakura and Otsubo4, Theorem 4.2]). This unit root can also be written in terms of $\mathscr {F}^{\mathrm {Dw}, f}_{\underline {a}}(t)$ (see [Reference Asakura3, Corollary 4.7]). Therefore, by using Proposition A.1 and comparing the unit root of the Artin L-function of $X_t$ with that of $X_{t^{-1}}$ , we can prove a special case of Corollary 3.2. Let $W=W(\mathbb {F}_q)$ be the Witt ring of $\mathbb {F}_q$ and $\mu _{q-1}$ be the group of $(q-1)$ th roots of unity in W. For $a \in \{1, \ldots , N-1\}$ and $\zeta \in \mu _{q-1}$ such that $h_{{a}/N, {a}/N}(\zeta ) \not \equiv 0 \pmod {pW}$ ,

$$ \begin{align*}\mathscr{F}^{\mathrm{Dw}, f}_{{a}/N, {a}/{N}}(\zeta)= \zeta^l \mathscr{F}^{\mathrm{Dw}, f}_{{a}/N, {a}/{N}}(\zeta^{-1}), \end{align*} $$

where $l \in \{0, \ldots , q-1\}$ is the unique integer such that $a/N + l \equiv 0 \pmod {q}$ . Here, the special value of Dwork’s p-adic hypergeometric function at $t=t_0$ is defined by

$$ \begin{align*}\mathscr{F}^{\mathrm{Dw}, f}_{a_0, a_1}(t_0)=\mathscr{F}^{\mathrm{Dw}, f}_{a_0, a_1}(t)|_{t=t_0} = \lim_{n \to \infty} \dfrac{[F_{a_0, a_1}(t)]_{<p^n} }{[F_{a^{(f)}_0, a^{(f)}_1}(t^q)]_{<p^n}} \bigg|_{t=t_0}. \end{align*} $$