On certain finite group related to cubic theta function

Nikolai V. Proskurin

Max-Planck-Institut für Mathematik.

Vivatsgasse 7, 53111 Bonn, Germany.

Sankt-Petersburg Branch of Mathematical Institute RAS. 191011, Fontanka 27, Sankt-Petersburg, Russia. e-mail: proskn@pdmi.ras.ru

ABSTRACT. Given the Kubota-Patterson cubic theta function, one can attach to it a family of 27 shifted theta functions. Then one can naturally define some group of permutations of shifted theta functions, which occurs to be isomorphic to a subgroup of the famous group of permutations of 27 lines on cubic surface.

0 Introduction. These notes give rise from some observation on the Kubota– Patterson cubic theta function. Let Γ(3) be the principal congruence subgroup mod3 in SL(2, Z[ω]), where ω = exp(2πi/3), and let Γ̃(3) be the subgroup of SL(2, Z[ω]) generated by the groups Γ(3) and SL(2, Z). Let us recall that the cubic theta function we mean, denoted Θ_{K-P}: H → C, is defined on the 3-dimensional hyperbolic space H ≃ SL(2, C)/SU(2) and satisfies the transformation low

Θ_{K-P}(γw) = κ(γ)Θ_{K-P}(w) for all γ ∈ Γ̃(3), w ∈ H,

where κ: Γ̃(3) → C^{∗} is the cubic Kubota homomorphism; we shall explain this in more details in sections 3 and 4.

One may notice that Γ̃(3) and Γ(3) are subgroups of indexes 27 and 648 in SL(2, Z[ω]), and that the numbers 27 and 648 give rise also in connection with configuration of lines on a smooth cubic surface. Given a smooth cubic surface, say X, in the 3-dimensional projective space over algebraically closed field, there are 27 lines l_{1}, l_{2}, …, l_{27} contained in X. In fact not only is the number of lines independent of particular choice of surface, but so is the configuration of these lines. By that we mean that the group P of permutations of the lines l_{1}, l_{2}, …, l_{27} leaving fixed the incidence relations does not depend (up to isomorphism) of particular choice of X. The P is the group of order 51840 = 2^{7} · 3^{4} · 5 isomorphic to the Weyl group W (E_{6}) of the root system E_{6} . We can refer to [11], [9], [17], [8], [14], and to many other sources. It was observed by A. Cayley one and half century ago that some finite number of lines must lie on the surface; then G. Salmon showed that this number is 27. Let us restrict ourselves by this remark, and to refer to [9] for further history.

Our aim is to define some group of order 648 = 2^{3} · 3^{4} in connection with the cubic theta function Θ_{K-P} , and to show that this group is isomorphic to a subgroup of P. It is worth to say that the group we define has a geometrical interpretation. Actually, several different realizations are known as for P itself as for its commutator subgroup, say P′, which is the simple group of order 25920 = 2^{6} ·3^{4} ·5. In particular, P′ is isomorphic to the projective symplectic group of rank 2 over the finite field F_{3} . Let us mention also that one can find these groups in the classical theory of theta functions of genus two with characteristics.

These notes are organized into several sections. To attach a finite group to the cubic theta function we adopt the well known general principle explained in Section 1. Then, in Section 2, we state our results. Basic properties of the Kubota’s κ and of the Θ_{K-P} are summarized in Section 3 and Section 4. In Section 5 we give the necessary definitions and explanations concerning the W (E_{6}) and the symplectic groups. Then, in sections 6,…, 10 we prove all the statements given in Section 2.

The author express sincere thanks to Max-Planck-Institut für Mathematik in Bonn and SFB-478 ‘Geometrische Strukturen in der Mathematik’ in Münster for hospitality and support of the work. This work was partially supported also by RFFR grant 99-01-00099.

1 Basic construction. Let Δ′ be a subgroup of finite index, say n, in a group Δ. Assume that Δ acts on some space, say S, and that λ: Δ′ → C^{∗} is a group homomorphism. Given any function f: S → C satisfying

f(γw) = λ(γ)f(w) for all γ ∈ Δ′ and w ∈ S,

we can define function F: S → C^{n} and a group homomorphism

ψ: Δ → \{ \begin{matrix}monomial \\ matrices n × n \end{matrix} \} ↪ GL(n, C)

in such a way that

F (γw) = ψ(γ)F (w) for all γ ∈ Δ and w ∈ S.

To this end, we take σ_{1}, σ_{2}, …, σ_{n} ∈ Δ such that

Δ = ⋃_{j=1,2,...,n}Δ′ σ_{j}  (1.1)

is a disjoint decomposition, and then we set

F (w) =\begin{cases}f(σ_{1}w) \\ f(σ_{2}w) \\ . \\ . \\ f(σ_{n}w) \end{cases}\} for all w ∈ S.

