1 Introduction
In [Reference Lehmer8], Lehmer remarks that the class of matrices for which one can explicitly determine the eigenvalues and the general kth power is very limited. Using the Legendre character on finite fields, Lehmer constructs two classes of matrices for which this is possible. More generally, using characters of arbitrary orders, Carlitz [Reference Carlitz2] and Lehmer [Reference Lehmer9] construct other classes of matrices for which they determine the characteristic polynomials and kth powers.
Here we consider a class of matrices, whose entries are roots of unity and zeroes, which arise in the transformation formulae for Gaussian hypergeometric functions over finite fields defined by Greene [Reference Greene3]. We first recall the definition of these functions. If p is a prime,
$q=p^r, n\geq 1,$
and
$A_1,\ldots ,A_n,B_2,\ldots ,B_{n}$
are complex-valued multiplicative characters over
${\mathbb {F}}_q^\times ,$
then the finite field hypergeometric functions are defined by
$$ \begin{align} {_{n}F_{n-1}}\left(\begin{array}{@{}cccc} A_1, & A_2,& \ldots, & A_n\\ ~& B_2, &\ldots, &B_{n} \end{array}\bigg|\, x\right)_q:=\frac{q}{q-1}\sum_{\chi}{A_1\chi\choose\chi}{A_2\chi\choose B_2\chi}\cdots{A_n\chi\choose B_{n}\chi}\,\chi(x), \end{align} $$
where the summation is over multiplicative characters
$\chi $
of
${\mathbb {F}}_q^\times $
and the binomial coefficient
$\binom {A}{B}$
is a normalised Jacobi sum, given by
$$ \begin{align} {A\choose B}:=\frac{B(-1)}{q}J(A,\overline{B}):=\frac{B(-1)}{q}\sum\limits_{x\in{\mathbb{F}}_q}A(x)\overline{B}\,(1-x). \end{align} $$
These functions have deep connections to étale cohomology [Reference Katz6] and often arise in geometry where they count the number of
${\mathbb {F}}_q$
-points on various algebraic varieties (see [Reference Beukers, Cohen and Mellit1, Theorem 1.5]). For example, if
$\lambda \in {\mathbb {F}}_q\setminus \{0,1\}$
and
$E_\lambda $
is the Legendre normal form elliptic curve
$$ \begin{align*} E_\lambda : y^2 = x(x-1)(x-\lambda), \end{align*} $$
then (see [Reference Koike7, Section 4] and [Reference Ono11, Theorem 1]),
$$ \begin{align*} \# E_\lambda({\mathbb{F}}_q) = 1+q+q\cdot\phi_q(-1)\cdot {_{2}F_{1}}\left(\begin{array}{@{}ccc} \phi_q, & \phi_q\\ ~& \varepsilon \end{array}\bigg|\, \lambda\right)_q, \end{align*} $$
where
$\phi _q$
and
$\varepsilon $
are respectively the Legendre symbol and the trivial character on
${\mathbb {F}}_q^\times .$
Moreover, these functions satisfy analogues of several transformation formulae of their classical counterparts, such as the generalised Euler integral transform (see [Reference Slater13, (4.1.1)]). More precisely (see [Reference Greene3, Theorem 3.13]),
$$ \begin{align} &{_{n+1}F_{n}}\left(\begin{array}{ccccc}A_1,&A_2,&\ldots,&A_n,&A_{n+1}\\&{B_2},&\ldots,&{B_{n}}&{B_{n+1}}\end{array}\bigg|\, x\right)_{q} \notag \\ &\quad = {\frac{A_{n+1}B_{n+1}(-1)}{q}}\sum\limits_{y\in{\mathbb{F}}_q}{_{n}F_{n-1}}\left(\begin{array}{@{}cccc} A_1,& A_2,&\ldots,&A_n\\ &{B_2},&\ldots,&{B_{n}}\end{array}\bigg|\, xy\right)_{{q}}\cdot{ A_{n+1}(y)\overline{A_{n+1}}B_{n+1}(1-y)}. \end{align} $$
