Hypergeometric functions over finite fields

Abstract

We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental properties and prove summation formulas, transformation formulas and product formulas. An application to zeta functions of K3-surfaces is given. In the appendix, we give an elementary proof of the Davenport–Hasse multiplication formula for Gauss sums.

Appendix A: A proof of the multiplication formula for Gauss sums

Here we give an elementary proof of Theorem 3.10, using a geometric construction of Terasoma in his proof of the same theorem [24, Theorem 3]. While he uses l-adic cohomology, we count the number of rational points, the two objects being related by the Lefschetz trace formula.

Let X be a variety over $\kappa$ equipped with a left action of a finite group G, and suppose that the quotient variety $G\backslash X$ exists. Fix an algebraic closure $\overline{\kappa }$ of $\kappa$. Let F denote the qth power Frobenius acting on $X=X(\overline{\kappa })$. For each $g\in G$, put

$$\begin{aligned} \Lambda (X,g)=\#\{x \in X \mid Fx=gx\}. \end{aligned}$$

For a character $\chi$ (of a $\mathbb {C}$-linear representation) of G, put

$$\begin{aligned} N(X,\chi )= \frac{1}{\# G} \sum _{g\in G} \chi (g) \Lambda (X,g). \end{aligned}$$

We have the following functorialities.

Lemma A.1

(cf. [21, 2.3])

• (i) If $H \subset G$ is a normal subgroup and $\chi$ is a character of G/H, then

  $$\begin{aligned} N(X,\chi |_G) = N(H\backslash X, \chi ). \end{aligned}$$

• (ii) If $H \subset G$ is a subgroup and $\chi$ is a character of H, then

  $$\begin{aligned} N(X,\chi )= N(X, {\text {Ind}}_H^G \chi ). \end{aligned}$$

Now we start the proof. Since the statement is obvious if $\alpha ^n=\varepsilon$, we suppose that $\alpha ^n\ne \varepsilon$. Then, by Proposition 2.2 (iv), we are reduced to prove

$$\begin{aligned} \alpha ^n(n) j(\underbrace{\alpha ,\dots , \alpha }_{n\ \textrm{times}}) = \prod _{\nu ^n=\varepsilon ,\nu \ne \varepsilon } j(\alpha ,\nu ). \end{aligned}$$

The following varieties, maps among them and group actions are all defined over $\kappa$. Let X be a twisted Fermat hypersurface of dimension $n-1$ defined by

$$\begin{aligned} t_1^{q-1} + \cdots + t_n^{q-1}= n, \quad t_1\cdots t_n \ne 0. \end{aligned}$$

Let C be a Fermat quotient curve defined by

$$\begin{aligned} x^{q-1}+y^n=1, \quad x \ne 0. \end{aligned}$$

Let $S\subset {\mathbb {A}}^n$ be a hyperplane defined by

$$\begin{aligned} s_1+\cdots +s_n=n, \quad s_1\cdots s_n \ne 0, \end{aligned}$$

and let $T\rightarrow S$ be a covering defined by $t^{q-1}=s_1\cdots s_n$. Define a map $X \rightarrow T$ by

$$\begin{aligned} s_j=t_j^{q-1} \quad (j=1,2,\dots , n), \quad t=\prod _{j=1}^n t_j. \end{aligned}$$

In this appendix, let $\mu _n$ denote the group of nth roots of unity in $\kappa$. Fix a primitive root $\zeta \in \mu _n$ and define a map $C^{n-1} \rightarrow T$ by

$$\begin{aligned} s_j=\prod _{i=1}^{n-1} (1-\zeta ^j y_i) \quad (j=1,2,\dots , n), \quad t=\prod _{i=1}^{n-1} x_i, \end{aligned}$$

where $(x_i,y_i)$ denotes the coordinates of the ith component of $C^{n-1}$. Then, X, T, $C^{n-1}$ are all Galois over S, and we have natural identifications

$$\begin{aligned} \textrm{Gal}(X/S)=\mu _{q-1}^n, \quad \textrm{Gal}(T/S)=\mu _{q-1}, \quad \textrm{Gal}(C^{n-1}/S)=\mu _{q-1}^{n-1} \rtimes S_{n-1}. \end{aligned}$$

The restriction maps $\textrm{Gal}(X/S)\rightarrow \textrm{Gal}(T/S)$ and $\textrm{Gal}(C^{n-1}/S)\rightarrow \textrm{Gal}(T/S)$ are identified respectively with the multiplication $\mu _{q-1}^n\rightarrow \mu _{q-1}$ and the first projection followed by the multiplication $\mu _{q-1}^{n-1}\rightarrow \mu _{q-1}$.