Given any γ ∈ Δ and j = 1, 2, …, n we have σ_{j}γ ∈ Δ, and there is a unique k such that σ_{j}γ ∈ Δ′ σ_{k}. With this k, we have f(σ_{j}γw) = f(σ_{j}γσ_{k}^{−1}σ_{k}w) = λ(σ_{j}γσ_{k}^{−1})f(σ_{k}w). Now we can define ψ(γ) to be the monomial matrix^{†} n × n with jk -th entry λ(σ_{j}γσ_{k}^{−1}). Clearly, ψ occurs to be a group homomorphism with desired properties.

Now we can attach to our date Δ′, Δ, λ, σ_{1}, σ_{2}, …, σ_{n}, and f the group

E = {ψ(γ) | γ ∈ Δ},

which is a subgroup of GL(n, C). We will interested also in the subgroup Z of all diagonal matrices in E, in the subgroup C of all scalar matrices in E, and especially in the factor group E/Z. (We should notice that Z is a normal subgroup of E just because E consists of monomial matrices only.)

Up to isomorphism, and even up to conjugation in GL(n, C), the groups C, Z, E, and E/Z do not depend on the particular choice of the cosets representatives σ_{j} in (1.1). More precisely, if the groups, say Ĉ, Ẑ, Ê, are attached as above to Δ′, Δ, λ, f and to some σ̂_{1}, σ̂_{2}, …, σ̂_{n} instead of σ_{1}, σ_{2}, …, σ_{n}, then there is a monomial matrix h ∈ GL(n, C) such that Ĉ = h^{−1}Ch, Ẑ = h^{−1}Zh, Ê = h^{−1}Eh.

2 Main theorem. Let us apply the construction described in Section 1 with the following date

Δ = B(O) = the subgroup of SL(2, C) generated by SL(2, O) and ι; Δ′ = B(3) = the subgroup of SL(2, C) generated by Γ̃(3) and ι; S = H that is the 3-dimensional hyperbolic space; (2.1) f = Θ_{K-P}, that is the Kubota-Patterson cubic theta function; λ = κ, that is the extended Kubota homomorphism B(3) → C^{∗}.

Here and in what follows we use the following notations.

Z, Q, C are the ring of rational integers, the field of rational

and the field of complex numbers; C^{∗} is the multiplicative group of C; O = Z[ω] is the integers ring of the field Q(\sqrt{−3} ); ω = exp(2πi/3) = (−1 + \sqrt{−3} )/2;

ι =\begin{cases}i & 0 \\ 0 & −i \end{cases}\}, i = \sqrt{−1} ;  (2.2)

diag(c_{1}, c_{2}, …, c_{n}) is the n × n diagonal matrix with c_{l} at

the intersection of l^{th} row and l^{th} column;

I_{n} is the identity matrix n × n. Also, if q is either an ideal of O or a non-zero element of O, then Γ(q) is the principal congruence subgroup modq in SL(2, O); ±Γ(q) is the subgroup of SL(2, O) generated by Γ(q) and −I_{2} ; Γ̃(q) is the subgroup of SL(2, O) generated by Γ(q) and SL(2, Z); B(q) is the subgroup of SL(2, C) generated by Γ̃(q) and ι.

† By a monomial matrix we mean a matrix with unique non-zero entry in every row and in every column.

In particular, B(O), i.e. the subgroup of SL(2, C) generated by SL(2, O) and ι, is known as the Bianchi extension of SL(2, O). The Bianchi group B(O) is a maximal discrete subgroup of SL(2, C), while SL(2, O) is its subgroup of index 2; see Elstrodt, Grunewald, Mennicke [7], 7.4, and the references therein.

Theorem. Let ψ: B(O) → GL(n, C) be the homomorphism attached to the date listed in (2.1) in accordance with the procedure described in Section 1. Let E be the image of ψ, and let Z be the subgroup of all the diagonal matrices in E. Then n = 27, Z is a normal subgroup of E, and the factor group E/Z is isomorphic to some subgroup of order 648 in the Weyl group W (E_{6}) of the root system E_{6} .

Some additional information on the groups involved and on the homomorphism ψ is given in the three propositions below.

Proposition 1. With the notations above one has

(a) \begin{matrix}[SL(2, O) : Γ(3)] = 648 \\ [Γ̃(3) : Γ(3)] = 24 \end{matrix} and [SL(2, O) : Γ̃(3)] = 27;

(b) [B(O) : SL(2, O)] = [B(3) : Γ̃(3)] = 2;

(c) \begin{matrix}[B(O) : ±Γ(3)] = 648 \\ [B(3) : ±Γ(3)] = 24 \end{matrix} and [B(O) : B(3)] = 27.

