TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 5, Pages 2101–2139 S 0002-9947(99)02469-1 Article electronically published on January 26, 1999

SYMPLECTIC GROUP LATTICES

RUDOLF SCHARLAU AND PHAM HUU TIEP

Abstract. Let p be an odd prime. It is known that the symplectic group Sp_{2n}(p) has two (algebraically conjugate) irreducible representations of degree (p^{n} +1)/2 realized over ℚ(\sqrt{ϵp}), where ϵ = (−1)^{(p−1)/2}. We study the integral lattices related to these representations for the case p^{n} ≡ 1 mod 4. (The case p^{n} ≡ 3 mod 4 has been considered in a previous paper.) We show that the class of invariant lattices contains either unimodular or p-modular lattices. These lattices are explicitly constructed and classified. Gram matrices of the lattices are given, using a discrete analogue of Maslov index.

1. Introduction

Let p be an odd prime, and set S_{n} = Sp_{2n}(p) for the symplectic group of degree 2n over 𝔽_{p}. Euclidean integral lattices in the space of the Weil representation of S_{n} have been investigated by several authors (see for instance [BaV], [Dum], [Gow], [Gro], [Tiep 1], [Tiep 2]). The Weil representation W of S_{n} is a complex representation of degree p^{n} that can be obtained from the action of S_{n} on the extraspecial group p_{+}^{1+2n} (as the outer automorphism group). See, for example, [Isa], [Sei], or [Ward 1] for a more general approach. W is a sum of two irreducible representations of degrees (p^{n} − 1)/2 and (p^{n} + 1)/2. ( These two characters seem to have been first investigated in [BRW].) One of these representations, which we shall denote by W_{1}, is faithful and has even degree, and the kernel of the other representation, W_{2}, is the center Z = C_{2} of S_{n}. Following [Gow], we shall refer to W_{1} and W_{2} as Weil representations. Weil representations have been characterized in several ways in [TZa 1], [TZa 2].

Set ϵ = (−1)^{(p−1)/2} and suppose that (dim W_{i}, ϵ) ≠ (\frac{p^{n}−1}{2}, 1). It is shown in [Gro] that, under this assumption, the character ψ_{i} of the representation W_{i} generates the field ℚ(\sqrt{ϵp}) over the rational field ℚ, and has Schur index 1 over ℚ. Hence, there exist an extension G_{n} of S_{n} and an absolutely irreducible ℚG_{n}- module V affording the S_{n}-character ψ_{i} + \overline{ψ}_{i}, where the bar denotes the algebraic conjugation of the field ℚ(\sqrt{ϵp}). The group G_{n} can be chosen as a homomorphic

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

Received by the editors December 10, 1996.

1991 Mathematics Subject Classification. Primary 20C10, 20C15, 20C20, 11E12, 11H31, 94B05.

Key words and phrases. Integral lattice, unimodular lattice, p-modular lattice, finite symplectic group, Weil representation, Maslov index, linear code, self-dual code.

Part of this work was done during the second author’s stay at the Department of Mathematics, Israel Institute of Technology. He is grateful to Professor D. Chillag and his colleagues at the Technion for stimulating conversations and their generous hospitality. His work was also supported in part by the DFG.

image of the conformal symplectic group

\begin{aligned}G_{n} = CSp_{2n}(p) = \{φ ∈ GL(W)| & ∃κ ∈ 𝔽_{p}^{•}, ∀u, v ∈ W, \\ & ⟨φ(u), φ(v)⟩ = κ⟨u, v⟩\}, \end{aligned}

where W denotes a natural 2n-dimensional S_{n}-module over 𝔽_{p}, with the symplectic form ⟨·, ·⟩. In what follows we shall be concerned with the following two homomorphic images of G_{n}: the factor-group G_{n}^{+} of G_{n} by its center C_{p−1} (consisting of the scalar matrices λE_{2n}, λ ∈ 𝔽_{p}^{•}), and G_{n}^{−}, the factor-group of G_{n} by the central group C_{(p−1)/2} (consisting of scalar matrices λE_{2n}, λ ∈ 𝔽_{p}^{•2}). Throughout the paper, C_{m} denotes the cyclic group of order m, and E_{m} denotes the identity matrix of order m (over any field).

The lattices for the Weil representations of degree ψ(1) = (p^{n} − 1)/2 have been investigated in [Gow] and [Gro]. Recall that in this case p ≡ 3 mod 4, according to our general assumption; see [Tiep 2] for the excluded case. If n is even, then every ℤS_{n}-lattice in V is even unimodular. If n is odd, V contains p-modular invariant lattices. Recall that an integral lattice Λ is said to be p-modular (or modular of level p) if the lattices ^{p}Λ^{#} (the dual lattice Λ^{#} rescaled by the scalar p) and Λ are isometric. p-modular lattices have been introduced and investigated in [CoS 1] and [Que]. In either of these cases, the corresponding representations are globally irreducible in the sense of Gross [Gro]. Some of the corresponding lattices have been realized as sublattices of the Mordell-Weil lattices of certain elliptic curves (cf. [Dum] and [Gro]).

The Weil representations of degree ψ(1) = (p^{n} + 1)/2 are the subject of our present work, begun in [SchT] and continued in this paper. Here the corresponding representation cannot be globally irreducible anymore; namely, ψ mod 2 = 1_{S}+η for some η ∈ IBr_{2}(S). In [SchT], the case p^{n} ≡ 3 mod 4 has been treated. The existence of unimodular ℤG-lattices in V has been established, where G = G_{n}^{−} ≃ S_{n} · C_{2}. All ℤG-lattices contained in V have been classified.

In this paper, we are concerned with the case p^{n} ≡ 1 mod 4. Then W_{2} viewed over ℚ is in fact a faithful representation of P Sp_{2n}(p) of degree p^{n} + 1. Moreover, if p ≡ 3 mod 4, this representation can be extended (in a unique way) to a rational representation of G_{n}^{+}. If p ≡ 1 mod 4, it can be extended to a rational representation for each of the two groups G_{n}^{+} and G_{n}^{−} (cf. Proposition 2.3). The reason is that when p ≡ 1 mod 4 the two groups C_{2} × G_{n}^{+} and G_{n}^{−} are isoclinic to each other. For more detail on isoclinic groups see [Atlas] and [Tiep 2], Lemma 2.11. When p ≡ 3 mod 4, it follows from this lemma that the rational representation of S_{n} of degree p^{n} + 1 is extendible to a rational faithful representation of G_{n}^{+} if n is even, and of G_{n}^{−} if n is odd, but not for its isoclinic variant.

From now on we keep the following notation: S_{n} = Sp_{2n}(p), G_{n} = CSp_{2n}(p), Z ≅ C_{p−1} the center of G_{n}^{+}, G_{n}^{+} = G_{n}/Z, G_{n}^{−} = G_{n}/Z^{2}, θ a fixed generating element of 𝔽_{p}^{•}. Clearly, G_{n} is generated by S_{n} and an element ϑ_{n} with matrix diag(E_{n}, θE_{n}) in a fixed symplectic basis (e_{1}, …, e_{n}, f_{1}, …, f_{n}) of the natural S_{n}- module W = W_{n} = 𝔽_{p}^{2n} (that is, a basis in which the symplectic form ⟨·, ·⟩ is given as follows: ⟨e_{i}, e_{j}⟩ = 0, ⟨f_{i}, f_{j}⟩ = 0, ⟨e_{i}, f_{j}⟩ = δ_{i,j}). V = V_{n} denotes an irreducible ℚG_{n}-module with character χ such that χ|_{S_{n}} = ψ + \overline{ψ}. Furthermore, either Ker χ = Z (and then V_{n} is a faithful G_{n}^{+}-module), or p ≡ 1 mod 4 and Ker χ = Z^{2} (and then V_{n} is a faithful G_{n}^{−}-module). Under these assumptions χ exists and is unique by Proposition 2.3. It is clear that there exists a unique (up to scalar) G_{n}-invariant positive definite symmetric bilinear form (·, ·) on V_{n}.

Our first main result is the following theorem which includes Theorem 1.1 of [SchT]:

Theorem 1.1. Let p be an odd prime. If p ≡ 3 mod 4, then suppose in addition that n is odd. Then V_{n} contains G_{n}^{−}-invariant odd unimodular Euclidean lattices (of rank p^{n} + 1). If n > 1, these lattices have no roots.

Actually, we provide an explicit construction of a G_{n}^{−}-stable odd unimodular lattice Δ = Δ(p, n) contained in V_{n} (cf. Theorem 3.9 and Corollary 5.4) for n odd, and a G_{n}^{−}-stable odd unimodular lattice Δ^{−}(p, n) for the case where n is even and p ≡ 1 mod 4 (cf. Corollary 5.9). In the case p ≡ 3 mod 4 this is just the construction exposed in [SchT]. The cases p^{n} = 27 and p^{n} = 25 have been considered by R. Bacher and B. B. Venkov [BaV], and G. Nebe, respectively. The corresponding lattices have minimum 3. In general, Theorem 7.1 yields min Δ(p, n) ≥ (p + 1)/2 for all n ≥ 3.

Our next results are concerned with p-modular lattices.

Theorem 1.2. Let p be any odd prime. If p ≡ 3 mod 4, then suppose in addition that n is even. Then V_{n} contains G_{n}^{+}-invariant p-modular Euclidean lattices (of rank p^{n} + 1).

Again, we provide an explicit construction of a G_{n}^{+}-stable p-modular lattice Δ = Δ^{−}(p, n) if n is odd and p ≡ 1 mod 4 (cf. Corollary 5.10), respectively Δ = Δ(p, n) if n is even (cf. Theorem 4.4 and Corollary 5.6). This result generalizes Theorem (V.2) of [NPl] dealing with the case n = 1 and p ≡ 1 mod 4. The case n = 2 and p = 5 has been considered by Nebe; the corresponding 5-modular lattice (Δ(5, 2) in our notation) has minimum 5. As before, Theorem 7.1 yields min Δ(p, n) ≥ (p+1)/2 for all n ≥ 2.

Gram matrices of the lattices Δ(p, n), Δ^{−}(p, n) are given in §5 using a discrete analogue of Maslov index. In a few words, our explicit constructions can be described as follows. First we start with the case n is odd and the group G_{n}^{−}. Using Lagrangians, an idea going back to [BaV] (cf. §3) and Maslov index (cf. §5), we explicitly construct the unimodular lattices Δ(p, n) (for any odd prime p). Then descending from n + 1 to n (cf. §4), we obtain the p-modular lattices Δ(p, n) which are stable under G_{n}^{+}, for any even n and any odd prime p. Finally, let p ≡ 1 mod 4. Then the isoclinism between C_{2} × G_{n}^{+} and G_{n}^{−} and Proposition 2.4 allow us to construct the lattices Δ^{−}(p, n), for any n.

We wish to point out that V cannot contain simultaneously invariant unimodular and p-modular lattices. Namely, the unimodular lattices are acted on faithfully by G_{n}^{−}, and the p-modular ones by G_{n}^{+}. Moreover, the invariant unimodular (pmodular) lattices are essentially unique (if they exist). More precisely, the classification of G_{n}-invariant lattices in V_{n} is provided by the following theorem. Given a lattice Γ, let Γ^{#} denote the dual lattice and Γ^{0} denote the sublattice consisting of all vectors of even norm in Γ; furthermore, Γ^{1} = Γ ∩ 2(Γ^{0})^{#}. Two integral lattices (Γ, (·, ·)) and (Γ′, (·, ·)′) are called similar if there exist a surjective homomorphism ϕ : Γ → Γ′ and a scalar λ ∈ ℚ such that (ϕ(u), ϕ(v))′ = λ(u, v) for any u, v ∈ Γ.

Theorem 1.3. Let p be any odd prime and n any integer. Suppose that G_{n} acts irreducibly on an integral lattice Γ of rank p^{n} + 1, with kernel K. If p^{n} = 3 or p^{n} = p ≡ 1 mod 6, then suppose in addition that S_{n} acts reducibly on Γ ⊗ ℂ. Then one of the following holds.

(i) G_{n}/K is equal to G_{n}^{−} for odd n, and G_{n}^{+} for even n. Furthermore, Γ is similar to one of the lattices Δ, Δ^{0}, Δ^{1}, where Δ = Δ(p, n).

(ii) p ≡ 1 mod 4, and G_{n}/K is equal to G_{n}^{+} for odd n, and G_{n}^{−} for even n. Furthermore, Γ is similar to one of the lattices Δ, Δ^{0}, Δ^{1}, where Δ = Δ^{−}(p, n).

The GL_{2}(p)-invariant (p+1)-dimensional lattices which are not covered by Theorem 1.3 (here p = 3 or p ≡ 1 mod 6) have been investigated in [NPl], Theorem (V.4).

The full automorphism groups of all G_{n}-invariant lattices Λ in V_{n} have been determined in [Tiep 1]. In particular, if n > 1, then either Aut(Λ) ∈ {C_{2}×G_{n}^{+}, G_{n}^{−}}, or p = 3 and Aut(Λ) = (C_{6} · P Sp_{2n}(3)) · C_{2}.

2. Implicit proofs

We recall the notations S_{n}, ψ of degree (p^{n} + 1)/2, G_{n}, Z, G_{n}^{+}, G_{n}^{−}. We start with the following simple observation:

Lemma 2.1. Let p be any odd prime and n any integer. Suppose χ is an irreducible complex character of G_{n} of degree p^{n} + 1 with the following properties:

(i) χ restricted to S_{n} is equal to ψ + \overline{ψ}; (ii) χ is rational-valued.

Then one of the following holds.

(a) χ is a faithful character, say χ^{+}, of G_{n}^{+}; furthermore, n is even if p ≡ 3 mod 4.

(b) χ is a faithful character, say χ^{−}, of G_{n}^{−}; furthermore, n is odd if p ≡ 3 mod 4.

Proof. Let K = Ker χ. Schur’s Lemma and (ii) imply that Z/(Z ∩ K) is a cyclic group of order at most 2. In particular, either K = Z, or K = Z^{2}. Observe that G_{n} permutes the two characters ψ and \overline{ψ} of S_{n} nontrivially. Denote Ḡ = G_{n}/K, S̄ = S_{n}/(S_{n} ∩ K).

First consider the case K = Z. Since K ⊆ Ker ψ, the degree ψ(1) is odd, i.e. p^{n} ≡ 1 mod 4. Here we have Ḡ = G_{n}^{+}, S̄ = P Sp_{2n}(p), and Ḡ = S̄ · C_{2}. Clearly, the desired character χ is now uniquely determined: χ = IndḠ ; S̄(ψ).

Next let K = Z^{2}. If p ≡ 1 mod 4, then Ḡ = (S̄×Z/K)·C_{2}. If p ≡ 3 mod 4, then in view of (i) n must be odd, and Ḡ = S̄ · C_{2}. Now the desired character χ exists and is unique: χ = IndḠ ; S̄(ψ) if p ≡ 3 mod 4, and χ = IndḠ ; S̄×Z/K(ψ̃) if p ≡ 1 mod 4 (where ψ̃ is equal to ψ on S̄ and to −ψ(1) on the unique nontrivial element of Z/K). Clearly, ℚ(χ) = ℚ.

As we have mentioned above, C_{2} × G_{n}^{+} and G_{n}^{−} are isoclinic to each other if p ≡ 1 mod 4. In this case, ind(ψ) = 1 (cf. [Gro], Corollary 13.7); hence by Lemma 2.11 of [Tiep 2] ind(χ^{+}) = ind(χ^{−}) = 1. If p ≡ 3 mod 4 and n is odd, then ind(ψ) = 0, and ind(χ^{−}) = 1 (see [SchT] or Proposition 2.3 below). Hence by Lemma 2.11 of [Tiep 2], the corresponding character of degree p^{n} +1 of the isoclinic variant of G_{n}^{−} (which now is not isomorphic to C_{2} × G_{n}^{+}) has Schur-Frobenius indicator −1 and so cannot be written over ℝ. Similarly, if p ≡ 3 mod 4, then ind(ψ) = 0, ind(χ^{+}) = 1, but the corresponding character of degree p^{n} + 1 of the isoclinic variant of C_{2} × G_{n}^{+} (which is no longer G_{n}^{−}) cannot be written over ℝ.

The next proposition is an analogue of [SchT], Lemma 5.1.

Proposition 2.2. Let p be an odd prime and n any integer. Let r be any prime and let χ be as in Lemma 2.1. Then the following assertions hold.

(i) The reduction χ mod r is irreducible if r ≠ 2, p.

(ii) χ mod 2 = 2 · 1_{G_{n}} + β for a certain β ∈ IBr_{2}(G_{n}). Furthermore, β is of symplectic type, if p ≡ 1 mod 4.

(iii) χ mod p = η_{1} + η_{2}, where η_{1}, η_{2} ∈ IBr_{p}(G_{n}) are distinct characters which can be written over 𝔽_{p}. Furthermore, for k = 1, 2 η_{k} is not self-dual if χ = χ^{−}, and η_{k} is of type + if χ = χ^{+}.

Proof. The case p^{n} ≡ 3 mod 4 has already been handled in [SchT] (cf. the proof of Theorem 1.1 and Lemma 5.1 therein). Hence in what follows we suppose that p^{n} ≡ 1 mod 4.

1) It is well-known (see e.g. [Gow], [Gro]) that ψ mod r ∈ IBr_{r}(S_{n}) for any odd prime r. Furthermore, ψ mod 2 = 1_{S_{n}} + α for some α ∈ IBr_{2}(S_{n}). If x is a regular unipotent element of S_{n}, then ψ(x) = (1 ± p^{n−1}\sqrt{ϵp})/2. Furthermore, ϑ_{n} interchanges the S_{n}-conjugacy classes of x and some power of x^{s} and ψ(x^{s}) = (1 ∓ p^{n−1}\sqrt{ϵp})/2. Therefore, χ mod r ∈ IBr_{r}(G_{n}) for any prime r, r ≠ 2, p. On the other hand, χ mod 2 = 2 · 1_{G_{n}} + β for some β ∈ IBr_{2}(G_{n}). If p ≡ 1 mod 4 , then the fact that α is of symplectic type has been established in [GoW]. From this it follows by [Tiep 2], Lemma 2.4 that β is of symplectic type.

2) Consider the reduction χ mod p. Recall that χ|_{S_{n}} = ψ+\overline{ψ}. It is shown in [Gro] that ψ mod p = \overline{ψ} mod p = η is obtained by restricting the irreducible algebraic representation of Sp_{2n}(\overline{𝔽_{p}}) with highest weight \frac{p−1}{2}ω_{n} to S_{n}. Furthermore, due to Lemma 2.6 [Tiep 3], η is invariant under the action of the distinguished element ϑ_{n}. Therefore, G_{n} has just two irreducible Brauer characters η_{1}, η_{2} with η_{k}|_{S_{n}} = η and η_{1} +η_{2} = 0 on G_{n} \S_{n}. In this case, χ mod p = η_{1} +η_{2}, since χ = 0 on G_{n} \S_{n}.

3) We can embed S_{n} into T = Sp_{2n}(p^{2}) in the following way. In a natural 2n-dimensional 𝔽_{p^{2}}-module W̃ of T consider a symplectic basis

(e_{1}, …, e_{n}, f_{1}, …, f_{n}).

In this basis we can set W = ⟨e_{1}, …, e_{n}, f_{1}, …, f_{n}⟩_{𝔽_{p}}, J = diag(ς^{−1}E_{n}, ςE_{n}). Here ς ∈ 𝔽_{p^{2}} is chosen with order 2(p − 1) such that θ = ς^{2}. Now we set S_{n} = T ∩End(W) ≃ Sp_{2n}(p), H = ⟨S_{n}, J⟩. Then J^{2} ∈ S_{n} and J normalizes S_{n}; therefore H ≃ S_{n} · C_{2}. Furthermore, H/Z(S_{n}) ≃ G_{n}^{+}.

Assume p ≡ 1 mod 4. Factoring the embedding S ↪ Sp_{2n}(\overline{𝔽_{p}}) through T = Sp_{2n}(p^{2}), one sees that η is extended to two absolutely irreducible Brauer characters μ_{1}, μ_{2} of H. We calculate the value of μ_{1}, μ_{2} at the element J. If one denotes σ = exp(\frac{πi}{p−1}), then μ_{1}(J) = ∑_{u∈I_{n}^{+}} σ^{|u|}, since n(p − 1)/2 is even in our case. Here

I_{n}^{+} = {u = (u_{1}, …, u_{n}) | u_{j} ∈ ℤ, |u_{j}| ≤ (p − 1)/2, ∑_{j=1}^{n} u_{j} ≡ 0 mod 2} ,

I_{n}^{−} = {u = (u_{1}, …, u_{n}) | u_{j} ∈ ℤ, |u_{j}| ≤ (p − 1)/2, ∑_{j=1}^{n} u_{j} ≡ 1 mod 2} ,

S_{n}^{+} = ∑_{u∈I_{n}^{+}} σ^{|u|}, S_{n}^{−} = ∑_{u∈I_{n}^{−}} σ^{|u|},

and |u| = ∑_{j} u_{j} for u ∈ I_{n}^{±}. Denote also τ = cot(\frac{π}{2(p−1)}). Then we have

S_{1}^{+} = \frac{τ − τ^{−1}}{2} , S_{1}^{−} = \frac{τ + τ^{−1}}{2} ,

S_{n+1}^{+} = S_{n}^{+}S_{1}^{+} + S_{n}^{−}S_{1}^{−}, S_{n+1}^{−} = S_{n}^{+}S_{1}^{−} + S_{n}^{−}S_{1}^{+}.