Remark A.2

Terasoma [24] also constructs a common covering of X and $C^{n-1}$ over S. Let $C' \rightarrow C$ be a covering given by $u_j^{q-1}=1-\zeta ^j y$ ($j=1,\dots , n$), $\prod _{j=1}^n u_j = x$. Then, as well as the map $C'^{n-1}\rightarrow C^{n-1}$, the map $C'^{n-1} \rightarrow X$ is given by $t_j= \prod _{i=1}^{n-1} u_{i,j}$ ($j=1,\dots , n$), and $C'^{n-1}$ is Galois over S with $\textrm{Gal}(C'^{n-1}/S)=(\mu _{q-1}^n)^{n-1} \rtimes S_{n-1}$.

For the given $\alpha \in \widehat{\kappa ^*}=\widehat{\mu _{q-1}}$, put $\alpha ^{(n)}=\alpha |_{\mu _{q-1}^n}$ and $\chi =\alpha |_{\mu _{q-1}^{n-1}\rtimes S_{n-1}}$. Then by Lemma A.1 (i),

$$\begin{aligned} N(X,\alpha ^{(n)}) =N(T,\alpha )= N(C^{n-1}, \chi ). \end{aligned}$$

By Weil [25] (cf. [17, (2.12)]), we have

$$\begin{aligned} N(X,\alpha ^{(n)}) = (-1)^{n-1}\alpha ^n(n) j(\underbrace{\alpha ,\dots , \alpha }_{n\,\,\textrm{times}}). \end{aligned}$$

Our task is to compute $N(C^{n-1}, \chi )$. Let $C_0\subset C$ be the subvariety defined by $y\ne 0$, and put $D=C\setminus C_0$. For $k \in \mathbb {N}$, put

$$\begin{aligned} H_k=\mu _{q-1}^k \rtimes S_k, \quad G_k=(\mu _{q-1}\times \mu _n)^k \rtimes S_k, \end{aligned}$$

($S_0=\{1\}$ by convention). An element of $G_k$ is written as $\xi \eta \sigma$ with $\xi =(\xi _i)_{i=1,\dots , k} \in \mu _{q-1}^k$, $\eta =(\eta _i)_{i=1,\dots , k}\in \mu _n^k$ and $\sigma \in S_k$. Then $G_k$ acts naturally on $C^k$, respecting $C_0^k$ and $D^k$. The support of $\sigma \in S_k$ is defined by ${\text {supp}}(\sigma )=\{i=1,\dots , k \mid \sigma (i)\ne i\}$. For $\xi \in \mu _{q-1}^k$ and $\sigma \in S_k$, put $p(\xi )=\prod _{i=1}^k \xi _i$, $p_\sigma (\xi )=\prod _{i\in {\text {supp}}(\sigma )} \xi _i$, and similarly $p(\eta )$, $p_\sigma (\eta )$ for $\eta \in \mu _n^k$. Define a character $\chi _k \in \widehat{H_k}$ by

$$\begin{aligned} \chi _k(\xi \sigma )=\alpha (p(\xi )). \end{aligned}$$

Note that $\chi =\chi _{n-1}$. For $\sigma \in S_k$, we write its cycle decomposition (unique up to ordering) as $\sigma =\sigma _1\cdots \sigma _r$, where $\sigma _j$ is a cyclic permutation of length $l_j$ with $\sum _{j=1}^r l_j=k$, and ${\text {supp}}(\sigma _j)$’s are all disjoint.

Lemma A.3

Let $\xi \eta \sigma \in G_k$ and $\sigma =\sigma _1\cdots \sigma _r$ be the cycle decomposition. Then

$$\begin{aligned} {\text {Ind}}^{G_k}_{H_k} \chi _k (\xi \eta \sigma )= \prod _{j=1}^r \sum _{\varphi \in \widehat{\mu }_n} \alpha (p_{\sigma _j}(\xi ))\varphi (p_{\sigma _j}(\eta )). \end{aligned}$$

Proof

Since $\mu _n^k$ represents $G_k/H_k$, we have by definition

$$\begin{aligned} {\text {Ind}}^{G_k}_{H_k} \chi _k (\xi \eta \sigma )= \sum _{\eta '\in \mu _n^k, \eta '^{-1}\eta \sigma (\eta ')=1} \chi _k(\xi ). \end{aligned}$$

