THETA FUNCTIONS AND A PRESENTATION OF 2^{1+(2g+1)}Sp(2g, 2)

DAVE BENSON

Abstract. We investigate a Curtis–Tits style group presentation based on the Dynkin diagram C_{g}, in which the short root generators have order two while the long root generators have order four. We prove that this describes a finite group with an almost extraspecial normal subgroup of order 2^{2g+2} and quotient isomorphic to the symplectic group Sp(2g, 2). Such an extension has to be non-split, and was abstractly proved to exist in a paper of Griess from 1973. Our presentation proves that it is a double cover of a finite quotient of Sp(2g, ℤ). We investigate a 2^{g} dimensional complex representation on a suitable space of theta functions, and produce some consequences for the signatures of 4-manifolds described as surface bundles over surfaces. In particular, we prove that if the monodromy is contained in the theta subgroup Sp^{q}(2g, ℤ) ≤ Sp(2g, ℤ) then the signature of the 4-manifold is divisible by eight.

1. Introduction

In this paper, which is a sequel to [1], we give a Curtis–Tits style presentation for a group ℌ̃ of shape 2^{1+(2g+1)}Sp(2g, 2), whose existence was proved abstractly by Griess [13]. This involves taking a root system of Dynkin type C_{g} and making the short root generators have order two, and the long root generators have order four.

The quotient ℌ of ℌ̃ by a central subgroup of order two appears in the paper [1] as a quotient of Sp(2g, ℤ) by a subgroup 𝔎 of finite index. We shall show that ℌ̃ has a representation of dimension 2^{g} over ℚ[i], by exhibiting explicit matrices and proving that they satisfy the relations. The subgroup 𝔎 contains the principal congruence subgroup of level four, and so this may also be regarded as a representation of Sp(2g, ℤ/4).

Next, we describe an action of ℌ̃ on a suitable space of theta functions, thereby providing an alternative proof that these matrices describe a projective representation of Sp(2g, ℤ) of dimension 2^{g} with kernel 𝔎.

For the matrices giving the representation, we form a 2^{g} dimensional vector space over ℚ[i] with basis elements e_{w} for w ∈ (ℤ/2)^{g}. In the following matrices, we regard det A, which is really an element of (ℤ/4)^{×} = \{1, −1\}, as being either +1 or −1 in ℂ, and \sqrt{det A} is either 1 or i.

(1.1)  σ (\begin{matrix}I & B \\ 0 & I \end{matrix}): e_{w} ↦ i^{w^{t}Bw}e_{w}
(1.2)  σ (\begin{matrix}A & 0 \\ 0 & (A^{t})^{−1} \end{matrix}): e_{w} ↦ \sqrt{det A} e_{(A^{t})^{−1}w}

Monday 9^{th} October, 2017

(1.3)  σ (\begin{matrix}0 & I \\ −I & 0 \end{matrix}): e_{w} ↦ \frac{1}{(1 − i)^{g}} ∑_{w′}(−1)^{w^{t}w′}e_{w′}.

It is easy to verify that (1.1)–(1.2) define a homomorphism from the Siegel parabolic subgroup 𝔓 of Sp(2g, ℤ) consisting of matrices of the form

(\begin{matrix}A & B \\ 0 & (A^{t})^{−1} \end{matrix})

to U(2^{g}, ℚ[i])/{±I}, but it is not at all obvious that (1.3) extends this to a homomorphism Sp(2g, ℤ) → U(2^{g}, ℚ[i])/{±I}. This will be verified using the presentation. The action on the space of theta functions also shows that this is a homomorphism up to scalars, but this approach only seems to show that the scalars are eighth roots of unity. So it seems best to combine the two approaches.

One consequence of comparing the two points of view is that the composite of restriction to the theta subgroup and the change of coefficients ℤ/2 → ℤ/8:

H^{2}(Sp(2g, ℤ), ℤ/2) → H^{2}(Sp^{q}(2g, ℤ), ℤ/2) → H^{2}(Sp^{q}(2g, ℤ), ℤ/8)

kills the metaplectic double cover of Sp(2g, ℤ), considered as an element of H^{2}(Sp(2g, ℤ), ℤ/2). Using this, we prove the following consequence for the signatures of surface bundles over surfaces.

Theorem 1.4. If Σ_{g} → M → Σ_{h} is a closed oriented surface bundle whose monodromy action on H^{1}(Σ_{g}, ℤ) lies in the theta subgroup Sp^{q}(2g, ℤ) ≤ Sp(2g, ℤ) then the signature of M is divisible by 8.

This implies a special case of the Klaus–Teichner conjecture (for the full formulation of this conjecture, see the introduction to the paper [14]), namely that if the monodromy action on H^{1}(Σ_{g}, ℤ/2) is trivial then the signature is divisible by eight. We should remark that the theorem above also follows from Lemma 7.5 of Galatius and Randal-Williams [8], which uses totally different methods.

Similar theta function methods allow us to prove the following.

Theorem 1.5. If Σ_{g} → M → Σ_{h} is a closed oriented surface bundle whose monodromy action on H^{1}(Σ_{g}, ℤ) lies in the subgroup 𝔎𝔓 ≤ Sp(2g, ℤ) then the signature of M is divisible by 8. This subgroup consists of the matrices ( \begin{matrix}A & B \\ C & D \end{matrix}) where the entries of C are even, and the diagonal entries of C are divisible by four.

Notation. If A is a g × g matrix, we write diag(A) for the length g vector of diagonal elements of A. Conversely, if v is a length g vector, we write Diag(v) for the matrix whose diagonal entries are the entries of v, and whose off diagonal entries are zero.

Let J be the 2g × 2g matrix ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ), where I denotes a g × g identity matrix. Then the symplectic group Sp(2g, ℤ) consists of the matrices X such that X^{t}JX = J. Equivalently, if X = ( \begin{matrix}A & B \\ C & D \end{matrix}) then X is in the symplectic group if and only if AB^{t} and CD^{t} are symmetric and AD^{t} − BC^{t} = I. Moreover, since J^{−1} = J^{t}, a matrix X is in the symplectic group if and only if X^{t} is. We write Sp(2g, 2) for the symplectic group defined in the same way over the field 𝔽_{2}.

We write Γ(2g, N) ≤ Sp(2g, ℤ) for the principal congruence subgroup consisting of symplectic matrices which are congruent to the identity modulo N. We write Γ(2g, N, 2N) for the Igusa subgroup [15] of Γ(2g, N) consisting of the matrices ( \begin{matrix}A & B \\ C & D \end{matrix}) where the entries of Diag(AB^{t}) and Diag(CD^{t}) are divisible by 2N, or equivalently where the entries of Diag(A^{t}C) and Diag(B^{t}D) are divisible by 2N. If N = 1, this is the theta subgroup, also known as the symplectic quadratic group, and denoted Sp^{q}(2g, ℤ). It is the inverse image in Sp(2g, ℤ) of the orthogonal subgroup O^{+}(2g, 2) ≤ Sp(2g, 2).

Acknowledgement. I would like to thank Caterina Campagnolo, Andrew Ranicki, and Carmen Rovi for stimulating my interest in this material, and for many thought provoking conversations.

2. Generators and relations

In this section we give a Curtis–Tits type presentation for the group ℌ̃. We begin by recalling Steinberg’s version of the Curtis–Tits theorem.

Theorem 2.1. Let Φ ⊆ ℝ^{n} be a finite indecomposable root system of rank at least two. Let G be a group of Lie type over a field k generated by root elements x_{r}(t) for r ∈ Φ, t ∈ k. For linearly independent roots r and s, let

[x_{r}(t), x_{s}(u)] = ∏_{i,j>0} x_{ir+js}(c_{ij,rs}t^{i}u^{j})

be the Chevalley commutation formula. Let G̃ be the abstract group generated by elements x̃_{r}(t) with relations

(2.2)  x̃_{r}(t)x̃_{r}(u) = x̃_{r}(t + u),
(2.3)  [x̃_{r}(t), x̃_{s}(u)] = ∏_{i,j>0} x̃_{ir+js}(c_{ij,rs}t^{i}u^{j}).

Then the kernel of the group homomorphism G̃ → G sending x̃_{r}(t) to x_{r}(t) is central in G̃.

Proof. This version is proved in Steinberg [25], Théorème 3.1. The proof is also given in Chapter 6 of Steinberg [26] and in Section 2.9 of Gorenstein, Lyons and Solomon [12]. The original version of Curtis [3] assumed that |k| > 4, while the version of Tits [27] had slightly different hypotheses. □

Remark 2.4. Unless the root system Φ has type G_{2}, the root elements in the product on the right hand side of the Chevalley commutator formula commute. In type G_{2} we have to stipulate that the product is taken in increasing order of i + j.

Next, we describe the root system Φ ⊂ ℝ^{g} of type C_{g} and its relation to the root subgroups of the symplectic group with respect to our basis. Let e_{1}, …, e_{g} be an orthonormal basis of ℝ^{g}, and let Φ consist of the long roots ±2e_{i} of length 2 and short roots ±e_{i} ± e_{j} (i ≠ j) of length \sqrt{2}. The total number of roots is 2g^{2}, consisting of 2g long roots and 2g(g − 1) short roots.

Letting E_{i,j} be a g × g matrix with 1 in the (i, j) place and 0 in the remaining places, the long root elements of Sp(2g, ℤ) are

x_{2e_{i}}(t) = (\begin{matrix}I & tE_{i,i} \\ 0 & I \end{matrix}) , x_{−2e_{i}}(t) = (\begin{matrix}I & 0 \\ tE_{i,i} & I \end{matrix})

(t ∈ ℤ), and the short root elements are

x_{e_{i}+e_{j}}(t) = (\begin{matrix}I & t(E_{i,j} + E_{j,i}) \\ 0 & I \end{matrix}) , x_{−e_{i}−e_{j}}(t) = (\begin{matrix}I & 0 \\ t(E_{i,j} + E_{j,i}) & I \end{matrix}) ,

x_{e_{i}−e_{j}}(t) = (\begin{matrix}I + tE_{i,j} & 0 \\ 0 & I − tE_{j,i} \end{matrix}) .

The Chevalley commutation relations are as follows (here, [x, y] denotes xyx^{−1}y^{−1}):

(2.5)  [x_{r}(t), x_{s}(t)] = 0  (r and s lin indep, r + s not a root)
(2.6)  [x_{e_{i}+e_{j}}(t), x_{−2e_{j}}(u)] = x_{e_{i}−e_{j}}(tu)x_{2e_{i}}(−t^{2}u)
(2.7)  [x_{−e_{i}−e_{j}}(t), x_{2e_{j}}(u)] = x_{e_{j}−e_{i}}(−tu)x_{−2e_{i}}(−t^{2}u)
(2.8)  [x_{e_{i}−e_{j}}(t), x_{−2e_{i}}(u)] = x_{−e_{i}−e_{j}}(−tu)x_{−2e_{j}}(t^{2}u) (i ≠ j)
(2.9)  [x_{e_{i}−e_{j}}(t), x_{2e_{j}}(u)] = x_{e_{i}+e_{j}}(tu)x_{2e_{i}}(t^{2}u)
(2.10)  [x_{e_{i}−e_{j}}(t), x_{e_{i}+e_{j}}(u)] = x_{2e_{i}}(2tu)
(2.11)  [x_{e_{i}−e_{j}}(t), x_{−e_{i}−e_{j}}(u)] = x_{−2e_{i}}(−2tu)
(2.12)  [x_{e_{i}+e_{j}}(t), x_{−e_{j}−e_{k}}(u)] = x_{e_{i}−e_{k}}(tu)
(2.13)  [x_{e_{i}−e_{j}}(t), x_{e_{j}−e_{k}}(u)] = x_{e_{i}−e_{k}}(tu) (i, j, k distinct)
(2.14)  [x_{e_{i}−e_{j}}(t), x_{e_{j}+e_{k}}(u)] = x_{e_{i}+e_{k}}(tu)
(2.15)  [x_{e_{j}−e_{i}}(t), x_{−e_{j}−e_{k}}(u)] = x_{−e_{i}−e_{k}}(−tu)

To ease notation, we write x_{r} for x_{r}(1). The argument in the following lemma is implicit in Section 7 of Steinberg [25].