Motivated by the transformation formula (1.3), Ono as well as Griffin and Rolen study the matrix corresponding to this transformation when
$q=p^r$
is odd,
$A_{n+1}=\phi _q$
and
$B_{n+1}=\epsilon $
. Consider the
$(q-2)\times (q-2)$
matrix
$M=(M_{ij})$
indexed by
${i,j\in {\mathbb {F}}_q\setminus \{0,1\},}$
where
$$ \begin{align*} M_{ij}=\phi_q(1-ij)\phi_q(ij) \end{align*} $$
and let
$f_q$
be its characteristic polynomial. In this notation, Griffin and Rolen [Reference Griffin and Rolen4] prove a conjecture by Ono that
$$ \begin{align*} f_q(x)=\begin{cases} (x+1)(x-1)(x+2)(x^2-q)^{(q-5)/2} & \text{if } \phi_q(-1)=1,\\ x(x^2-3)(x^2-q)^{(q-5)/2} & \text{if }\phi_q(-1)=-1. \end{cases} \end{align*} $$
The purpose of this paper is to study, à la Lehmer, a more general analogue of the matrix M that arises when the characters
$A_{n+1}$
and
$B_{n+1}$
are arbitrary. More precisely, we consider the
$({q}-1)\times ({q}-1)$
matrix
$M_q=(M_q)_{ij}$
indexed by
$i,j\in {\mathbb {F}}_q^\times ,$
where
$$ \begin{align*} (M_q)_{ij}:= A(ij)\overline{A}B(1-ij). \end{align*} $$
We first determine the characteristic polynomial
$f_q$
of
$M_q.$
Theorem 1.1. If p is an odd prime,
$q=p^r$
and
$\omega $
is a character of order
$q-1$
of
${\mathbb {F}}_q^\times $
, then
$$ \begin{align*} f_q(x) = (x-J(\overline{A}B, A))(x - J(\overline{A}B,\overline{A}\phi)) \prod_{l = 1}^{{(q-3)}/{2}}(x^2 - J(\overline{A}B, A\omega^l)J(\overline{A}B, A\overline{\omega^l})). \end{align*} $$
Our proof explicitly determines the eigenvectors of
$M_q.$
Furthermore, when
$B=\varepsilon $
and
$k\geq 1,$
we explicitly determine the entries of
$M_q^k.$
Theorem 1.2. If
$k\geq 1,$
we write
$k=2l$
if k is even and
$k=2l+1$
if k is odd. In this notation, if p is an odd prime,
$q=p^r$
and
$B=\varepsilon ,$
then
$$ \begin{align*} (M_q^k)_{ij}=A^l(-1)\cdot q^{k-1}\cdot{_kF_{k-1}}\left(\begin{array}{@{}cccc} A_1, & A_2,& \ldots, & A_k\\ ~& B_2, &\ldots, &B_k \end{array}\bigg|\, \frac{j^{(-1)^k}}{i}\right)_q, \end{align*} $$
where
$$ \begin{align*} A_n = \begin{cases} A & \text{if } 1\leq n \leq l ,\\ \varepsilon & \text{otherwise}, \end{cases}\quad\text{and}\quad B_n = \begin{cases} \varepsilon & \text{if } 2\leq n \leq l, \\ \overline{A} & \text{otherwise}. \end{cases} \end{align*} $$
Remark 1.3. If
$B\neq \varepsilon ,$
the entries of
$M_q^k$
can be written in terms of more general finite field hypergeometric functions, such as those given by McCarthy [Reference McCarthy10, Definition 2.4] and Otsubo [Reference Otsubo12, Definition 2.7]. The proof is analogous to the proof of Theorem 1.2.
The paper is organised as follows. In Section 2, we recall facts concerning characters and finite field hypergeometric functions and determine the action of
$M_q$
on an appropriate basis. In Section 3, we prove Theorems 1.1 and 1.2.
2 Nuts and Bolts
Here we recall facts about characters on finite fields and hypergeometric functions. We also determine the behaviour of
$M_q$
on an appropriate set of vectors.