Proposition 2. Let σ_{1}, σ_{2}, …, σ_{27} represent cosets in Γ̃(3)\SL(2, O), so that

SL(2, O) = ⋃_{j=1,2,...,27} Γ̃(3) σ_{j}  (2.3)

is a disjoint decomposition. Then the same matrices σ_{1}, σ_{2}, …, σ_{27} represent cosets in B(3)\B(O), so that

B(O) = ⋃_{j=1,2,...,27} B(3) σ_{j}  (2.4)

is a disjoint decomposition too.

Proposition 3. Under the assumption of Theorem, for the homomorphism ψ: B(O) → GL(27, C) one has

(a) ψ(γ) is the identity matrix for γ = ±I_{2} ;

(b) ψ(γ) is a scalar matrix for γ ∈ ±Γ(9);

(c) ψ(γ) is a diagonal matrix for γ ∈ ±Γ(3);

(d) ψ(γ) is not a diagonal matrix for γ ∉ ±Γ(3).

3 Kubota homomorhpism. Let ( \frac{·}{·} ) be the cubic residue symbol in O. Given

γ =\begin{cases}a & b \\ c & d \end{cases}\}∈ Γ(3), we set κ(γ) = \{ \begin{matrix}( \frac{c}{d} )_{−1} & if c ≠ 0, \\ 1 & if c = 0. \end{matrix}  (3.1)

According to Kubota [12], the so defined κ: Γ(3) → C^{∗} is a group homomorphism. It is known, see [15], [16], it can be extended to homomorphism Γ̃(3) → C^{∗} by setting κ(γ) = 1 for γ ∈ SL(2, Z); recall that the group Γ̃(3) is generated by Γ(3) and SL(2, Z). We would like to notice here that

the Kubota homomorphism κ can be extended to homomorphism B(3) → C^{∗} (which we still denote by κ ) just by setting κ(ι) = 1.

One can prove this statement by elementary computation using (6.1), (6.3) in Section 6. Another proof is given in Section 4 below. We will need the following:

κ(δ^{−1}γδ) = κ(γ) for all γ ∈ Γ(9), δ ∈ SL(2, O).  (3.2)

For this we can refer to [1].

4 Cubic metaplectic forms. Let H be the 3-dimensional hyperbolic space realized as the set of pairs (z, v) with z ∈ C, v ∈ R, v > 0. The group SL(2, C) acts on the H in such a way that for the image γw of the point

w = (z, v) ∈ H under γ =\begin{cases}a & b \\ c & d \end{cases}\}∈ SL(2, C) one has

γw = (\frac{(az + b)\overline{(cz + d)} + a\overline{c}v^{2}}{cz + d ^{2} + c ^{2}v^{2}} , \frac{v}{cz + d ^{2} + c ^{2}v^{2}} ).

The center of SL(2, C), consisting of ±I_{2} only acts trivially on H, so there is the induced action of PSL(2, C). According to Poincaré, PSL(2, C) is just the group of all orientation preserving isometrices of H. There is the second order differential operator whose action on the smooth functions H → C commute with the action of SL(2, C) on H, that is the Laplace-Beltrami operator

D_{L-B} = −v^{2}(\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} + \frac{∂^{2}}{∂v^{2}} ) + v \frac{∂}{∂v} , x = ℜ(z), y = ℑ(z).

Let Γ be a cofinite discrete subgroup of SL(2, C), such that the Kubota homomorphism is well defined on Γ. We say that f: H → C is a cubic metaplectic form on H under Γ if f is the D_{L-B} -eigenfunction of moderate growth satisfying

f(γw) = κ(γ)f(w) for all γ ∈ Γ, w ∈ H.

The moderate growth condition is not essential for our present goals. Anyway, the explicit statement is as follows: there is a polynomial P with real coefficients such that |f(z, v)| < P (|z|, v, v^{−1}) for all (z, v) ∈ H.

We shall deal with the Kubota-Patterson cubic theta function Θ_{K-P} , which is known to be a cubic metaplectic form under the group Γ̃(3). The Θ_{K-P} has been defined and studied originally by Kubota [13]. One has the Fourier expansion

Θ_{K-P}(z, v) = v^{2/3} + (6πv)^{2/3} ∑_{ν∈(\sqrt{−3} )^{−3}O\\{0\}} τ̃(ν)Ai((6π|ν|v)^{2/3})e(νz)

with the Fourier coefficients τ̃(ν) given by Patterson [15]; here Ai is the Airy function and e(t) = exp(2πi(t + t̄)). See also the authors exposition in [16]. We would like to show now that

the Kubota-Patterson cubic theta function Θ_{K-P} is a cubic metaplectic form under the Bianchi group B(3).

