数理解析研究所講究録
第 1715 巻 2010 年 32-36

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Orbital Gauss sums associated with the space of binary
cubic forms over a finite field
森 伸吾 (Shingo Mori)
京都大学理学部数学教室
Department of Mathematics, Kyoto University\S 0 Introduction
We consider an orbital L-function associated with the space of binary cubic forms over rational
integer ring. The orbital L-function satisfy a functional equation. The functional equation
may be expressed in terms of an orbital Gauss sum. In this paper, we shall evaluate theorbital Gauss sum.

matrices
is its group of units and $M_{n}(K)$ is the ring ofNotation. If $K$ is a field,$K$
which arematrices overover $K$ . When $K$ is commutative, $GL_{n}(K)$ is the group of$GL_{n}(K)$$N(K)$$B(K)$of matrices offor the subgroups ofandinvertible. We use the notation
the form$n\cross n$$K^{x}$$n\cross n$$(_{0}^{*}$$**)$,$(\begin{array}{l}l*01\end{array})$

respectively. Unless otherwise specified, $G_{K}=G(K)=GL_{2}(K)$ .
Let be a Dirichlet character of conductor . An usual Gauss sum is defined by$\chi$$f$$\tau(\chi)=\sum_{a=1}^{f}\chi(a)\exp(\frac{2\pi\sqrt{-1}}{f})$.

\S 1 The space of binary cubic forms over a finite field.
of binaryFirst, a review of the basic theory is in order. Let $K$ be a field. The space$K$
identify
ais of four dimensional, and we shallcubic forms with coefficients in the field
4-tuple $x=(x_{1}, x_{2}, x_{3}, x_{4})\in K^{4}$ with the form given by:$V_{K}$$F_{x}(u, v)=x_{1}u^{3}+x_{2}u^{2}v+x_{3}uv^{2}+x_{4}v^{3}$

We shall define an action of the group $G_{K}=GL_{2}(K)$ on$V_{K}$equation:$F_{g\cdot x}=(\det g)^{-1}F_{x}((u, v)(\begin{array}{ll}a bc d\end{array}))$.

by the following functional

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where

$x$

is any element of$(\begin{array}{ll}a 00 a\end{array})\cdot x=ax$$V_{K}$

and$g=(\begin{array}{ll}a bc d\end{array})$

is any element of. Let $P(x)$ denote the discriminant of the form

$G_{K}$$F_{x}$. This is arranged so that, explicitly given by$P(x)=x_{2}^{2}x_{3}^{2}+18x_{1}x_{2}x_{3}x_{4}-4x_{2}^{3}x_{4}-4x_{1}x_{3}^{3}-27x_{1}^{2}x_{4}^{2}$.

denote the set of. Let
The hypersurface $S_{K}=\{x\in V|P(x)=0\}$ is invariant under$V_{K}’=\{x\inV_{K}|P(x)\neq0\}=V_{K}-S_{K}$. A basic feature of thisall nonsingular forms in
representation is that
$G_{K}$$V_{K}’$$V_{K},$$P(g\cdot x)=(\det g)^{2}P(x)$.

-invariant if there exists ais called a relative
A non zero rational function $R(x)$ on$g\in G_{K}$ . The discriminant$x\inV_{K}$$R(g\cdotx)=\chi(g)R(x)$andforallofsuchthatcharacter
generates the ring of relative invariants of this representation of $GL_{2}(K)$ .
$G_{K}$$V_{K}$$G_{K}$$\chi$\S 2
be the finite field of prime
Let be a prime number. We shall assume that $p\neq 2,3$ . Let
decomposes
and nonsingular set
power of order . We put $K=F_{q}$ . The hypersurface
orbits.
into three$F_{q}$$p$$S_{K}$$q$$V_{K}^{f}$$G_{K}$

Lemma 1. We put $s_{1}=(1,0,0,0)$ and $s_{2}=(0,1,0,0)$ . The$S_{0}=\{0\}$$S_{2}=G_{K}\cdot s_{2}=$$x$-orbits in$S_{K}$

are preciously;$S_{1}=G_{K}\cdot s_{1}=$For a form

$G_{K}$

in$V_{K}’${{has a triple root};$x\in V_{K}|F_{x}$ has a double root and a distinct simple root}.