We denote by
${\widehat {\mathbb {F}}_q^{\times }}$
the group of characters on
${\mathbb {F}}_q^\times .$
It is well known (see [Reference Ireland and Rosen5, Proposition 8.1.2]) that if
$\chi \in {\widehat {\mathbb {F}}_q^{\times }},$
then
$$ \begin{align*} \sum\limits_{x\in{\mathbb{F}}_q}\chi(x) = \begin{cases} q-1 & \text{if }\chi=\varepsilon, \\ 0 & \text{otherwise}, \end{cases} \end{align*} $$
and that if
$x\in {\mathbb {F}}_q,$
then
$$ \begin{align} \sum\limits_{\chi\in{\widehat{\mathbb{F}}_q^{\times}}}\chi(x) = \begin{cases} q-1 & \text{if }x=1, \\ 0 & \text{otherwise.} \end{cases} \end{align} $$
Furthermore, if
$A,B\in {\widehat {\mathbb {F}}_q^{\times }},$
then the following properties of binomial coefficients are known [Reference Greene3, (2.6)–(2.8)]:
$$ \begin{align} \binom{A}{B} &= \binom{A}{A\overline{B}},\qquad\quad \end{align} $$
$$ \begin{align} \binom{A}{B} &= B(-1)\cdot \binom{B\overline{A}}{B}, \notag \\ \binom{A}{B} &= \overline{AB}(-1)\cdot\binom{\overline{B}}{\overline{A}}. \end{align} $$
To state our results, we fix a generator
$\omega $
of
${\widehat {\mathbb {F}}_q^{\times }}.$
For
$1\leq l\leq q-1,$
we define the vectors
$\boldsymbol {w}^l$
indexed by
$i\in {\mathbb {F}}_q^\times ,$
where
$$ \begin{align*} \boldsymbol{w}^{l}_i=\omega^l(i). \end{align*} $$
The following lemma determines
$M_q\boldsymbol {w}^l.$
Lemma 2.1. If
$1\leq l\leq q-1,$
then
$$ \begin{align*} M_q \boldsymbol{w}^l=J(\overline{A}B, A\omega^l){\boldsymbol{w}}^{q-1-l}. \end{align*} $$
Proof. Fix
$l.$
Then, for
$i\in {\mathbb {F}}_q^\times ,$
$$ \begin{align*} (M_q\boldsymbol{w}^l)_i=\sum\limits_{j\in{\mathbb{F}}_q^\times} A(ij)\overline{A}B(1-ij)\omega^l(j). \end{align*} $$
Replacing j by
$j/i$
gives
$$ \begin{align*} (M_q\boldsymbol{w}^l)_i&=\sum\limits_{j\in{\mathbb{F}}_q^\times} A(j)\overline{A}B(1-j)\omega^l\bigg(\frac{j}{i}\bigg) \\ &=\overline{\omega^l}(i)\sum\limits_{j\in{\mathbb{F}}_q^\times} (A\omega^l)(j)\overline{A}B(1-j) \\ &=J(\overline{A}B, A\omega^l){\boldsymbol{w}^{q-1-l}_i}. \\[-2.6pc] \end{align*} $$
Remark 2.2. Recall that the Fourier transform of
$f:{\mathbb {F}}_q\to {\mathbb {C}}$
is a function
${\widehat {f}:{\widehat {\mathbb {F}}_q^{\times }}\to {\mathbb {C}}}$
defined by
$$ \begin{align*} \widehat{f}(\nu)=\sum_{\lambda\in {\mathbb{F}_q}}f(\lambda){\overline{\nu}}(\lambda). \end{align*} $$
By a similar argument to the proof of Lemma 2.1, the Fourier transforms of the components of
$M_q^2$
are products of two Jacobi sums.
To determine the quadratic terms in Theorem 1.1, we make use of the following lemma which follows from a direct computation.