One knows τ̃(−ν) = τ̃(ν) for all ν, that yields Θ_{K-P}((−z, v)) = Θ_{K-P}((z, v)) for all (z, v) ∈ H. On the other hand, (−z, v) = ι(z, v). It follows that not only Θ_{K-P}(γw) = κ(γ)Θ_{K-P}(w) for all γ ∈ Γ̃(3), but also Θ_{K-P}(γw) = Θ_{K-P}(w) for γ = ι. This observation gives the extension of the Kubota homomorphism κ described in Section 3, and proves the statement above.

5 Symplectic groups and W (E_{6}). Recall, that by W (E_{6}) we understand the Weyl group of the root system E_{6} . One knows, W (E_{6}) is a group of order 51840 = 2^{7} · 3^{4} · 5 isomorphic to the group of incidence preserving permutations of 27 lines on smooth cubic surface in the 3-dimensional projective space over algebraically closed field. We have the exact sequence

1 → W (E_{6})^{+} → W (E_{6}) → {1, R} → 1,  (5.1)

in which W (E_{6})^{+} is the commutator subgroup of W (E_{6}). That is the simple group of order 25920 = 2^{6} · 3^{4} · 5. By R we mean any element in W (E_{6}) \ W (E_{6})^{+} of order 2, so that R generates the cyclic subgroup {1, R} of W (E_{6}); for instance, one can take R to be one of the reflections which generate W (E_{6}). In other words, W (E_{6}) is a semidirect product of its subgroups W (E_{6})^{+} and {1, R}, so that (5.1) is a split extension. We should notice also that the mapping which takes g ∈ W (E_{6})^{+} to R^{−1}gR ∈ W (E_{6})^{+} is an outer involutory automorphism of W (E_{6})^{+} . For the group W (E_{6}) we can refer to Bourbaki [2], Manin [14], Coxeter [3]. We have two group isomorphisms

W (E_{6})^{+} ≃ PSp(4, F_{3}) = \{ \begin{matrix}projective symplectic \\ group of rank 2 over F_{3} \end{matrix} \},  (5.2)
W (E_{6}) ≃ PGSp(4, F_{3}) = \{ \begin{matrix}extended projective \\ symplectic group of rank 2 over F_{3} \end{matrix} \},  (5.3)
the exact sequence 1 → PSp(4, F_{3}) → PGSp(4, F_{3}) → C_{2} → 1,  (5.4)
and the cosets decomposition PGSp(4, F_{3}) = PSp(4, F_{3}) ∪ ϱ PSp(4, F_{3}), (5.5)

where F_{3} is the field consisting of 3 elements, and C_{2} is the cyclic subgroup of order 2 in PGSp(4, F_{3}) generated by ϱ = ± diag(−1, 1, 1, −1).

For further convenience, let us describe the symplectic groups in more details. Let us agree to write 4 × 4 matrices like

\begin{cases}A & B \\ C & D \end{cases} \}, where A, B, C, D are 2 × 2 matrices,  (5.6)

and let Ω = \begin{cases}0_{2} & −I_{2} \\ I_{2} & 0_{2} \end{cases} \}, where 0_{2} and I_{2} are 2 × 2 zero and identity matrices. Then the symplectic group of rank 2 over F_{3} is

Sp(4, F_{3}) = {γ ∈ GL(4, F_{3}) | γΩ^{t}γ = Ω},

where ^{t}γ is the matrix transpose to γ. For γ written as in (5.6), the condition γΩ^{t}γ = Ω is equivalent to

A^{t}B = B^{t}A, C^{t}D = D^{t}C, A^{t}D − B^{t}C = I_{2}.  (5.7)

By the extended symplectic group we mean

GSp(4, F_{3}) = {γ ∈ GL(4, F_{3}) | γΩ^{t}γ = ±Ω}.