$x\in V_{K}|F_{x}$, let $K(x)$ denote the cubic ring of$x$

areLemma 2. Two nonsingular binary cubic forms overpreciouslyin
are-orbits
cubic ring are same. The$F_{q}$$G_{K}$

over $K$ . The degree of $K(x)$ is 3.
$G_{K}$-equivalent if and only if their$V_{K}’$$V_{K,1}’=\{x\in V_{K}’|F_{q}(x)=F_{q}\cross F_{q}\cross F_{q}\}$$V_{K,2}’=\{x\in V_{K}’|F_{q}(x)=F_{q^{2}}\cross F_{q}\}$$V_{K,3}’=\{x\in V_{K}’|F_{q}(x)=F_{q^{3}}\}$;;.

of nonsingular binary cubic forms with cubic ring
The order of stabilizer inlmod3, there are three nonsingular
is 6, 2 and 3, respectively. Ifandorbits with representatives:$F_{q}\cross F_{q}\cross F_{q}$$G_{K}$$F_{q^{2}}\cross F_{q}$$p\equiv$$F_{q^{3}}$$x_{I}=(1,0, -1,0),$ $x_{II}=(r, 0, -1,0),$ $x_{III}=(s, 0,0, -1)$ ,where$r$

is any element of$F_{q}^{\cross}$that is not a square and$s$is any element that is not a cube.

\S 3 The orbital Gauss sum.

,

$G_{K^{-}}$

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For simplicity, we shall assume that $K=F_{p}$ . Let be a character of multiplicative groupby the convention $\psi(0)=0$ . The alternating
ofof nonzero elements of . Extend toform:$\psi$$F_{p}$$F_{p}^{\cross}$$F_{p}$$\psi$$[x, y]=x_{1}y_{4}- \frac{1}{3}x_{2}y_{3}+\frac{1}{3}x_{3}y_{2}-x_{4}y_{1}$has the property that $[g\cdot x, \det(g)^{-1}g\cdot y]=[x, y]$ for all
we put$\langle x,$,$x,$ $y\in V_{K}$and $g\in G_{K}$ . For$x,$ $y\in V_{K}$,.

$y \rangle=\exp(\frac{2\pi\sqrt{-1}}{p}[x, y])$We define the orbital Gauss sum.
Definition 1. For$a,$ $b\in V_{K}$, we define$g\cdot y\rangle$$W( \psi, a, b)=\sum_{g\in G_{K}}\psi(\det(g))\langle x,$

After basic calculation, we find that$W(\psi, g\cdot a, g’\cdot b)=\psi(\det g)^{-1}\psi(\det g^{f})^{-1}W(\psi, a, b)$

where$g,$$g’\in G(K)$ .

We can take the following set:.

$V(F_{p})=\{y_{0}|y_{0}\in S_{0}\}u\{y_{1}|y_{1}\in S_{1}\}u\{y_{2}|y_{2}\in S_{2}\}u\{y_{3}|y_{3}\in V_{1,K}’\}u\{y_{4}|y_{4}\in V_{1,K}’\}u\{ys|y_{5}\in V_{1,K}’\}$$0\leq i,j\leq 5$ , we define a matrix valued Gauss sum $W(\psi)$ as a
For positive integers$(i,j)$.
component is given bymatrix whose$i,$$j,$$6\cross 6$$\frac{1}{\# G(K)_{y_{j}}}W(\psi, y_{i}, y_{j})$We shall assume that $\psi^{3}=1$ . Our main result is as follows.
Theorem 1. Let$W(1)=(1I1111$$\psi$

be a trivial character. If$-p-1p^{2}-12p-1p-1-1-1p(p^{2_{0}}-1)p(p-1)p(p-2)-3p-p$$p\equiv$