The condition $\eta '^{-1}\eta \sigma (\eta ')=1$ is satisfied only when $p_{\sigma _j}(\eta )=1$ for all j, and then the number of such $\eta '$’s is $n^r$. Therefore, for any $\eta$, the number of $\eta '$’s satisfying the condition is $\prod _{j=1}^r \sum _{\varphi \in \widehat{\mu _n}} \varphi (p_{\sigma _j}(\eta ))$. Since $\chi _k(\xi )=\prod _j \alpha (p_{\sigma _j}(\xi ))$, the statement follows. $\square$

For an integer $l\ge 1$, let $\kappa _l$ denote the degree l extension of $\kappa$ contained in $\overline{\kappa }$, and let $N_l:\kappa _l^* \rightarrow \kappa ^*$ denote the norm map.

Lemma A.4

Let $\xi \eta \sigma \in G_k$ and $\sigma =\sigma _1\cdots \sigma _r$ be the cycle decomposition. Then

$$\begin{aligned}&\Lambda (C_0^k,\xi \eta \sigma )= ((q-1)n)^r\\&\times \prod _{j=1}^r \#\left\{ (u,v) \in (\kappa _{l_j}^*)^2 \bigm | u+v=1, N_{l_j}(u)=p_{\sigma _j}(\xi ), N_{l_j}(v)^\frac{q-1}{n}=p_{\sigma _j}(\eta ) \right\} . \end{aligned}$$

Proof

It reduces to the case $r=1$. Let $(x_i,y_i)_{i=1,\dots , k} \in C_0^k$. Then $\xi \eta \sigma (x_i,y_i)_i=F(x_i,y_i)_i$ happens only when $F^k(x_1,y_1)=(p(\xi )x_1,p(\eta )y_1)$, i.e. $x_1^{q^k-1}=p(\xi )$, $y_1^{q^k-1}=p(\eta )$. If we put $u=x_1^{q-1}$, $v=y_1^n$, then $u, v \in \kappa _{l_j}^*$, $u+v=1$ and the condition above becomes $N_k(u)=p(\xi )$, $N_k(v)^\frac{q-1}{n}=p(\eta )$. To each (u, v) as above correspond $(q-1)n$ points $(x_1,y_1)$, hence the lemma. $\square$

Proposition A.5

We have

$$\begin{aligned} N(C_0^k,\chi _k)=\frac{(-1)^k}{k!} \sum _{\nu _1,\dots , \nu _k} \prod _{i=1}^k j(\alpha ,\nu _i), \end{aligned}$$

where the sum is taken over all distinct $\nu _1,\dots , \nu _k\in \widehat{\kappa ^*}$ with $\nu _1^n=\cdots =\nu _k^n=\varepsilon$.

Proof

By Lemma A.1 (ii), we have $N(C_0^k,\chi _k)=N(C_0^k, {\text {Ind}}_{H_k}^{G_k} \chi _k)$. First, fix $\sigma =\sigma _1\cdots \sigma _r \in S_k$. By Lemmas A.3 and A.4,

$$\begin{aligned}&\sum _{\xi ,\eta } {\text {Ind}}_{H_k}^{G_k} \chi _k(\xi \eta \sigma ) \Lambda (C_0^k,\xi \eta \sigma )\\&=((q-1)n)^r \prod _j \sum _{\varphi } ((q-1)n)^{l_j-1} \sum _{u_j,v_j \in \kappa _{l_j}^*, u_j+v_j=1} \alpha (N_{l_j}(u))\varphi (N_{l_j}(v)^\frac{q-1}{n}). \end{aligned}$$

Note that, for each $\xi _0\in \mu _{q-1}$, the number of $\xi \in \mu _{q-1}^{l_j}$ such that $p(\xi )=\xi _0$ is $(q-1)^{l_j-1}$, and similarly for $\eta \in \mu _n^{l_j}$. We identify $\varphi \in \widehat{\mu _n}$ with $\nu \in \widehat{\kappa ^*}$ satisfying $\nu ^n=\varepsilon$ by $\nu (v)=\varphi (v^\frac{q-1}{n})$. Then, the last sum is written as $-j(\alpha \circ N_{l_j}, \nu \circ N_{l_j})$, the Jacobi sum over $\kappa _{l_j}$. We have another well-known formula of Davenport–Hasse [7] (cf. [25, (5)])

$$\begin{aligned} j(\alpha \circ N_{l_j}, \nu \circ N_{l_j})=j(\alpha ,\nu )^{l_j}. \end{aligned}$$