From this it follows that

S_{n}^{+} = \frac{τ^{n} + (−τ)^{−n}}{2} , S_{n}^{−} = \frac{τ^{n} − (−τ)^{−n}}{2} .

In particular, μ_{1}(J) = (τ^{n} + (−τ)^{−n})/2. Since p ≥ 5, τ > 1, and so μ_{1}(J) is a positive real number. Moreover, the Frobenius endomorphism σ ↦ σ^{p} sends σ to −σ, i = σ^{(p−1)/2} to i, μ_{1}(J) to μ_{1}(J) = −μ_{2}(J). We have shown that μ_{k}^{(p)} = μ_{k} = \overline{μ_{k}} for k = 1, 2.

If p ≡ 3 mod 4, then n is even. Under the above notation, the computation in the proof of [SchT], Lemma 5.1 shows that μ_{1}(J) = (τ^{n} + τ^{−n})/2. In particular, μ_{1}(J) is again a positive real number. Moreover, the Frobenius endomorphism σ ↦ σ^{p} sends σ to −σ, i = σ^{(p−1)/2} to −i, μ_{1}(J) to μ_{1}(J) = −μ_{2}(J). Therefore, μ_{k}^{(p)} = μ_{k} = \overline{μ_{k}} for k = 1, 2.

4) Here we consider the case χ = χ^{+}. Since Ker μ_{k} = Z(S_{n}) and H/Z(S_{n}) ≃ G_{n}^{+}, we have η_{k} = μ_{k}, k = 1, 2. Thus η_{k} can be written over 𝔽_{p}. Furthermore, since η_{k} is real-valued and η_{k}|_{S_{n}} = η is of quadratic type, η_{k} itself is of quadratic type.

5) Next let χ = χ^{−}. Then p ≡ 1 mod 4. Consider a representation Φ : H → GL_{(p^{n}+1)/2}(\overline{𝔽_{p}}) with Brauer character μ_{1}. Put ω = ς^{(p−1)/2}, and set G̃ = {±Φ(g), ±ωΦ(h) | g ∈ S_{n}, h ∈ H \S_{n}}. Since ω^{2} = −1, G̃ is a group. We claim that G̃ ≃ Ḡ = G_{n}/K = G_{n}^{−}, where K = Ker χ ≃ Z^{2}. For the proof, we first observe that G̃ is generated by the subgroup G′ = {±Φ(g) | g ∈ S_{n}} and the element ωΦ(J). Observe that the representation Φ is not faithful: its kernel is equal to the center Z(S_{n}) of S_{n}. But the factor-group H/Z(S_{n}) ≃ P Sp_{2n}(p) · C_{2} has trivial center; therefore Φ(H) also has trivial center. In particular, G′ ≃ S_{n}/Z(S_{n}) × C_{2}. The subgroup C_{2} here is generated by j, the multiplication by −1 (on the representation space of Φ); hence we can identify j with the central element θE_{2n} in Ḡ. Now one has:

ϑ_{n}^{2} = diag(E_{n}, θ^{2}E_{n}) = θE_{2n} · diag(θ^{−1}E_{n}, θE_{n}) = jJ^{2},
(ωΦ(J))^{2} = −Φ(J^{2}) = jΦ(J^{2}).

Modulo Z(S_{n}) = Ker(Φ|_{S_{n}}) one can identify the two elements ϑ_{n}^{2} and (ωΦ(J))^{2}. Furthermore, the actions of ωΦ(J) and of ϑ_{n} on S_{n} via conjugation are obviously the same. This means that G̃ ≃ Ḡ.

The isomorphism Ḡ ≃ G̃ gives us a representation Ψ : G_{n} → GL_{(p^{n}+1)/2}(\overline{𝔽_{p}}) with kernel K. One may suppose that this representation affords the Brauer character η_{1}. Then η_{1}(ϑ_{n}) = \sqrt{−1}μ_{1}(J). The computations in item 3) show that η_{1}(ϑ_{n}) is purely imaginary, and that the Frobenius endomorphism ^{(p)} leaves η_{1}(ϑ_{n}) fixed. Consequently, for k = 1, 2 the Brauer character η_{k} can be realized over 𝔽_{p} but it is not real.

Proposition 2.3. Let p be any odd prime and n any integer. Let χ be as in Lemma 2.1. Then χ is afforded by a ℚG_{n}-module (of dimension p^{n} + 1).

Proof. 1) First we give an argument settling the case where n is odd and χ = χ^{−}. Let q = p^{n}. Then we can identify W with 𝔽_{q}^{2}, and endow W with the symplectic form ⟨u, v⟩ = tr(ad − bc), where u = (a, b), v = (c, d), a, b, c, d ∈ 𝔽_{q}, and tr stands for the trace form tr : 𝔽_{q} → 𝔽_{p}. Make the group

R = {φ ∈ H = GL_{2}(q) | det φ ∈ 𝔽_{p}^{•}} ≃ SL_{2}(q) · C_{p−1}

act on W in a natural way. Clearly, this action embeds R in G_{n} = CSp(W). Let T denote the central subgroup {diag(λ, λ) | λ ∈ 𝔽_{q}^{•2}} ≃ C_{(q−1)/2} of H. Then the assumption that n is odd implies that T ∩R = K, where K = Ker χ = C_{(p−1)/2}, and RT = H. Hence, χ|_{R} can be viewed as a faithful character of R/K = R/(T ∩ R) ≃ H/T and so as a character, say ρ, of H (with kernel T). Beside that, the restriction of χ to the subgroup R′ = SL_{2}(q) is the sum of two irreducible Weil characters of degree (q + 1)/2 of R′. Inspecting the character table of H (cf. [DiM]) we see that ρ = Ind_{B}^{H}(μ), where

B = \{( \begin{matrix}a & b \\ 0 & d \end{matrix} ) | a, d ∈ 𝔽_{q}^{•}, c ∈ 𝔽_{q}\}

is a Borel subgroup of H, and the linear character μ sends ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) to δ(a), δ the

quadratic character of 𝔽_{q}. In particular, ρ is rational and absolutely irreducible. The same is true for χ|_{R}. Now a standard lemma (see for instance [KoT], Lemma 8.3.1) says that χ is also rational.

2) Now we give another argument handling the case p^{n} ≡ 1 mod 4. The equality ℚ(χ) = ℚ implies by the Brauer-Speiser theorem that the Schur index m_{ℚ}(χ) is either 1 or 2. If m_{ℚ}(χ) = 1, we are done. Assuming m_{ℚ}(χ) = 2, we get an irreducible ℚG_{n}-module V with character 2χ. Clearly, the commuting algebra 𝕂 = End_{G_{n}}(V) is a quaternion algebra over ℚ. If a prime r is ramified in 𝕂, then there exists a Brauer character μ such that χ mod r = μ + μ^{(r)}. By Proposition 2.2, this cannot occur for any prime r. Thus 𝕂 is unramified at any prime r, and by Hasse’s principle we get a contradiction.

Having established Proposition 2.3, we are given a ℚG-module V = V_{n} with character χ such that χ|_{S_{n}} = ψ + \overline{ψ}, where (G, χ) = (G_{n}^{+}, χ^{+}) or (G_{n}^{−}, χ^{−}). We shall maintain this notation in what follows.

Proof of Theorem 1.1. Proposition 2.2 shows that the pair (G, χ) = (G_{n}^{−}, χ^{−}) satisfies the conditions (i), (ii) of Proposition 2.4 from [SchT]. Below (cf. (7) and Corollary 4.3) we shall see that condition (iii) is also fulfilled: det V = ℚ^{•2}. Applying that proposition, we obtain a G_{n}^{−}-invariant unimodular lattice Γ. Standard arguments show that Γ is odd, and min Γ ≥ 3 if n > 1.

The link between lattices invariant under isoclinic groups is indicated in the following statement. We make use of the following observation: for any prime p ≡ 1 mod 4, there exist a, b ∈ ℕ such that a^{2} − pb^{2} = −1 (cf. [Coh], pp. 105, 106). Henceforth we fix such a pair a, b.

Proposition 2.4. Let p ≡ 1 mod 4 be a prime. Let G^{+} ≃ H · C_{2} and G^{−} = (H · C_{2})^{−} be isoclinic groups, where H is a finite group with center of order 2. Suppose V is an absolutely irreducible ℚG^{+}-module with character χ^{+}, and ℚ(ψ) = ℚ(\sqrt{p}) where χ^{+}|_{H} = ψ + \overline{ψ}. Then the following assertions hold.

(i) V can be viewed as an absolutely irreducible ℚG^{−}-module. In particular, for each sign ε ∈ {+, −}, V has a unique (up to scalar) G^{ε}-invariant scalar product (·, ·)^{ε}.

(ii) V has an endomorphism σ with the following properties: σ centralizes the group H, σ^{2} = p · id_{V}, σ is a self-adjoint similarity of norm p w.r.t. both scalar products (·, ·)^{±}, and

gσg^{−1} = −σ for g ∈ G^{±} ∖ H . (1)

(iii) V contains σ-stable lattices which are invariant under both groups G^{+} and G^{−}.

(iv) Let Λ be a lattice as in (iii), and denote by Λ^{ε}, ε ∈ {+, −}, its dual lattice with respect to (·, ·)^{ε}. After suitably rescaling one of (·, ·)^{ε}, we have the equality

Λ^{−} = σ^{−1}(Λ^{+}) .

On this scale, Λ is unimodular w.r.t. (·, ·)^{+} if and only if it is p-modular w.r.t. (·, ·)^{−}. Similarly, Λ is p-modular w.r.t. (·, ·)^{+} if and only if it is unimodular w.r.t. \frac{1}{p}(·, ·)^{−}.

Proof. The equality χ|_{H} = ψ + \overline{ψ} implies that the commuting algebra

𝕂 = End_{H}(V) = {φ ∈ End_{ℚ}(V) | ∀h ∈ H, φ · h = h · φ}

is isomorphic to the field ℚ(ψ) = ℚ(\sqrt{p}). Denoting by σ the (unique up to sign) element σ ∈ 𝕂 with σ^{2} = p·id_{V}, we have to show that it satisfies all other properties stated. We begin with (1). Fix an element g ∈ G^{+} ∖ H. For h ∈ H and λ ∈ 𝕂 it is readily checked that h · gλg^{−1} = gλg^{−1} · h; in other words, gλg^{−1} ∈ 𝕂. Thus, conjugation by g induces an automorphism of 𝕂. If λ is fixed by this automorphism, then λ centralizes H and g, and thus G^{+}, that is, λ ∈ ℚ. Thus, λ ↦ gλg^{−1} is the unique nontrivial field automorphism of 𝕂. Property (1) is a special case of this (σ corresponds to \sqrt{p}).

If Γ is any G^{+}-invariant lattice, then Γ + σ(Γ) is clearly σ-invariant, and by (1) still G^{+}-invariant, which proves (iii) in the “+”-case. The scalar product (σ(x), σ(y)), x, y ∈ V, is also G^{+}-invariant and therefore of the form (σ(x), σ(y)) = c(x, y) for some c ∈ ℝ. From σ^{2} = p · id_{V} it follows that c = p. The self-adjointness

(σ(x), y)^{+} = (x, σ(y))^{+}

now is a formal consequence, and part (i) is proved for (·, ·)^{+}.

Recall that we have fixed positive integers a, b with a^{2} − pb^{2} = −1. Denote by ρ ↦ ρ̄ the non-trivial automorphism of 𝕂, that is, gρ = ρ̄g for any g ∈ G^{+} ∖ H (see above), and by μ the particular element

μ = a + bσ ∈ 𝕂 .

Since μ∘\overline{μ} = \overline{μ}∘μ = − id_{V}, \overline{μ} = a−bσ, this element induces in fact an automorphism of any G^{+}-stable lattice. If g ∈ G^{+} ∖ H and g′ := gμ, then g^{′2} = gμgμ = g^{2}\overline{μ}μ = −g^{2}. Thus ⟨H, g′⟩ ≅ G^{−}. Observe that the particular representation of G^{−} thus constructed is obviously absolutely irreducible, and an H-invariant σ-stable lattice is G^{+}-invariant if and only if it is G^{−}-invariant.

For a given choice of (·, ·), consider the bilinear form

(x, y)^{−} := (x, σμ(y))^{+}

which is clearly H-invariant. Since σμ is self-adjoint w.r.t. (·, ·)^{+}, this bilinear form is symmetric. From the fact that σμ is a totally positive element in 𝕂 it easily follows that (·, ·)^{−} is also positive definite. The following computation shows that (·, ·)^{−} is invariant under g′ = gμ and thus under all of G^{−}:

(g′x, g′y)^{−} = (gμ(x), (σμ)gμ(y))^{+}
= (gμ(x), g\overline{σμ}μ(y))^{+}
= (μ(x), \overline{σμ}μ(y))^{+}
= (μ(x), σ(y))^{+}
= (x, μσ(y)) = (x, y)^{−}

(since \overline{μ}μ = − id_{V}).

For the dual lattices, we clearly have

Λ^{−} = (σμ)^{−1}Λ^{+} = σ^{−1}μ^{−1}Λ^{+} = σ^{−1}Λ^{+} .

If Λ is unimodular w.r.t. (·, ·)^{+}, then Λ^{+} = Λ; therefore Λ^{−} = σ^{−1}(Λ), i.e. σ is the desired similarity between Λ and Λ^{−}. Conversely, if Λ is p-modular w.r.t. (·, ·)^{−}, then

p^{rank Λ/2} = (Λ^{−} : Λ) = (σ^{−1}(Λ^{+}) : Λ)
= (Λ^{+} : σ(Λ)) = (Λ^{+} : Λ)(Λ : σ(Λ)) = p^{rank Λ/2}(Λ^{+} : Λ),

yielding Λ^{+} = Λ. The last assertion follows from the previous one by considering (x, y)^{−−} := (x, σ(−\overline{μ})(y))^{−}.

Clearly, Proposition 2.4 applies to {G^{+}, G^{−}} = {C_{2} × G_{n}^{+}, G_{n}^{−}} and the module V = V_{n}, if p ≡ 1 mod 4. Also, the endomorphism σ is uniquely determined up to sign. Therefore, in what follows we can speak about σ-stable lattices in V.

We shall need the following supplement to Proposition 2.4:

Lemma 2.5. Keep all the notation of Proposition 2.4. Suppose that, as 𝔽_{p}Hmodule, U := Λ/pΛ is the direct sum of two copies M, M′ of an absolutely irreducible 𝔽_{p}H-module. Then the 𝔽_{p}G^{+}-module U is indecomposable if and only if the 𝔽_{p}G^{−}-module U is indecomposable.

Proof. By our assumption, in a suitably chosen basis of U = M ⊕M′ the commuting

algebra K := End_{H}(U) consists of matrices of the form ( \begin{matrix}aE_{n} & bE_{n} \\ cE_{n} & dE_{n} \end{matrix} ), where

a, b, c, d ∈ 𝔽_{p} and n = dim M. Clearly, the endomorphism σ (cf. Proposition 2.4 (ii)) belongs to K, and without loss of generality one may suppose that σ =

( \begin{matrix}0 & E_{n} \\ 0 & 0 \end{matrix} ), because σ^{2} = 0 on U. Recall that G^{+} = ⟨H, g⟩ and G^{−} = ⟨H, g′⟩ with

g′ = g(a + bσ). Since gσ = −σg, g has the matrix ( \begin{matrix}A & B \\ 0 & −A \end{matrix} ), and g′ has the

matrix ( \begin{matrix}aA & bA + aB \\ 0 & −aA \end{matrix} ). Denoting L^{+} = End_{G^{+}}(U), we see that L^{+} = C_{K}(g).

In particular, if f = ( \begin{matrix}xE_{n} & yE_{n} \\ zE_{n} & tE_{n} \end{matrix} ) belongs to L^{+}, then either f = xE_{2n}, or x ≠ t

and B = \frac{2y}{x−t}A. Now observe that: the 𝔽_{p}G^{+}-module U is decomposable if and only if L^{+} contains two nonzero idempotents f, g such that fg = gf = 0 if and only if B ∈ ⟨A⟩_{𝔽_{p}}. The same applies to the 𝔽_{p}G^{−}-module U. But

B ∈ ⟨A⟩_{𝔽_{p}} ⟺ bA + aB ∈ ⟨aA⟩_{𝔽_{p}};

hence our statement follows.

Corollary 2.6. Let p ≡ 1 mod 4 and n any integer. Then V_{n} contains σ-stable G_{n}^{−}-invariant odd unimodular lattices.

Proof. By Proposition 2.4, V_{n} contains σ-stable G_{n}^{−}-invariant lattices. Choose such a lattice Γ with minimal possible determinant, and suppose that det Γ > 1. Clearly, the symmetry of σ implies that the dual lattice Γ^{#} is σ-stable. In particular, taking the sum Γ + mΓ^{#}, m ∈ ℤ, produces again a σ-stable lattice. Hence Lemma 2.1 [SchT] holds inside the class of σ-stable lattices. Now the arguments in the proof of [SchT], Proposition 2.4, show that A = Γ^{#}/Γ = (C_{2})^{2}. Consider the form q : A → \frac{1}{2}ℤ/ℤ, q(v + Γ) = (v, v) + ℤ. Clearly, σ acts on A as an endomorphism of order 1 or 2, and σ preserves q. If σ = 1 on A, then we are done by Lemma 2.2 [SchT]. If σ ≠ 1 on A, then σ has a unique nonzero fixed vector v̄ in A. This vector is obviously fixed by G_{n}^{−}, and one can check that q(v̄) = 0. Thus ⟨Γ, v⟩ is a σ-stable G_{n}^{−}-invariant unimodular lattice, contrary to the choice of Γ.

A σ-stable G_{n}^{−}-invariant odd unimodular lattice will be explicitly constructed in Theorem 3.9 (for odd n) and Corollary 5.9 (for even n and p ≡ 1 mod 4).

Proof of Theorem 1.2 for the case p ≡ 1 mod 4. By Corollary 2.6, V_{n} contains a σstable G_{n}^{−}-invariant unimodular lattice ∇. Applying Proposition 2.4, we obtain a G_{n}^{+}-invariant scalar product on ∇ that converts it into a p-modular lattice which is acted on by G_{n}^{+}.

Remark 2.7. Let n = 1 and p ≡ 1 mod 4. Then Proposition 2.3 and its proof tell us that the group G_{1}^{−} = GL_{2}(p)/C_{(p−1)/2} has a (unique) faithful, absolutely irreducible, rational representation of degree p + 1, which is monomial. On the other hand, G_{1}^{+} = P GL_{2}(p) has a (unique) faithful, absolutely irreducible, rational representation of degree p + 1, which is non-monomial. These observations were mentioned in [NPl], Lemma (V.3) and its proof. Theorem (V.2) of [NPl] exposes a G_{1}^{+}-invariant lattice of rank p+1 and determinant p^{(p+1)/2}, called M_{p+1,2}. Observe that M_{p+1,2} is obtained from the G_{1}^{−}-invariant lattice with Gram matrix E_{p+1} by means of the procedure indicated in Proposition 2.4. Hence by this proposition M_{p+1,2} is p-modular.

An explicit construction of G_{n}^{−}-invariant odd unimodular lattices is exposed in Theorem 3.9 for any odd n. Combined with the procedure indicated in Proposition 2.4, this yields an explicit construction of G_{n}^{+}-invariant p-modular lattices Δ^{−}(p, n) for any odd n (cf. Corollary 5.10), which generalizes Theorem (V.2) of [NPl].