lmod3, then$\frac{}{6}p(p-1)(2p-1)\frac{1}{f}p(p^{2}-1)(p-1)-\frac{1}{2}p(p-1)-\frac{1}{6,p}p(p-1)\frac{1}{6}(p-1)\frac{1}{6}p(p+5)$$\frac{1}{2}p(p-1)(p^{2}-1)-\frac{1}{2}p(p-1)-\frac p(p-1)-\frac p(p-1)-\frac{\not\in\not\in 1}{2}p(p-1)\frac{1}{2}p(p+1)$$\frac{1}{3}p(p-1)(p^{2}-1)-\frac{1}{3}p(p^{2}-1)-\frac{1}{3}p(p-1)\frac{1}{3}p(p-1)\frac{1}{3}p(p+2)0)\cdot$$\frac p(p-1)(2p-1)\frac{1}{6f}p(p^{2}-1)(p-1)-\frac{1}{2}p(p-1)\frac{1}{6}p(\frac{}{(p}p+5)-\frac{1}{6}p(p+1)\frac{1}{6}p+1)$$\frac{1}{2}p(p-1)(p^{2}-1)-\frac{1}{}p(p-1)-\frac{\not\in}{2}p(p-1)\frac{1}{2}(\frac{(p}{(p}p+1)\frac{1}{}p+1)\frac{2p1}{2}p+1)$$\frac{1}{3}p(p-1)(p^{2}-1)-\frac{1}{3}p(p^{2}-1)-\frac{1}{3,p}p(p+1)\frac{}{3}p(p-2)\frac{1}{?}(p+1)0)\cdot$

If $p\equiv 2mod 3$ , then$W(1)=(111111$$-p-1p^{2}-12p-1p-1-1-1p(p^{2_{0}}-1)p(p-1)p(p-2)-3p-p$

35Theorem 2. Let$W(\psi)=(000000$$\psi$be a nontrivial cubic character. If$-\psi(4r)\tau^{2}(\psi)\overline{\psi}(s)\tau^{2}(\psi)p\tau_{0}^{0}(\overline{\psi})\tau^{2}(\psi)$$000000$$\frac{1}{6}p(p-1)\tau^{2}(\psi)\frac{\frac{1}{f}}{\frac{?}{6}}Xo_{B}o_{A}$$p\equiv$

lmod3, then$- \frac{1}{2}\psi(4r)p(p-1)\tau^{2}(\psi)\frac{1}{\frac{\frac{3}{\int}}{2}}Xo_{Y}0D$$\frac{1}{3}\overline{\psi}(s)p(p_{o_{B}^{-}}^{o_{C}}1)\tau^{2}(\psi)\frac{\frac{1}{3}}{\frac{3}{3}}D)$

where$A= \tau^{4}(\overline{\psi})+4\tau^{2}(\psi)-\frac{\tau^{5}(\psi)}{p},$$B= \overline{\psi}(s)(\tau^{4}(\overline{\psi})-2\tau^{2}(\psi)p-\frac{\tau^{5}(\psi)}{p})$$C= \psi(s)(\tau^{4}(\overline{\psi})+\tau^{2}(\psi)p-\frac{\tau^{5}(\psi)}{p}),$$X= \psi(4r)(\tau^{4}(\overline{\psi})+\frac{\tau^{5}(\psi)}{p})$and$D= \psi(4rs^{2})(\tau^{4}(\overline{\psi})+\frac{\tau^{5}(\psi)}{p})$$Y= \tau^{4}(\overline{\psi})-\frac{\tau^{5}(\psi)}{p}$Proofs. For simplicity we assume $a=b=s_{1}$ . We put,,.

$w=(\begin{array}{ll}0 11 0\end{array})$. Elementary methods oflinear algebra give the Bruhat decomposition$G(K)=B(K)uB(K)wN(K)$

where$B(K)=\{(\begin{array}{ll}a 00 c\end{array})(\begin{array}{ll}1 n0 1\end{array})|a,$and$c\in K^{\cross},$$n\in K\}$

$B(K)wN(K)=\{(\begin{array}{ll}a 00 c\end{array})(\begin{array}{ll}1 n0 l\end{array})(\begin{array}{ll}0 11 0\end{array})(\begin{array}{ll}1 m0 l\end{array})|a,$$c\in K^{\cross},$ $n,$$m\in K\}$ .