Lemma 2.16. As long as g ⩾ 2, the relations above imply the braid relation

x_{2e_{i}}x_{−2e_{i}}^{−1}x_{2e_{i}} = x_{−2e_{i}}^{−1}x_{2e_{i}}x_{−2e_{i}}^{−1}.

Proof. Since g ⩾ 2, we may choose j different from i. Setting w_{i} = x_{2e_{i}}x_{−2e_{i}}^{−1}x_{2e_{i}}, the Chevalley commutation relations imply that w_{i} conjugates x_{e_{i}+e_{j}} to x_{e_{j}−e_{i}}:

w_{i}x_{e_{i}+e_{j}}w_{i}^{−1} = x_{2e_{i}}x_{−2e_{i}}^{−1}x_{2e_{i}}x_{e_{i}+e_{j}}x_{2e_{i}}^{−1}x_{−2e_{i}}x_{2e_{i}}^{−1}
= x_{2e_{i}}x_{−2e_{i}}^{−1}x_{e_{i}+e_{j}}x_{−2e_{i}}x_{2e_{i}}^{−1}
= x_{2e_{i}}[x_{−2e_{i}}(−1), x_{e_{i}+e_{j}}(1)]x_{e_{i}+e_{j}}x_{−2e_{i}}x_{2e_{i}}^{−1}
= x_{2e_{i}}x_{2e_{j}}^{−1}x_{e_{j}−e_{i}}x_{e_{i}+e_{j}}x_{2e_{i}}^{−1}
= x_{2e_{j}}^{−1}x_{2e_{i}}x_{e_{j}−e_{i}}x_{2e_{i}}^{−1}x_{e_{i}+e_{j}}
= x_{2e_{j}}^{−1}x_{e_{j}−e_{i}}[x_{e_{j}−e_{i}}(−1), x_{2e_{i}}(1)]x_{e_{i}+e_{j}}
= x_{2e_{j}}^{−1}x_{e_{j}−e_{i}}x_{e_{i}+e_{j}}^{−1}x_{2e_{j}}x_{e_{i}+e_{j}}

= x_{e_{j}−e_{i}}.

Similarly, w_{i} conjugates x_{e_{i}−e_{j}} to x_{−e_{i}−e_{j}}^{−1}, and it centralises x_{−2e_{j}}. By relation (2.6) we have

x_{2e_{i}} = [x_{−2e_{j}}, x_{e_{i}+e_{j}}]x_{e_{i}−e_{j}}

and so w_{i} conjugates x_{2e_{i}} to

[x_{−2e_{j}}, x_{e_{j}−e_{i}}]x_{−e_{i}−e_{j}}^{−1} = x_{−2e_{i}}^{−1}.

Finally, using the definition of w_{i} we have

x_{−2e_{i}}^{−1} = w_{i}x_{2e_{i}}w_{i}^{−1}
= x_{2e_{i}}x_{−2e_{i}}^{−1}x_{2e_{i}}x_{2e_{i}}x_{2e_{i}}^{−1}x_{−2e_{i}}x_{2e_{i}}^{−1}
= x_{2e_{i}}x_{−2e_{i}}^{−1}x_{2e_{i}}x_{−2e_{i}}x_{2e_{i}}^{−1}.

The lemma follows. □

Lemma 2.17. Let x and y be elements of a group, satisfying the braid relation xyx = yxy. Then

z = xy^{2}xy^{2} = y^{2}xy^{2}x = x^{2}yx^{2}y = yx^{2}yx^{2} = (xy)^{3} = (yx)^{3}

commutes with x and y. If also x^{4} = y^{4} = 1 then

z^{2} = x^{2}y^{2}x^{2}y^{2} = y^{2}x^{2}y^{2}x^{2}

also commutes with x and y, and z^{4} = 1.

Proof. If xyx = yxy then

xy^{2}xy^{2} = xy(yxy)y = (xy)^{3} = (xyx)(yxy) = (yxy)(xyx) = (yx)^{3} = y(yxy)yx = y^{2}xy^{2}x

so z = xy^{2}xy^{2} commutes with x. Similarly with x and y interchanged. If x^{4} = y^{4} = 1 then

z^{2} = xy^{2}xy^{2}y^{2}xy^{2}x = xy^{2}x^{2}y^{2}x = x^{2}y^{2}x^{2}y^{2} = y^{2}x^{2}y^{2}x^{2}

since z commutes with x. Finally, z^{4} = (x^{2}y^{2}x^{2}y^{2})(y^{2}x^{2}y^{2}x^{2}) = 1. □

Definition 2.18. We summarise some notation from [1]. We write 𝔎 for the normal subgroup of Sp(2g, ℤ) consisting of the matrices ( \begin{matrix}I+2a & 2b \\ 2c & I+2d \end{matrix} ) such that the trace of a is even, and the diagonal elements of b and c are even. We write ℌ for the quotient Sp(2g, ℤ)/𝔎, a group of shape 2^{2g+1} ^{·}Sp(2g, 2), with the property that the normal subgroup O_{2}(ℌ) is an elementary abelian 2-group on which the quotient Sp(2g, 2) acts as the orthogonal module via the isomorphism Sp(2g, 2) ≅ O(2g + 1, 2). The group ℌ has a double cover ℌ̃ such that the normal subgroup E = O_{2}(ℌ̃) is an almost extraspecial group of order 2^{2g+2}. The pullback of ℌ̃ → ℌ and Sp(2g, ℤ) → ℌ is the metaplectic double cover Mp(2g, ℤ) of Sp(2g, ℤ).

Theorem 2.19. Let G be the group presented by generators x_{r} for r in the root system of Dynkin type C_{g}, with the Chevalley commutator relations (2.5)–(2.15) together with the relations x_{r}^{2} = 1 if r is a short root and x_{r}^{4} = 1 if r is a long root. Then there is a surjective homomorphism from G → ℌ given by the matrices above. Provided g ⩾ 2, the kernel of this homomorphism has order two. It is contained in Z(G), which is cyclic of order four.

Proof. Let G̃ be the group generated by x̃_{r} for r in the root system of type C_{g}, with just the Chevalley commutator relations. Then there is a surjective homomorphism from G̃ to Sp(2g, ℤ) since the matrices in Sp(2g, ℤ) satisfy these relations. The elements x_{r}^{2} for r a short root and x_{r}^{4} for r a long root are in the subgroup 𝔎, and so this induces a homomorphism from G to ℌ.

Now consider the elements x_{r}^{2} for r a long root. These elements x_{r}^{2} all commute except for opposite pairs, and have order two. By Lemma 2.16, the relations in the hypothesis of Lemma 2.17 hold with x = x_{r} and y = x_{−r}^{−1}, r a long root. So

z_{i} = (x_{2e_{i}}x_{−2e_{i}}^{2})^{2} = (x_{2e_{i}}^{2}x_{−2e_{i}}^{−1})^{2}

satisfies z_{i}^{2} = (x_{2e_{i}}^{2}x_{−2e_{i}}^{2})^{2} and z_{i}^{4} = 1, and z_{i} commutes with x_{2e_{i}} and x_{−2e_{i}}. In particular, the subgroup generated by x_{2e_{i}}^{2} and x_{−2e_{i}}^{2} has order at most eight.

Next we assume that g ⩾ 2 and we show that the z_{i} are all equal. Let i ≠ j. Then x_{e_{i}−e_{j}} conjugates x_{2e_{j}} to x_{2e_{j}}x_{e_{i}+e_{j}}x_{2e_{i}} and x_{−2e_{i}} to x_{−2e_{i}}x_{−e_{i}−e_{j}}x_{−2e_{j}} (remember that short root elements square to the identity), so it conjugates x_{2e_{j}}^{2} to x_{2e_{j}}^{2}x_{2e_{i}}^{2} and x_{−2e_{i}}^{2} to x_{−2e_{i}}^{2}x_{−2e_{j}}^{2}, and it centralises x_{−2e_{j}} and x_{2e_{i}}. So conjugating the equation

(x_{2e_{i}}x_{−2e_{i}}^{2})^{2} = (x_{2e_{i}}^{2}x_{−2e_{i}}^{−1})^{2} = (x_{2e_{j}}^{2}x_{2e_{i}}^{2}x_{−2e_{i}}^{−1})^{2}

by x_{e_{i}−e_{j}}, we get

(x_{2e_{i}}x_{−2e_{i}}^{2}x_{−2e_{j}}^{2})^{2} = (x_{2e_{j}}^{2}x_{−2e_{i}}^{−1}x_{−e_{i}−e_{j}}x_{−2e_{j}}^{−1})^{2}

The left side is (x_{2e_{i}}x_{−2e_{i}}^{2})^{2} = z_{i}, while the right side is

(x_{2e_{j}}^{2}x_{−2e_{j}}^{−1})^{2}x_{−2e_{i}}^{−2}x_{−e_{i}−e_{j}}^{2}[x_{2e_{j}}^{2}, x_{−e_{i}−e_{j}}].

Since [x_{2e_{j}}^{2}, x_{−e_{i}−e_{j}}] = x_{−2e_{i}}^{2} and x_{−e_{i}−e_{j}}^{2} = 1, all terms cancel except (x_{2e_{j}}^{2}x_{−2e_{j}}^{−1})^{2}, so this is equal to z_{j}.

From now on, we write z for the common value of the z_{j}. It is now easy to check that the subgroup N generated by the x_{2e_{i}}^{2}, the x_{−2e_{i}}^{2} (1 ≤ i ≤ g), and z is normal in G and satisfy the relations for an almost extraspecial group of order 2^{2g+2}. It is a central product of the g dihedral groups ⟨x_{2e_{i}}^{2}, x_{−2e_{i}}^{2}⟩ each of order eight, and the cyclic group ⟨z⟩ of order four. Note that z^{2} is in the kernel of G → ℌ, and the image of z is the central element of order two in ℌ.

The group G/N is generated by the images of the root elements. These all have order two and satisfy the Chevalley commutator relations, and so G/N is isomorphic to a central extension of Sp(2g, 2) by Theorem 2.1. For g ⩾ 4 this implies that it is Sp(2g, 2). For g = 2 and g = 3 we need to use the fact that the non-trivial central extension of Sp(2g, 2) by ℤ/2 does not satisfy the Steinberg relations since these are among the exceptions in Steinberg’s theorem. □

Remark 2.20. The presentation given in the theorem makes sense for any perfect field k of characteristic two. The short root subgroups are isomorphic to the additive group k^{+}, while the long root subgroups are isomorphic to the additive group of the length two Witt vectors over k.

Corollary 2.21. For g ⩾ 4, the group G described in Theorem 2.19 is the universal central extension of ℌ.

Proof. This follows from Theorem 2.19 together with the computation of H_{2}(ℌ) in [1]. □

3. Matrices for quadratic forms

We now turn to the matrix representation given by (1.1)–(1.3). We begin with a discussion of (1.1), which depends on regarding B as a quadratic form on (ℤ/2)^{g} taking values in ℤ/4. These were investigated in Brown [2].

Let Q be the set consisting of g × g symmetric matrices B whose diagonal entries are in ℤ/4 and whose off diagonal entries are in ℤ/2. If B ∈ Q then B represents a quadratic form (ℤ/2)^{g} → ℤ/4 sending w to

w^{t}Bw = ∑_{i} B_{ii}w_{i}^{2} + 2 ∑_{i<j} B_{ij}w_{i}w_{j} ∈ ℤ/4.

The ℤ/2-valued quadratic forms 𝔮 correspond to matrices B with the diagonal entries even, and the form 𝔮 is determined by 2𝔮(w) = w^{t}Bw. On the other hand, the reduction modulo two of B gives a g × g symmetric matrix B̄ with entries in ℤ/2, representing a symmetric bilinear form on (ℤ/2)^{g}.

If A ∈ GL(g, ℤ/2) then a change of basis of B via A gives the matrix A^{t}BA, an equivalent matrix in Q. In particular, if B̄ is invertible then its inverse is also symmetric, and B̄^{−1}BB̄^{−1} can be thought of as B^{−1}, another matrix in Q.