The rest of the section is devoted to proving Theorem 1.2 for the case p ≡ 3 mod 4. Denote θ = \sqrt{−p}, π = (1 + θ)/2, 𝕂 = ℚ(θ), 𝔬 = ⟨1, π⟩_{ℤ} the maximal order in 𝕂. By Proposition 5.2 [SchT], whose proof does not use the oddness of n, V = V_{n} has an endomorphism σ such that

\begin{matrix}σ^{2}(v) = −pv, (σ(u), v) = −(u, σ(v)), (σ(u), σ(v)) = p(u, v), \\ sσs^{−1} = σ, gσg^{−1} = −σ \end{matrix} (2)

for any u, v ∈ V, s ∈ S̄_{n} = P Sp_{2n}(p), g ∈ G_{n}^{+} \ S̄_{n}. Following the proof of that proposition, it is not difficult to show that V contains a G_{n}^{+}-invariant lattice Λ, which is stable under the endomorphism (1 + σ)/2. For u, v ∈ V we set

u ∘ v = \frac{(u, v)}{2} + θ \frac{(u, σ(v))}{2p} ∈ 𝕂. (3)

Using (2) it is straightforward to check that v ∘ u = \overline{u ∘ v}, u ∘ σ(v) = −θ(u ∘ v), su ∘ sv = u ∘ v, gu ∘ gv = v ∘ u, and

(u, v) = Tr(u ∘ v), (4)

where s ∈ S̄_{n}, g ∈ G_{n}^{+} \ S̄_{n}, and Tr denotes the trace of 𝕂 over ℚ. Thus, if we set θ ·v = σ(v), then V is a 𝕂-space of dimension (p^{n} +1)/2, with S̄_{n}-invariant positive definite Hermitian scalar product u ∘ v. Multiplying (·, ·) by a suitable scalar, for instance by 2p, we can ensure that Λ ∘ Λ ⊆ 𝔬. Thus Λ is a G_{n}^{+}-invariant integral 𝔬-lattice, that is contained in its Hermitian dual,

Γ^{⊥} = {u ∈ V | u ∘ Γ ⊆ 𝔬}.

The property (4) by the way characterizes the Hermitian form uniquely, that is, implies (3). Clearly, both Λ^{⊥} and Λ^{#}, the Euclidean dual, are stable under 𝔬 and G_{n}^{+}. Using (3), one readily checks that

Λ^{⊥} = θΛ^{#}.

(This is actually well known from (4), since (θ) is the different of 𝔬 over ℤ.) We shall use this formula frequently in what follows. We shall also need the following two simple statements.

Lemma 2.8. Let p ≡ 3 mod 4 as above, and let G be a finite group. Suppose that Γ is a G-invariant integral Hermitian 𝔬-lattice such that Γ ⊇ 2Γ^{⊥}. Then the 𝔽_{2}G-module Γ^{⊥}/Γ supports a non-degenerate G-invariant alternating form, namely b(u, v) = Tr(2u∘v) mod 2 for any u, v ∈ Γ^{⊥}. In particular, the index (Γ^{⊥} : Γ) differs from 2 for any integral hermitian 𝔬-lattice Γ.

Proof. Since Γ ⊇ 2Γ^{⊥}, 2u ∈ Γ and so 2u∘v ∈ 𝔬 and Tr(2u∘v) ∈ ℤ for all u, v ∈ Γ^{⊥}. If v ∈ Γ, then 2u ∘ v ∈ 2𝔬 and Tr(2u ∘ v) ∈ 2ℤ. Thus b is well defined. Clearly, it is 𝔽_{2}-bilinear and G-invariant. If u ∈ Γ^{⊥}, then 2u∘u ∈ ℝ∩𝔬 = ℤ, yielding b(u, u) = 0, i.e. b is alternating. Finally, assume that v ∈ Γ^{⊥} such that Tr(u ∘ v) ∈ ℤ for any u ∈ Γ^{⊥}. Then v ∈ (Γ^{⊥})^{#} = θ^{−1}Γ^{⊥⊥} = θ^{−1}Γ and θ^{−1}Γ ∩ (1/2)Γ = Γ. In other words, b is non-degenerate.

Lemma 2.9. Let Γ be an S̄_{n}-invariant 𝔬-lattice in V. Suppose that the index (Γ^{⊥} : Γ) is divisible by p. Then in fact Γ ⊆ pΓ^{⊥}.

Proof. Consider the 𝔽_{p}S̄_{n}-module U = Γ/θΓ. Here we are identifying 𝔬/θ𝔬 with 𝔽_{p} (and π with 1/2). First we show that Γ ⊆ θΓ^{⊥}. We know that U is a simple 𝔽_{p}S_{n}-module with character ψ mod p. Furthermore, U′ = (Γ ∩ θΓ^{⊥})/θΓ is an S_{n}- submodule of U. Suppose that U′ = 0. Then Γ∩θΓ^{⊥} = θΓ. As (Γ^{⊥} : Γ) is divisible by p, one can find a vector v ∈ θΓ^{⊥} \ θΓ such that pv ∈ θΓ. Then

θv ∈ pΓ^{⊥} ∩ Γ ⊆ θΓ^{⊥} ∩ Γ = θΓ,

i.e. v ∈ Γ. Hence, v ∈ Γ ∩ θΓ^{⊥} = θΓ, contradicting the choice of v. Therefore, U′ ≠ 0, which implies that U′ = U, Γ ⊆ θΓ^{⊥}.

Now we can define on U an S_{n}-invariant form:

f(x̄, ȳ) = \frac{1}{θ}x ∘ y mod θ𝔬,

where x̄ = x + θΓ, ȳ = y + θΓ. Clearly, f is well defined and bilinear. But f is alternating: f(x̄, x̄) = 0 because x ∘ x ∈ ℝ ∩ θ𝔬 = θ^{2}ℤ for any x ∈ Γ. Suppose the kernel of f is zero. Then U carries a non-degenerate alternating bilinear form (namely f) and so ψ mod p is of symplectic type, contrary to Proposition 2.2 (iii). We have shown that the kernel of f is nonzero. Since U is irreducible, f is zero, i.e. Γ ⊆ pΓ^{⊥}.

Now we choose a G_{n}^{+}-invariant 𝔬-lattice Λ lying in V such that det Λ = (Λ^{#} : Λ) is minimal. We also suppose that Λ is not integral for any rescaled Hermitian form \frac{1}{λ}u ∘ v with λ ∈ ℝ and λ > 1. (If such a λ exists, we simply divide the Hermitian scalar product to λ and get an invariant Hermitian lattice with strictly smaller determinant.)

First we observe that det Λ cannot be divisible by any odd prime r ≠ p. Suppose the contrary. Then consider the form (x̄, ȳ)_{r} = (x, y) mod r on Λ/rΛ, where x̄ = x+rΛ, ȳ = y +rΛ. As r divides det Λ, this G_{n}^{+}-invariant symmetric bilinear form is degenerate, and so its kernel (Λ ∩ rΛ^{#})/rΛ is nonzero. By Proposition 2.2 (i) this means simply that Λ ⊆ rΛ^{#}. Then for any u, v ∈ Λ one has (u, v), (u, πv) ∈ rℤ. Denote u ∘ v = a + πb for some a, b ∈ ℤ. Then (u, v) = 2a + b and (u, πv) = a − b(p − 1)/2 belong to rℤ. This implies a, b ∈ rℤ. In particular, u ∘ v ∈ r𝔬 for any u, v ∈ Λ. Thus, one can divide the form u ∘ v by r, a contradiction.

We have seen that det Λ can be divisible only by the primes 2 and p. Furthermore, if p divides (Λ^{⊥} : Λ), then by Lemma 2.9 one can divide the form u ∘ v by p, a contradiction. Therefore, there exists a non-negative integer k such that (Λ^{⊥} : Λ) = 2^{k}. In this case,

det Λ = (Λ^{#} : Λ^{⊥})(Λ^{⊥} : Λ) = 2^{k}p^{N},

where N = (p^{n} + 1)/2.

It is obvious that Λ ⊇ 2^{k}Λ^{⊥}. Suppose that Λ ⊉ 2Λ^{⊥}. Let l denote the minimal integer such that Λ ⊇ 2^{l}Λ^{⊥}. Then l ≥ 2. Set Γ = Λ + 2^{l−1}Λ^{⊥}. One readily checks that Γ is a G_{n}^{+}-invariant 𝔬-lattice with det Γ strictly smaller than det Λ, contradicting the choice of Λ. Hence, Λ ⊇ 2Λ^{⊥}. This implies that Λ ⊇ 2pΛ^{#}, i.e. the discriminant group Λ^{#}/Λ has exponent 2p (and order 2^{k}p^{N}).

The inclusion Λ ⊇ 2pΛ^{#} also implies k ≤ 2N. Setting Γ = \sqrt{2}Λ^{⊥} (which is equivalent to considering Λ^{⊥} and multiplying the form u ∘ v by 2), we have

Γ ∘ Γ = \sqrt{2}Λ^{⊥} ∘ \sqrt{2}Λ^{⊥} = 2Λ^{⊥} ∘ Λ^{⊥} ⊆ Λ ∘ Λ^{⊥} ⊆ 𝔬,

i.e. Γ is an G_{n}^{+}-invariant integral 𝔬-lattice. Furthermore,

2Γ^{⊥} = \sqrt{2}Λ ⊆ \sqrt{2}Λ^{⊥} = Γ,

and

(Γ^{⊥} : Γ) = (\frac{1}{\sqrt{2}}Λ : \sqrt{2}Λ^{⊥}) = (Λ : 2Λ^{⊥}) = \frac{(Λ^{⊥} : 2Λ^{⊥})}{(Λ^{⊥} : Λ)} = 2^{2N−k}.

This computation shows that, after replacing Λ by Γ if necessary, one may suppose that 0 ≤ k ≤ N. Claim that the last condition implies k = 0, 1, 2. For F = (Λ ∩ 2Λ^{#})/2Λ is a G_{n}^{+}-submodule of Λ/2Λ, and |F | = 2^{k}. By Proposition 2.2 (ii), k = 0, 1, or 2.

We have arrived at the situation where (Λ^{⊥} : Λ) = 2^{k}, k = 0, 1, 2. By Lemma 2.8, k ≠ 1. If k = 0, then Λ = Λ^{⊥} = θΛ^{#}, and Λ is a p-modular Euclidean lattice of rank 2N, and we are done. Suppose k = 2. Then the discriminant group Λ^{#}/Λ is isomorphic to (C_{2})^{2} ⊕ (C_{p})^{N}, and (Λ^{⊥} : Λ) = (C_{2})^{2}. By Lemma 2.2 of [SchT], there exists a vector v ∈ Λ^{⊥}\Λ with 2v ∈ Λ such that Δ = ⟨Λ, v⟩_{ℤ} is a G_{n}^{+}- invariant Euclidean lattice and Δ^{#}/Δ ≃ (C_{p})^{N}. Remark that θΔ ⊆ Δ. Indeed, θv = π · 2v − v ∈ Δ. Furthermore, θΔ ⊆ pΔ^{#}. (For

(θΔ, Δ) ⊆ (θΛ, Λ) + (θΛ, v) + (Λ, θv) + (θv, v).

Here, θΛ ⊆ θΛ^{⊥} = pΛ^{#}, so (θΛ, Λ) ⊆ pℤ. As v ∈ Λ^{⊥}, we have Λ ∘ v ⊆ 𝔬, and so

(θΛ, v) = (Λ, θv) ⊆ θ𝔬 ∩ ℤ = pℤ.

Finally, (θv, v) = θv ∘ v − θv ∘ v = 0.) Now we have

(θΔ : pΔ) = p^{N} = (Δ^{#} : Δ) = (pΔ^{#} : pΔ);

therefore in fact θΔ = pΔ^{#}, Δ = θΔ^{#}. The map f : x ↦ θx, where x ∈ Δ^{#}, maps Δ^{#} onto Δ and preserves (·, ·) up to the scalar p: (f(x), f(y)) = p(x, y). This means the lattice Δ is p-modular, as desired.

The proof of Theorem 1.2 is finished.

Remark 2.10. Observe that all the lattices Δ = Δ(p, n), Δ^{−}(p, n), are symplectic (for the definition of symplectic lattices cf. [BuS]). For suppose first that Δ is

invariant under G_{n}^{−}. Then det Δ = 1. Taking τ = ( \begin{matrix}0 & E_{n} \\ θE_{n} & 0 \end{matrix} ) (in the chosen

symplectic basis of W), 𝔽_{p}^{•} = ⟨θ⟩, one sees that τ ∈ G_{n}^{−} and τ^{2} = θE_{2n} acts on Δ as −1. According to [BuS], this means that Δ is symplectic. Suppose now that Δ is invariant under G_{n}^{+}. If p ≡ 3 mod 4, then we put τ = \frac{1}{\sqrt{p}}σ. Considered under the new scalar product (u, v)′ = (u, v)/\sqrt{p}, the dual lattice of Δ equals τ(Δ). Also, τ preserves (·, ·)′ and τ^{2} = −1. Hence Δ is symplectic. Finally, let p ≡ 1 mod 4.

Taking g = ( \begin{matrix}0 & E_{n} \\ θE_{n} & 0 \end{matrix} ), one sees that g ∈ G_{n}^{+} \ S_{n}. Put τ = \frac{1}{\sqrt{p}}gσ and consider

Δ w.r.t. the new scalar product (·, ·)′ introduced above. Clearly, τ preserves (·, ·)′ and sends Δ to its dual (w.r.t. the new scalar product). Besides, g ∈ G_{n}^{+} \ S_{n} implies that τ^{2} = −\frac{1}{p}g^{2}σ^{2} = −1. Consequently, Δ is symplectic.

Remark 2.11. One could formalize Lemmas 2.8, 2.9 and the above arguments in order to get an analogue of Proposition 2.4 in [SchT] for the existence of p-modular lattices. Here is one more well known example. Let G = ^{2}G_{2}(3) = SL_{2}(8) · C_{3} and χ an irreducible complex character of G of degree 7 with ℚ(χ) = ℚ(\sqrt{−3}). Then for any prime r ≠ 2, 3, χ mod r is an irreducible Brauer character, which is not of quadratic type. Furthermore, χ mod 3 is irreducible and of quadratic type. Finally, χ mod 2 is a sum of a character of degree 1 and an irreducible Brauer character of degree 6 which is of symplectic type. From this it follows that G has an irreducible ℚ-module V with character χ + \overline{χ}. The above arguments show that G stabilizes a 3-modular lattice Λ in V with Aut(Λ) = C_{2} × G_{2}(3). The lattice Λ occurs in [Atlas] and was investigated in detail in [KoT], Chapter 8. It is the unique extremal 3-modular lattice in dimension 14, after [SchHem].

3. Explicit construction. I: n odd

We maintain the notation W for the natural 𝔽_{p}S_{n}-module 𝔽_{p}^{2n} endowed with a non-degenerate symplectic form ⟨·, ·⟩. Throughout this section we suppose that n is odd. We use the ideas of [SchT], §3 to explicitly construct G_{n}^{−}-invariant lattices in V = V_{n}, for any odd prime p.

Consider an arbitrary G_{n}^{−}-invariant (integral) lattice Λ in V. Fix a symplectic basis (e_{1}, …, e_{n}, f_{1}, …, f_{n}) of W. Recall that G_{n} is generated by S_{n} and the element ϑ_{n} with matrix diag(E_{n}, θE_{n}) in this basis. We shall view V as a G_{n}- module with kernel Z^{2} ≃ C_{(p−1)/2}. A Lagrangian is a maximal totally isotropic subspace in W. Following [BaV], we consider them oriented, i.e. equipped with an appropriate equivalence class of bases. Two bases (l_{1}, …, l_{n}) and (l′_{1}, …, l′_{n}) of a Lagrangian L are equivalent, i.e. define the same orientation, if the element g ∈ GL_{n}(L) defined by g(l_{i}) = l′_{i}, 1 ≤ i ≤ n, has det g ∈ 𝔽_{p}^{•2}. We denote by L(W) the set of all oriented Lagrangians contained in W.

To each Lagrangian L of W we now associate the following two subgroups:

G(L) = {φ ∈ G_{n} | φ(L) = L} , S(L) = {φ ∈ G(L) | det(φ|_{L}) ∈ 𝔽_{p}^{•2}} . (5)

Since the determinant of λE_{2n} (λ ∈ 𝔽_{p}^{•2}) acting on any Lagrangian L is a square in 𝔽_{p}, the definition (5) of S(L) factors through the kernel Z^{2} of χ. Let ξ_{L} denote the linear character of G(L) (and S(L)) which sends g ∈ G(L) to (\frac{det(φ|_{W/L})}{p}). Here

and hereafter, (\frac{.}{p}) stands for the Legendre symbol.

Proposition 3.1. Let n be odd. For any Lagrangian L in W, the sets

Λ(L) = {v ∈ Λ | ∀φ ∈ S(L), φ(v) = v},
Λ^{−}(L) = {v ∈ Λ | ∀φ ∈ S(L), φ(v) = ξ_{L}(g)v}

are 1-dimensional ℤ-modules.

Proof. 1) Without loss of generality, one can take L = ⟨e_{1}, …, e_{n}⟩_{𝔽_{p}} with the basis (e_{1}, …, e_{n}). Denote

P = St_{S}(L) = E · H, Q = S(L), R = P ∩ Q = E · H^{•},

where E = (C_{p})^{n(n+1)/2}, H = GL_{n}(p), H^{•} = {g ∈ GL_{n}(p) | det g ∈ 𝔽_{p}^{•2}}. A model for the Weil representation of S with character ψ is described in [Gro]. From this description it follows that ψ|_{P} = δ +ζ, where ζ is a P -character of degree (p^{n} −1)/2 and

δ(φ) = (\frac{det(φ|_{L})}{p})

for φ ∈ P. In particular, ψ|_{R} = 1_{R} + ζ|_{R}.

If n = 1, one directly checks that the trivial character of S(L) and the character ξ_{L} each enter into χ|_{S(L)} with multiplicity 1.

2) In this paragraph we suppose n > 1. We claim that ζ|_{R} ∈ Irr(R). Indeed, one can identify E with the space of symmetric matrices of degree n over 𝔽_{p}. Furthermore, P/E = H acts on E by the rule:

A ∘ X = A · X · ^{t}A

for A ∈ H viewed as an element of GL_{n}(p) and X ∈ E. Obviously E ⊈ Ker ζ. So it is sufficient to show that every R/E-orbit on the set Irr(E) \ {1_{E}} has length ≥ (p^{n} − 1)/2, or equivalently, every H^{•}-orbit on the set E^{∗} \ {0} has length ≥ (p^{n} − 1)/2. Here E is viewed as a 𝔽_{p}-space, and E^{∗} stands for the dual space. Actually, one can identify the GL_{n}(p)-module E^{∗} with E itself, but endowed with the action A • X = ^{t}A^{−1} · X · A^{−1}, where A ∈ GL_{n}(p), X ∈ E. (Indeed, each element f ∈ E^{∗} can be realized as the map f = f_{M} : X ↦ Tr(X · M) for a uniquely determined M ∈ E. Now we can write down the action of A ∈ GL_{n}(p) on E^{∗}:

\begin{aligned}(A • f)(X) & = f_{M}(A^{−1} ∘ X) = Tr(A^{−1} · X · ^{t}A^{−1} · M) \\ & = Tr(X · ^{t}A^{−1} · M · A^{−1}) = f_{A•M}(X).) \end{aligned}

Consider a GL_{n}(p)-orbit O on E\{0} and X ∈ O. Then the stabilizer H(X) of X in GL_{n}(p) is nothing else but the isometry group of the symmetric bilinear form on 𝔽_{p}^{n} with the matrix X. It is not difficult to show that the cardinality of O is (p^{n} − 1)/2 if rank X = 1, and strictly greater than (p^{n} − 1)/2 if rank X > 1 (and greater than p^{n} − 1 if rank X = n > 1). On the other hand, if rank X ≤ n − 1, then H(X) contains an element A not contained in H^{•}, whence O is also an H^{•}-orbit. Therefore, an H^{•}-orbit in E \ {0} can have length less than (p^{n} − 1)/2 only in case n = 1. (When n = 1, any H^{•}-orbit in E \ {0} has length (p^{n} − 1)/4.) Now n > 1 by our assumption, so our claim has been proved.

Decompose V ⊗_{ℚ}ℂ into a sum U ⊕U_{1}⊕U_{2} of three R-submodules, with character 2 · 1_{R}, ζ and \overline{ζ}, respectively. Remark that R contains a regular unipotent element x and ζ(x) = (−1 ± p^{n−1}\sqrt{ϵp})/2. Furthermore, Q = ⟨R, ϑ_{n}⟩, and ϑ_{n} normalizes R. Therefore ϑ_{n} fixes U, and either leaves both U_{1}, U_{2} invariant or interchanges them. But ϑ_{n} interchanges the S-conjugacy classes of x and x^{...} (some power of x), and ζ(x^{...}) = \overline{ζ(x)} ≠ ζ(x). This means ϑ_{n} interchanges U_{1} and U_{2}. The construction of χ (see the proof of Lemma 2.1) ensures that χ(ϑ_{n}) = 0. As a consequence, ϑ_{n} acting on U has trace 0. Observe that ϑ_{n}^{2} leaves U pointwise fixed. (Indeed, ϑ_{n}^{2} is the product of α = diag(θ^{−1}E_{n}, θE_{n}) and β = θE_{2n}. Clearly, α belongs to P and acts on U as multiplication by δ(α) = (\frac{θ^{−n}}{p} ) = −1, because n is odd. Furthermore, β acts as multiplication by −1 on the whole of V.) We have shown that both of the subspaces

F̃ = {v ∈ V ⊗_{ℚ} ℂ | ∀ φ ∈ Q, φ(v) = v} = U ∩ Ker(ϑ_{n} − 1),
F̃^{−} = {v ∈ V ⊗_{ℚ} ℂ | ∀ φ ∈ Q, φ(v) = ξ_{L}(g)v} = U ∩ Ker(ϑ_{n} + 1)

have dimension 1.