For $g_{1}\in B(K)$ and $g_{2}\in B(K)wN(K)$ , we define$W_{1}( \psi, s_{1}, s_{1})=\sum_{g_{1}\in B(K)}\psi(\det g_{1})\langle[s_{1}, g_{1}\cdot s_{1}]\rangle$

and$W_{2}( \psi, s_{1}, s_{1})=\sum_{g_{2}\in B(K)wN(K)}\psi(\det g_{2})\langle[s_{1}, g_{2}\cdot s_{1}]\rangle$.

For $1\leq i\leq 2$ , the twisted action of on the element is given by $g_{1}\cdot s_{1}=(a^{2}c^{-1},0,0,0),$$s_{1}=$ $(a^{2}c^{-1}n^{3},3an^{2},3an, a^{-1}c^{2})$ . A straightforward calculation shows that$g_{i}$$s_{1}$$W_{1}( \psi, s_{1}, s_{1})=\sum_{g\in B(K)}\psi(\det g)\langle[s_{1}, g_{i}\cdot s_{1}]\rangle$$= \sum_{a,c\in K^{\cross},n\in K}\psi(ac)\langle 0\rangle$$=\{\begin{array}{ll}(p-1)^{2}p if \psi=1,0 otherwise.\end{array}$$g_{2}$.

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We deduce the analogous equality for$W_{2}(\psi, s_{1}, s_{1})$$W_{2}( \psi, s_{1}, s_{1})=\sum_{g\in B(K)wN(K)}\psi(\det g)\langle[s_{1}, g_{i}\cdot s_{1}]\rangle$$= \sum_{a,c\in K^{\cross},nm\in K},\psi(ac)\langle a^{-1}c^{2}\rangle$$= \sum_{a,c\in K^{\cross},nm\in K},\psi(ac^{3})\langle a^{-1}\rangle$$= \sum_{a,c\in K^{x},nm\in K},\overline{\psi}(a)\langle a\rangle$$=p^{2}(p-1)\tau(\overline{\psi})$.

Combining all these equalities, we obtain$W(\psi, s_{1}, s_{1})=W_{1}(\psi, s_{1}, s_{1})+W_{2}(\psi, s_{1}, s_{1})=\{\begin{array}{ll}-p(p-1) if \psi=1,p^{2}(p-1)\tau(\overline{\psi}) otherwise.\end{array}$More precious proof will be shown in [SM].

References[B] M. Bhargava; Higher composition laws II: On cubic analogues of Gauss composition Ann. of Math 159(2004), 865-886[G] A. Gyoja; Character sums and intersection cohomology complexes associated to the space of squarematrices, Indag. Math. (N.S.)8 (1997), 371-385[GGS] W.-T. Gan, B. H. Gross and G. Savin; Fourier coefficients of modular forms on115 (2002), no. 1, 105-169$G_{2}$, Duke. Math. J.

[SH] H. Saito; A generalization of Gauss sums and its applications to Siegel modular forms and L-functions
associated with the vector space of quadratic forms, J. Reine Angew. Math. 416 (1991), 91-142[S-S] M. Sato and T. Shintani; On zeta functions associated with prehomogeneous vector spaces, Ann. ofMath. (2) 100 (1974), 131-170[SM] S. Mori; Orbital Gauss sum for the space of binary cubic forms over a finite field. in preparation.

[T] T. Taniguchi; Orbital L-functions for the space of binary cubic forms. in preparation.

[Dl] D.J. Wright; The adelic zeta function associated to the space of binary cubic forms part I: Global theory,
Math. Ann270 (1985), 503-534[D2] D.J. Wright; Cubic Character sums of cubic polynomial, Amer. Math. Soc 100 No. 3, (1987), 409-413

Department of Mathematics
Kyoto University
Kyoto 606-8502, Japansin-mori@math.kyoto-u.ac.jp