The symplectic group Sp(4, F_{3}) is a normal subgroup of index 2 in GSp(4, F_{3}). We have the disjoint decomposition and the exact sequence

GSp(4, F_{3}) = Sp(4, F_{3}) ∪ ϱ′ Sp(4, F_{3}) with ϱ′ = diag(−1, 1, 1, −1),
1 → Sp(4, F_{3}) → GSp(4, F_{3}) → C′_{2} → 1,

(5.8)

where C′_{2} is the cyclic subgroup of order 2 in GSp(4, F_{3}) generated by ϱ′. So, GSp(4, F_{3}) occurs to be semidirect product of its normal subgroup Sp(4, F_{3}) and cyclic subgroup C′_{2} . The centers of both Sp(4, F_{3}) and GSp(4, F_{3}) consist of the ±I_{4} only. By definition,

PSp(4, F_{3}) = Sp(4, F_{3})/{±I_{4}}, PGSp(4, F_{3}) = GSp(4, F_{3})/{±I_{4}},

and passing to quotient in (5.8) we obtain the desired cosets decomposition (5.5) and the exact sequence (5.4) for the projective groups.

The isomorphism (5.2) is given by Dieudonné [6], ch.4, with references to C. Jordan [11] and L. Dickson [4]. It seems very likely that the isomorphism (5.3) is known also, but we can not give reference. So let us give a short explanation on (5.3). By Dieudonné [5], every automorphism of the projective group PSp(4, F_{3}) is induced (by passing to quotient) by an automorphism of Sp(4, F_{3}). Loo-Keng Hua [10] and Dieudonné [5] showed also that every automorphism of Sp(4, F_{3}) is conjugation by some element of GSp(4, F_{3}). Hence, every automorphism of PSp(4, F_{3}) is conjugation by some element of PGSp(4, F_{3}). Then one can easily find that (up to isomorphism) PGSp(4, F_{3}) is the only split extension of PSp(4, F_{3}) by cyclic group of order 2 related to an outer involutory automorphism of PSp(4, F_{3}). With this observation, we obtain isomorphism (5.3) just comparing (5.4) with (5.1).

6 Proof of Proposition 1. Let us begin with (a). The principal congruence subgroup Γ(q), being the kernel of the homomorphism SL(2, O) → SL(2, O/q) induced by O → O/q, is a normal subgroup of SL(2, O). One has the exact sequence

1 → Γ(q) → SL(2, O) → SL(2, O/q) → 1,

that yields isomorphism

SL(2, O)/Γ(3) ≃ SL(2, O/3O).

Hence, the index [SL(2, O) : Γ(3)] equals to the order of SL(2, O/3O). Next, one can easily find that the mapping 1 ↦ 1, \sqrt{−3} ↦ x induces a ring isomorphism O/3O → F_{3}[x]/(x^{2}), so that we have a group isomorphism

SL(2, O/3O) ≃ SL(2, F_{3}[x]/(x^{2})).

Also we have a ring epimorphism F_{3}[x]/(x^{2}) → F_{3} , which takes a + bx (mod x^{2}) to a ∈ F_{3} , and which henceforth induces an exact sequence

1 → Γ(x) → SL(2, F_{3}[x]/(x^{2})) → SL(2, F_{3}) → 1,

where Γ(x) is the principal congruence subgroup mod x in SL(2, F_{3}[x]/(x^{2})) consisting of the matrices

\begin{cases}1 + ax & bx \\ cx & 1 − ax \end{cases}\}, a, b, c ∈ F_{3}.

The SL(2, F_{3}) is known to be a group of order 24. The order of Γ(x) is 27. Hence, the order of SL(2, F_{3}[x]/(x^{2})) occurs to be 27 · 24 = 648, and so are the order of SL(2, O/3O) and the index [SL(2, O) : Γ(3)], as required.

Now, given any γ ∈ SL(2, O), one can easily find that γ ∈ Γ̃(q) if and only if γ ≡ δ (mod q) for some δ ∈ SL(2, Z), so if and only if γ = δγ′ for some δ ∈ SL(2, Z), γ′ ∈ Γ(q). For q = 3, this remark yields

Γ̃(3)/Γ(3) ≃ SL(2, Z)/(SL(2, Z) ∩ Γ(3)) ≃ SL(2, Z/3Z),

that is a group of order 24, and so is the index [Γ̃(3) : Γ(3)]. To conclude the proof of (a) notice that [SL(2, O) : Γ̃(3)] = 648/24 = 27, just in view of the inclusions Γ(3) ⊂ Γ̃(3) ⊂ SL(2, O).

Now, to prove (b), we notice that, according to the definition (2.2),

ι =\begin{cases}i & 0 \\ 0 & −i \end{cases}\}, so that ι^{2} = −I_{2}, and

ι^{−1}\begin{cases}a & b \\ c & d \end{cases}\}ι =\begin{cases}a & −b \\ −c & d \end{cases}\} for all \begin{cases}a & b \\ c & d \end{cases}\}∈ SL(2, C).