Definition 3.1. For B ∈ Q we define B^{⊥} to be the subspace consisting of those v ∈ (ℤ/2)^{g} such that B̄v = 0 ∈ ℤ/2. We say that B is non-singular if B^{⊥} = {0}.

If v, v′ ∈ B^{⊥} then v^{t}Bv ∈ ℤ/4 is either 0 or 2 and (v + v′)^{t}B(v + v′) = v^{t}Bv + v^{′t}Bv′. So the subset consisting of those v ∈ B^{⊥} such that v^{t}Bv = 0 ∈ ℤ/4 is a linear subspace of codimension zero or one in B^{⊥}. We write Rad(B) for this subspace. Thus Rad(B) ⊆ B^{⊥}, with equality if and only if there does not exist v ∈ (ℤ/2)^{g} such that B̄v = 0 ∈ ℤ/2 and v^{t}Bv = 2 ∈ ℤ/4. We say that B is non-degenerate if Rad(B) = 0. This is weaker than being non-singular.

Finally, if v ∈ (ℤ/2)^{g} then we define v^{⊥} to be the subspace consisting of those u ∈ (ℤ/2)^{g} such that u^{t}v = 0 ∈ ℤ/2.

Lemma 3.2. Let B ∈ 𝔔. For v ∈ (ℤ/2)^{g} we have v^{⊥} ⊇ B^{⊥} if and only if there exists u ∈ (ℤ/2)^{g} with v = B̄u.

Proof. The first condition is equivalent to the statement that v^{t} is a linear combination of the rows of B̄, while the second is equivalent to the condition that v is a linear combination of the columns of B̄. So the lemma follows from the fact that B̄ is symmetric. □

4. The Brown–Kervaire invariant

Following Brown [2], we define

λ(B) = ∑_{w∈(ℤ/2)^{g}} i^{w^{t}Bw} ∈ ℂ.

Thus for example

λ(0) = 2, λ(1) = 1 + i, λ(2) = 0, λ(3) = 1 − i,

λ (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) = 2, λ (\begin{matrix}2 & 1 \\ 1 & 2 \end{matrix}) = −2.

Furthermore, we have λ(B ⊕ B′) = λ(B)λ(B′) and λ(−B) = \overline{λ(B)}. It is proved in [2] that if B is non-singular then

λ(B) = \sqrt{2}^{g}ζ_{8}^{BK(B)}

where ζ_{8} = e^{2πi/8} and BK(B) is the Brown–Kervaire invariant of B in ℤ/8. This is invariant under equivalence, additive under direct sum, and displays the fact that the Witt group of non-singular ℤ/4-valued quadratic forms on elementary abelian 2-groups is is isomorphic to ℤ/8 with (1) as generator.

Lemma 4.1. (i) If Rad(B) is a proper subspace of B^{⊥} then λ(B) = 0.

(ii) If Rad(B) = B^{⊥} then

λ(B) = \sqrt{2}^{n(B)+g}ζ_{8}^{BK(B/Rad(B))}

where n(B) is the nullity of the matrix B.

Proof. (i) If Rad(B) is a proper subspace of B^{⊥} then there exists a vector v with Bv = 0 ∈ ℤ/2 and v^{t}Bv = 2 ∈ ℤ/4. Then

(w + v)^{t}B(w + v) = w^{t}Bw + 2w^{t}Bv + v^{t}Bv = w^{t}Bw + 2

and so

i^{(w+v)^{t}B(w+v)} + i^{w^{t}Bw} = 0.

Pairing each w with w + v, we see that λ(B) = ∑_{w∈(ℤ/2)^{g}} i^{w^{t}Bw} = 0.

(ii) If Rad(B) = B^{⊥} then B is equivalent to B′ ⊕ B′′ where B′ is non-singular and B′′ is a zero matrix of size n(B). Thus

λ(B) = λ(B′)λ(B′′) = \sqrt{2}^{g−n(B)}ζ_{8}^{BK(B/Rad(B))}.2^{n(B)} = \sqrt{2}^{g+n(B)}ζ_{8}^{BK(B/Rad(B))}.  □

Proposition 4.2. The following conditions on B ∈ Q, v ∈ (ℤ/2)^{g} are equivalent: (i) v^{⊥} ∩ B^{⊥} = Rad(B), (ii) B^{⊥} = Rad(B + 2diag(v)), (iii) (B + 2diag(v))^{⊥} = Rad(B + 2diag(v)), (iv) λ(B + 2diag(v)) ≠ 0.

Proof. It is clear that (i) ⟺ (ii) ⟺ (iii). By Lemma 4.1, (iii) ⟺ (iv). □

Proposition 4.3. Let B ∈ Q. For v ∈ (ℤ/2)^{g}, we have

∑_{w∈(ℤ/2)^{g}}(−1)^{w^{t}v}i^{w^{t}Bw} = λ(B + 2diag(v)).

If v = B̄u with u ∈ (ℤ/2)^{g} then we have

∑_{w∈(ℤ/2)^{g}}(−1)^{w^{t}v}i^{w^{t}Bw} = λ(B + 2diag(v)) = (−i)^{u^{t}Bu}λ(B).

Proof. If v = Bu then w^{t}diag(v)w is congruent modulo two to w^{t}v = w^{t}Bu. So working modulo four we have

w^{t}(B + 2diag(v))w = w^{t}Bw + 2w^{t}diag(v)w
= w^{t}Bw + 2w^{t}v
= w^{t}Bw + 2w^{t}Bu
= (u + w)^{t}B(u + w) − u^{t}Bu.

Thus

∑_{w∈(ℤ/2)^{g}} i^{w^{t}(B+2diag(v))w} = (−i)^{u^{t}Bu} ∑_{w∈(ℤ/2)^{g}} i^{(u+w)^{t}B(u+w)}

= (−i)^{u^{t}Bu} ∑_{w∈(ℤ/2)^{g}} i^{w^{t}Bw}

= (−i)^{u^{t}Bu}λ(B).  □

5. Matrices for the lower unipotent subgroup

In this section, we describe the matrix representation σ on elements of the form

(\begin{matrix}I & 0 \\ C & I \end{matrix}) ∈ Sp(2g, ℤ).

Since we have not yet shown that σ is a homomorphism, we proceed as follows. We have

(\begin{matrix}0 & I \\ −I & 0 \end{matrix})^{−1} (\begin{matrix}I & −C \\ 0 & I \end{matrix}) (\begin{matrix}0 & I \\ −I & 0 \end{matrix}) = (\begin{matrix}I & 0 \\ C & I \end{matrix}) ,

and so we define

σ (\begin{matrix}I & 0 \\ C & I \end{matrix}) = σ (\begin{matrix}0 & I \\ −I & 0 \end{matrix})^{−1} σ (\begin{matrix}I & −C \\ 0 & I \end{matrix}) σ (\begin{matrix}0 & I \\ −I & 0 \end{matrix}) .

This way, we have defined the matrices representing at least all the root elements of Sp(2g, ℤ), and it is easy to see that the long root elements have order four and the short ones have order two. So what remains is to verify the Chevalley commutator relations (2.5)–(2.15). What makes this task easier is that most of the commutator relations either only involve the Siegel parabolic subgroup 𝔓 or only involve the negative Siegel parabolic subgroup 𝔓^{−}, which is the conjugate of 𝔓 by ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ).

Lemma 5.1. We have

σ (\begin{matrix}0 & I \\ −I & 0 \end{matrix})^{−1} : e_{w} ↦ \frac{1}{(1 + i)^{g}} ∑_{w′}(−1)^{w^{t}w′}e_{w′}.

Proof. Composing with σ ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ), we get the identity map:

e_{w} ↦ \frac{1}{2^{g}} ∑_{w′,w′′}(−1)^{w^{t}w′}(−1)^{w^{′t}w′′}e_{w′′} = e_{w}.  □

Theorem 5.2. Let C ∈ Q. Choose v ∈ (ℤ/2)^{g} with v^{⊥} ∩ C^{⊥} = Rad(C) (we allow the possibility that v = 0, so this is always possible). Then

σ (\begin{matrix}I & 0 \\ C & I \end{matrix}) : e_{w} ↦\frac{1}{2^{g}} ∑_{w′∈(ℤ/2)^{g}} λ(−C + 2diag(w + w′))e_{w′}

= \frac{λ(−C + 2diag(v))}{2^{g}} ∑_{Cu=w+w′+v} i^{u^{t}(C+2diag(v))u}e_{w′}.

In this last expression, the sum is over those w′ for which there exists u with Cu = w+w′+v, and the value of the summand is independent of the choice of such u.

Proof. Let B = −C (which is equivalent to C modulo two). Using Lemma 5.1, the effect of

σ (\begin{matrix}I & 0 \\ C & I \end{matrix})

is as follows.

e_{w} ⟼^{σ(\begin{matrix}0 & I \\ −I & 0 \end{matrix})} \frac{1}{(1 − i)^{g}} ∑_{w′′}(−1)^{w^{t}w′′}e_{w′′}

⟼^{σ(\begin{matrix}I & B \\ 0 & I \end{matrix})} \frac{1}{(1 − i)^{g}} ∑_{w′′}(−1)^{w^{t}w′′}i^{w^{′′t}Bw′′}e_{w′′}

⟼^{σ(\begin{matrix}0 & I \\ −I & 0 \end{matrix})^{−1}} \frac{1}{(1 − i)^{g}} ∑_{w′′}(−1)^{w^{t}w′′}i^{w^{′′t}Bw′′} (\frac{1}{(1 + i)^{g}} ∑_{w′}(−1)^{w^{′t}w′′}e_{w′})

= \frac{1}{2^{g}} ∑_{w′,w′′} i^{w^{′′t}Bw′′+2(w+w′)^{t}w′′}e_{w′}

= \frac{1}{2^{g}} ∑_{w′} (∑_{w′′} i^{w^{′′t}(B+2diag(w+w′))w′′}) e_{w′}

= \frac{1}{2^{g}} ∑_{w′} λ(B + 2diag(w + w′))e_{w′}.

This proves the first expression in the theorem. Now using Proposition 4.2, the expression in the sum vanishes unless (w + w′)^{⊥} ∩ B^{⊥} = Rad(B). Since we’ve chosen v so that v^{⊥} ∩ B^{⊥} = Rad(B), we have (w+w′)^{⊥} ∩B^{⊥} = Rad(B) if and only if (w+w′ +v)^{⊥} ⊇ B^{⊥}. By Lemma 3.2, this happens if and only if there exists u ∈ (ℤ/2)^{g} such that Bu = w + w′ + v. In this case, by Proposition 4.3 we have

λ(B + 2diag(w + w′)) = (−i)^{u^{t}(B+2diag(v))u}λ(B + 2diag(v))
= i^{u^{t}(C+2diag(v))u}λ(−C + 2diag(v)).  □

Remark 5.3. If C^{⊥} = {0} then we may take v = 0. The expressions C̄^{−1} and C^{−1} make sense and we have u = C̄^{−1}(w + w′). The formula then becomes

σ (\begin{matrix}I & 0 \\ C & I \end{matrix}) : e_{w} ↦↦ \frac{λ(−C)}{2^{g}} ∑_{w′} i^{(w+w′)^{t}C^{−1}(w+w′)}e_{w′}.

On the other hand, the larger C^{⊥} is, the fewer terms survive in the sum in the theorem. The total number of non-zero terms is 2^{rank(C̄)}. Example 5.4. Let g = 3, let C = (\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) = −C and let α = (\begin{matrix}1 \\ 0 \\ 0 \end{matrix}), β = (\begin{matrix}0 \\ 1 \\ 0 \end{matrix}). Then α^{t}Cα = β^{t}Cβ = 0, (α + β)^{t}C(α + β) = 2 and λ(C) = 4. So σ (\begin{matrix}I & 0 \\ C & I \end{matrix}) sends

e_{w} ↦ \frac{1}{2}(e_{w} + e_{w+α} + e_{w+β} − e_{w+α+β}).