3) For any odd n, Lemma 3 of [CoT] now implies that both of the subspaces

\begin{matrix}F = \{v ∈ V | ∀ φ ∈ Q, φ(v) = v\}, \\ F^{−} = \{v ∈ V | ∀ φ ∈ Q, φ(v) = ξ_{L}(g)v\} \end{matrix}

also have dimension 1 (over ℚ). Since V = Λ ⊗_{ℤ} ℚ, we arrive at the conclusion that Λ(L) and Λ^{−}(L) are 1-dimensional ℤ-modules.

Keeping Proposition 3.1 in mind, we denote by v(L) (resp. u(L)) a generating element of the ℤ-module Λ(L) (resp. Λ^{−}(L)) for Lagrangian L. Then v(L) (resp. u(L)) is determined uniquely up to sign. It is clear that Λ(L) and Λ^{−}(L) are stabilized by G(L). Namely,

φ(v(L)) = (\frac{det(φ|_{L})}{p}) · v(L), φ(u(L)) = (\frac{det(φ|_{W/L})}{p}) · u(L) (6)

for φ ∈ G(L). Since we consider Lagrangians oriented, we can set v(−L) = −v(L), u(−L) = −u(L) for the opposite Lagrangian −L corresponding to a given oriented Lagrangian L. We fix an oriented Lagrangian L_{0} with a basis (e_{1}, …, e_{n}), and fix a generating vector v(L_{0}) of Λ(L_{0}) (resp. u(L_{0}) of Λ^{−}(L_{0})). For an arbitrary oriented Lagrangian M with a basis (f_{1}, …, f_{n}), we find an element ν_{M} ∈ G such that ν_{M}(e_{i}) = f_{i} for all i, and set v(M) = ν_{M}(v(L_{0})), u(M) = ν_{M}(u(L_{0})). This definition is independent of the choice of ν_{M}. Moreover, for any h ∈ G with h(L_{0}) = M, we have

h(v(L_{0})) = (\frac{det((ν_{M}^{−1}h)|_{L_{0}})}{p}) · v(M), h(u(L_{0})) = (\frac{det((ν_{M}^{−1}h)|_{W/L_{0}})}{p}) · u(M).

Lemma 3.2. Let L and M be arbitrary Lagrangians. Then |(v(L), v(M))| (resp. |(u(L), u(M))|, |(u(L), v(M))|) depends only on the dimension of L∩M (and on the choice of the norm (v(L), v(L))). In other words, there exist non-negative constants a_{k}, b_{k}, c_{k}, k = 0, 1, …, n, such that |(v(L), v(M))| = a_{k}, |(u(L), u(M))| = b_{k}, |(u(L), v(M))| = c_{k} whenever dim(L ∩ M) = k.

Proof. Consider Lagrangians L′, M′ with dim(L ∩ M) = dim(L′ ∩ M′). It is clear that there exists an element φ ∈ S mapping L into L′ and M into M′. One readily verifies that φS(L)φ^{−1} = S(L′). Taking g ∈ S(L) and applying (6) we have

gφ^{−1}(v(L′)) = φ^{−1} · φgφ^{−1}(v(L′)) = φ^{−1}(v(L′))

for each g ∈ S(L). By Proposition 3.1 this implies that φ^{−1}(v(L′)) = ±v(L), i.e. φ(v(L)) = ±v(L′). Similarly, φ(v(M)) = ±v(M′). In particular, (v(L′), v(M′)) = ±(v(L), v(M)). Next we have ξ_{L}(g) = ξ_{L′}(φgφ^{−1}), and

gφ^{−1}(u(L′)) = φ^{−1} · φgφ^{−1}(u(L′)) = φ^{−1}(ξ_{L′}(φgφ^{−1})u(L′)) = ξ_{L}(g)φ^{−1}(u(L′)).

By Proposition 3.1 this implies that φ^{−1}(u(L′)) = ±u(L), i.e. φ(u(L)) = ±u(L′). Similarly, φ(u(M)) = ±u(M′). Hence,

(u(L′), u(M′)) = ±(u(L), u(M)), (u(L′), v(M′)) = ±(u(L), v(M)).

Lemma 3.3. If k is even, then a_{k} = 0. If k is odd, then c_{k} = 0.

Proof. Again consider the symplectic basis (e_{1}, …, e_{n}, f_{1}, …, f_{n}). If the intersection of given Lagrangians L, L′ has dimension k, k a non-negative integer, then without loss of generality one can suppose that

L = ⟨e_{1}, …, e_{n}⟩_{𝔽_{p}}, L′ = ⟨e_{1}, …, e_{k}, f_{k+1}, …, f_{n}⟩_{𝔽_{p}}.

Clearly that ϑ_{n} is contained in both of G(L), G(L′). Furthermore, det(ϑ_{n}|_{L}) = 1, det(ϑ_{n}|_{W/L}) = θ^{n}, and det(ϑ_{n}|_{L′}) = θ^{n−k}.

First suppose that k is even. Due to (6) one then has ϑ_{n}(v(L)) = v(L), ϑ_{n}(v(L′)) = −v(L′). Therefore,

(v(L), v(L′)) = (ϑ_{n}(v(L)), ϑ_{n}(v(L′))) = −(v(L), v(L′)),

i.e. (v(L), v(L′)) = 0.

Next suppose that k is odd. Then due to (6) one has ϑ_{n}(u(L)) = −u(L), ϑ_{n}(v(L′)) = v(L′). Now we get

(u(L), v(L′)) = (ϑ_{n}(u(L)), ϑ_{n}(v(L′))) = −(u(L), v(L′)),

i.e. (u(L), v(L′)) = 0.

Our next goal is to determine a_{k} for k odd, and c_{k} for k even. Recall that a symplectic spread of W is a collection π = {W_{i} | 1 ≤ i ≤ p^{n} + 1} consisting of p^{n} + 1 maximal totally isotropic subspaces such that ⋃_{i=1}^{p^{n}+1} W_{i} = W. The socalled standard, or desarguesian, symplectic spread of W can be constructed in the following way. Identify W with 𝔽_{q}^{2}, q = p^{n}, and endow W with the symplectic form ⟨u, v⟩ = tr(αδ − βγ), where u = (α, β), v = (γ, δ), α, β, γ, δ ∈ 𝔽_{q}, and tr stands for the trace form tr : 𝔽_{q} → 𝔽_{p}. Then

π_{D} = {W^{λ} | λ ∈ 𝔽_{q} ∪ {∞}} ,

where W^{∞} = {(0, α) | α ∈ 𝔽_{q}}, W^{λ} = {(α, λα) | α ∈ 𝔽_{q}} for λ ∈ 𝔽_{q} is the desired spread. One may suppose that

W^{0} = ⟨e_{1}, …, e_{n}⟩_{𝔽_{p}}, W^{∞} = ⟨f_{1}, …, f_{n}⟩_{𝔽_{p}}.

For a given symplectic spread π = {W_{i}}, its automorphism group Aut (π) is defined as the group {φ ∈ CSp_{2n}(p) | ∀i ∃j s.t. φ(W_{i}) = W_{j}}. For example (see [KoT], Lemma 1.2.6),

Aut (π_{D}) = SL_{2}(q) · C_{n} · C_{p−1},

the extension of SL_{2}(q) first by the Galois group of the extension 𝔽_{q}/𝔽_{p} and then by the element ϑ_{n}. Set

Λ(π) = ⟨v(L) | L ∈ π⟩_{ℤ}.

Then, by Lemma 3.3, Λ(π) is a sublattice of Λ of determinant (a_{n})^{p^{n}+1}, where a_{n} = (v(L), v(L)) as in Lemma 3.2. In particular,

det V = ℚ^{•2}, (7)

the fact we used in the proof of Theorem 1.1. Also, it shows that V contains no p-modular lattices (if p^{n} ≡ 1 mod 4).

Now we consider the standard symplectic spread π_{D}, and project v(M) and u(M), M a fixed Lagrangian, to the orthogonal basis (v(W^{λ})):

v(M) = ∑_{λ∈𝔽_{q}∪{∞}} z_{λ}v(W^{λ}), u(M) = ∑_{λ∈𝔽_{q}∪{∞}} y_{λ}v(W^{λ}).

It is obvious that z_{λ} = a_{n}^{−1}(v(M), v(W^{λ})), y_{λ} = a_{n}^{−1}(u(M), v(W^{λ})), and so

∑_{λ∈𝔽_{q}∪{∞}}(v(M), v(W^{λ}))^{2} = a_{n}^{2}, ∑_{λ∈𝔽_{q}∪{∞}}(u(M), v(W^{λ}))^{2} = a_{n}b_{n}. (8)

Proposition 3.4. In the notation of Lemma 3.2 one has (i) a_{k} = p^{−(n−k)/2} · a_{n} for odd k, 1 ≤ k ≤ n; (ii) (c_{k})^{2} = p^{k−n} · a_{n}b_{n} for even k, 0 ≤ k < n.

Proof. We shall proceed by induction on n = 1, 3, . . ..

1) Applying (8) to M = W^{∞} we get a_{n}b_{n} = p^{n}(c_{0})^{2}. Next we take M = ⟨e_{1}, f_{2}, …, f_{n}⟩_{𝔽_{p}} and write e_{1} = (e, 0) for e ∈ 𝔽_{q}^{•}. Then M ∩ W^{∞} = ⟨f_{2}, …, f_{n}⟩ has dimension n − 1. Furthermore, for an arbitrary λ ∈ 𝔽_{q} one has

\begin{aligned}M ∩ W^{λ} & = \{(xe, λxe) | x ∈ 𝔽_{p}, ⟨(0, λe), e_{1}⟩ = 0\} \\ & = \{(xe, λxe) | x ∈ 𝔽_{p}, tr(λe^{2}) = 0\} . \end{aligned}

Therefore, dim(M ∩ W^{λ}) is equal to 1 for just p^{n−1} values of λ ∈ 𝔽_{q}, and 0 for the other λ’s. Applying (8), one has p^{n−1}a_{1}^{2} = a_{n}^{2}, i.e. a_{1} = p^{−(n−1)/2}a_{n}. Thus we have proved Proposition 3.4 for a_{1} and c_{0} with n ≥ 1. In particular, the induction base n = 1 has been established.

2) For the induction step we suppose n ≥ 3. We already proved the desired relations for a_{1} and c_{0}. Put

W′ = ⟨e_{1}, …, e_{n−2}, f_{1}, …, f_{n−2}⟩_{𝔽_{p}}, W′′ = ⟨e_{n−1}, e_{n}, f_{n−1}, f_{n}⟩_{𝔽_{p}},

U = ⟨e_{n−1}, e_{n}⟩_{𝔽_{p}}, and introduce the following subgroups in S: B = St_{S}(W′), S′ = {φ ∈ B | φ|_{W′′} = 1_{W′′}}, S′′ = {φ ∈ B | φ|_{W′} = 1_{W′}}, C = S′′ ∩ St_{S}(U), K = S′ × C. Then S′ ≃ Sp(W′) = Sp_{2n−4}(p), S′′ ≃ Sp(W′′) = Sp_{4}(p), C ≃ (C_{p})^{3} · GL_{2}(p), B = S′ × S′′. We also set G′ = ⟨S′, ϑ_{n}⟩, H = ⟨K, ϑ_{n}⟩ = G′ · C. It is well known that ψ|_{B} = ψ′ ⊗ ψ′′ + τ′ ⊗ τ′′, where ψ′ (resp. τ′) is an irreducible Weil character of S′ of degree (p^{n−2} + 1)/2 (resp. (p^{n−2} − 1)/2). Furthermore, ψ′′ (resp. τ′′) is an irreducible Weil character of S′′ of degree (p^{2} + 1)/2 (resp. (p^{2} − 1)/2). Arguing as in the proof of Proposition 3.1, we are convinced that α := τ′′|_{C} ∈ Irr(C), and ψ′′|_{C} = δ + β, where β ∈ Irr(C) and δ(φ) = (\frac{det(φ|_{U})}{p}) for φ ∈ C. (In particular, δ(1) = 1.) Thus

ψ|_{K} = ψ′ ⊗ δ + ψ′ ⊗ β + τ′ ⊗ α

is a sum of three (pairwise distinct) irreducible constituents. From this it follows that

χ|_{K} = (ψ′ + \overline{ψ′}) ⊗ δ + (ψ′ ⊗ β + \overline{ψ′} ⊗ \overline{β}) + (τ′ ⊗ α + \overline{τ′} ⊗ \overline{α}).

Observe that ϑ_{n} acts on S′ as an outer automorphism, and ϑ_{n} interchanges the characters ψ′ and \overline{ψ′}. Furthermore, C ⊲ H. Consequently, χ|_{H} has a unique irreducible constituent in which C acts by scalars. This constituent affords Kcharacter (ψ′ + \overline{ψ′}) ⊗ δ. Also,

(χ|_{C}, δ)_{C} = p^{n−2} + 1. (9)

3) Next we consider the following ℤ-submodule:

Λ′ = ⟨v(L) | L = L′ ⊕ U, L′ Lagrangian in W′⟩_{ℤ}

in Λ. (The symplectic form on W′ is inherited from the one on W.) Clearly, H leaves Λ′ fixed. Moreover, let L = L′ ⊕ U, L′ a Lagrangian in W′ and φ ∈ C. Then φ(L) = L. Hence, due to (6) the subgroup C acts on Λ′ as scalars (and the corresponding character is dim_{ℤ} Λ′ · δ). By the result of 2), dim_{ℤ} Λ′ = p^{n−2} + 1. Recall that we chose G′ to be generated by S′ = Sp(W′) and ϑ_{n}. Considering the natural action of G′ on W′, we conclude that G′ ≃ CSp(W′). We want to find the kernel of G′ acting on Λ′. To this end, consider a generating element z = θE_{2n−4} of the center C_{p−1} of CSp(W′). Then z acting on W has the following matrix: diag(θE_{2n−4}, E_{2}, θ^{2}E_{2}) in the basis (e_{1}, …, e_{n−2}, f_{1}, …, f_{n−2}, e_{n−1}, e_{n}, f_{n−1}, f_{n}). If L = L′ ⊕ U (L′ any Lagrangian in W′), then due to (6) z(v(L)) = −v(L), as n is odd. Thus z acts on Λ′ as multiplication by −1. We have shown that the lattice Λ′ is in fact acted on by CSp_{2n−4}(p)/C_{(p−1)/2} = G_{n−2}^{−}, and this action affords S′-character ψ′ + \overline{ψ′}. If we denote G′(L′) = St_{G′}(L′), and define S′(L′) similarly to (5), then of course G′(L′) = G(L)∩G′, S′(L′) = S(L)∩G′ for L = L′ ⊕U. In other words, W′, Λ′, L′, v′(L′) and u′(L′) (generating vectors of Λ′(L′) and Λ^{′−}(L′), cf. Proposition 3.1) play the same roles for G′ as W, Λ, L, v(L) and u(L) do for G.

Observe that there are nonzero rational scalars s and t such that v′(L′) = ±sv(L), u′(L′) = ±tu(L). Indeed, v(L) ∈ Λ′ by the definition of Λ′, and v(L) is obviously fixed by S′(L′); hence v(L) ∈ Λ′(L′), and v′(L′) = ±sv(L) for some s ∈ ℚ^{•}. Next, ⟨u(L)⟩_{ℤ} is a C-module with character δ (cf. (6)). On the other hand, Λ′ affords C-character (p^{n−2} + 1)δ. Hence by (9) we have u(L) ∈ Λ′ ⊗_{ℤ} ℂ. From this it follows that u(L) ∈ Λ^{′−}(L′) ⊗_{ℤ} ℂ, i.e. u′(L′) = ±tu(L) for a certain t ∈ ℂ^{•}. Observe that

st = ±(u′(L′), v′(M′))/(u(L), v(M))

is a rational number, where L′, M′ are Lagrangians inside W′ with dim(L′ ∩ M′) = n − 3. Hence t is rational. We may suppose that s, t > 0.

Now we can apply the induction hypothesis to G′ and Λ′. In doing so we consider two arbitrary Lagrangians L′, M′ of W′ with dim(L′ ∩ M′) = k. Then for L = L′ ⊕ U, M = M′ ⊕ U one has dim(L ∩ M) = k + 2, which implies that a′_{k} = |(v′(L′), v′(M′))| = s^{2}|(v(L), v(M))| = s^{2}a_{k+2}, b′_{k} = |(u′(L′), u′(M′))| = t^{2}|(u(L), u(M))| = t^{2}b_{k+2}, c′_{k} = |(u′(L′), v′(M′))| = st|(u(L), u(M))| = stc_{k+2}. By the induction hypothesis, for k odd we have

s^{2}a_{k+2} = a′_{k} = p^{(n−2−k)/2}a′_{n−2} = s^{2}p^{(n−(k+2))/2}a_{n},

i.e. a_{k+2} = p^{(n−(k+2))/2}a_{n}. Thus we have proved the desired relation for a_{l} with l = 3, 5, …, n. Similarly, if k is even, then

s^{2}t^{2}(c_{k+2})^{2} = (c′_{k})^{2} = p^{k−n+2}a′_{n−2}b′_{n−2} = s^{2}t^{2}p^{k−n+2}a_{n}b_{n},

i.e. (c_{k+2})^{2} = p^{k+2−n}a_{n}b_{n}. Thus we have proved the desired relation for c_{l} with l = 2, 4, …, n − 1. The induction step is over.

Corollary 3.5. Rescale the v(L)’s such that (v(L), v(L)) = p^{(n−1)/2}. Then

(v(L), v(M)) = \{ \begin{matrix}±p^{(k−1)/2}, & dim(L ∩ M) = k ≡ 1 mod 2, \\ 0, & dim(L ∩ M) ≡ 0 mod 2. \end{matrix} □

The signs ± involved in this corollary will be determined in §5, cf. Corollary 5.4. Now we consider the endomorphism σ of V (constructed in Proposition 2.4 for

p ≡ 1 mod 4 and in [SchT], §5 for p ≡ 3 mod 4. Recall that Λ is a G-invariant lattice in V. Let Γ be the sublattice of Λ generated by v(L) with L running over all Lagrangians in W. Clearly, one can rescale the scalar product on V such that ∇ = Γ + σ(Γ) is an integral G-invariant σ-stable lattice lying in V. Also, Γ(L) = Λ(L) for any Lagrangian L. We can now apply Propositions 3.1, 3.4 and Lemmas 3.2, 3.3 to the lattice ∇. Let ṽ(L), ũ(L) be generating vectors of ∇(L), ∇^{−}(L).

Lemma 3.6. For the lattice ∇ = Γ + σ(Γ) we have ṽ(L) = ±v(L) and ũ(L) = ±σ(v(L)). In particular, the parameters a_{k}, b_{k}, c_{k} of ∇ satisfy the following relations:

(i) b_{k} = pa_{k} for any k;

(ii) c_{k} = p^{(k+1−n)/2} · a_{n} for any even k.

Proof. Since Γ ⊆ ∇, v(L) = mṽ(L) for some integer m. As Γ is generated by the v(L)’s, Γ ⊆ m∇, and so ∇ = Γ + σ(Γ) ⊆ m(∇ + σ(∇)) = m∇, yielding m = ±1, i.e. ṽ(L) = ±v(L).

Recall that gσ = (\frac{det(g|_{W})}{p}) σg for any g ∈ G. If g ∈ G(L), then det(g|_{W}) =

det(g|_{L}) · det(g|_{W/L}), and so (\frac{det(g|_{W})}{p}) = (\frac{det(g|_{L})}{p}) · ξ_{L}(g). Therefore, by (6) one has

gσ(v(L)) = (\frac{det(g|_{W})}{p}) σg(v(L))
= (\frac{det(g|_{W})}{p}) · (\frac{det(g|_{L})}{p}) σ(v(L)) = ξ_{L}(g)σ(v(L)).