(6.1)

It follows immediately that SL(2, O) is a normal subgroup of B(O), as well as Γ̃(3) is a normal subgroup of B(3), and that one has the cosets decompositions

B(O) = SL(2, O) ∪ ι SL(2, O),  (6.2)
B(3) = Γ̃(3) ∪ ι Γ̃(3),  (6.3)

which imply (b). At last, (c) follows easily from (a) and (b), let us omit details.

7 Proof of Proposition 2. Assume that we have a disjoint decomposition (2.3). Substituting (2.3) into (6.2) and applying then (6.3) we obtain

\begin{matrix}B(O) = \{ ⋃_{j=1,...,27}Γ̃(3) σ_{j}\} ∪ \{ ⋃_{j=1,...,27}ι Γ̃(3) σ_{j}\} = \\ ⋃_{j=1,...,27}\{ Γ̃(3) ∪ ι Γ̃(3) \} σ_{j} = ⋃_{j=1,...,27}B(3) σ_{j}, \end{matrix}  (7.1)

that is required decomposition (2.4), and it remains only to prove that it is disjoint. To this end, assume that B(3)σ_{j} = B(3)σ_{k} for some k ≠ j. Then σ_{j}σ_{k}^{−1} ∈ B(3). In view of (6.3) this means that either σ_{j}σ_{k}^{−1} ∈ Γ̃(3) or σ_{j}σ_{k}^{−1} ∈ ιΓ̃(3). The first case is impossible because of (2.1) is disjoint. In the second case, we find σ_{j}σ_{k}^{−1}δ = ι for some δ ∈ Γ̃(3) ⊂ SL(2, O). With this δ we have σ_{j}σ_{k}^{−1}δ ∈ SL(2, O), while ι ∉ SL(2, O). Hence, we have that all the cosets in (7.1) are different, and so (7.1) is a disjoint decomposition.

8 Proof of Proposition 3. Point (a) is obvious, because of I_{2} and −I_{2} act trivially on H. For (b) and (c) we should notice firstly, that both Γ(3) and Γ(9) are normal subgroups of B(O). Then, given γ ∈ Γ(3) and any j = 1, 2, …, 27 we have σ_{j}γσ_{j}^{−1} ∈ Γ(3) ⊂ B(3), and so

f(σ_{j}γw) = f(σ_{j}γσ_{j}^{−1}σ_{j}w) = κ(σ_{j}γσ_{j}^{−1})f(σ_{j}w).

This yields ψ(γ) = diag(κ(σ_{1}γσ_{1}^{−1}), κ(σ_{2}γσ_{2}^{−1}), …, κ(σ_{27}γσ_{27}^{−1})), that is (c). To prove (b) we only have to notice that κ(δγδ^{−1}) = κ(γ) for all γ ∈ Γ(9) and δ ∈ SL(2, O) (see (3.2)), as well as for δ = ι, and hence for all δ ∈ B(O). This yields κ(σ_{j}γσ_{j}^{−1}) = κ(γ), and hence ψ(γ) is a scalar matrix for γ ∈ Γ(9).

It is not so easy to prove (d). The statement to be proved do not depend on particular choice of σ_{j} in (1.1), (2.3), (2.4). Let us take the σ_{j} from the list given in [15]. This list includes

σ_{1} =\begin{cases}1 & 0 \\ 0 & 1 \end{cases}\}, σ_{2} =\begin{cases}ω & 0 \\ 0 & ω^{2} \end{cases} \}, σ_{3} =\begin{cases}1 & ω \\ 0 & 1 \end{cases} \}.

We do not have to specify all of σ_{j} in [15]. Let us notice only that σ_{j} ∉ B(3) for j = 2, 3, …, 27, and that σ_{j} ∈ SL(2, O) for j = 1, 2, …, 27.

Let us define E to be the set of all matrices γ ∈ B(O) such that

w ↦ \frac{Θ_{K-P}(γw)}{Θ_{K-P}(w)} is a constant function H → C  (8.1)

(depending on γ only). Clearly, E is a subgroup of B(O), and B(3) ⊂ E. We would like to begin with proving that E = B(3). Assume that γ ∈ E, γ ∉ B(3). We have the cosets decomposition (2.4) in Proposition 2, from which it follows that γ can be factored as γ = γ′σ_{k} with some k = 1, 2, …, 27 and γ′ ∈ B(3) ⊂ E. It follows that σ_{k} ∈ E, σ_{k} ∉ B(3). So, if (8.1) has place for some γ ∉ B(3) then (8.1) has place for at least one γ = σ_{k}, k = 2, 3, …, 27. The author have used Maple V to write the procedure which returns numerical values of Θ_{K-P}(w). This procedure being applied to the functions (8.1) with γ = σ_{k}, k = 2, 3, …, 27, shows that no one of them is a constant. We conclude that E = B(3). Now, given γ ∈ B(O), the matrix ψ(γ) is a diagonal one if and only if

w ↦ \frac{Θ_{K-P}(σ_{k}γw)}{Θ_{K-P}(σ_{k}w)} is a constant function H → C

for every k = 1, 2, …, 27 (depending on k and γ only). That is equivalent to σ_{k}γσ_{k}^{−1} ∈ E for all k = 1, 2, …, 27. Since E = B(3), we conclude that ψ(γ) is a diagonal matrix if and only if