Example 5.5. Let g = 3, let C = (\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) and v = 0. Let α be the vector (\begin{matrix}1 \\ 0 \\ 0 \end{matrix}). Then α^{t}Cα = 1 and λ(−C) = 4(1 − i). So σ (\begin{matrix}I & 0 \\ C & I \end{matrix}) sends

e_{w} ↦ \frac{4(1 − i)}{8}(e_{w} + ie_{w+α}) = \frac{1}{2}((1 − i)e_{w} + (1 + i)e_{w+α}).

Example 5.6. Let g = 3, let C = (\begin{matrix}2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}), v = (\begin{matrix}2 \\ 0 \\ 0 \end{matrix}), and α = (\begin{matrix}1 \\ 0 \\ 0 \end{matrix}). Then α^{t}Cα = 2 and λ(−C + 2diag(v)) = 8. So σ (\begin{matrix}I & 0 \\ C & I \end{matrix}) sends

e_{w} ↦ \frac{8}{8}(e_{w+α}) = e_{w+α}.

It is easy to check that this is equivalent to squaring the operation in the previous example.

Generalising these examples, we have the following corollary. Corollary 5.7. Let α_{i} be the ith basis vector of (ℤ/2)^{g}.

(i) If C is the elementary matrix α_{i}α_{i}^{t} then

σ (\begin{matrix}I & 0 \\ C & I \end{matrix}) : e_{w} ↦ \frac{1}{2}((1 − i)e_{w} + (1 + i)e_{w+α_{i}})
σ (\begin{matrix}I & 0 \\ 2C & I \end{matrix}) : e_{w} ↦ e_{w+α_{i}}.

(ii) If C is the elementary symmetric matrix α_{i}α_{j}^{t} + α_{j}α_{i}^{t} (i ≠ j) then

σ (\begin{matrix}I & 0 \\ C & I \end{matrix}) : e_{w} ↦ \frac{1}{2}(e_{w} + e_{w+α_{i}} + e_{w+α_{j}} − e_{w+α_{i}+α_{j}}).

We now describe the effect of σ on the normal subgroup O_{2}(Sp(2g, ℤ)/𝔎) = Γ(2g, 2)/𝔎.

This subgroup is generated by the following: (\begin{matrix}I & 2Diag(x) \\ 0 & I \end{matrix}), (\begin{matrix}I & 0 \\ 2Diag(y) & I \end{matrix}), (\begin{matrix}A & 0 \\ 0 & (A^{t})^{−1} \end{matrix}),

where A is any matrix congruent to the identity modulo two and with determinant −1, such as for example Diag(−1, 1, …, 1); all such matrices are equivalent modulo 𝔎. We have

σ (\begin{matrix}I & 2Diag(x) \\ 0 & I \end{matrix}): e_{w} ↦ (−1)^{x^{t}w}e_{w}
σ (\begin{matrix}I & 0 \\ 2Diag(y) & I \end{matrix}): e_{w} ↦ e_{w+y}
σ (\begin{matrix}A & 0 \\ 0 & (A^{t})^{−1} \end{matrix}): e_{w} ↦ ie_{w}

(with A as described above). These are the matrices for an irreducible complex 2^{g} dimensional representation of an almost extraspecial group of order 2^{2g+2}.

6. Verification of the relations in the matrix representation

We now verify that the relations given in Section 2 hold for the matrices of the projective representation σ: Sp(2g, ℤ) → U(2^{g}, ℚ[i])/{±I}, and that therefore there is an associated representation ℌ̃ → U(2^{g}, ℚ[i]) fitting into a diagram

\xymatrix{Mp(2g, ℤ) \ar[r] \ar[d] & ℌ̃ \ar[r] \ar[d] & U(2^{g}, ℚ[i]) \ar[d] \\ Sp(2g, ℤ) \ar[r] & ℌ \ar[r] & U(2^{g}, ℚ[i])/\{±I\}}

where Mp(2g, ℤ) is the metaplectic double cover of Sp(2g, ℤ).

Recall that the basis elements of ℚ[i]^{2^{g}} are e_{w}, w ∈ (ℤ/2)^{g}. The root subgroups act as follows:

σ(x_{2e_{i}})(e_{w}) = i^{w_{i}^{2}}e_{w}  σ(x_{−2e_{i}})(e_{w}) = \frac{1}{2}((1 − i)e_{w} + (1 + i)e_{w+α_{i}})
σ(x_{e_{i}+e_{j}})(e_{w}) = (−1)^{w_{i}w_{j}}e_{w}  σ(x_{−e_{i}−e_{j}})(e_{w}) = \frac{1}{2}(e_{w} + e_{w+α_{i}} + e_{w+α_{j}} − e_{w+α_{i}+α_{j}})
σ(x_{e_{i}−e_{j}})(e_{w}) = e_{w+w_{i}α_{j}}

It is straightforward to check that the long root elements act with order four and the short root elements act with order two. So it remains to check the Chevalley commutator identities (2.5)–(2.15). It suffices to check these with t = 1 and u = 1 by Remark 2.4. One way to perform most these checks at once is to use the fact that most of the commutator relations hold in either the Siegel parabolic subgroup 𝔓 or the negative Siegel parabolic subgroup 𝔓^{−}. The ones in 𝔓 are easy. The matrices for 𝔓^{−} were obtained by conjugating by ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ), and so the relations hold there too. So the real task is to compute the commutators of the root subgroups in the upper right quadrant with those in the lower left quadrant. Most of these commute, as is easy to check. The relations that need some work to check are (2.6), (2.7), and (2.12). We first check (2.6) with t = u = 1:

[σ(x_{e_{i}+e_{j}}), σ(x_{−2e_{j}})](e_{w}) = σ(x_{e_{i}+e_{j}})σ(x_{−2e_{j}})σ(x_{e_{i}+e_{j}})^{−1}σ(x_{−2e_{j}})^{−1}(e_{w})

= σ(x_{e_{i}+e_{j}})σ(x_{−2e_{j}})σ(x_{e_{i}+e_{j}})^{−1} \frac{1}{2}((1 + i)e_{w} + (1 − i)e_{w+α_{j}})

= σ(x_{e_{i}+e_{j}})σ(x_{−2e_{j}})\frac{1}{2}((−1)^{w_{i}w_{j}}(1 + i)e_{w} + (−1)^{w_{i}(1+w_{j})}(1 − i)e_{w+α_{j}})

= σ(x_{e_{i}+e_{j}})\frac{1}{4}((−1)^{w_{i}w_{j}}(1 + i)((1 − i)e_{w} + (1 + i)e_{w+α_{j}})

+ (−1)^{w_{i}(1+w_{j})}(1 − i)((1 − i)e_{w+α_{j}} + (1 + i)e_{w}))

= σ(x_{e_{i}+e_{j}})((−1)^{w_{i}w_{j}}i^{w_{i}^{2}}e_{w+w_{i}α_{j}})

= (−1)^{w_{i}(w_{i}+w_{j})}(−1)^{w_{i}w_{j}}i^{w_{i}^{2}}e_{w+w_{i}α_{j}}

= i^{−w_{i}^{2}}e_{w+w_{i}α_{j}}

= σ(x_{e_{i}−e_{j}})(i^{−w_{i}^{2}}e_{w})

= σ(x_{e_{i}−e_{j}})σ(x_{2e_{i}}^{−1})(e_{w}).

Next, we check (2.7) with t = u = 1:

[σ(x_{−e_{i}−e_{j}}), σ(x_{2e_{j}})](e_{w}) = σ(x_{−e_{i}−e_{j}})σ(x_{2e_{j}})σ(x_{−e_{i}−e_{j}})^{−1}σ(x_{2e_{j}})^{−1}(e_{w})

= σ(x_{−e_{i}−e_{j}})σ(x_{2e_{j}})(\frac{1}{2}i^{−w_{j}^{2}}(e_{w} + e_{w+α_{i}} + e_{w+α_{j}} − e_{w+α_{i}+α_{j}}))

= σ(x_{−e_{i}−e_{j}})(\frac{1}{2}i^{−w_{j}^{2}}(i^{w_{j}^{2}}e_{w} + i^{w_{j}^{2}}e_{w+α_{i}} + i^{1−w_{j}^{2}}e_{w+α_{j}} − i^{1−w_{j}^{2}}e_{w+α_{i}+α_{j}}))

= σ(x_{−e_{i}−e_{j}})(\frac{1}{2}(e_{w} + e_{w+α_{i}} + (−1)^{w_{j}}ie_{w+α_{j}} − (−1)^{w_{j}}ie_{w+α_{i}+α_{j}}))

= \frac{1}{4}(e_{w} + e_{w+α_{i}} + e_{w+α_{j}} − e_{w+α_{i}+α_{j}} + e_{w+α_{i}} + e_{w} + e_{w+α_{i}+α_{j}} − e_{w+α_{j}}

+ (−1)^{w_{j}}i(e_{w+α_{j}} + e_{w+α_{i}+α_{j}} + e_{w} − e_{w+α_{i}})
− (−1)^{w_{j}}i(e_{w+α_{i}+α_{j}} + e_{w+α_{j}} + e_{w+α_{i}} − e_{w}))

= \frac{1}{2}((1 + (−1)^{w_{j}}i)e_{w} + (1 − (−1)^{w_{j}}i)e_{w+α_{i}})

= \frac{1}{2}((1 − i)e_{w+w_{j}α_{i}} + (1 + i)e_{w+(1+w_{j})α_{i}})

= σ(x_{e_{j}−e_{i}})(\frac{1}{2}((1 − i)e_{w} + (1 + i)e_{w+α_{i}})) = σ(x_{e_{j}−e_{i}})σ(x_{−2e_{i}})(e_{w}).

Finally, we check (2.12) with t = u = 1:

[σ(x_{e_{i}+e_{j}}), σ(x_{−e_{j}−e_{k}})](e_{w}) = σ(x_{e_{i}+e_{j}})σ(x_{−e_{j}−e_{k}})σ(x_{e_{i}+e_{j}})^{−1}σ(x_{−e_{j}−e_{k}})^{−1}(e_{w})

= σ(x_{e_{i}+e_{j}})σ(x_{−e_{j}−e_{k}})σ(x_{e_{i}+e_{j}})^{−1} \frac{1}{2}(e_{w} + e_{w+α_{j}} + e_{w+α_{k}} − e_{w+α_{j}+α_{k}})

= σ(x_{e_{i}+e_{j}})σ(x_{−e_{j}−e_{k}})\frac{1}{2}((−1)^{w_{i}w_{j}}e_{w} + (−1)^{w_{i}(w_{j}+1)}e_{w+α_{j}}

+ (−1)^{w_{i}w_{j}}e_{w+α_{k}} − (−1)^{w_{i}(w_{j}+1)}e_{w+α_{j}+α_{k}})

= σ(x_{e_{i}+e_{j}})\frac{1}{4}((−1)^{w_{i}w_{j}}(e_{w} + e_{w+α_{j}} + e_{w+α_{k}} − e_{w+α_{j}+α_{k}})

+ (−1)^{w_{i}(w_{j}+1)}(e_{w+α_{j}} + e_{w} + e_{w+α_{j}+α_{k}} − e_{w+α_{k}})
+ (−1)^{w_{i}w_{j}}(e_{w+α_{k}} + e_{w+α_{j}+α_{k}} + e_{w} − e_{w+α_{j}})
− (−1)^{w_{i}(w_{j}+1)}(e_{w+α_{j}+α_{k}} + e_{w+α_{k}} + e_{w+α_{j}} − e_{w}))

= σ(x_{e_{i}+e_{j}})\frac{1}{2}(−1)^{w_{i}w_{j}}((1 + (−1)^{w_{i}})e_{w} + (1 − (−1)^{w_{i}})e_{w+α_{k}})

= σ(x_{e_{i}+e_{j}})(−1)^{w_{i}w_{j}}e_{w+w_{i}α_{k}}
= e_{w+w_{i}α_{k}}
= σ(x_{e_{i}−e_{k}})(e_{w}).

This completes the proof that the action of the root subgroups given at the beginning of this section satisfies the Chevalley relations. In order to check that the matrices given before define a representation, it remains to verify that the matrix representing ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) is as given. This element is the product over i of the commuting elements

x_{2e_{i}}(1)x_{−2e_{i}}(−1)x_{2e_{i}}(1) = x_{2e_{i}}x_{−2e_{i}}^{−1}x_{2e_{i}}.