This means: σ(v(L)) ∈ ∇^{−}(L); hence σ(v(L)) = k · ũ(L) for some k ∈ ℤ. Similarly,

gσ(ũ(L)) = (\frac{det(g|_{W})}{p}) σg(ũ(L))
= (\frac{det(g|_{W})}{p}) · ξ_{L}(g)σ(ũ(L)) = (\frac{det(g|_{L})}{p}) σ(ũ(L)),

which implies that σ(ũ(L)) ∈ ∇(L). From this it follows that σ(ũ(L)) = l · v(L) for some l ∈ ℤ. In this case we have

ϵp · v(L) = σ^{2}(v(L)) = σ(k · ũ(L)) = kσ(ũ(L)) = kl · v(L),

i.e. kl = ±p. Assume k ≠ ±1. Then k = ±p, l = ±1, and v(L) = ±σ(ũ(L)) belongs to σ(∇). Since ∇ is generated by the vectors v(L) and the sublattice σ(Γ) which is contained in σ(∇), we conclude that ∇ ⊆ σ(∇). Applying σ once more again, we get ∇ ⊆ σ^{2}(∇) = p∇, a contradiction. Hence k = ±1, i.e. ũ(L) = ±σ(v(L)).

Next we take L, M such that dim(L ∩ M) = k. Then

b_{k} = |(ũ(L), ũ(M))| = |(σ(v(L)), σ(v(M)))| = p|(v(L), v(M))| = pa_{k}.

Furthermore, by Proposition 3.4 for even k one has

(c_{k})^{2} = p^{k−n}a_{n}b_{n} = p^{k+1−n}(a_{n})^{2},

i.e. c_{k} = p^{(k+1−n)/2} · a_{n}.

Remark 3.7. The assumption Γ = ⟨v(L) | L any Lagrangian⟩ is essential for the conclusions of Lemma 3.6. For example, the parameters a_{k}, b_{k} of the lattice σ(∇) satisfy a_{k} = pb_{k}.

A key ingredient in our further arguments is the following observation:

Proposition 3.8. Let (e_{1}, …, e_{n}, f_{1}, …, f_{n}) be any arbitrary symplectic basis of W, and let M, L^{λ}, λ ∈ 𝔽_{p}, be Lagrangians with bases (f_{1}, …, f_{n}), (e_{1}+λf_{1}, f_{2}, …, f_{n}), respectively. Then in the notation of Lemma 3.6 one has

ũ(M) = ∑_{λ∈𝔽_{p}} d_{λ}v(L^{λ}), pv(M) = ∑_{λ∈𝔽_{p}} d′_{λ}ũ(L^{λ})

with d_{λ}, d′_{λ} = ±1.

Proof. Observe that dim(M ∩ L^{λ}) = n − 1. Hence in accordance with Lemma 3.6 we have (ũ(M), v(L^{λ})) = d_{λ}a_{n} with d_{λ} = ±1. Besides, (ũ(M), ũ(M)) = pa_{n} and (v(L^{λ}), v(L^{λ′})) = a_{n}δ_{λ,λ′}. Hence, for v = ũ(M) − ∑_{λ∈𝔽_{p}} d_{λ}v(L^{λ}) we have (v, v) = 2pa_{n} − 2pa_{n} = 0, yielding ũ(M) = ∑_{λ∈𝔽_{p}} d_{λ}v(L^{λ}). Applying σ to this identity, we obtain pv(M) = ∑_{λ∈𝔽_{p}} d′_{λ}ũ(L^{λ}).

Now we are in a position to explicitly exhibit a G-invariant odd unimodular lattice in V.

Theorem 3.9. Let p be any odd prime and n any odd integer. For every Lagrangian L in W, choose a vector v(L) in V ⊗_{ℚ} ℝ fixed by S(L) and such that (v(L), v(L)) = p^{(n−1)/2}. Then the lattice Δ = Δ(p, n) generated by all v(L)’s

Δ = ⟨v(L) | L ∈ L(W)⟩_{ℤ}

is a σ-stable G_{n}^{−}-invariant odd unimodular lattice.

Proof. We start with some G-invariant integral lattice Λ and choose v′(L) to be a generating vector of the ℤ-module Λ(L), L any Lagrangian. Then according to Lemma 3.3 and Proposition 3.4, (v′(L), v′(M)) = 0 if k = dim(L ∩ M) is even, and (v′(L), v′(M)) = ±p^{(k−1)/2}a_{1} if k is odd. Here a_{1} is some natural integer. Now we set v(L) = a_{1}^{−1/2}v′(L) for all Lagrangians L. Clearly, v(L) ∈ V ⊗_{ℚ} ℝ, (v(L), v(L)) = p^{(n−1)/2} and v(L) is S(L)-stable. (We could assume v(L) ∈ V by means of rescaling the scalar product on V by the scalar a_{1}^{−1}.) Furthermore, (v(L), v(M)) ∈ ℤ for any L, M. We see that Δ as defined in the theorem is a Ginvariant integral lattice. Moreover, if π_{D} denotes the standard symplectic spread, then Δ contains the sublattice

Δ(π_{D}) = ⟨v(L) | L ∈ π_{D}⟩_{ℤ}

of determinant p^{(n−1)(p^{n}+1)/2}. In particular, det Δ is a power of p: det Δ = p^{m} for some non-negative integer m.

If m = 0, we are done. Suppose that m ≥ 1. Then consider the form (x̄, ȳ)_{p} = (x, y) mod p on Δ/pΔ, where x̄ = x + pΔ, ȳ = y + pΔ. As p divides det Δ, (·, ·)_{p} is degenerate on Δ/pΔ. This means that pΔ is a proper sublattice of Δ ∩ pΔ^{#}. If Δ ∩ pΔ^{#} = Δ, then Δ ⊆ pΔ^{#}; in particular, (v(L), v(M)) ∈ pℤ for all L, M, contrary to the equality (v(L), v(M)) = ±1 for dim(L ∩ M) = 1. Therefore, Δ ⊃ Δ ∩ pΔ^{#} ⊃ pΔ. One may then suppose that Δ/(Δ ∩ pΔ^{#}) affords the G-character η_{1} mentioned in Proposition 2.2 (iii). Since Δ/(Δ∩pΔ^{#}) supports the G-invariant non-degenerate symmetric bilinear form (·, ·)_{p}, η_{1} is of quadratic type, contrary to Proposition 2.2 (iii).

2) By Lemma 3.6 and Proposition 3.8, σ(v(L)) belongs to Δ for any L. Hence Δ is σ-stable.

Corollary 3.10. For the lattice Δ = Δ(p, n) and generating vectors v(L), u(L) of Δ(L), Δ^{−}(L), we have u(L) = ±σ(v(L)). In particular, the parameters a_{k}, b_{k}, c_{k} of Δ satisfy the following relations:

(i) b_{k} = pa_{k} for any k;

(ii) c_{k} = p^{(k+1−n)/2} · a_{n} for any even k.

4. Explicit construction. II: n is even

Let p be an odd prime and n any even integer. In this section we exploit the results of §3 to describe an explicit construction of G_{n}^{+}-invariant p-modular lattices in V = V_{n}. Setting S′ = S_{n+1} = Sp_{2n+2}(p), we consider a natural 𝔽_{p}S′-module W′ = W_{n+1} = 𝔽_{p}^{2n+2} endowed with a non-degenerate symplectic form ⟨·, ·⟩. Fix some symplectic basis (e_{1}, …, e_{n+1}, f_{1}, …, f_{n+1}) of W′. Consider the endomorphism ϑ_{n+1} of W′ with the matrix diag(E_{n+1}, θE_{n+1}), and set G′ = G_{n+1} = ⟨S′, ϑ′_{n}⟩ ≃ CSp_{2n+2}(p). Now we can embed W into W′, S = S_{n} into S′, G = G_{n} into G′ by means of setting W = ⟨e_{1}, …, e_{n}, f_{1}, …, f_{n}⟩_{𝔽_{p}}, S = St_{S′}(e_{n+1}, f_{n+1}), G = ⟨S, ϑ_{n+1}⟩. Clearly, G ≃ CSp_{2n}(p), and one can identify ϑ_{n+1} with ϑ_{n}. Choose an irreducible Weil character ψ′ of S′ of degree (p^{n+1} + 1)/2 such that (ψ′|_{S}, ψ)_{S} > 0. Let χ′ be the rational irreducible character of G′ of degree p^{n+1} +1 and with kernel C_{(p−1)/2}, and let V′ = V_{n+1} be an irreducible ℚG′-module with character χ′ (cf. Proposition 2.3). Thus V′ is a faithful G_{n+1}^{−}-module.

We collect several facts from [SchT] and §3. For any Lagrangian L′ in W′ set

G′(L′) = {φ ∈ G′ | φ(L′) = L′} , S′(L′) = {φ ∈ G′(L′) | det(φ|_{L′}) ∈ 𝔽_{p}^{•2}} .

The subspace {v ∈ V′ | ∀φ ∈ S′(L′), φ(v) = v} has dimension 1. Therefore, one can choose an S′(L′)-stable vector v(L′) such that (v(L′), v(L′)) = p^{n/2}. Then the lattice

Δ′ = Δ(p, n + 1) = ⟨v(L′) | L′ ∈ L(W′)⟩_{ℤ}

is an odd unimodular G′-invariant lattice in V′. Moreover, Δ′ has a ℤ-linear endomorphism σ with the following properties:

(a) σ commutes with S′, and σϑ_{n+1} = −ϑ_{n+1}σ;

(b) σ^{2}(v) = ϵpv, (σ(u), v) = ϵ(u, σ(v)), (σ(u), σ(v)) = p(u, v) for any u, v ∈ V′, where ϵ = (−1)^{(p−1)/2}.

Let L′, M′ are any Lagrangians in W′, and set u(L′) = σ(v(L′)). If k = dim_{𝔽_{p}}(L′ ∩ M′) is odd, then a_{k} = |(v(L′), v(M′))| = p^{(k−1)/2}, b_{k} = |(u(L′), u(M′))| = p^{(k+1)/2}, c_{k} = |(u(L′), v(M′))| = 0. If k is even, then a_{k} = b_{k} = 0, and c_{k} = p^{k/2}.

The descent from G_{n+1}^{−} to G_{n}^{+} is provided by the following statement. Denote U = ⟨e_{n+1}⟩_{𝔽_{p}}, W′′ = ⟨e_{n+1}, f_{n+1}⟩_{𝔽_{p}}.

Proposition 4.1. The subspace V = ⟨v(L′) | L′ = L ⊕ U, L ∈ L(W)⟩_{ℚ} of V′ is a faithful absolutely irreducible ℚG_{n}^{+}-module of dimension p^{n} + 1. Moreover, V is σ-stable.

Proof. 1) We introduce the following subgroups in S′: B = St_{S′}(W), S′′ = {φ ∈ B | φ_{W} = 1_{W} }, C = S′′ ∩St_{S′}(U), K = S ×C. Then S′′ ≃ Sp(W′′) = Sp_{2}(p), C ≃ C_{p} · GL_{1}(p), B = S × S′′. By our definition, G = ⟨S, ϑ_{n+1}⟩. We also set H = ⟨K, ϑ_{n+1}⟩ = G · C. It is well known that ψ′|_{B} = ψ ⊗ ψ′′ + τ ⊗ τ′′, where ψ′′ (resp. τ′′) is an irreducible Weil character of S′′ of degree (p+1)/2 (resp. (p−1)/2). Furthermore, τ is an irreducible Weil character of S of degree (p^{n} − 1)/2 (ψ_{1} in the notation of §1). It is easy to check that α := τ′′|_{C} ∈ Irr(C), and ψ′′|_{C} = δ + β, where β ∈ Irr(C) and δ(φ) = (\frac{det(φ|_{U})}{p}) for φ ∈ C. (In particular, δ(1) = 1.) Observe that β ≠ δ. It is so if p > 3, since in this case β(1) = (p − 1)/2 > 1. If p = 3, then ℚ(β) = ℚ(\sqrt{−3}) ≠ ℚ = ℚ(δ). Thus

ψ′|_{K} = ψ ⊗ δ + ψ ⊗ β + τ ⊗ α

is a sum of three (pairwise distinct) irreducible constituents. From this it follows that

χ|_{K} = (ψ + \overline{ψ}) ⊗ δ + (ψ ⊗ β + \overline{ψ} ⊗ \overline{β}) + (τ ⊗ α + \overline{τ} ⊗ \overline{α}).

Observe that ϑ_{n+1} acts on S as an outer automorphism, and ϑ_{n+1} interchanges the characters ψ and \overline{ψ}. Furthermore, C ⊲ H. Consequently, χ|_{H} has a unique irreducible constituent, say γ, in which C acts via a multiple of the character δ. This constituent γ affords K-character (ψ + \overline{ψ}) ⊗ δ.

2) Next we observe that H leaves V fixed. Moreover, let L′ = L⊕U, let L be a Lagrangian in W and φ ∈ C. Then φ(L) = L. Hence, due to (6) the subgroup C acts on V as scalars (and the corresponding character is dim_{ℚ} V · δ). By the result of 1), dim_{ℚ} V = p^{n}+1. Recall that we chose G ≃ CSp(W) to be generated by S = Sp(W) and ϑ_{n+1}. We want to find the kernel of G acting on V. For consider a generating element z = θE_{2n} of the center C_{p−1} of CSp(W). Then z acting on W′ has the following matrix: diag(θE_{2n}, 1, θ^{2}) in the basis (e_{1}, …, e_{n}, f_{1}, …, f_{n}, e_{n+1}, f_{n+1}). If L′ = L ⊕ U (L any Lagrangian in W), then due to (6) z(v(L′)) = v(L′), as n is even. Thus z acts trivially on V. We have shown that V is in fact acted on by CSp_{2n}(p)/C_{p−1} = G_{n}^{+}, and this action affords G_{n}^{+}-character χ^{+} (cf. Proposition 2.3).

3) Finally, we show that σ(V) = V. Recall that the endomorphism σ centralizes S′. In particular, σ centralizes K. Hence, the subspace σ(V) affords the same Kcharacter as of V. Since ϑ_{n+1}(V) = V and ϑ_{n+1}σ = −σϑ_{n+1}, σ(V) is ϑ_{n+1}-stable, that is, σ(V) is an H-module. By the results of 1), σ(V) also affords the H-character γ. As γ is irreducible and it enters χ′|_{H} with multiplicity 1, σ(V) = V.

Now we are in a position to give some more explicit lattice constructions. We start with (p^{n} + 1)/2-dimensional lattices. Let R = Sp_{2m}(q), where q = p^{2f}. Then R has two irreducible Weil characters ϱ, ϱ^{∗} of degree (q^{m} + 1)/2. These characters are conjugate under some outer automorphism of R. Both of them are rational, as shown in [Gro]. We want to expose an explicit construction for ℤR-lattices of dimension (q^{m} + 1)/2. To this end, put n = mf. Consider the natural 𝔽_{q}Rmodule W^{(f)} = 𝔽_{q}^{2m} endowed with a non-degenerate 𝔽_{q}-valued symplectic form ⟨·, ·⟩_{(f)}. Then we can identify W with W^{(f)} viewed as 𝔽_{p}-space and assume that ⟨u, v⟩ = Tr_{𝔽_{q}/𝔽_{p}}⟨u, v⟩_{(f)}. This identification embeds R = Sp(W^{(f)}) canonically in S_{n} = Sp(W). One may also suppose that ϱ = ψ|_{R}. Clearly, any Lagrangian in W^{(f)} (that is, an m-dimensional 𝔽_{q}-subspace in W^{(f)} which is totally isotropic w.r.t. ⟨·, ·⟩_{(f)}) is also a Lagrangian in W. We call these special Lagrangians 𝔽_{q}- Lagrangians in W.

Theorem 4.2. Keep the above notation. Set

Δ(q, m) = ⟨v(L′) | L′ = L ⊕ U, L any 𝔽_{q}-Lagrangian in W ⟩_{ℤ}.

Then Δ(q, m) is an R-invariant integral lattice affording the Weil character ϱ.

Proof. In addition to Γ := Δ(q, m) we consider

Γ′ = ⟨u(L′) | L′ = L ⊕ U, L an 𝔽_{q}-Lagrangian in W ⟩_{ℤ}.

Clearly, Γ and Γ′ are invariant under R. We have mentioned that the restriction ψ|_{R} is equal to ϱ and so it is irreducible. Hence χ|_{R} = 2ϱ. From this it follows that dim_{ℤ} Γ and dim_{ℤ} Γ′ are at least ϱ(1) = (p^{n} + 1)/2. Observe that Γ⊥Γ′. For, if L, M are 𝔽_{q}-Lagrangians in W, then dim_{𝔽_{p}}(L ∩ M) = 2f · dim_{𝔽_{p}}(L ∩ M) is always even. This implies that dim_{𝔽_{p}}(L′ ∩ M′) is odd, and so (v(L′), u(M′)) = 0 by Lemma 3.3. By Proposition 4.1, Γ and Γ′ are contained in the ℚ-space V of dimension p^{n} + 1, and the scalar product on V is positive definite. Hence we must have dim_{ℤ} Γ = ϱ(1), and Γ = Δ(q, m) is an R-invariant integral lattice affording the Weil character ϱ.

Corollary 4.3. In the notation of Proposition 4.1, det V = pℚ^{•2}. On the other hand, if p ≡ 1 mod 4 and V is considered as a G_{n}^{−}-module by means of Proposition 2.4, then det V = ℚ^{•2}.

Proof. The proof of Theorem 4.2 shows that V contains the lattice Γ ⊕ σ(Γ) of determinant det Γ · det σ(Γ) = p^{(p^{n}+1)/2}(det Γ)^{2} ∈ pℚ^{•2}. The other claim follows from the oddness of (p^{n} + 1)/2.

Theorem 4.4. Keep the above notation. Set

Δ = Δ(p, n) = ⟨v(L′) | L′ = L ⊕ U, L ∈ L(W)⟩_{ℤ}.

Then Δ is a G_{n}^{+}-invariant p-modular lattice.

Proof. Recall that the scalar product on V is inherited from the one on V′, and the dual Δ^{#} to Δ is taken under this scalar product. Clearly, Δ is fixed by G_{n}^{+}. Applying Proposition 3.8, we see that u(L′) = σ(v(L′)) is contained in Δ for any L, and Δ is σ-stable.

1) First assume that det Δ is divisible by some prime r ≠ 2, p. Consider the form (x̄, ȳ)_{r} = (x, y) mod r on Δ/rΔ, where x̄ = x + rΔ, ȳ = y + rΔ. As r divides det Δ, this G-invariant symmetric bilinear form is degenerate, and so its kernel (Δ ∩ rΔ^{#})/rΔ is nonzero, i.e. Δ ⊇ Δ ∩ rΔ^{#} ⊃ rΔ. By Proposition 2.2 (i), this means Δ = Δ ∩ rΔ^{#}. Hence (u, v) ∈ rℤ for any u, v ∈ Δ. In the meantime, (v(L′), v(M′)) = ±1 for L′ = L ⊕ U, M′ = M ⊕ U with dim(L ∩ M) = 0, a contradiction.

2) At this point we show that det Δ is odd. Suppose the contrary: 2 divides det Δ. Consider the form (x̄, ȳ)_{2} = (x, y) mod 2 on Δ/2Δ, where x̄ = x + 2Δ, ȳ = y + 2Δ. As 2| det Δ, Δ′ =: Δ ∩ 2Δ^{#} contains properly 2Δ. Since Δ is an odd lattice, its even part Δ^{0} = {v ∈ Δ | (v, v) ∈ 2ℤ} is a sublattice of index 2 in Δ. Moreover, Δ^{0} ⊃ Δ′. (For Δ′ is clearly contained in Δ^{0}. On the other hand,

(v(L′) + u(L′), v(L′) + u(L′)) ∈ 2ℤ, (v(L′) + u(L′), v(L′)) = p^{n/2} ∉ 2ℤ,

i.e. v(L′) + u(L′) ∈ Δ^{0} \ Δ′.) Applying Proposition 2.2 (ii), we see that A ⊃ B ⊃ C ⊃ 0 is a composition series for the 𝔽_{2}G-module A = Δ/2Δ, where B = Δ^{0}/2Δ, C = Δ′/2Δ.