γ ∈ P, where P = ⋂_{k=1,...,27} σ_{k}^{−1}B(3)σ_{k}.  (8.2)

For (d) we only have to prove P = ±Γ(3). We have σ_{1} = I_{2} , so that P ⊂ σ_{1}^{−1}B(3)σ_{1} = B(3). We know that ±Γ(3) is a normal subgroup of B(O), that yields ±Γ(3) = ±σ_{k}^{−1}Γ(3)σ_{k} ⊂ σ_{k}^{−1}B(3)σ_{k} for all k, so that ±Γ(3) ⊂ P. On the other hand, we have

P ⊂ B(3) ∩ σ_{2}^{−1}B(3)σ_{2} ∩ σ_{3}^{−1}B(3)σ_{3} ⊂ ±Γ(3),  (8.3)

that implies P = ±Γ(3). The first inclusion in (8.3) is obvious. To prove the second one we need (6.3). Given γ ∈ B(3), it follows from (6.3) that either its entries ∈ O and then γ ∈ Γ̃(3), or its entries ∉ O and then γ ∈ ι Γ̃(3). A similar alternative takes place for γ ∈ σ_{k}^{−1}B(3)σ_{k} because of σ_{k} ∈ SL(2, O) for all k. Hence, given γ ∈ P, we have either

γ ∈ Γ̃(3) ∩ σ_{2}^{−1}Γ̃(3)σ_{2} ∩ σ_{3}^{−1}Γ̃(3)σ_{3} or  (8.4)
γ ∈ ιΓ̃(3) ∩ σ_{2}^{−1}ιΓ̃(3)σ_{2} ∩ σ_{3}^{−1}ιΓ̃(3)σ_{3}  (8.5)
= ι{Γ̃(3) ∩ ι^{−1}σ_{2}^{−1}ιΓ̃(3)σ_{2} ∩ ι^{−1}σ_{3}^{−1}ιΓ̃(3)σ_{3}}.

We should notice that ι^{−1}σ_{k}^{−1}ι ∈ SL(2, O) for both k = 2 and k = 3, and that Γ̃(3) consists of the matrices in SL(2, O) whose entries ≡ 0, ±1 (mod 3). Then an elementary computation, which we omit for short, shows that the set in the right hand side in (8.4) equals to ±Γ(3), while the set in the right hand side in (8.5) is empty. This concludes the proof.

9 On monomorphism SL(2, F_{3}[x]/(x^{2})) → Sp(4, F_{3}). The conditions (5.7) with C = 0 yield A = ^{t}D^{−1} , B = ^{t}D^{−1}S with ^{t}S = S. In particular, det D ≠ 0. Let

G = \{\begin{cases}A & B \\ 0_{2} & D \end{cases} \}∈ Sp(4, F_{3}) | det D = 1\},

that is a subgroup of index 2 in the Siegel maximal parabolic subgroup of Sp(4, F_{3}). Given D ∈ SL(2, F_{3}) and a symmetric 2 × 2 matrix S with entries in F_{3} , we have

\{ \begin{matrix}^{t}D^{−1} & 0_{2} \\ 0_{2} & D \end{matrix} \}∈ G and \begin{cases}I_{2} & S \\ 0_{2} & I_{2} \end{cases} \}∈ G.  (9.1)

Moreover, every γ ∈ G can be factored uniquely as

γ =\{ \begin{matrix}^{t}D^{−1} & 0_{2} \\ 0_{2} & D \end{matrix} \}\begin{cases}I_{2} & S \\ 0_{2} & I_{2} \end{cases} \}  (9.2)

with D and S as in (9.1). The number of a symmetric 2 × 2 matrices with entries in F_{3} is 3^{3} = 27, while the order of the group SL(2, F_{3}) equals 24. Hence, G is the group of order 27 · 24 = 648. We would like to prove that G is isomorphic to the group SL(2, F_{3}[x]/(x^{2})) already considered in Section 6. It is clear from our considerations in Section 6 that every matrix δ ∈ SL(2, F_{3}[x]/(x^{2})) can be factored in SL(2, F_{3}[x]/(x^{2})) uniquely as

δ = D\begin{cases}1 + ax & bx \\ cx & 1 − ax \end{cases}\}, where D ∈ SL(2, F_{3}), a, b, c ∈ F_{3}.  (9.3)

We can define h: SL(2, F_{3}[x]/(x^{2})) → G assining, for δ in (9.3), h(δ) to be the matrix γ in (9.2) with

S =\begin{cases}−c & a \\ a & b \end{cases}\}.