We have

σ(x_{2e_{i}})σ(x_{−2e_{i}})^{−1}σ(x_{2e_{i}})(e_{w}) = σ(x_{2e_{i}})σ(x_{−2e_{i}})^{−1}(i^{w_{i}^{2}}e_{w})

= σ(x_{2e_{i}})\frac{1}{2}i^{w_{i}^{2}}((1 + i)e_{w} + (1 − i)e_{w+α_{i}})

= \frac{1}{2}i^{w_{i}^{2}}((1 + i)i^{w_{i}^{2}}e_{w} + (1 − i)i^{1−w_{i}^{2}}e_{w+α_{i}})

= \frac{1}{2}((−1)^{w_{i}}(1 + i)e_{w} + i(1 − i)e_{w+α_{i}})

= \frac{1}{2}(1 + i)((−1)^{w_{i}}e_{w} + e_{w+α_{i}})

= \frac{1}{1 − i}((−1)^{w^{t}α_{i}}e_{w} + e_{w+α_{i}}).

Composing these, we have

(∏_{i} σ(x_{2e_{i}})σ(x_{−2e_{i}}^{−1})σ(x_{2e_{i}})) (e_{w}) = \frac{1}{(1 − i)^{g}} ∑_{w′}(−1)^{w^{t}w′}e_{w′}

as required.

7. Theta functions, g = 1

The purpose of this and the next section is to describe how the representation

σ: Sp(2g, ℤ) → U(2^{g}, ℚ[i])/{±I}

arises from the action on a suitable space of theta functions.

References for the representation on theta functions discussed here include Funar and Pitsch [7], Gocho [10, 11], Runge [22, 23, 24], Tsushima [28]. Our group ℌ̃ is explicitly described in Lemma 2.1 of [24]. Other descriptions of the representation can be found in Glasby [9], and in Nebe, Rains and Sloane [19].

We begin with some background. For further details, the reader should consult the books of Mumford [18], Igusa [16], and Freitag [6]. It is easier to follow the general case after seeing the genus one case first, so that’s where we begin.

Consider expressions of the form

∑_{n∈ℤ} e^{πif(n)}

where f is a polynomial with complex coefficients. If f has odd degree, this cannot converge. If f has even degree, then the sum converges when the imaginary part of the leading coefficient is positive. So the simplest case is when f is quadratic. Adding a constant to f merely multiplies the sum by a constant, so we might as well ignore the constant term. We regard the other two coefficients of the quadratic polynomial as complex variables.

The classical Jacobi theta function is the function of two complex variables z and τ defined via

ϑ(z, τ) = ∑_{n∈ℤ} e^{πin^{2}τ+2πinz}.

This sum converges provided that Im(τ) > 0, and defines an analytic function for z ∈ ℂ and τ in the upper half plane H of ℂ. Its transformational properties are as follows. As a function of z it is symmetric and quasiperiodic with periods 1 and τ:

ϑ(z + 1, τ) = ϑ(z, τ) = ϑ(−z, τ), ϑ(z + τ, τ) = e^{−πi(τ+2z)}ϑ(z, τ).

As a function of τ, it satisfies the following:

(7.1)  ϑ(z, τ + 2) = ϑ(z, τ),
(7.2)  ϑ(z/τ, −1/τ) = \sqrt{−iτ} e^{πiz^{2}/τ}ϑ(z, τ).

Here, −iτ has positive real part, and the square root is also taken to have positive real part. The second of these is called the functional equation for the theta function, and is the only difficult one to prove. This was used by Riemann in his investigations of the zeta function [21].

To interpret this group theoretically, we make the semidirect product ℤ^{2} ⋊ SL(2, ℤ) act on ℂ × H. An element (m, n) ∈ ℤ^{2} acts via

(z, τ) ↦ (z + mτ + n, τ)

while ( \begin{matrix}a & b \\ c & d \end{matrix}) acts by

(z, τ) ↦ (\frac{z}{cτ + d}, \frac{aτ + b}{cτ + d}) .

This corresponds to an interpretation where τ determines an elliptic curve ℂ^{+}/(1, τ), and z corresponds to a point on this elliptic curve. With this action, the subgroup

ℤ^{2} ⋊ Sp^{q}(2, ℤ) ≤ ℤ^{2} ⋊ Sp(2, ℤ) = ℤ^{2} ⋊ SL(2, ℤ)

transforms ϑ(z, τ) to itself, up to a manageable factor. Combining the actions of the two generators ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) of Sp^{q}(2, ℤ) given by (7.1) and (7.2), for α = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ Sp^{q}(2, ℤ) we have

(7.3)  ϑ(α(z, τ)) = m(α)\sqrt{cτ + d} e^{πicz^{2}/(cz+d)}ϑ(z, τ)

where m(α) is a suitable eighth root of unity. The function m is called the theta multiplier system on Sp^{q}(2, ℤ).

But for example, the matrix ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ), which is not in Sp^{q}(2, ℤ), sends τ to τ + 1, and we have

ϑ(z, τ + 1) = ϑ(z + \frac{1}{2}, τ) = ∑_{n∈ℤ}(−1)^{n}e^{πin^{2}τ+2πinz}.

Now ϑ(z, 2τ) has better transformation properties than ϑ(z, τ) under the matrix ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). This induces us to set

ϑ_{0}(z, τ) = ∑_{n∈2ℤ} e^{\frac{πi}{2} (n^{2}τ+2nz)} = ϑ(z, 2τ)

ϑ_{1}(z, τ) = ∑_{n∈2ℤ+1} e^{\frac{πi}{2} (n^{2}τ+2nz)} = e^{\frac{πi}{2} (τ+2z)}ϑ(z + τ, 2τ).

(note: ϑ_{0} and ϑ_{1} are not standard notation; the traditional notation for these functions is ϑ_{0,0}(z, 2τ) and ϑ_{1,0}(z, 2τ), but the notations ϑ_{0} and ϑ_{1} fit better with the notation we use in the higher genus case in the next section). We have

ϑ_{0}(z, τ) + ϑ_{1}(z, τ) = ϑ(z/2, τ/2)
ϑ_{0}(z, τ) − ϑ_{1}(z, τ) = ϑ((z + 1)/2, τ/2).

The matrix ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) fixes ϑ_{0} and multiplies ϑ_{1} by a factor of e^{πi/2} = i. Using the functional equation (7.2), an elementary computation shows that ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) sends ϑ_{0} to

ϑ(−z/τ, −2/τ) = \sqrt{−iτ/2} e^{πiz^{2}/2τ}(ϑ_{0}(z, τ) + ϑ_{1}(z, τ))

and ϑ_{1} to

e^{−πi(2z+1)/2τ}ϑ(−z/τ − 1/τ, −2/τ) = \sqrt{−iτ/2} e^{πiz^{2}/2τ}(ϑ_{0}(z, τ) − ϑ_{1}(z, τ)).

Combining these, we see that

(7.4)  α = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL(2, ℤ): (\begin{matrix}ϑ_{0} \\ ϑ_{1} \end{matrix}) ↦ ±\sqrt{cτ + d} e^{πicz^{2}/2(cτ+d)}σ(α) (\begin{matrix}ϑ_{0} \\ ϑ_{1} \end{matrix})

where σ: SL(2, ℤ) → U(2, ℚ[i])/{±1} is the projective representation

(\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) ↦ (\begin{matrix}1 & 0 \\ 0 & i \end{matrix})
(\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) ↦ \frac{1}{1 − i} (\begin{matrix}1 & 1 \\ 1 & −1 \end{matrix}) .

These matrices only multiply well up to a factor of ±1, because of the choices of square roots. It is clear that no choice of signs makes it an actual representation, because an element of order four has been represented by a matrix of order eight. It follows from (7.4) that the element of H^{2}(SL(2, ℤ), ℤ/2) defined by this projective representation is equal to the cohomology class defined by the signs of the square roots \sqrt{cτ + d}.

8. Higher genus

The generalisation to higher genus is by no means obvious, but it goes as follows. Consider the lattice ℤ^{g} of integer column vectors, and consider expressions of the form

∑_{n∈ℤ^{g}} e^{πif(n)}

where f is a polynomial function of the vector n with complex coefficients. Again the simplest case where the sum can converge is when f is a quadratic function, and again the constant term only leads to a constant factor outside the sum. Moreover, the sum converges if and only if the leading quadratic term has positive definite imaginary part. Therefore we define the Siegel upper half space H to be the space of symmetric g ×g matrices Ω whose imaginary part is positive definite. Note that such matrices are automatically invertible. Then for z ∈ ℂ^{g}, we define

ϑ(z, Ω) = ∑_{n∈ℤ^{g}} e^{πin^{t}Ωn+2πin^{t}z}.

This sum converges for z ∈ ℂ and Ω ∈ H, and defines an analytic function. As a function of z, for m ∈ ℤ^{g} it satisfies

ϑ(z + m, Ω) = ϑ(z, Ω) = ϑ(−z, Ω), ϑ(z + Ωm, Ω) = e^{−πim^{t}Ωm−2πim^{t}z}ϑ(z, Ω).

As a function of Ω, it satisfies the following.

(i) For A ∈ GL(g, ℤ),

ϑ(Az, AΩA^{t}) = ϑ(z, Ω).

(ii) If B ∈ Mat(g, ℤ) is a symmetric matrix with even diagonal then

ϑ(z, Ω + B) = ϑ(z, Ω).

(iii) The functional equation is

ϑ(Ω^{−1}z, −Ω^{−1}) = \sqrt{det(−iΩ)} e^{πiz^{t}Ω^{−1}z}ϑ(z, Ω).

This time, the appropriate group is the semidirect product ℤ^{2g} ⋊ Sp(2g, ℤ), acting on ℂ^{g} × H. An element (m, n), with m, n ∈ ℤ^{g} acts via

(z, Ω) ↦ (z + Ωm + n, Ω)

while ( \begin{matrix}A & B \\ C & D \end{matrix}) acts by

(z, Ω) ↦ (((CΩ + D)^{t})^{−1}z, (AΩ + B)(CΩ + D)^{−1}).

This corresponds to an interpretation where Ω determines an abelian variety

(ℂ^{+})^{2g}/(ℤ^{g} ⊕ Ωℤ^{g}),

and z corresponds to a point on this abelian variety. With this action, the subgroup

ℤ^{2g} ⋊ Sp^{q}(2g, ℤ) ≤ ℤ^{2g} ⋊ Sp(2g, ℤ)

transforms ϑ(z, Ω) to itself, up to a manageable factor.

Lemma 8.1. The group Sp^{q}(2g, ℤ) is generated by the matrices

(\begin{matrix}A & 0 \\ 0 & (A^{t})^{−1} \end{matrix}) ,  (\begin{matrix}I & B \\ 0 & I \end{matrix}) ,  (\begin{matrix}0 & I \\ −I & 0 \end{matrix})

where B is symmetric with Diag(B) even.

Proof. See Proposition A.4 in Appendix A of Mumford [18]. Note that Mumford’s notation for Sp^{q}(2g, ℤ) is Γ(1, 2). □

Lemma 8.2. For α = ( \begin{matrix}A & B \\ C & D \end{matrix}) ∈ Sp^{q}(2g, ℤ) we have

(8.3)  ϑ(α(z, Ω)) = m(α)\sqrt{det(CΩ + D)} e^{πiz^{t}(CΩ+D)^{−1}Cz}ϑ(z, Ω)

where m(α) is a suitable eighth root of unity.

Proof. It suffices to prove this for a set of generators of Sp^{q}(2g, ℤ). A suitable set of generators is given in Lemma 8.1. The action of these generators on θ(z, Ω) is given by (i), (ii) and (iii) above. □

Definition 8.4. The function m in Lemma 8.2 is called the theta multiplier system on Sp^{q}(2g, ℤ).

Instead of the two functions ϑ_{0} and ϑ_{1} for the genus one case, for higher genus g we introduce 2^{g} functions ϑ_{w} indexed by vectors w ∈ {0, 1}^{g}. The definition is

ϑ_{w}(z, Ω) = ∑_{n∈2ℤ^{g}+w} e^{\frac{πi}{2} (n^{t}Ωn+2n^{t}z)} = e^{\frac{πi}{2} (w^{t}Ωw+2w^{t}z)}ϑ(z + Ωw, 2Ω).