Lemma 2.3. If M is an
$n\times n$
matrix,
$\lambda _1,\lambda _2\in {\mathbb {C}},$
and
$v_1\neq \pm v_2\in {\mathbb {C}}^n$
such that
$$ \begin{align*} Mv_1 = \lambda_1 v_2, \quad Mv_2=\lambda_2v_1, \end{align*} $$
then the vectors
$v_1\pm \sqrt {\lambda _1/\lambda _2}v_2$
are eigenvectors of M corresponding to the eigenvalues
$\pm \sqrt {\lambda _1\lambda _2}.$
Finally, we need to determine the inverse change-of-basis matrix for the basis
$\{\boldsymbol {w}^l\}_{1\leq l\leq q-1}.$
Lemma 2.4. If P is the matrix given by
$P_{ij}=\omega ^j(i),$
where
$i\in {\mathbb {F}}_q^\times $
and
$1\leq j\leq q-1,$
then
$$ \begin{align*} (P^{-1})_{ij}=\frac{1}{q-1}\overline{\omega^i(j)}. \end{align*} $$
Remark 2.5. Note that the indices for rows and columns are inverted in
$P^{-1}.$
In other words, for
$P^{-1}, 1\leq i\leq q-1$
and
$j\in {\mathbb {F}}_q^\times .$
Proof. Note that
$$ \begin{align*} \sum\limits_{k\in{\mathbb{F}}_q^\times}\omega^k(i)\cdot\frac{1}{q-1}\overline{\omega^k(j)}= \frac{1}{q-1}\sum\limits_{k\in{\mathbb{F}}_q^\times}\omega^k\bigg(\frac{i}{j}\bigg). \end{align*} $$
Since
$\omega $
is a generator of
${\widehat {\mathbb {F}}_q^{\times }},$
the lemma follows by (2.1).
3 Proofs of Theorems 1.1 and 1.2
Proof of Theorem 1.1.
Applying Lemma 2.1 with
$l = (q-1)/2$
and
$l = q-1$
shows that
$x-J(\overline {A}\phi ,\overline {A})$
and
$x-J(\overline {A},A)$
divide
$f_q(x).$
Similarly, applying Lemma 2.1 with
$1\leq l\leq (q-3)/2$
and Lemma 2.3 to the vectors
$\boldsymbol {w}^l$
and
$\boldsymbol {w}^{q-1-l}$
shows that
$x^2-J(\overline {A}B, A\omega ^l)J(\overline {A}B,A\overline {\omega ^l})$
divides
$f_q(x).$
Proof of Theorem 1.2.
We give the proof of this theorem when
$k=2l$
is even. Applying Lemma 2.1 twice shows
$$ \begin{align*} M_q^2 = PDP^{-1}, \end{align*} $$
where
$$ \begin{align*} D_{mn}=\begin{cases} J(\overline{A}, A\omega^m)J(\overline{A}, A\overline{\omega^m}) & \text{if } m=n, \\ 0 & \text{otherwise}, \end{cases} \end{align*} $$
and
$P_{ij}=\omega ^j(i)$
for
$i\in {\mathbb {F}}_q^\times $
and
$1\leq j\leq q-1.$
By Lemma 2.4 and a direct computation,
$$ \begin{align*} (M_q^{2l})_{ij}=\frac{1}{q-1}\sum\limits_{m=1}^{q-1}\omega^m \bigg(\frac{i}{j}\bigg)J(\overline{A},A\omega^m)^lJ(\overline{A},A\overline{\omega^m})^l. \end{align*} $$
By applying (1.2), (2.2) and (2.3),
$$ \begin{align*} (M_q^k)_{ij}=A^{l}(-1)\cdot \frac{q^{m}}{q-1}\sum\limits_{m=1}^{q-1}\binom{\overline{\omega^m}}{\overline{A}\overline{\omega^m}}^l\binom{A\overline{\omega^m}}{\overline{\omega^m}}^l \overline{\omega^m}\bigg(\frac{j}{i}\bigg). \end{align*} $$
Since
$\overline {\omega }$
generates
${\widehat {\mathbb {F}}_q^{\times }},$
the theorem follows from (1.1).
The proof is similar when k is odd.
Acknowledgements
The authors would like to thank Ken Ono for introducing the paper by Griffin and Rolen to them and for many valuable comments. This paper was written while the first author was visiting the University of Virginia. He would like to deeply thank Ken Ono for his hospitality and support during his visit.