We exhibit one more nonzero proper submodule inside B. Set Γ = ⟨v + σ(v) | v ∈ Δ⟩_{ℤ} + 2Δ. Since gσ = ±σg for all g ∈ G, Γ is G-stable. Furthermore,

(u + σ(u), v + σ(v)) = (p + 1)(u, v) + (1 + ϵ)(u, σ(v)) ∈ 2ℤ

due to the properties of the endomorphism σ. Thus D = Γ/2Δ is a G-submodule of B and D is totally isotropic w.r.t. (·, ·)_{2}. Since (v(L′)+u(L′), v(L′)) = p^{n/2}, we see that v(L′) + u(L′) ∈ Γ \ Δ′. From this it follows that 0 ≠ D ≠ C. Since B ⊃ C ⊃ 0 is a composition series for B, we must have B = C + D. But C = Ker(·, ·)_{2}; hence we come to the conclusion that B is totally isotropic w.r.t. (·, ·)_{2}. On the other hand, choosing

L_{1} = ⟨e_{1}, …, e_{n}, e_{n+1}⟩_{𝔽_{p}}, L_{2} = ⟨f_{1}, …, f_{n}, e_{n+1}⟩_{𝔽_{p}},
L_{3} = ⟨e_{1}, …, e_{n−2}, e_{n−1} + f_{n−1}, f_{n}, e_{n+1}⟩_{𝔽_{p}},

we get v(L_{1}) + v(L_{2}), v(L_{1}) + v(L_{3}) ∈ Δ^{0} with

(v(L_{1}) + v(L_{2}), v(L_{1}) + v(L_{3})) ≡ p^{n/2} + p^{(n−2)/2} + 1 + 0 ≡ 1 mod 2,

a contradiction.

3) Observe that pΔ^{#} ⊇ σ(Δ). (Indeed, Δ is generated by the vectors v(L′), and σ(Δ) is generated by the vectors u(M′), with L′ = L ⊕ U, M′ = M ⊕ U, L, M arbitrary Lagrangians in W. It is obvious that k = dim(L′ ∩ M′) ≥ 1. But |(v(L′), u(M′))| = c_{k} is 0 if k is odd, and p^{k/2} if k is even. Hence c_{k} is divisible by p.) In fact we have

Δ ∩ pΔ^{#} = σ(Δ). (10)

For, assume the contrary. Then Δ ⊇ Δ ∩ pΔ^{#} ⊃ σ(Δ) ⊃ pΔ. By Proposition 2.2 (iii) Δ/σ(Δ) is an irreducible 𝔽_{p}G-module. Hence Δ ∩ pΔ^{#} = Δ, Δ ⊆ pΔ^{#}. The last inclusion contradicts the equality (v(L′), v(M′)) = ±1 for dim(L′ ∩ M′) = 1.

In addition to (10) we show that

Δ ∩ p^{2}Δ^{#} ⊆ pΔ. (11)

To this end we denote Λ = Δ ∩ pΔ^{#}. Then Λ ∩ p^{2}Λ^{#} is a proper sublattice of Λ, because u(L′), u(M′) ∈ Λ and (u(L′), u(M′)) = ±p provided that dim(L′∩M′) = 1. Furthermore, (Δ ∩ pΔ^{#}, pΔ) ⊆ p^{2}ℤ. Thus we have

pΔ ⊆ Λ ∩ p^{2}Λ^{#} ⊂ Λ = σ(Δ).

Now the irreducibility of the 𝔽_{p}G-module σ(Δ)/pΔ implies that pΔ = Λ ∩ p^{2}Λ^{#}. Keeping in mind that

Λ ∩ p^{2}Λ^{#} = (Δ ∩ pΔ^{#}) ∩ (pΔ + p^{2}Δ^{#}) ⊇ Δ ∩ p^{2}Δ^{#},

one obtains (11).

4) By the results of 1) and 2), det Δ is not divisible by any prime r other than p. Hence det Δ = p^{m} and so Δ ⊇ p^{m}Δ^{#} for some non-negative integer m. Choose the minimal non-negative integer ℓ such that Δ ⊇ p^{ℓ}Δ^{#}. If ℓ = 0, then by (10) one has σ(Δ) = pΔ, a contradiction. Assume that ℓ ≥ 2. Then applying (11) we have

p^{ℓ}Δ^{#} ⊆ Δ ∩ p^{2}Δ^{#} ⊆ pΔ,

i.e. p^{ℓ−1}Δ^{#} ⊆ Δ, contrary to the choice of ℓ. Hence ℓ = 1. In this case (10) yields pΔ^{#} = σ(Δ), Δ = σ(Δ^{#}). In other words, Δ is p-modular.

From now on, when considering Δ(p, n) with n even, we denote v(L ⊕ U) by v(L) (L a Lagrangian in W) and then forget the initial descent n + 1 ⇝ n. In particular, (v(L), v(M)) = ±p^{k/2} if k = dim(L ∩ M) is even, and 0 otherwise. The signs involved in this formula will be determined in the next section.

5. Maslov index and Gram matrix

Let k be any field of characteristic other than 2 and S(k) = Sp_{2n}(k). If k = ℂ or k is a finite field (and (n, |k|) ≠ (1, 9)), then it is well known that S(k) is simply connected. However, if k is ℝ or any local field, then S(k) is not simply connected, and S(k) has a double covering group called the metasymplectic group. An important role in physics is played by a faithful complex representation of the metasymplectic group called the Shale-Weil representation. A key ingredient of constructing this representation is Maslov index (or Maslov-Kashiwara index), which is defined on triples of Lagrangians inside the symplectic space k^{2n}. For more detail the reader is referred to [LiV].

Remarkably, we can define a discrete analogue of Maslov index for Sp_{2n}(p), which enables one to completely determine the Gram matrices of the lattices Δ(p, n), n any integer and p any odd prime (cf. Theorems 3.9, 4.4), and the lattices Δ^{−}(p, n) (in the case p ≡ 1 mod 4). Here, Δ^{−}(p, n) is obtained from Δ(p, n) by means of Proposition 2.4 (with G^{+} = G_{n}^{−} if n is odd and G^{+} = C_{2} × G_{n}^{+} if n is even). Throughout this section, Lagrangians are considered oriented.

First we deal with the lattices Δ(p, n). Fix an oriented Lagrangian L_{0} with an ordered basis (u_{1}, …, u_{n}) (for short: L_{0} = (u_{1}, …, u_{n})), and a generating vector v(L_{0}) of Δ(L_{0}). For an arbitrary oriented Lagrangian M = (v_{1}, …, v_{n}) we find an element ν_{M} ∈ Sp_{2n}(p) such that ν_{M}(u_{i}) = v_{i} for all i, and set v(M) = ν_{M}(v(L_{0})). It is easy to see that this definition does not depend on the choice of ν_{M}. Finally, we put u(L) = σ(v(L)) (cf. Lemma 3.6).

Definition 5.1. Let p be any odd prime and n any integer. Let L, M be arbitrary oriented Lagrangians in W = 𝔽_{p}^{2n}. Then the index [L, M] of the ordered pair

(L, M) is defined to be (\frac{det F}{p} ), where the matrix F is defined as follows. Let dim(L ∩ M) = k, choose ordered bases

(u_{1}, …, u_{k}, v_{1}, …, v_{n−k}), (u_{1}, …, u_{k}, w_{1}, …, w_{n−k})

of L, M, respectively; set F := F (L, M) := (⟨v_{i}, w_{j}⟩)_{1≤i,j≤n−k}.

Proposition 5.2. The index is well defined. It is symmetric and G_{n}-invariant on pairs (L, M) with n − dim(L ∩ M) even. Moreover, if L, M, L′, M′ are oriented Lagrangians and dim(L ∩ M) = dim(L′ ∩ M′), then

[L, M](v(L), v(M)) = [L′, M′](v(L′), v(M′)),
[L, M](u(L), v(M)) = [L′, M′](u(L′), v(M′)),
[L, M](u(L), u(M)) = [L′, M′](u(L′), u(M′)).

Proof. First we show that (\frac{det F}{p} ) is independent of the bases chosen. For, suppose

(u′_{1}, …, u′_{k}, v′_{1}, …, v′_{n−k}), (u′_{1}, …, u′_{k}, w′_{1}, …, w′_{n−k})

are other ordered bases of

L = (u_{1}, …, u_{k}, v_{1}, …, v_{n−k}), M = (u_{1}, …, u_{k}, w_{1}, …, w_{n−k}).

Then the transition matrices (from the old bases to the new bases) are ( \begin{matrix}A & X \\ 0 & B \end{matrix} )

and ( \begin{matrix}A & Y \\ 0 & C \end{matrix} ), where A ∈ GL_{k}(p), X, Y ∈ M_{k,n−k}(𝔽_{p}), B, C ∈ GL_{n−k}(p). Since

L, M are oriented, det A · det B and det A · det C belong to 𝔽_{p}^{•2}. Clearly, F is

changed to ^{t}BF C and (\frac{det F}{p} ) = (\frac{det ^{t}BF C}{p} ).

If g ∈ G_{n} and ⟨gu, gv⟩ = λ · ⟨u, v⟩ for all u, v ∈ V, then [g(L), g(M)] =

(\frac{λ}{p})^{n−k} [L, M]. Furthermore, [M, L] = ϵ^{n−k}[L, M]. In particular, [L, M] is symmetric and G_{n}-invariant on pairs (L, M) with n − dim(L ∩ M) even.

Finally, assume

\begin{matrix}L = (u_{1}, …, u_{k}, v_{1}, …, v_{n−k}), M = (u_{1}, …, u_{k}, w_{1}, …, w_{n−k}), \\ L′ = (u′_{1}, …, u′_{k}, v′_{1}, …, v′_{n−k}), M′ = (u′_{1}, …, u′_{k}, w′_{1}, …, w′_{n−k}) \end{matrix}

are oriented Lagrangians in W. Then there exists g ∈ Sp_{2n}(p) such that

g(L) = ±L′, g(M) = ±M′. Let ( \begin{matrix}A & X \\ 0 & B \end{matrix} ) (resp. ( \begin{matrix}A & Y \\ 0 & C \end{matrix} )) be the transition

matrix from the basis (g(u_{1}), …, g(u_{k}), g(v_{1}), …, g(v_{n−k})) of g(L) to the basis (u′_{1}, …, u′_{k}, v′_{1}, …, v′_{n−k}) of L′ (resp. from the basis (g(u_{1}), …, g(u_{k}), g(w_{1}), …, g(w_{n−k})) to (u′_{1}, …, u′_{k}, w′_{1}, …, w′_{n−k})). Then

v(L′) = (\frac{det A · det B}{p} ) g(v(L)), v(M′) = (\frac{det A · det C}{p} ) g(v(M))

and so (v(L′), v(M′)) = (\frac{det B·det C}{p} ) (v(L), v(M)). On the other hand, one can

show that F (L′, M′) = ^{t}B · F (L, M) · C, yielding [L′, M′] = (\frac{det B·det C}{p} ) [L, M]. Hence [L, M](v(L), v(M)) = [L′, M′](v(L′), v(M′)). The identities

\begin{matrix}[L, M](u(L), v(M)) = [L′, M′](u(L′), v(M′)), \\ [L, M](u(L), u(M)) = [L′, M′](u(L′), u(M′)) \end{matrix}

are proved in the same way.

Theorem 5.3. Let p be any odd prime and n any integer, and let ϵ = (−1)^{(p−1)/2}. Under the above notation one has

(v(L), v(M)) = (ϵ/p)^{(n−k)/2}p^{[n/2]}[L, M]

for any oriented Lagrangians L, M with k = dim(L ∩ M) and n − k even.

Proof. By Corollary 3.5, Theorem 4.4 and Proposition 5.2, there are constants C_{k} = ±1 such that [L, M](v(L), v(M)) = p^{[n/2]−(n−k)/2}C_{k} for any oriented Lagrangians L, M with k = dim(L ∩ M) and n − k even. We want to show that

C_{k} = ϵ^{(n−k)/2}. (12)

Clearly, (12) holds for k = n.

1) At this point we prove (12) for k = n − 2 (and n ≥ 2). Because of the descent n ⇝ n − 1 used in Theorem 4.4, we can restrict ourselves to the case n is odd (and so n ≥ 3). In order to determine C_{n−2}, we use the standard spread {W^{∞}, W^{λ} | λ ∈ 𝔽_{q}} of W = 𝔽_{p}^{2n} (see the discussion before (7)). As usual, we assume that ⟨e_{i}, f_{j}⟩ = δ_{i,j}, where W^{0} = (e_{1}, …, e_{n}), W^{∞} = (f_{1}, …, f_{n}). Due to our identification of W with 𝔽_{q}^{2}, q = p^{n}, we have e_{i} = (α_{i}, 0), f_{i} = (0, β_{i}) for any i and some α_{i}, β_{i} ∈ 𝔽_{q}. Observe that there is a map from Sp_{2n}(p) which sends the oriented Lagrangian L_{0} := W^{0} to W^{∞} (resp. to W^{λ} = ((α_{1}, λα_{1}), …, (α_{n}, λα_{n})), λ ∈ 𝔽_{q}). Now take

L = (e_{1}, f_{2}, f_{3}, …, f_{n}), M = (f_{1}, e_{2}, f_{3}, …, f_{n}).

Then dim(L∩M) = n−2, [L, M] = ϵ. Since dim(L∩W^{∞}) = dim(M ∩W^{∞}) = n−1, we have

v(L) = ∑_{λ∈𝔽_{q}} a_{λ}v(W^{λ}), v(M) = ∑_{λ∈𝔽_{q}} b_{λ}v(W^{λ})

and so

ϵp^{(n−3)/2}C_{n−2} = (v(L), v(M)) = p^{(n−1)/2} ∑_{λ∈𝔽_{q}} a_{λ}b_{λ}. (13)

One easily sees that a_{λ} ≠ 0 if and only if tr(λ(α_{1})^{2}) = 0. Similarly, b_{λ} ≠ 0 if and only if tr(λ(α_{2})^{2}) = 0. Observe that (α_{1})^{2} and (α_{2})^{2} are linearly independent over 𝔽_{p}; otherwise 𝔽_{q} would contain 𝔽_{p}(α_{1}/α_{2}) = 𝔽_{p^{2}}, contrary to the assumption that n is odd. Hence a_{λ}b_{λ} ≠ 0 for exactly p^{n−2} values of λ ∈ 𝔽_{q}. Moreover, if a_{λ}b_{λ} ≠ 0, then a_{λ}b_{λ} = p^{1−n}, since in this case dim(L ∩ W^{λ}) = dim(M ∩ W^{λ}) = 1 and [W^{λ}, L] = [W^{λ}, M] = 1. Bearing (13) in mind, we obtain C_{n−2} = ϵ, as stated.

2) Here we show that

C_{n−2[n/2]} = C_{n+2−2[n/2]}C_{n−2} (14)

for any n ≥ 2.

2a) Because of the descent n ⇝ n − 1 used in Theorem 4.4, we can restrict ourselves to the case n is even. In order to prove (14): C_{0} = C_{2}C_{n−2}, we consider the standard spread {W^{∞}, W^{λ} | λ ∈ 𝔽_{q}} of W = 𝔽_{p}^{2n}. As above, we assume that ⟨e_{i}, f_{j}⟩ = δ_{i,j}, where W^{0} = (e_{1}, …, e_{n}), W^{∞} = (f_{1}, …, f_{n}). Due to our identification of W with 𝔽_{q}^{2}, q = p^{n}, we have e_{i} = (α_{i}, 0), f_{i} = (0, β_{i}) for any i and some α_{i}, β_{i} ∈ 𝔽_{q}. Since n is even, without loss of generality we may suppose that α_{1} = 1 and α_{2} = e ∈ 𝔽_{p^{2}} \ 𝔽_{p}. Observe that there is a map from Sp_{2n}(p) which sends the oriented Lagrangian L_{0} := W^{0} to W^{∞} (resp. to W^{λ} = ((α_{1}, λα_{1}), …, (α_{n}, λα_{n})), λ ∈ 𝔽_{q}). Our identification of W with 𝔽_{q}^{2} embeds R = SL_{2}(q) naturally in Sp_{2n}(p). Consider the following elements

r_{a} = ( \begin{matrix}a^{−1} & 0 \\ 0 & a \end{matrix} ) , s_{b} = ( \begin{matrix}1 & 0 \\ b & 1 \end{matrix} ) , t = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) , a ∈ 𝔽_{q}^{•}, b ∈ 𝔽_{q}

of R. Then they act on the vectors v(W^{λ}) as follows:

\begin{matrix}r_{a} : & v(W^{∞}) ↦ (\frac{a}{q})v(W^{∞}), & v(W^{λ}) ↦ (\frac{a}{q})v(W^{a^{2}λ}), \\ s_{b} : & v(W^{∞}) ↦ v(W^{∞}), & v(W^{λ}) ↦ v(W^{λ+b}), \\ t : & v(W^{∞}) ↔ μv(W^{0}), & v(W^{λ}) ↦ (\frac{λ}{q})v(W^{−1/λ}). \end{matrix} (15)

Here, (\frac{λ}{q}) = λ^{(q−1)/2} and μ = (\frac{det T}{q} ), where T = (tr(α_{i}β_{j}))_{1≤i,j≤n} and tr := tr_{𝔽_{q}/𝔽_{p}}. (For instance, μ = −1 if n = 2.) Indeed, the relation (15) is evident for s_{b}. Furthermore, the factor (\frac{a}{q}) appears in (15) for r_{a}, since the map sending each f_{i} to af_{i} has determinant N_{𝔽_{q}/𝔽_{p}}(a). Similarly, for any λ ∈ 𝔽_{q}, the map sending each (a^{−1}α_{i}, aλα_{i}) to (α_{i}, a^{2}λα_{i}) has determinant N_{𝔽_{q}/𝔽_{p}}(a). By the same reason the factor (\frac{λ}{q}) appears in the formula for t. Finally, t(e_{i}) = (0, α_{i}), and the map sending each (0, α_{i}) to f_{i} = (0, β_{i}) has matrix T^{−1}.

2b) Using the action of s_{b}, we see that there is γ = ±1 such that (v(W^{∞}), v(W^{λ})) = γ for all λ ∈ 𝔽_{q}. Next, t acting on this relation yields (v(W^{0}), v(W^{λ})) = μγ if λ ∈ 𝔽_{q} is a square, and −μγ otherwise. Finally, using the action of s_{b} once more, we see that for λ ≠ λ′ ∈ 𝔽_{q}, (v(W^{λ}), v(W^{λ′})) = μγ if λ − λ′ is a square, and −μγ otherwise. Since [W^{∞}, W^{0}] = 1, γ = C_{0}.

2c) The proof of Theorem 4.2 shows that v(W^{∞}) and v(W^{λ}), λ ∈ 𝔽_{q}, are linearly independent: v(W^{∞}) = ∑_{λ∈𝔽_{q}} a_{λ}v(W^{λ}) for a_{λ} ∈ ℂ. Averaging this relation by means of s_{b}, b ∈ 𝔽_{q}, we get v(W^{∞}) = a ∑_{λ∈𝔽_{q}} v(W^{λ}) for a ∈ ℂ. Hence

γ = (v(W^{∞}), v(W^{0}))
= a\sqrt{q} + a ∑_{λ∈𝔽_{q}^{•2}}(v(W^{0}), v(W^{λ})) + a ∑_{λ∈𝔽_{q}^{•}\𝔽_{q}^{•2}}(v(W^{0}), v(W^{λ}))
= a\sqrt{q} + \frac{q−1}{2}aμγ − \frac{q−1}{2}aμγ = a\sqrt{q},

i.e. a = γp^{−n/2}. We have shown that

v(W^{∞}) = C_{0}p^{−n/2} ∑_{λ∈𝔽_{q}} v(W^{λ}). (16)

2d) Now we consider the oriented Lagrangian M = (e_{1}, e_{2}, f_{3}, …, f_{n}). Since dim(M ∩ W^{∞}) = n − 2 and [M, W^{∞}] = 1, (v(M), v(W^{∞})) = p^{n/2−1}C_{n−2}. We compute this scalar product in another way using (16). For λ ∈ 𝔽_{q}, it is clear that dim(M ∩ W^{λ}) ≤ 2. Moreover, dim(M ∩ W^{λ}) = 2 if and only if

tr(λ) = tr(λe) = tr(λe^{2}) = 0. (17)

(Recall that we have chosen α_{1} = 1 and α_{2} = e ∈ 𝔽_{p^{2}} \ 𝔽_{p}.) Since 𝔽_{p}(e) = 𝔽_{p^{2}}, (17) holds for exactly p^{n−2} values of λ ∈ 𝔽_{q}. Denote X = {λ ∈ 𝔽_{q} | dim(M ∩ W^{λ}) = 2}, Y = 𝔽_{q} \ X. By the choice of α_{1} and α_{2}, M ∩ W^{λ} = 0 for any λ ∈ Y.

2e) It is easy to check that [M, W^{λ}] = 1 for any λ ∈ X. In particular,

∑_{λ∈X}(v(M), v(W^{λ})) = p^{n−1}C_{2}.

Similarly, [M, W^{λ}] = (\frac{det A(λ)}{p}) for any λ ∈ Y, where

A(λ) := ( \begin{matrix}tr(λ) & tr(λe) \\ tr(λe) & tr(λe^{2}) \end{matrix} ) .