For w ∈ {0, 1}^{g} we have

∑_{w′∈{0,1}^{g}} w^{t}w′ϑ_{w′}(z, Ω) = ϑ((z + w)/2, Ω/2).

We are now ready to examine the action of Sp(2g, ℤ). The matrix ( \begin{matrix}I & B \\ 0 & I \end{matrix}) multiplies ϑ_{w}(z, Ω) by a factor of i^{w^{t}Bw}. Remembering to invert when we take a transformation inside a function, the matrix ( \begin{matrix}A & 0 \\ 0 & (A^{t})^{−1} \end{matrix} ) sends

ϑ_{w}(z, Ω) ↦ ϑ_{w}(A^{−1}z, A^{−1}Ω(A^{−1})^{t}) = ϑ_{(A^{t})^{−1}w}(z, Ω).

Finally, using the functional equation, the matrix ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) sends ϑ_{w}(z, Ω) to

\sqrt{det(−iΩ/2)} e^{πiz^{t}Ω^{−1}z/2}ϑ((z + w)/2, Ω/2)

= \frac{1}{(1 − i)^{g}} \sqrt{det(Ω)} e^{πiz^{t}Ω^{−1}z/2} ∑_{w′∈\{0,1\}^{g}} w^{t}w′ϑ_{w′}(z, Ω).

Note that here, we have used Lemma 5.1 to perform the inversion when taking the transformation inside the function.

Write Θ for the column vector whose entries are the ϑ_{w}(z, Ω). Then putting together the above, we find that

(8.5)  α = (\begin{matrix}A & B \\ C & D \end{matrix}) ∈ Sp(2g, ℤ): Θ ↦ ±\sqrt{det(CΩ + D)} e^{πiz^{t}(CΩ+D)^{−1}Cz/2}σ(α)(Θ)

where σ: Sp(2g, ℤ) → U(2^{g}, ℚ[i])/{±1} is the projective representation given by the matrices (1.1)–(1.3). The subgroup 𝔎 acts trivially, so it gives a projective representation of ℌ and an ordinary representation of ℌ̃. This completes our second proof that the matrices (1.1)– (1.3) do indeed define a projective representation of Sp(2g, ℤ), although this proof does not address the problem of what scalar multiples to choose. In particular, the factor of \sqrt{det A} in (1.1), which is necessary in order to get us down from eighth roots of unity to ±1, is not apparent from this discussion.

Remark 8.6. It is worth examining Equation (8.5) carefully. It not only defines the projective representation (1.1)–(1.3) in terms of theta functions, but it gives us an insight into the cohomology class. Namely, the exponential term on the right of this equation transforms well with respect to the Sp(2g, ℤ), without signs. So the signs involved in composing the values of σ are the same as the signs involved in taking the square root of the determinant. Therefore, a particular choice of square roots gives a cocycle defining the central extension of Sp(2g, ℤ) by ℤ/2 given by the projective representation on theta functions. In the next section, we describe a particular choice of signs for the square root of the determinant, and we introduce the concept of a multiplier system.

9. Multiplier systems

Parts of the background material for this section are based on Chapter 1 of Freitag [6]. For a matrix α = ( \begin{matrix}A & B \\ C & D \end{matrix}) ∈ Sp(2g, ℝ), consider the expression det(CΩ+D). This is a holomorphic function of Ω ∈ H without zeros, and H is a convex set. So there exists a holomorphic square root, well defined up to sign.

We make a choice of signs for the square root so that the value at i times the identity is equal to e^{πig/4} (whose square is e^{πig/2} = i^{g} = det(iI)), as follows. Consider the function f: [0, 1] → ℂ given by

f(t) = det(t(CΩ + D) + (1 − t)iI).

In other words, f(t) is the determinant of a matrix which linearly interpolates between CΩ + D and iI. Then

∫_{0}^{1} \frac{f′(t)}{f(t)} dt

is a holomorphic function of Ω, and

\frac{det(CΩ + D)}{e^{πig/2}} = \frac{f(1)}{f(0)} = e^{∫_{0}^{1}(f′(t)/f(t))dt}.

We define

\sqrt{det(CΩ + D)} = e^{πig/4+ \frac{1}{2} ∫_{0}^{1}(f′(t)/f(t))dt},

with the notational caveat that det(CΩ+D) can equal det(CΩ′+D) without \sqrt{det(CΩ + D)} equalling \sqrt{det(CΩ′ + D)}.

Let us write j(α, Ω) for this choice of square root. Then

(9.1)  j(α_{1}α_{2}, Ω) = s(α_{1}, α_{2})j(α_{1}, α_{2}Ω)j(α_{2}, Ω),

where s(α_{1}, α_{2}) = ±1. The function

s: Sp(2g, ℝ) × Sp(2g, ℝ) → {±1}

is a 2-cocycle, since by associativity it satisfies

s(α_{1}, α_{2})s(α_{1}α_{2}, α_{3}) = s(α_{1}, α_{2}α_{3})s(α_{2}, α_{3}).

The fact that it defines the same cohomology class in H^{2}(Sp(2g, ℤ), ℤ/2) as the projective representation σ: Sp(2g, ℤ) → U(2^{g}, ℚ[i])/{±1} follows from equation (8.5). We have already seen that this is the metaplectic double cover Mp(2g, ℤ), and so s is a 2-cocycle defining Mp(2g, ℝ).

A multiplier system (of weight \frac{1}{2}) for a congruence subgroup Γ ≤ Sp(2g, ℝ) is a function m: Γ → ℂ^{×} such that

(9.2)  m(α_{1}α_{2}) = s(α_{1}, α_{2})m(α_{1})m(α_{2}).

In other words, it is a 1-cochain whose coboundary is s.

Lemma 9.3. If m is a multiplier system for Γ then m(α)j(α, Ω) behaves consistently with respect to composition, inasmuch as it satisfies

m(α_{1}α_{2})j(α_{1}α_{2}, Ω) = (m(α_{1})j(α_{1}, α_{2}Ω))(m(α_{2})j(α_{2}, Ω)).

Proof. This follows from equations (9.1) and (9.2). □

Proposition 9.4. If m is a multiplier system for a subgroup Γ ≤ Sp(2g, ℤ) then the map

α = (\begin{matrix}A & B \\ C & D \end{matrix}) ↦ m(α)σ(α)

is a homomorphism from Γ to U(2^{g}, ℂ).

Proof. This follows from Lemma 9.3 and equation (8.5). □

Corollary 9.5. If there is a multiplier system for a subgroup Γ ≤ Sp(2g, ℤ) then the composite of the restriction map and the change of coefficients

H^{2}(Sp(2g, ℤ), {±1}) → H^{2}(Γ, {±1}) → H^{2}(Γ, ℂ^{×})

kills the cohomology class represented by s. □

Theorem 9.6. There is a multiplier system for the theta subgroup Sp^{q}(2g, ℤ) with values in the eight roots of unity.

Proof. Lemma 8.2 shows that the theta multiplier system is a multiplier system for the theta subgroup. □

Corollary 9.7. The composite of the restriction map and change of coefficients ℤ/2 → ℤ/8,

H^{2}(Sp(2g, ℤ), ℤ/2) → H^{2}(Sp^{q}(2g, ℤ), ℤ/2) → H^{2}(Sp^{q}(2g, ℤ), ℤ/8)

kills the cohomology class represented by the cocycle s. □

Proposition 9.8. Let m: Sp^{q}(2g, ℤ) → ℂ^{×} be the theta muliplier system. Then the map α ↦ m(α)^{2} is a group homomorphism from Sp^{q}(2g, ℤ) to the group of fourth roots of unity.

Proof. This follows by squaring equation (9.2), since s(α_{1}, α_{2})^{2} = 1. □

The following puts this in context.

Proposition 9.9. Suppose that g ⩾ 3. We have H_{1}(Sp^{q}(2g, ℤ)) ≅ ℤ/4. The square of the theta multiplier system provides a surjective homomorphism Sp^{q}(2g, ℤ) → ℤ/4. This takes value 1 on the matrix ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ∈ Sp^{q}(2, ℤ) ⊆ Sp^{q}(2g, ℤ). The kernel is the derived subgroup Sp^{q}(2g, ℤ)′.

Thus the quotient Sp^{q}(2g, ℤ)/Sp^{q}(2g, ℤ)′ is cyclic of order four, and the element ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ∈ Sp^{q}(2, ℤ) ⊆ Sp^{q}(2g, ℤ) (embedded via the first and (g + 1)st coordinates) generates a complementary subgroup of order four.

Proof. See Theorem 2 of Endres [4]. See also Theorem 1.1 (i) of Johnson and Millson [17] and the comments at the end of Section 1 of that paper.

The value of m on the given element is e^{2πi/8}, as is easy to check. So m^{2} takes value i, which corresponds to the generator 1 for ℤ/4, on this element. □

Corollary 9.10. The restriction to the derived subgroup

H^{2}(Sp(2g, ℤ), ℤ/2) → H^{2}(Sp^{q}(2g, ℤ)′, ℤ/2)

kills the cohomology class represented by the cocycle s.

Another example of a multiplier system is the following. Recall that 𝔎𝔓 ≤ Sp(2g, ℤ) is the subgroup consisting of the matrices X = ( \begin{matrix}A & B \\ C & D \end{matrix}) where the entries of C are even, and the diagonal entries are divisible by four.

Theorem 9.11. There is a multiplier system for the subgroup 𝔎𝔓 ≤ Sp(2g, ℤ) with values in the fourth roots of unity. It takes value one on the derived subgroup (𝔎𝔓)′.

Proof. Without the factor of \sqrt{det A} in (1.2), the assignments (1.1)–(1.2) describe a representation 𝔓 → U(2^{g}, ℚ[i]) with kernel 𝔎∩𝔓. This extends in an obvious way to a representation 𝔎𝔓 → U(2^{g}, ℚ[i]) with kernel 𝔎. It follows that m(X) = \sqrt{det A}^{−1} is a multiplier system for this subgroup. It vanishes on commutators. □

10. The subgroup 𝔎_{1} ≤ Sp^{q}(2g, ℤ)

In this and the next section, we investigate the cohomology of Sp^{q}(2g, ℤ) enough so that we can explain group theoretically what is going on in Corollary 9.7. This will enable us to offer two proofs of Theorem 1.4, one using group theory and the other using theta functions. The reader only interested in the theta function proof may safely skip this and the next section.

We begin by studying the inverse image in the metaplectic double cover Mp(2g, ℤ) of the subgroup Sp^{q}(2g, ℤ) ≤ Sp(2g, ℤ). For this purpose, we introduce a subgroup 𝔎_{1} of index two in 𝔎, and prove that it is normal in Sp^{q}(2g, ℤ).

Recall that 𝔎 is the subgroup of Γ(2g, 2) consisting of the matrices

(\begin{matrix}I + 2a & 2b \\ 2c & I + 2d \end{matrix}) ∈ Sp(2g, ℤ)

such that Diag(b), Diag(c) are even, and det(I + 2a) ≡ 1 (mod 4). This has index two in Γ(2g, 2, 4). Recall also that such a matrix is in Sp(2g, ℤ) if and only if (I + 2a)b^{t} and c(I + 2d^{t}) are symmetric and a + d^{t} + 2(ad^{t} − bc^{t}) = 0. We define a subgroup 𝔎_{1} ⊴ 𝔎 of index two by refining this last condition to

det(I + 2a) ≡ 1 (mod 8).

To show that this is a subgroup, we begin with a lemma.

Lemma 10.1.

(i) If X and Y are symmetric modulo two, with Diag(X) and Diag(Y) even, then Tr(XY) is even.

(ii) If X and Y are symmetric modulo four, with Diag(X) and Diag(Y) even, then det(I + 2XY) is congruent to one modulo eight.

Proof. (i) We have Tr(XY) = ∑_{i,j} X_{ij}Y_{ji}. A term with i ≠ j cancels modulo two with the term with i and j interchanged, while the terms with i = j are even.