Clearly, h is a bijection. To prove that h is a group homomorphism one need the following: let D ∈ SL(2, F_{3}), a, b, c ∈ F_{3} , and a′, b′, c′ ∈ F_{3} ; then

tD\begin{cases}−c & a \\ a & b \end{cases}\}D =\begin{cases}−c′ & a′ \\ a′ & b′ \end{cases} \} if and only if
D^{−1}\begin{cases}1 + ax & bx \\ cx & 1 − ax \end{cases}\}D =\begin{cases}1 + a′x & b′x \\ c′x & 1 − a′x \end{cases}\}.

With this observation, a simple computation shows that the bijection h is a group homomorphism. Hence, h is an isomorphism. Let us omit the details.

10 Proof of Theorem. We have n = [B(O) : B(3)] = 27, by Proposition 1 (c). Then, ±Γ(3) is a normal subgroup of B(O) and, according to Proposition 3 (c), (d), Z is the image of ±Γ(3) under the homomorphism ψ. In its turn, E is the image of B(O) under the ψ. Hence, Z is a normal subgroup of E,

E/Z ≃ B(O)/ ±Γ(3),

and that is a group of order 648, by Proposition 1 (c). On the other hand, we know that

W (E_{6}) ≃ PGSp(4, F_{3}).

So, the assertion of Theorem concerning the group E/Z will be proved if we shall find some group monomorphism

B(O)/ ±Γ(3) \underrightarrow{Q̃} PGSp(4, F_{3}).  (10.1)

To this end, let us look at the diagramm

PSp(4, F_{3}) ↩ \{±\begin{cases}A & B \\ 0 & D \end{cases} \}∈ PSp(4, F_{3}) | det D = 1\}
↑
SL(2, O)/ ±Γ(3) → PSL(2, O/3O) → PSL(2, F_{3}[x]/(x^{2}))

(10.2)

Here ↩ is the inclusion of subgroup, ↑ is the isomorphism obtained by passing to quotient from the one constructed in Section 9, → are the isomorphisms obtained by passing to quotients from the ones constructed in Section 6. The composition

SL(2, O)/ ±Γ(3) \underrightarrow{Q} PSp(4, F_{3})

of all the homomorphisms in (10.2) is a group monomorphism, and we can use Q to define Q̃ in (10.1).

We find in [7], 7.4, that three matrices

p_{1} =\begin{cases}1 & 0 \\ 1 & 1 \end{cases}\}, p_{2} =\begin{cases}0 & −1 \\ 1 & 0 \end{cases}\}, p_{3} =\begin{cases}1 & 0 \\ ω & 1 \end{cases}\} (ω = (−1 + \sqrt{−3} )/2)

generate SL(2, O); these matrices and ι in (2.2) generate B(O). The canonical epimorphism B(O) → B(O)/±Γ(3) takes p_{1}, p_{2}, p_{3}, ι to some elements, say p′_{1}, p′_{2}, p′_{3}, ι′, which generate the group B(O)/±Γ(3). Clearly, p′_{1}, p′_{2}, p′_{3} generate SL(2, O)/±Γ(3), and we have the disjoint decomposition

B(O)/ ±Γ(3) = SL(2, O)/ ±Γ(3) ∪ ι′ SL(2, O)/ ±Γ(3)

(compare with (6.2)). Let us notice that the above defined monomorphism Q satisfies

Q(ι^{′−1}γι′) = ϱ^{−1}Q(γ)ϱ for all γ ∈ SL(2, O)/ ±Γ(3).  (10.3)

Clearly, it is sufficient to verify (10.3) for generators of SL(2, O)/ ±Γ(3) only, say for γ = p′_{1}, p′_{2}, p′_{3} , and this can be done easily by inspection the homomorphisms involved into (10.2). Now, let us define Q̃ by

Q̃(γ) = Q(γ), Q̃(ι′γ) = ϱQ(γ) for all γ ∈ SL(2, O)/ ±Γ(3).

Recall that ι^{′2} = 1 and ϱ^{2} = 1. It follows easily from (10.3) that Q̃ is a group monomorphism, that concludes the proof of Theorem.

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