Now we fix a non-square element σ in 𝔽_{p^{2}}. Then λ satisfies (17) if and only if λσ does. This means that the multiplication by σ leaves Y fixed. On the other hand, observe that [M, W^{λ}] = −[M, W^{λσ}] for any λ ∈ Y. (Indeed,

det A(λ) = ∑_{i,j=0}^{n−1} λ^{p^{i}+p^{j}} (e^{2p^{j}} − e^{p^{i}+p^{j}}) .

Clearly, if i − j is even, then e^{2p^{j}} = e^{p^{i}+p^{j}}. If i − j is odd, then σ^{p^{i}+p^{j}} = σ^{p+1}. This argument shows that det A(λσ) = σ^{p+1} det A(λ). Now σ^{p+1} is a non-square in 𝔽_{p}; hence the claim follows.) Consequently,

∑_{λ∈Y}(v(M), v(W^{λ})) = \frac{1}{2} ∑_{λ∈Y} ((v(M), v(W^{λ})) + (v(M), v(W^{λσ})))

= \frac{C_{0}}{2} ∑_{λ∈Y} ([M, W^{λ}] + [M, W^{λσ}]) = 0.

2f) As a result of the computations in pp. 2c), 2d) and 2e), we obtain

p^{n/2−1}C_{n−2} = (v(M), v(W^{∞}))

= (∑_{λ∈X}(v(M), v(W^{λ})) + ∑_{λ∈Y}(v(M), v(W^{λ}))) = p^{n/2−1}C_{0}C_{2},

i.e. C_{0} = C_{2}C_{n−2}, as stated.

3) Finally, we prove (12) by induction on n. Because of the descent n ⇝ n − 1 used in Theorem 4.4, we can restrict ourselves to the case n is odd. The induction base n = 1, 3 has already been established, since we have proved (12) for k = n, n−2. For the induction step, observe that the descent n ⇝ n − 2 used in the proof of Proposition 3.4 allows us to state that C_{k} = ϵ^{(n−k)/2} for any odd k ≥ 3. According to (14),

C_{1} = C_{3} · C_{n−2} = ϵ^{(n−3)/2} · ϵ = ϵ^{(n−1)/2},

and the induction step is over.

Recall that (v(L), v(M)) is 0 if n − dim(L ∩ M) is odd. Therefore, Theorem 5.3 completely determines the Gram matrix of the lattice Δ(p, n).

Corollary 5.4. Let p be any odd prime and n any odd integer. Relative to the generating system {v(L) | L ∈ L(W)} the unimodular lattice Δ(p, n) has the following Gram matrix:

(v(L), v(M)) = \{ \begin{matrix}ϵ^{(n−k)/2}p^{(k−1)/2}[L, M], & dim(L ∩ M) = k ≡ 1 mod 2, \\ 0, & dim(L ∩ M) ≡ 0 mod 2. \end{matrix} □

Example 5.5. Let p = n = 3. Then the Gram matrix for Δ(3, 3) produced by Corollary 5.4 is the same as given in [BaV].

Corollary 5.6. Let p be any odd prime and n any even integer. Relative to the generating system {v(L) | L ∈ L(W)} the p-modular lattice Δ(p, n) has the following Gram matrix:

(v(L), v(M)) = \{ \begin{matrix}ϵ^{(n−k)/2}p^{k/2}[L, M], & dim(L ∩ M) = k ≡ 0 mod 2, \\ 0, & dim(L ∩ M) ≡ 1 mod 2. \end{matrix} □

Example 5.7. Let p be any odd integer. Then the p-modular (p^{2} +1)-dimensional lattice Δ(p, 2) is generated by 2(p + 1)(p^{2} + 1) vectors v(L), L any oriented Lagrangian in 𝔽_{p}^{4}. Here, v(−L) = −v(L), (v(L), v(L)) = p, and (v(L), v(M)) equals ϵ[L, M] if L ∩ M = 0 and 0 if dim(L ∩ M) = 1. Taking p = 5, we get the 26dimensional 5-modular lattice with minimum 5 constructed by Nebe.

Clearly, (u(L), u(M)) = p(v(L), v(M)). Now we want to compute the scalar products (u(L), v(M)). Since σ is determined up to sign, the scalar products (u(L), v(M)) are determined also up to sign. Recall that (u(L), v(M)) = ±p^{[n/2]} if dim(L ∩ M) = n − 1 (cf. Lemma 3.6). For definiteness, we choose σ such that

(u(L), v(M)) = p^{[n/2]}[L, M] (18)

for oriented Lagrangians L, M with dim(L ∩ M) = n − 1.

Theorem 5.8. Let p be any odd prime and n any integer. Under the convention (18) one has

(u(L), v(M)) = (ϵ/p)^{(n−1−k)/2}p^{[n/2]}[L, M]

for any oriented Lagrangians L, M with k = dim(L ∩ M) and n − k odd.

Proof. 1) By Lemma 3.6, Theorem 4.4 and Proposition 5.2, there are constants D_{k} = ±1 such that [L, M](u(L), v(M)) = p^{[n/2]−(n−1−k)/2}D_{k} for any oriented Lagrangians L, M with k = dim(L ∩ M) and n − k odd. We want to show that

D_{k} = ϵ^{(n−1−k)/2}. (19)

Clearly, (19) holds for k = n − 1, due to (18). Because of the descent n ⇝ n − 1 used in Theorem 4.4, it suffices to prove (19) for odd n.

2) Consider the standard spread of W = 𝔽_{p}^{2n} (recall n is odd). In the notation of p. 1) of the proof of Theorem 5.3 we set L = W^{∞}, M = (e_{1}, f_{2}, …, f_{n}). Since L ∩ W^{∞} has odd dimension, (u(L), v(W^{∞})) = 0. Furthermore, for any λ ∈ 𝔽_{q}, L ∩ W^{λ} = 0 and [L, W^{λ}] = ϵ^{n}. Therefore, (u(L), v(W^{λ})) = ϵ^{n}D_{0}, yielding

u(L) = ϵ^{n}D_{0}p^{(1−n)/2} ∑_{λ∈𝔽_{q}} v(W^{λ}).

On the other hand, by Theorem 5.3 we have

v(M) = ϵ^{(n−1)/2}p^{(1−n)/2} ∑_{λ∈𝔽_{q},tr(λ(α_{1})^{2})=0} v(W^{λ}).

Hence (u(L), v(M)) = ϵ^{(n+1)/2}p^{(n−1)/2}D_{0}. But dim(L∩M) = n−1 and [L, M] = ϵ; therefore we get D_{0} = ϵ^{(n−1)/2}. This establishes (19) for k = 0.

Now we can prove (19) by induction on odd n. The induction base n = 1, 3 has already been established, since we have proved (19) for k = n − 1, 0. For the induction step, observe that the descent n ⇝ n − 2 used in the proof of Proposition 3.4 allows us to state that D_{k} = ϵ^{(n−1−k)/2} for any even k ≥ 2, and so the induction step is over.

Next, let p ≡ 1 mod 4. We determine the Gram matrices for the lattices Δ^{−}(p, n). Recall (cf. §2) that Δ^{−}(p, n) has the same generating system as of Δ(p, n). But if (·, ·) is the scalar product on Δ(p, n), then Δ^{−}(p, n) is endowed with the scalar product (·, ·)^{−}, where (u, v)^{−} = pb(u, v) + a(σu, v). Here a, b are integers such that a^{2} − pb^{2} = −1. Also, here we have ϵ = 1. Bearing this in mind, from the above results we immediately obtain:

Corollary 5.9. Let p ≡ 1 mod 4 be a prime and n any even integer. Relative to the generating system {v(L) | L ∈ L(W)} the unimodular lattice Δ^{−}(p, n) has the following Gram matrix:

(v(L), v(M)) = \{ \begin{matrix}bp^{k/2}[L, M], & dim(L ∩ M) = k ≡ 0 mod 2, \\ ap^{(k−1)/2}[L, M], & dim(L ∩ M) = k ≡ 1 mod 2. \end{matrix} □

Corollary 5.10. Let p ≡ 1 mod 4 be a prime and n any odd integer. Relative to the generating system {v(L) | L ∈ L(W)} the p-modular lattice Δ^{−}(p, n) has the following Gram matrix:

(v(L), v(M)) = \{ \begin{matrix}bp^{(k+1)/2}[L, M], & dim(L ∩ M) = k ≡ 1 mod 2, \\ ap^{k/2}[L, M], & dim(L ∩ M) = k ≡ 0 mod 2. \end{matrix} □

Example 5.11. Let p ≡ 1 mod 4. Then the p-modular (p + 1)-dimensional lattice Δ^{−}(p, 1) has a basis consisting of the vectors e_{λ}, λ ∈ 𝔽_{p} ∪ {∞}. These vectors are of norm pb; furthermore, (e_{λ}, e_{∞}) = a for any λ ∈ 𝔽_{p}. Finally, for λ ≠ μ ∈ 𝔽_{p} we have (e_{λ}, e_{μ}) = a if λ − μ is a square, and −a otherwise. Thus Δ^{−}(p, 1) is just the lattice M_{p+1,2} constructed in Theorem (V.2) of [NPl].

Example 5.12. Let p ≡ 1 mod 4. Then the unimodular (p^{2} + 1)-dimensional lattice Δ^{−}(p, 2) is generated by 2(p+1)(p^{2}+1) vectors v(L), L any oriented Lagrangian in 𝔽_{p}^{4}, with the following Gram matrix:

(v(L), v(M)) = \{ \begin{matrix}bp[L, M], & dim(L ∩ M) = 2, \\ a[L, M], & dim(L ∩ M) = 1, \\ b[L, M], & dim(L ∩ M) = 0. \end{matrix}

Taking p = 5 (and a = 2, b = 1), we get the 26-dimensional unimodular lattice with minimum 3 constructed by Nebe.

As we have mentioned above, the lattices Δ = Δ(p, n) (n even), Δ^{−}(p, n) (p ≡ 1 mod 4 and n even) are p-modular. But they are not (self-dual) 𝔬-lattices (where 𝔬 = ⟨1, (1 + θ)/2⟩_{ℤ} and θ = \sqrt{ϵp}) by the following reason. The multiplication by θ should be given as θ(v) = σ(v), v ∈ Δ. If Δ is an 𝔬-lattice, then Δ contains \frac{1+θ}{2} v(L) = (v(L) + u(L))/2, L a Lagrangian. On the other hand, ((v(L) + u(L))/2, v(L)) = p^{[n/2]}/2 is not integral, a contradiction.

However, if we restrict ourselves to the S_{n}-stable lattices, then in some cases we can get self-dual 𝔬-lattices. Recall that an integral lattice Γ is called a 2-neighbour of a given integral lattice Λ if the intersection Γ ∩ Λ has index 2 in both of Λ and Γ.

Proposition 5.13. Let Δ denote any of the lattices Δ(p, n), Δ^{−}(p, n). Then the following assertions hold.

(i) If p ≡ 1 mod 4, then Δ has no 2-neighbours.

(ii) Let p ≡ 3 mod 4. Then Δ has exactly two 2-neighbours, namely Δ^{δ} = ⟨Δ^{0}, \frac{1}{2}(v(L) + δu(L))⟩_{ℤ}, where δ = ±1, Δ^{0} the even part of Δ and L a fixed Lagrangian. These neighbours are S_{n}-stable. If p ≡ 7 mod 8, then they are self-dual 𝔬-lattices (w.r.t. the Hermitian form u ∘ v defined in (3)) if n is even, and even unimodular (Euclidean) 𝔬-stable lattices if n is odd.

Proof. It is easy to see (cf. Lemma 6.5) that Δ^{0} is the unique sublattice of index 2 in Δ. Hence, if Γ is an arbitrary 2-neighbour of Δ, then Γ = ⟨Δ^{0}, w⟩_{ℤ} with 2w ∈ Δ. By definition, 2w ∈ Δ ∩ 2(Δ^{0})^{#} = Δ^{1} (see also the discussion before Lemma 6.5).

Now Δ^{1}/2Δ^{0} contains exactly 3 nontrivial cosets, namely those with representatives v(L), and w^{δ}(L) := v(L)+δu(L), δ = ±1 and L a fixed Lagrangian. The first coset is contained in Δ; hence we can avoid it. Thus we can take w = \frac{1}{2}w^{δ}(L) and then Γ = Δ^{δ}. Observe that this w has (squared) norm p^{[n/2]}(p + 1)/4 if Δ = Δ(p, n), (b(p + 1) + 2a)p^{(n+1)/2} if Δ = Δ^{−}(p, n) and n is odd, and (b(p + 1) + 2a)p^{n/2} if Δ = Δ^{−}(p, n) and n is even (cf. Theorems 5.3 and 5.8). Therefore, Γ is integral (w.r.t. (·, ·)) if and only if p ≡ 3 mod 4, and even if and only if p ≡ 7 mod 8.

Clearly, G_{n} permutes the two cosets with representatives w^{δ}(L) in Δ^{1}/2Δ^{0}. But S_{n} has no subgroups of index 2; hence S_{n} stabilizes each of Δ^{δ}. On the other hand, G_{n} permutes the lattices Δ^{δ} transitively. (For recall that u(L) = σ(v(L)). Choose g ∈ G_{n} \ S_{n} such that g(L) = L. Then g(w^{δ}(L)) = g(v(L)) − δσ(g(v(L))) = ±w^{−δ}(L).)

Observe that Δ^{0}, Δ^{1} are always 𝔬-stable. Indeed, put w^{δ}(M) = v(M) + δu(M) for any oriented Lagrangian M. For any s ∈ S_{n} with s(L) = M one has s(w^{δ}(L)) = ±w^{δ}(M). But we already know that s fixes the coset w^{δ}(L) + 2Δ^{0}. Hence w^{δ}(L) + w^{δ}(M) ∈ 2Δ^{0}. In other words, (1 + σ)(v(L) + v(M)) ∈ 2Δ^{0}. This means Δ^{0} is 𝔬-stable, since we have θv = σ(v) by definition and Δ^{0} is generated by the v(L) + v(M)’s. Next, \frac{1−δσ}{2} (w^{δ}(L)) = \frac{1−ϵp}{2}v(L) ∈ Δ^{1}, as ϵp ≡ 1 mod 4. This implies that Δ^{1} is 𝔬-stable. This computation also convinces us that Δ^{δ} is 𝔬-stable if and only if p ≡ ±1 mod 8.

Finally, assume p ≡ 3 mod 4. If n is odd, then Δ is unimodular; hence Δ^{δ} is unimodular, and even if p ≡ 7 mod 8. Suppose n is even and p ≡ 7 mod 8. Then direct computation shows that Δ^{δ} ∘ Δ^{δ} ⊆ 𝔬. On the other hand, Δ = θΔ^{#} = Δ^{⊥} and Δ, Δ^{δ} are neighbours. Consequently, Δ^{δ} is a self-dual 𝔬-lattice.

6. Classification of invariant lattices

The aim of this section is to prove Theorem 1.3. The case p^{n} = 3 is trivial (see [SchT], §5), so throughout this section we suppose that p^{n} > 3.

Let H, Γ, Δ be as in Theorem 1.3. Let ρ denote the H-character afforded by Γ and θ any irreducible constituent of ρ restricted to S := Sp_{2n}(p). If n ≥ 2, then the condition 1 < θ(1) ≤ p^{n} + 1 implies by Theorem 5.2 [TZa 1] that θ ∈ \{ψ, \overline{ψ}\}; hence ρ is absolutely irreducible. The same is true if n = 1, except for the cases p = 3 or p ≡ 1 mod 6, where ρ|_{S} can be irreducible. Thus, under the assumptions of Theorem 1.3, ρ satisfies the assumptions of Lemma 2.1. Hence, G := H/K is as defined in Theorem 1.3, and ρ is afforded by Δ, i.e. ρ = χ, Γ ⊗ ℂ = Δ ⊗ ℂ. By the Deuring-Noether Theorem, the ℚH-modules Γ ⊗ ℚ and Δ ⊗ ℚ are equivalent. Therefore, without loss of generality one may suppose that Γ is a G-invariant sublattice in Δ. Thus the proof of Theorem 1.3 reduces to the classification of G-invariant sublattices Γ in Δ.

For any Lagrangian L, let S(L) be as defined in (5) and R(L) = S ∩ S(L). We start with the following observation.

Lemma 6.1. Let n ≥ 2. Then the restriction of χ mod p to R(L) contains the trivial character with multiplicity ≤ 4.

Proof. Consider the standard embedding T := SL_{n}(p) ↪ R(L) ⊆ Sp_{2n}(p). It suffices to show that 1_{T} enters (χ mod p)|_{T} with multiplicity at most 4. Let θ denote the S-character of the Weil representation W (then θ is the sum of ψ and another character of degree (p^{n} − 1)/2). If n ≥ 3, then Zalesskii’s formula for (θ mod p)|_{T} [Zal] tells us that this character contains 1_{T} with multiplicity 2. Since χ|_{S} = ψ + \overline{ψ}, we are done. Now let n = 2. Then due to [Tiep 4], §3,

θ|_{T} = 2 · 1_{T} + ξ_{1} + ξ_{2} + St + 2(χ_{1} + … + χ_{(p−3)/2}),

where ξ_{s}, St, χ_{s} are irreducible characters of T of degree (p + 1)/2, p, and p + 1, respectively. All the nontrivial characters occurring in this formula remain absolutely irreducible, being reduced modulo p. Hence (θ mod p)|_{H} contains 1_{T} with multiplicity 2, and so we are done.

Lemma 6.2. The module V_{p} = Δ/pΔ has a unique nonzero proper G-submodule, and this submodule coincides with ϕ(Δ)/pΔ.

Proof. 1) Let A be any nonzero proper submodule in V_{p}. By Proposition 2.2, the Brauer character afforded by A is η_{i} for some i = 1, 2. In particular, A is absolutely irreducible. An example of such a submodule A is ϕ(Δ)/pΔ. Therefore, the lemma is equivalent to saying that V_{p} is indecomposable. Assume the contrary: V_{p} is decomposable: V_{p} = A ⊕ B. Clearly, A and B are isomorphic as S-modules. Hence, due to Lemma 2.5, in the case p ≡ 1 mod 4, it suffices to prove the lemma for one of the isoclinic groups C_{2} × G_{n}^{+} and G_{n}^{−}. In what follows, we take G = G_{n}^{−} if n is odd, and G = G_{n}^{+} if n is even; furthermore, Δ = Δ(p, n).

If n = 1, then due to [Ward 2], A is a unique nonzero submodule of V_{p} (and A is called the modular quadratic residue code). This forces V_{p} to be indecomposable, a contradiction. Therefore from now on we suppose that n ≥ 2.

2) Let L be any Lagrangian. By Proposition 3.1, the subgroup S(L) fixes the vector v(L). Observe that v(L) ∉ pΔ; hence one can view v(L) as a nonzero vector in V_{p}. Set

W (L) = {v ∈ V_{p} | ∀φ ∈ S(L), φ(v) = v}.

Without loss of generality one may suppose that ϑ_{n} ∈ S(L), and so S(L) = ⟨R(L), ϑ_{n}⟩. For brevity, we denote by χ_{S} the restriction of χ mod p = η_{1} + η_{2} to S(L), by χ_{R} the restriction of χ mod p to R(L), by α the trivial character of S(L), by β the nontrivial character of degree 1 of S(L) with Ker β = ⟨R(L), ϑ_{n}^{2}⟩. Write v(L) = a + b for a ∈ A, b ∈ B. Remark that a, b ≠ 0. (Assume the contrary: a = 0. Then v(L) ∈ B for any Lagrangian L. As Δ is generated by the vectors v(M), which are acted on transitively by S, B must be equal to the whole of V_{p}, a contradiction.) Now S(L) fixes each of the subspaces A, B; therefore in fact a, b ∈ W (L).

3) First consider the case n is odd. Then η_{i} is not self-dual by Proposition 2.2; hence A and B are totally singular relative to (·, ·)_{p}, the reduction modulo p of the scalar product. In particular, (a, a)_{p} = (b, b)_{p} = 0. As n ≥ 3, we have:

0 = (v(L), v(L))_{p} = (a + b, a + b)_{p} = (a, a)_{p} + (b, b)_{p} + 2(a, b)_{p} = 2(a, b)_{p},

which implies that (a, b)_{p} = 0. We have just shown that C := ⟨a, b⟩_{𝔽_{p}} ⊆ W (L) is totally singular with respect to (·, ·)_{p}: C ⊆ C^{⊥}. Besides, the S(L)-modules V_{p}/C^{⊥} and C^{∗} are isomorphic. (Recall that det Δ = 1 in the case n is odd.) From this it follows that V_{p}/C^{⊥} affords the S(L)-character 2α. Thus χ_{S} contains α with multiplicity at least 4. In the proof of Proposition 3.1 we have singled out some subspace U of V, which is acted on by S(L) with character α + β. From this it follows that χ_{S} contains 4α + β, and so χ_{R} contains 1_{R(L)} with multiplicity at least 5, contrary to Lemma 6.1.