(ii) Expanding out the determinant and ignoring terms divisible by eight, we have

det(I + 2XY) = 1 + 2 ∑_{i,j} X_{ij}Y_{ji} + 4 ∑_{\begin{matrix}i,j,k,ℓ \\ i<k \end{matrix}}((X_{ij}Y_{ji})(X_{kℓ}Y_{ℓk}) − (X_{ij}Y_{jk})(X_{kℓ}Y_{ℓi}))

In the second sum, any term with i = j or k = ℓ, as well as any second term with j = k or ℓ = i, is divisible by two, and therefore does not contribute to the sum modulo eight.

The terms in the second sum with j = ℓ are zero, so they do not contribute to the sum. To deal with the terms in the second sum where all four of i, j, k and ℓ are different, we

rewrite 4 ∑_{i<k} … as 2 ∑_{i≠k} . . .. Each term is equal to the terms obtained by applying the permutations (i, j)(k, ℓ), (i, k)(j, ℓ) and (i, ℓ)(j, k), and therefore they come in sets of four adding to a contribution which is zero modulo eight (recall that X and Y are symmetric modulo four).

What remains is to show that

2 ∑_{i,j} X_{ij}Y_{ji} + 2(∑_{\begin{matrix}j=k \\ i=ℓ \end{matrix}} + ∑_{j=k} + ∑_{i=ℓ})((X_{ij}Y_{ji})(X_{kℓ}Y_{ℓk}) − (X_{ij}Y_{jk})(X_{kℓ}Y_{ℓi}))

is divisible by eight, where no other indices are equal in the last three sums. The terms in the first sum with i = j have both X_{ij} and Y_{ji} even, so the result is divisible by eight, and we can delete these terms. For the second terms in the second sum, i.e.,

2 ∑_{\begin{matrix}j=k \\ i=ℓ \end{matrix}}(−(X_{ij}Y_{jk})(X_{kℓ}Y_{ℓi})),

both Y_{jk} and Y_{ℓi} are even, so the result is divisible by eight. For the second terms in the third sum, i.e.,

2 ∑_{j=k}(−(X_{ij}Y_{jk})(X_{kℓ}Y_{ℓi})),

we can swap i and ℓ to get an equal term. They both have Y_{jk} even, so the result is again divisible by eight. Similarly for the second terms in the fourth sum. So what remains is

2 ∑_{i≠j} X_{ij}Y_{ji} + 2(∑_{\begin{matrix}j=k \\ i=ℓ \end{matrix}} + ∑_{j=k} + ∑_{i=ℓ})((X_{ij}Y_{ji})(X_{kℓ}Y_{ℓk})).

The first and second of these sums give

2 ∑_{i≠j}(X_{ij}Y_{ji} + (X_{ij}Y_{ji})^{2}).

Every integer is congruent to modulo four to its square, so this is divisible by eight.

In the third sum, swapping i and ℓ gives another copy of the same term. Similarly, in the fourth sum, swapping j and k gives another copy of the same term. Finally, applying (ij)(kℓ) to a term in the third sum gives an equal term in the fourth sum. So every term occurs four times, and the whole sum is then doubled to give a contribution divisible by eight. □

Proposition 10.2. 𝔎_{1} is a subgroup of Sp(2g, ℤ).

Proof. If

(\begin{matrix}I + 2a & 2b \\ 2c & I + 2d \end{matrix}) , (\begin{matrix}I + 2a′ & 2b′ \\ 2c′ & I + 2d′ \end{matrix})

are in 𝔎_{1} then the top left block of the product is

(I + 2a)(I + 2a′) + 4bc′,

whose determinant is congruent to

det(I + 2a) det(I + 2a′) + 4Tr(bc′)

modulo eight. Now b and c′ are symmetric modulo two, with Diag(b) and Diag(c′) even, and so by Lemma 10.1, 4Tr(bc′) is divisible by eight. □

Theorem 10.3. The subgroup 𝔎_{1} is normal in Sp^{q}(2g, ℤ).

Proof. It suffices to show that 𝔎_{1} is normalised by each of the matrices in Lemma 8.1. This is obvious for the first and third type, so it remains to show that 𝔎_{1} is normalised by matrices of the form ( \begin{matrix}I & Y \\ 0 & I \end{matrix} ) with Y symmetric and Diag(Y) even. The top left block of

(\begin{matrix}I & Y \\ 0 & I \end{matrix} ) (\begin{matrix}I + 2a & 2b \\ 2c & I + 2d \end{matrix}) (\begin{matrix}I & −Y \\ 0 & I \end{matrix} )

is equal to I + 2a + 2Y c. We must therefore show that this has determinant one modulo eight.

Now working modulo eight, I + 2a and I − 2a + 4a^{2} are inverse and have determinant one, so multiplying on the right by this, we see that we must show that I + 2Y c(I − 2a) has determinant one.

Next, we note that (I + 2a)^{t}c is symmetric with even diagonal, so we have

c = c^{t} + 2c^{t}a − 2a^{t}c,

and hence the matrix c^{t} − 2a^{t}c is also symmetric with even diagonal. Substituting, and still working modulo eight, we must prove that I + 2Y (c^{t} − 2a^{t}c) has determinant one. But this now follows by applying part (ii) of Lemma 10.1 to the matrices Y and c^{t} − 2a^{t}c. □

11. Cohomology of Sp^{q}(2g, ℤ)

In this section, we prove that the restriction of the double cover of Sp(2g, ℤ) to Sp^{q}(2g, ℤ) is inflated from its abelianisation. To be more precise, it is the pullback of Sp^{q}(2g, ℤ) → ℤ/4 and ℤ/8 → ℤ/4.

We need to name several subgroups of Sp^{q}(2g, ℤ). Let Q_{1} be the unique subgroup of index two in Sp^{q}(2g, ℤ), namely the inverse image of

Ω^{+}(2g, 2) ⊆ O^{+}(2g, 2) = Sp^{q}(2g, ℤ)/Γ(2g, 2).

Let Q_{2} = Sp^{q}(2g, ℤ)′, of index two in Q_{1}. Let Q_{3} = Q_{2} ∩ Γ(2), of index two in Γ(2g, 2) which contains 𝔎 but not Γ(2g, 2, 4).

[figure: Sp(2g, ℤ); Sp^{q}(2g, ℤ); 2; Q_{1}; 2; Ω^{+}(2g,2); Q_{2}; Γ(2g, 2); 2; (ℤ/2)^{2g}; Q_{3}; Γ(2g, 2, 4); 2; 𝔎; 2; 𝔎_{1}; Γ(2g, 4); Γ(2g, 8)]

The following lemma shows that although Γ(2g, 2)/𝔎_{1} is not isomorphic to the double cover E of Γ(2g, 2)/𝔎 (cf. Definition 2.18), the subgroup Q_{3}/𝔎_{1} of index two in Γ(2g, 2)/𝔎_{1} is isomorphic to the inverse image of Q_{3}/𝔎, of index two in E.

Lemma 11.1. The quotient Q_{3}/𝔎_{1} is an extraspecial group of type 2_{+}^{1+2g}. The quadratic form Q_{3}/𝔎 → 𝔎/𝔎_{1} ≅ ℤ/2 is given by ⟨Diag(B), Diag(C)⟩.

The quotient Γ(2g, 2)/𝔎_{1} is a direct product of Q_{3}/𝔎_{1} with ℤ/2.

Proof. To compute this quadratic form, let e_{i} be a matrix whose entries are all zero except

the (i, i) entry, which is equal to one. Then (\begin{matrix}I & 2e_{i} \\ 0 & I \end{matrix})^{2} and (\begin{matrix}I & 0 \\ 2e_{i} & I \end{matrix})^{2} are in 𝔎_{1}, while

[(\begin{matrix}I & 2e_{i} \\ 0 & I \end{matrix}) , (\begin{matrix}I & 0 \\ 2e_{j} & I \end{matrix})] = (\begin{matrix}I + 4δ_{ij} & 0 \\ 0 & I \end{matrix})

which is in 𝔎_{1} if and only if i ≠ j. The quadratic form is therefore non-singular, with Arf invariant zero.

Complementary to Q_{3}/𝔎 in Γ(2g, 2)/𝔎 is any matrix ( \begin{matrix}A & 0 \\ 0 & (A^{t})^{−1} \end{matrix} ) where A is congruent to the identity modulo two but does not have determinant one modulo eight. The square of such a matrix does have determinant one modulo eight, and therefore is in 𝔎_{1}. □

It follows from this lemma that there is a commutative diagram

[figure: 1; ℤ/2; Q_{3}/𝔎_{1}; Q_{3}/𝔎; 1; 1; ℤ/2; E; Γ(2)/𝔎; 1.]

Proposition 11.2. For g ⩾ 5 we have H^{2}(Q_{2}/𝔎, ℤ/2) ≅ ℤ/2. The restriction of the nonzero element of this group to H^{2}(Q_{3}/𝔎, ℤ/2) represents the extraspecial extension class given by Q_{3}/𝔎_{1} as in the lemma.

Proof. We examine the spectral sequence

H^{∗}(Q_{2}/Q_{3}, H^{∗}(Q_{3}/𝔎, ℤ/2)) ⇒ H^{∗}(Q_{2}/𝔎, ℤ/2).

We have

H^{2}(Q_{2}/Q_{3}, ℤ/2) = H^{2}(Ω^{+}(2g, 2), ℤ/2) = 0

for g ⩾ 5, see for example Feit [5]. Next, we have

H^{1}(Q_{2}/Q_{3}, H^{1}(Q_{3}/𝔎)) = H^{1}(Ω^{+}(2g, 2), (ℤ/2)^{2g}) = 0

for g ⩾ 4 by Theorem 4.2 and Corollary 4.3 of Pollatsek [20]. Finally, we have

H^{0}(Q_{2}/Q_{3}, H^{2}(Q_{3}/𝔎)) = ℤ/2

because there is a unique quadratic form fixed by Ω^{+}(2g, 2). Using this information, we see that H^{2}(Q_{2}/𝔎, ℤ/2) is either ℤ/2 or zero. It follows from the existence of the non-split extension

1 → ℤ/2 → Q_{2}/𝔎_{1} → Q_{2}/𝔎 → 1

that it is not zero. This also proves that the restriction of this class to Q_{3}/𝔎 is the extraspecial extension class given by Q_{3}/𝔎_{1}. □

Corollary 11.3. For g ⩾ 5 we have H^{2}(Q_{2}/𝔎_{1}, ℤ/2) = 0. □

Theorem 11.4. For g ⩾ 5 the restriction map H^{2}(Sp(2g, ℤ), ℤ/2) → H^{2}(Q_{2}, ℤ/2) is zero.

Proof. We have the following diagram of restriction and inflation maps:

\xymatrix{H^{2}(Sp(2g, ℤ)/𝔎, ℤ/2) \ar[r] \ar[dd]^{≅} & H^{2}(Q_{2}/𝔎, ℤ/2) \ar[d] \\  & H^{2}(Q_{2}/𝔎_{1}, ℤ/2) \ar[d] \\ H^{2}(Sp(2g, ℤ), ℤ/2) \ar[r] & H^{2}(Q_{2}, ℤ/2)}

The theorem now follows from Corollary 11.3. □

Next, we recall that if H is a subgroup of index two in a group G then there is an exact sequence

⋯ → H^{1}(G, ℤ/2) → H^{2}(G, ℤ/2) → H^{2}(H, ℤ/2) → H^{2}(G, ℤ/2) → H^{3}(G, ℤ/2) → ⋯

where the first map is multiplication by the element of H^{1}(G, ℤ/2) ≅ Hom(G, ℤ/2) corresponding to the homomorphism with kernel H, the second map is restriction, and the third map is transfer.

Proposition 11.5. For g ⩾ 5 the restriction map

ℤ/2 ≅ H^{2}(Sp(2g, ℤ), ℤ/2) → H^{2}(Sp^{q}(2g, ℤ), ℤ/2)

is not zero, but its image is the same as the image of inflation in degree two from

Sp^{q}(2g, ℤ)/Sp^{q}(2g, ℤ)′ ≅ ℤ/4.