Hence it follows

$$\begin{aligned} \frac{1}{((q-1)n)^k} \sum _{\xi ,\eta } {\text {Ind}}_{H_k}^{G_k} \chi _k(\xi \eta \sigma ) \Lambda (C_0^k,\xi \eta \sigma ) = (-1)^r \prod _{j=1}^r \sum _{\nu \in \widehat{\kappa ^*}, \nu ^n=\varepsilon } j(\alpha ,\nu )^{l_j}. \end{aligned}$$

Let us say that a k-tuple $(\nu _1,\dots , \nu _k)$ is $\sigma$-admissible if for each $j=1,\dots , r$, $\nu _i$’s agree for all $i\in {\text {supp}}(\sigma _j)$. Then the right-hand side is written as

$$\begin{aligned} (-1)^r \sum _{(\nu _1,\dots , \nu _k): \sigma \text {-admissible}} \prod _{i=1}^k j(\alpha ,\nu _i). \end{aligned}$$

Now we let $\sigma$ vary and write $r=r(\sigma )$. Then,

$$\begin{aligned} N(C_0^k, {\text {Ind}}_{H_k}^{G_k}\chi ) =\frac{1}{k!} \sum _{\nu _1,\dots , \nu _k} \sum _{\sigma } (-1)^{r(\sigma )}\prod _{i=1}^k j(\alpha ,\nu _i), \end{aligned}$$

where the last sum is taken over $\sigma$ for which $(\nu _1,\dots , \nu _k)$ is $\sigma$-admissible. This sum vanishes unless $\nu _1,\dots , \nu _k$ are all distinct, since

$$\begin{aligned} \sum _{\sigma \in S_l} (-1)^{r(\sigma )} = (-1)^l \sum _{\sigma \in S_l} {\text {sgn}} \sigma = 0\quad (l \ge 2). \end{aligned}$$

Hence the proposition is proved. $\square$

Proposition A.6

For any $k\ge 0$, we have $N(D^k,\chi _k)=1.$

Proof

Since $D(\overline{\kappa })=\{(x,0)\mid x\in \kappa ^*\}$, it is fixed by F. For any $\xi \sigma \in H_k$ with $\sigma =\sigma _1\cdots \sigma _r$ as before, $\Lambda (D^k,\xi \sigma )=(q-1)^{r(\sigma )}$ if $p_{\sigma _j}(\xi )=1$ for all $j=1,\dots , r$, and $\Lambda (D^k,\xi \sigma )=0$ otherwise. The number of $\xi$’s such that $p_{\sigma _j}(\xi )=1$ for all j is $\prod _j (q-1)^{l_j-1}$, and for such $\xi$, we have $\chi (\xi \sigma )=\prod _j \alpha (p_{\sigma _j}(\xi ))=1$. Hence $N(D^k,\chi _k)=(\#H_k)^{-1} \sum _{\sigma \in S_k} (q-1)^k =1.$ $\square$

Now, let $(C^{n-1})_k \subset C^{n-1}$ denote the $S_{n-1}$-orbit of $D^k \times {C_0}^{n-1-k}$ ($k=0,\dots , n-1$). Then,

$$\begin{aligned} N((C^{n-1})_k,\chi )&= \frac{1}{\# H_{n-1}} \left( {\begin{array}{c}n-1\\ k\end{array}}\right) \sum _{\xi _1\sigma _1\in H_k, \xi _2\sigma _2\in H_{n-1-k}}\\&\qquad \qquad \chi _k(\xi _1) \chi _{n-1-k}(\xi _2) \Lambda (D^k,\xi _1\sigma _1)\Lambda (C_0^{n-1-k},\xi _2\sigma _2)\\&= N(D^k,\chi _k) N(C_0^{n-1-k},\chi _{n-1-k}). \end{aligned}$$

By Propositions A.5 and A.6, noting $j(\alpha ,\varepsilon )=1$, it follows

$$\begin{aligned} N(C^{n-1},\chi ) =\sum _{k=0}^{n-1} N((C^{n-1})_k,\chi ) =(-1)^{n-1} \prod _{\nu ^n=\varepsilon , \nu \ne \varepsilon } j(\alpha ,\nu ). \end{aligned}$$

Hence the theorem is proved.