Proof. First we restrict to Q_{1}. Since the restriction to Q_{2} is zero, it follows from the exact sequence that the restriction to Q_{1} is in the image of multiplication by the degree one element corresponding to the homomorphism Q_{1} → ℤ/2 with kernel Q_{2}. Since this is the unique non-zero degree one element, there are two possibilities. The restriction to Q_{1} is either zero or the square of the non-zero degree one element.

Now it follows from Lemma 9.9 that the element of order two

(\begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix}) ∈ Sp^{q}(2, ℤ) ⊆ Sp^{q}(2g, ℤ)

generates a complement to Q_{2} in Q_{1}. Every inverse image of this element in { \atop Sp(2, ℤ)} (and hence also in { \atop Sp(2g, ℤ)}) has infinite order, and so the restriction of the generator of

H^{2}(Sp(2g, ℤ), ℤ/2) ≅ ℤ/2

to the subgroup of order two generated by this element is non-zero. The restriction to Q_{1} is therefore also non-zero, and is hence equal to x^{2}, where x is the non-zero element of H^{1}(Q_{1}, ℤ/2).

Next, we restrict to Sp^{q}(2g, ℤ). Again apply the exact sequence, and note that the image in H^{2}(Sp^{q}(2g, ℤ), ℤ/2) of multiplication by the degree one element is zero, since this is true in the cohomology of the commutator quotient ℤ/4. Therefore restriction map H^{2}(Sp^{q}(2g, ℤ), ℤ/2) → H^{2}(Q_{1}, ℤ/2) is injective. Since the inflation from H^{2}(ℤ/4, ℤ/2) of the non-zero degree two element restricts to x^{2}, the proposition is proved. □

For the corresponding theorem in homology, we need the following lemma.

Lemma 11.6. For g ⩾ 3, the coinflation map H_{2}(Sp^{q}(2g, ℤ)) → H_{2}(O^{+}(2g, 2)) is surjective.

Proof. The five term sequence of the short exact sequence

1 → Γ(2g, 2) → Sp^{q}(2g, ℤ) → O^{+}(2g, 2) → 1

is

H_{2}(Sp^{q}(2g, ℤ)) → H_{2}(O^{+}(2g, 2)) → H_{0}(O^{+}(2g, 2), H_{1}(Γ(2g, 2)))

→ H_{1}(Sp^{q}(2g, ℤ)) → H_{1}(O^{+}(2g, 2)) → 0.

The end of this sequence is ℤ/2 → ℤ/4 → ℤ/2 → 0, so the map

H_{2}(O^{+}(2g, 2)) → H_{0}(O^{+}(2g, 2), H_{1}(Γ(2g, 2)))

is zero. □

Theorem 11.7. For g ⩾ 4, the image of the corestriction map

H_{2}(Sp^{q}(2g, ℤ)) → H_{2}(Sp(2g, ℤ)) ≅ ℤ

is contained in 2ℤ.

For g = 2 and g = 3 we have H_{2}(Sp(2g, ℤ)) ≅ ℤ ⊕ ℤ/2, and the image of corestriction is 2ℤ ⊕ ℤ/2. For g = 1, H_{2}(Sp(2, ℤ)) = 0 and the image of corestriction is zero.

Proof. First assume that g ⩾ 5. Examine the map of universal coefficient sequences

\xymatrix{0 \ar[r] & Ext(H_{1}(Sp(2g, ℤ)), ℤ/2) \ar[r] \ar[d] & H^{2}(Sp(2g, ℤ), ℤ/2) \ar[r] \ar[d] & Hom(H_{2}(Sp(2g, ℤ)), ℤ/2) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ext(H_{1}(Sp^{q}(2g, ℤ)), ℤ/2) \ar[r] & H^{2}(Sp^{q}(2g, ℤ), ℤ/2) \ar[r] & Hom(H_{2}(Sp^{q}(2g, ℤ)), ℤ/2) \ar[r] & 0}

It follows from Proposition 11.5 that the image of the middle vertical map is equal to the image of the lower left hand horizontal map. It therefore goes to zero in the lower right hand side:

\xymatrix{0 \ar[r] & 0 \ar[r] \ar[d] & ℤ/2 \ar[r] \ar[d] & ℤ/2 \ar[r] \ar[d]^{0} & 0 \\ 0 \ar[r] & ℤ/2 \ar[r] & ℤ/2 ⊕? \ar[r] & ? \ar[r] & 0.}

The upper right hand map is an isomorphism, so the theorem follows.

For 2 ≤ g ≤ 4, we have the diagram

\xymatrix{H_{2}(Sp^{q}(4, ℤ)) \ar[r] \ar[d] & H_{2}(Sp(4, ℤ)) \ar[d] \\ H_{2}(Sp^{q}(6, ℤ)) \ar[r] \ar[d] & H_{2}(Sp(6, ℤ)) \ar@{->>}[d] \\ H_{2}(Sp^{q}(8, ℤ)) \ar[r] \ar[d] & H_{2}(Sp(8, ℤ)) \ar[d]^{≅} \\ H_{2}(Sp^{q}(10, ℤ)) \ar[r] & H_{2}(Sp(10, ℤ))}

For g = 4, the result follows from the case g = 5.

For g = 3, using the same arguments, all that remains is to see whether corestriction hits the ℤ/2 summand of H_{2}(Sp(6, ℤ)). We have a diagram

\xymatrix{H_{2}(Sp^{q}(6, ℤ)) \ar[r] \ar[d] & H_{2}(Sp(6, ℤ)) \ar[d] \\ H_{2}(O^{+}(6, 2)) \ar[r] & H_{2}(Sp(6, 2)).}

The ℤ/2 summand of H_{2}(Sp(6, ℤ)) has non-zero image in H_{2}(Sp(6, 2)), so it remains to examine the corestriction map H_{2}(O^{+}(6, 2)) → H_{2}(Sp(6, 2)). Looking at the character tables of these groups, it is easy to see that this is an isomorphism (note that O^{+}(6, 2) is isomorphic to the symmetric group of degree eight). So using Lemma 11.6, we are done.

For g = 2, the map H_{2}(Sp(6, ℤ)) → H_{2}(Sp(4, ℤ)) kills the ℤ/2 summand and is an isomorphism on the ℤ summand. So again we are left with the task of examining the corestriction map H_{2}(O^{+}(4, 2)) → H_{2}(Sp(4, 2)). The group O^{+}(4, 2) is a wreath product of a symmetric group of degree three with ℤ/2, while Sp(4, 2) is the symmetric group of degree six. The corestriction map in this case is surjective in degree two. Again using Lemma 11.6, we are done.

Finally, for g = 1, we have H_{2}(Sp(2, ℤ)) = 0 and there is nothing to prove. □

12. Signatures of surface bundles

Theorem 12.1. The restriction of the Meyer cohomology class τ̄ ∈ H^{2}(Sp(2g, ℤ), ℤ) to H^{2}(Sp^{q}(2g, ℤ), ℤ) is divisible by eight.

Proof. Recall that H^{2}(Sp(2g, ℤ), ℤ) ≅ ℤ, and τ̄ is four times a generator.

Examine the universal coefficient sequences

\xymatrix{0 \ar[r] & Ext(H_{1}(Sp(2g, ℤ)), ℤ) \ar[r] \ar[d] & H^{2}(Sp(2g, ℤ), ℤ) \ar[r] \ar[d] & Hom(H_{2}(Sp(2g, ℤ)), ℤ/2) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ext(H_{1}(Sp^{q}(2g, ℤ)), ℤ) \ar[r]^{i} & H^{2}(Sp^{q}(2g, ℤ), ℤ) \ar[r] & Hom(H_{2}(Sp^{q}(2g, ℤ)), ℤ) \ar[r] & 0}

Since τ̄ is divisible by 4, we write τ̄ = 4u with u ∈ H^{2}(Sp(2g, ℤ), ℤ). By Theorem 11.7, the image of the right hand vertical map is divisible by two. So the image of u in Hom(H_{2}(Sp^{q}(2g, ℤ)), ℤ) is divisible by 2. Since the universal coefficient sequence is (noncanonically) split, it follows that the image of u in H^{2}(Sp^{q}(2g, ℤ), ℤ) is 2v + i(x) for some x ∈ Ext(H_{1}(Sp^{q}(2g, ℤ)), ℤ). By Lemma 9.9, we have 4x = 0. It follows that the image of τ̄ in H^{2}(Sp^{q}(2g, ℤ), ℤ) is 4(2v + i(x)) = 8v. □

We now offer two proofs of Theorem 1.4.

First proof of Theorem 1.4. By Theorem 12.1, there is a class μ ∈ H^{2}(Sp^{q}(2g, ℤ), ℤ) such that the restriction of τ̄ ∈ H^{2}(Sp(2g, ℤ), ℤ) is 8μ. If the monodromy action on ℤ^{2g} lies in the subgroup Sp^{q}(2g, ℤ) of Sp(2g, ℤ) then there is a map ψ making the following diagram commute.

\xymatrix{π_{1}(Σ_{h}) \ar[r]^{χ} \ar[dr]_{ψ} & Sp(2g, ℤ) \\  & Sp^{q}(2g, ℤ). \ar@{^{(}->}[u]}

Thus we have

σ(M) = ⟨χ^{∗}(τ̄), [Σ_{h}]⟩ = 8⟨ψ^{∗}(μ), [Σ_{h}]⟩.  □

Second proof of Theorem 1.4. Alternatively, we can prove this theorem using the theory of theta functions, as follows. The recipe for computing the signature modulo eight given in [1] is to use the composite

ϕ: π_{1}(Σ_{h}) → Sp(2g, ℤ) → U(2^{g}, ℚ[i])/{±I}.

The relation in π_{1}(Σ_{h}) takes the form [a_{1}, b_{1}] … [a_{h}, b_{h}] = 1. The element [ϕ(a_{i}), ϕ(b_{i})] is well defined in U(2^{g}, ℚ[i]), and the product

[ϕ(a_{1}), ϕ(b_{1})] … [ϕ(a_{h}), ϕ(b_{h})]

is equal to I if the signature of M is zero modulo eight, and −I if the signature is four modulo eight. If the image of π_{1}(Σ_{h}) lies in Sp^{q}(2g, ℤ) then the image of the commutator subgroup π_{1}(Σ_{h})′ lies in Sp^{q}(2g, ℤ)′. Theorem 9.6 shows that there is a multiplier system for Sp^{q}(2g, ℤ), and Proposition 9.8 shows that on Sp^{q}(2g, ℤ)′ it squares to the identity, so it takes values ±1. Thus by Proposition 9.4, the restriction of σ to

Sp^{q}(2g, ℤ)′ → U(2^{g}, ℚ[i])/{±I}

lifts to a group homomorphism Sp^{q}(2g, ℤ)′ → U(2^{g}, ℚ[i]). Pulling back to π_{1}(Σ_{h})′, it follows that

[ϕ(a_{1}), ϕ(b_{1})] … [ϕ(a_{h}), ϕ(b_{h})] = I

and so the signature of M is divisible by eight. □

Corollary 12.2. If Σ_{g} → M → Σ_{h} is a closed oriented surface bundle whose monodromy action on H^{1}(Σ_{g}; ℤ/2) is trivial then the signature σ(M) is divisible by 8.

Proof. If the monodromy action on (ℤ/2)^{2g} is trivial, then the map χ: π_{1}(Σ_{h}) → Sp(2g, ℤ) factors through Γ(2g, 2), which is contained in Sp^{q}(2g, ℤ). Now apply Theorem 1.4. □

Proof of Theorem 1.5. This follows the same lines as the second proof of Theorem 1.4. Theorem 9.11 shows that there is a multiplier system for 𝔎𝔓 which takes value one on commutators. So by Proposition 9.4, the restriction of σ to (𝔎𝔓)′ → U(2^{g}, ℚ[i])/{±I} lifts to U(2^{g}, ℚ[i]). The rest of the proof is the same. Note that the theory of theta functions is not necessary for this proof, as it already follows easily from what we have proved in Section 6 that σ: Sp(2g, ℤ) → U(2^{g}, ℚ[i])/{±I} restricted to (𝔎𝔓)′ lifts to U(2^{g}, ℚ[i]). □

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Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland, United Kingdom