4) Finally, let n be even. Then both η_{1}, η_{2} are of type +. Namely, the form (·, ·)_{p} is non-degenerate on B, since A = pΔ^{#}/pΔ is the radical of the form (·, ·)_{p}. Also, A carries the non-degenerate symmetric form (x + pΔ, y + pΔ)′_{p} = \frac{1}{p}(x, y) mod p. Now (b, b)_{p} = (v(L), v(L))_{p} = 0. Thus C := ⟨b⟩_{𝔽_{p}} ⊆ W (L) is totally singular with respect to (·, ·)_{p}|_{B}: C ⊆ C^{⊥}. Besides, the S(L)-modules B/C^{⊥} and C^{∗} are isomorphic. From this it follows that B/C^{⊥} and of course ⟨a⟩_{𝔽_{p}} afford the S(L)- character α. Thus χ_{S} contains α with multiplicity at least 3.

On the other hand, χ_{S} contains β with multiplicity at least 2. (For set D = ⟨u(L)⟩_{𝔽_{p}}. Since (u(L), u(L)) = p^{n/2+1}, D is totally singular with respect to (·, ·)′_{p}: D ⊆ D^{⊥}. Besides, the S(L)-modules A/D^{⊥} and D^{∗} are isomorphic. From this it follows that A/D^{⊥} affords the S(L)-character β.)

As a consequence, χ_{R} contains 1_{R(L)} with multiplicity at least 5, contradicting Lemma 6.1.

Lemma 6.3. Let r be a prime, G a finite group, and Λ an integral G-invariant lattice with the following properties:

(i) The 𝔽_{r}G-module module U = Λ/rΛ is uniserial, that is, it has a unique composition series U = U_{0} ⊃ U_{1} ⊃ … ⊃ U_{m} = 0;

(ii) Let Λ_{i} be the inverse image of U_{i} in Λ, 0 ≤ i ≤ m−1. Then the 𝔽_{r}G-module Λ_{i}/rΛ_{i} is also uniserial for any i > 0. Suppose that Γ is any G-invariant sublattice in Λ whose index is an r-power. Then Γ is similar to one of the lattices Λ_{i}, 0 ≤ i ≤ m − 1.

Proof. Denote Λ_{m} = rΛ_{0}, Λ_{m+1} = rΛ_{1} and, more generally, Λ_{k+m} = rΛ_{k}, k = 2, 3, . . .. Our assumptions imply that

Λ_{k}/rΛ_{k} ⊃ Λ_{k+1}/rΛ_{k} ⊃ … ⊃ Λ_{k−1+m}/rΛ_{k} ⊃ 0

is the unique composition series of the 𝔽_{r}G-module Λ_{k}/rΛ_{k}. In particular, if Γ lies between Λ_{k} and rΛ_{k}, then Γ = Λ_{k+j} for some j, 0 ≤ j ≤ m, and our claim follows.

Since (Λ : Γ) is an r-power,

Λ_{0} = Λ ⊇ Γ ⊇ r^{n}Λ = Λ_{nm}

for some non-negative integer n. Let ℓ be the minimal non-negative integer such that Λ_{i} ⊇ Γ ⊇ Λ_{i+ℓ} for some i. We prove by induction on ℓ that Γ is equal to some Λ_{k}. If ℓ ≤ m, we are done due to the above observation. Assume ℓ > m. Without loss of generality we may suppose that i = 0. Since Λ_{0} ⊇ Γ + Λ_{ℓ−m} ⊇ Λ_{ℓ−m}, by the induction hypothesis we get Γ + Λ_{ℓ−m} = Λ_{k} for some k, 0 ≤ k ≤ ℓ − m. Now it is clear that Λ_{k} ⊇ Γ ⊇ Λ_{ℓ}. By the minimality of ℓ we must have k = 0, i.e., Γ + Λ_{ℓ−m} = Λ_{0}. This implies

Λ_{0} ⊇ Γ ⊇ p(Γ + Λ_{ℓ−m}) = pΛ_{0} = Λ_{m},

contrary to the choice of ℓ. The induction step is over.

Corollary 6.4. If Γ is any G_{n}-invariant sublattice of Δ with the index (Δ : Γ) being a power of p, then there exists an integer k ≥ 0 such that Γ = ϕ^{k}(Δ).

Proof. By Lemma 6.2, ϕ(Δ)/pΔ is the unique nonzero proper submodule of V_{p}; hence V_{p} is uniserial. Suppose that ϕ(Δ) ⊃ Λ ⊃ pϕ(Δ) for some G_{n}-invariant sublattice Λ. Since gϕg^{−1} = ±ϕ for all g ∈ G_{n}, ϕ^{−1}(Λ) is a G_{n}-stable sublattice lying between Δ and pΔ, which implies that ϕ^{−1}(Λ) = ϕ(Δ), Λ = pΔ. Thus the module ϕ(Δ)/pϕ(Δ) is also uniserial. Now we can apply Lemma 6.3.

Since Δ is an odd lattice, the even part Δ^{0} is a G-invariant sublattice of index 2 containing 2Δ. Also, Δ^{1} = Δ∩2(Δ^{0})^{#} is another G-invariant sublattice containing 2Δ.

Lemma 6.5. The 𝔽_{2}G-module V_{2} = Δ/2Δ has precisely two nontrivial proper submodules, namely, Δ^{i}/2Δ with i = 0, 1. Moreover, if Γ is any G-invariant sublattice of Δ with the index (Δ : Γ) being a power of two, then there exists an integer k ≥ 0 such that

Γ ∈ {2^{k}Δ, 2^{k}Δ^{0}, 2^{k}Δ^{1}}.

Proof. Observe that a is even and b is odd. Hence the lattices Δ(p, n) and Δ^{−}(p, n) have the same Gram matrix modulo 2. In particular, in calculating scalar products modulo 2 we can restrict ourselves to Δ(p, n).

1) At this point we show that S := Sp_{2n}(p) fixes a unique nonzero vector w in V_{2}, and w = v(L) + u(L) for any Lagrangian L. To this end, we first observe that det Δ is odd; hence the reduction (·, ·)_{2} of the scalar product is non-degenerate on V_{2}. Next, putting w(L) = v(L) + u(L), by Theorems 5.3 and 5.8 we see that (w(L), v(M))_{2} = 1 for any arbitrary Lagrangian M. If φ ∈ G and φ(L) = L′, then φ(w(L)) = w(L′) (in V_{2}). Hence (φ(w(L))−w(L), v(M))_{2} = 0. But V_{2} is generated by the vectors v(M) and (·, ·)_{2} is non-degenerate. Therefore, φ(w(L)) = w(L). Thus w := w(L) is G-stable. Conversely, let w′ ∈ V_{2} be a nonzero vector which is fixed by S. Since S acts transitively on the vectors v(M), M any Lagrangian, there exists λ ∈ 𝔽_{2} such that (w′, v(M))_{2} = λ for all M. If λ = 0, then the nondegeneracy of (·, ·)_{2} implies that w′ = 0, contrary to the choice of w′. If λ = 1, then (w − w′, v(M))_{2} = 0, yielding w′ = w.

2) Set U_{0} = Δ^{0}/2Δ, U_{1} = ⟨w⟩_{𝔽_{2}}. Clearly, Δ^{0} and so U_{0} are generated by the vectors of the form v(L) + v(M), L, M any Lagrangians. Since (w, v(L) + v(M))_{2} = 0, we see that U_{1} = Δ^{1}/2Δ. Also, w ∈ U_{0}. By Proposition 2.2 (ii), 0 ⊂ U_{1} ⊂ U_{0} ⊂ V_{2} is a composition series of the 𝔽_{2}G-module V_{2}, with two trivial composition factors and one (absolutely) irreducible factor of dimension p^{n} − 1. Clearly, U_{0} and U_{1} are dual to each other w.r.t. (·, ·)_{2}.

Now let U be any nonzero proper G-submodule in V_{2}. Then dim U ∈ {1, 2, p^{n} − 1, p^{n}}. If dim U = 1, then U must be generated by a nonzero G-stable vector; hence U = U_{1} due to 1). If dim U = p^{n}, then the dual module U^{⊥} has dimension 1; therefore U^{⊥} = U_{1}, which implies that U = U_{0}. Assume dim U = 2. Then the action of S on U induces a homomorphism from S to GL(U) = GL_{2}(2) ≃ 𝕊_{3}. But S = Sp_{2n}(p) is perfect (as p^{n} > 3); therefore this homomorphism is trivial, i.e. S acts trivially on U. In this case, V_{2} has at least three (distinct) S-stable vectors, contrary to 1). If dim U = p^{n} − 1, then the dual module U^{⊥} has dimension 2, again a contradiction.

We have shown that V_{2} has just two nontrivial proper submodules: U_{0} and U_{1}.

3) Next we consider any nontrivial proper submodule U in V_{4} = Δ/4Δ, and suppose that U ⊈ 2V_{4}. Then (U +2V_{4})/2V_{4} is a nonzero submodule in V_{4}/2V_{4} ≃ V_{2}. By the results of 2), (U +2V_{4})/2V_{4} contains U_{1}. From this it follows that U contains a vector w′ = w + 2x for a certain x ∈ V_{4}. Pick an element φ ∈ G(L) such that φ : v(L) ↦ v(L), u(L) ↦ −u(L). Then w′ +φ(w′) = 2y, where y = v(L)+x+φ(x). Since x + φ(x) ∈ U_{0} = ⟨w⟩^{⊥}, we get (w, y)_{2} = (w, v(L))_{2} = 1, which means that y ∉ U_{0}. We have seen that U′ = (U ∩2V_{4})/4V_{4} is a G-submodule in 2V_{4}/4V_{4} ≃ 2V_{2}, which contains a vector 2y ∉ 2U_{0}. By the results of 2), U′ = 2V_{4}/4V_{4}, i.e. U ⊇ 2V_{4}. This means: if U is any G-submodule of V_{4}, then either U ⊇ 2V_{4}, or U ⊆ 2V_{4}.

4) Finally, let Γ be any G-invariant sublattice of Δ with (Δ : Γ) = 2^{m}. Then Δ ⊇ Γ ⊇ 2^{m}Δ. We prove by induction on m ≥ 0 that there exists an integer k ≥ 0 such that Γ ∈ {2^{k}Δ, 2^{k}Δ^{0}, 2^{k}Δ^{1}}. This claim is obvious if m = 0 or 1 (see item 2)). Now assume m ≥ 2. Then Δ ⊇ Γ + 4Δ ⊇ 4Δ. Due to 3), either Γ + 4Δ ⊆ 2Δ, or Γ + 4Δ ⊇ 2Δ. In the former case, 2Δ ⊇ Γ ⊇ 2^{m}Δ; therefore Δ ⊇ \frac{1}{2}Γ ⊇ 2^{m−1}Δ, and one can now use the induction hypothesis. In the latter case,

2^{m−1}Δ ⊆ 2^{m−2}(Γ + 4Δ) = 2^{m−2}Γ + 2^{m}Δ ⊆ Γ,

and one can again use the induction hypothesis.

Proof of Theorem 1.3. Consider any G-invariant lattice Γ lying in Δ. We may suppose that Γ ⊈ kΔ for any integer k > 1. Clearly, Γ ⊇ lΔ for some natural l. Choose minimal natural ℓ with the property Γ ⊇ ℓΔ. If ℓ = 1, then Γ = Δ. Assume that ℓ > 1. Claim that ℓ = 2^{a}p^{b} for some non-negative integers a, b. (For assume the contrary: ℓ is divisible by an odd prime r, r ≠ p. Observe that (Γ + rΔ)/rΔ is a nonzero G-module in V_{r} = Δ/rΔ. By Proposition 2.2 (i), (Γ + rΔ)/rΔ = V_{r}, Γ + rΔ = Δ. Hence,

\frac{ℓ}{r} Δ = \frac{ℓ}{r} (Γ + rΔ) = \frac{ℓ}{r} Γ + ℓΔ ⊆ Γ,

contradicting the minimality of ℓ.)

Setting Γ̃ = Γ + p^{b}Δ, one has

Δ ⊃ Γ̃ ⊇ p^{b}Δ, Γ̃ ⊇ Γ ⊇ 2^{a}Γ + ℓΔ = 2^{a}Γ̃.

By Corollary 6.4, Γ̃ = ϕ^{k}(Δ). Replacing Γ by ϕ^{−k}(Γ), which is isometrically similar to Γ, we can suppose that k = 0, i.e. Γ̃ = Δ. In this case, Δ ⊃ Γ ⊇ 2^{a}Δ. By Lemma 6.5, Γ is similar to one of the lattices Δ, Δ^{0}, Δ^{1}.

7. Properties of Δ(p, n)

This section is very sketchy, because a detailed exposition has been given in [SchT], §§4, 6. It turns out that the arguments, given there for the case p^{n} ≡ 3 mod 4, are also applicable to the case p^{n} ≡ 1 mod 4. Hence we restrict ourselves to exposing the results, which hold for any odd prime p, but omitting the proofs.

For short we denote G = G_{n}^{−} if n is odd, and G = G_{n}^{+} if n is even. Furthermore, p is any odd prime and Δ = Δ(p, n).

First we consider the G-invariant odd unimodular lattice Δ = Δ(p, 3) obtained in Theorem 3.9. The generating vectors v(L) now have norm (v(L), v(L)) = p, and Δ contains a p-scaled unit lattice Γ, spanned by N := p^{3} + 1 pairwise orthogonal vectors of norm p (for instance, the v(L), where L runs over a symplectic spread). Therefore, Δ can be described (non-canonically) by a subspace C := Δ/Γ ⊂ Γ^{#}/Γ = \frac{1}{p}Γ/Γ ≃ 𝔽_{p}^{N}, that is, by a linear code over 𝔽_{p}. In this way we obtain an injective mapping π ↦ C = C(π) from the set S of all isomorphism classes of symplectic spreads π of W = 𝔽_{p}^{6} to the set C of all equivalence classes of self-dual codes C of length p^{3}+1 over 𝔽_{p}. Moreover, Aut(C(π)) = Aut(π)/C_{(p−1)/2}. Observe that the definition of Aut(π) used in this paper differs from the one given in [SchT]. In particular, the central subgroup C_{(p−1)/2} of Aut(π) acts trivially on every vector v(L), hence on C(π).

Now we turn to the case n ≥ 5 and n is odd. Let

π = {W_{i} | 1 ≤ i ≤ p^{n} + 1}

be a symplectic spread of W = 𝔽_{p}^{2n}. Set

v_{i} = v(W_{i}), Γ = Δ(π) = ⟨v_{i} | 1 ≤ i ≤ p^{n} + 1⟩_{ℤ}.

For brevity we denote ℓ = (n − 1)/2. Then

Γ ⊂ Δ = Δ^{#} ⊂ Γ^{#} = p^{−ℓ}Γ.

For each j, 1 ≤ j ≤ ℓ + 1, one can view H_{j} = p^{j−1}Γ^{#}/p^{j}Γ^{#} as standard orthogonal space over 𝔽_{p}, with the basis (p^{j−1−ℓ}v_{i} | 1 ≤ i ≤ p^{n} + 1) and with the form (x, y)_{(j)} = p^{ℓ+2−2j}(x, y) mod p. (Here and below, we identify the coset x + p^{j}Γ^{#} with x.) Clearly, the H_{j}’s are isometric to each other, and so one can identify them canonically with H = H_{ℓ+1}. Keeping this identification in mind, we can view every factor-group

C_{j} = ((Δ ∩ p^{j−1}Γ^{#}) + p^{j}Γ^{#})/p^{j}Γ^{#}

as a linear code of length p^{n} + 1 over 𝔽_{p}, with the ambient space H. It is obvious that C_{1} ⊆ C_{2} ⊆ … ⊆ C_{ℓ}. One shows that C_{j}^{⊥} = C_{ℓ+1−j} for 1 ≤ j ≤ ℓ. In particular, C_{j} is self-orthogonal if 1 ≤ j ≤ (ℓ + 1)/2; and C_{(n+1)/4} is self-dual if n ≡ 3 mod 4.

Now we take π to be the standard symplectic spread π_{D}. Then the same arguments as in the proof of Proposition 4.6 [SchT] assure that all the codes C_{j}, 1 ≤ j ≤ ℓ, are among the GL_{2}(q)-codes having a 𝔽_{p}-form, which have been introduced by Ward in [Ward 2]. (Actually, Ward uses an irreducible representation of H = GL_{2}(q) with kernel T = C_{(q−1)/2}, where q = p^{n}. But H/T ≃ R/K, where R = SL_{2}(q)·C_{p−1} and K = R∩T ≃ C_{(p−1)/2}, cf. page 1 of the proof of Proposition 2.3. Now Ward’s representation coincides with the action of R/K on Δ.) He has shown that the lattice of his GL_{2}(q)-codes is inversely isomorphic to the lattice of the so-called closed subsets of 𝔽_{2}^{n}. He has also distinguished the following analogues of Reed-Muller codes. View elements of 𝔽_{2}^{n} as binary words of length n and take B_{w} to be the set of all binary words of length n and weight ≤ w. Then B_{w} is closed and cyclic (in the sense of [Ward 2]), and Ward’s correspondence gives us a GL_{2}(q)-code C_{n,w} over 𝔽_{p}, 0 ≤ w ≤ n − 1. The middle code is just C_{n,(n−1)/2}; more generally, C_{n,w}^{⊥} = C_{n,n−1−w}. We conjecture that the above codes C_{j} are equal to C_{n,n−2j} for j, 1 ≤ j ≤ ℓ = (n − 1)/2. Without this conjecture, we can only give the following lower bound for the minimum of Δ which is unfortunately independent of n. A proof of the conjecture would lead to a lower bound (p^{[n/2]} + 1)/2 instead of (p + 1)/2.

Proposition 7.1. Let p be any odd prime and n ≥ 2 arbitrary. Then

max \{3, \frac{p + 1}{2}\} ≤ min Δ(p, n) ≤ p^{[n/2]}.

For the proof, observe that Δ(p, n) with even n is a sublattice of Δ(p, n + 1); hence it suffices to prove Theorem 7.1 for odd n. The inequality min Δ(p, n) ≥ 3 has been mentioned in Theorem 1.1. Now one repeats the proof of Proposition 6.4 [SchT].

Remark 7.2. Observe that the lattices Δ(p, n), n > 1 odd, are unimodular lattices with relatively short shadow. More precisely, recall that a characteristic vector of a unimodular lattice Λ is any vector w ∈ Λ such that (v, w) ≡ (v, v) mod 2 for all v ∈ Λ, and the coset \frac{1}{2}w + Λ is called the shadow of Λ in [CoS 2]. It is known that (w, w) ≡ rank Λ mod 8 for any characteristic vector w. Define

e(Λ) = \frac{1}{8}(rank Λ − min\{w | w any characteristic vector of Λ\}).

Clearly, e(Λ) = \frac{1}{8} rank Λ if and only if Λ is an even (unimodular) lattice. Elkies [Elk] has shown that Λ = ℤ^{m} is the unique unimodular lattice with e(Λ) = 0; all other lattices have e ≥ 1. Moreover, he has described all the unimodular lattices Λ with e(Λ) = 1.

Clearly, e(Δ(p, 1)) = 0. We observe that e(Δ) = 2, if Δ = Δ(3, 3) or Δ^{−}(5, 2). More generally, we claim that

\frac{1}{8}(p^{k} − 1)(p^{k+1} − 1) ≤ e(Δ) ≤ \frac{1}{8}p^{k+1}(p^{k} − 1)

if Δ := Δ(p, 2k+1), which means in particular that Δ has a relatively short shadow. (For, from Theorems 5.3 and 5.8, it follows that v(L)+u(L) is a characteristic vector of norm p^{k}(p + 1), L any Lagrangian. On the other hand, if w is any characteristic vector, then (w, v(M)) ≡ 1 mod 2 for any Lagrangian M. Hence, if π is a symplectic spread in 𝔽_{p}^{2k+1} and w = ∑_{M∈π} a_{M}v(M), then a_{M} ≠ 0. But p^{k}a_{M} = (v(M), w) ∈ ℤ; hence a_{M} ≥ p^{−k}. As a consequence, (w, w) ≥ p^{−2k} ∑_{M∈π}(v(M), v(M)), and so (w, w) ≥ p^{k+1} + 1.) Specializing p = 3 and k = 1, one gets e(Δ(3, 3)) = 2. Next, let p = 5 and k = 2. Then again v(L) − u(L) is a characteristic vector, of norm 10, yielding e(Δ) ≥ 2 for Δ = Δ^{−}(5, 2). On the other hand, if e(Δ) > 2, then e(Δ) = 3 and Δ would have a (characteristic) vector of norm 26 − 3 · 8 = 2, contrary to the fact that min Δ = 3.

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Department of Mathematics, University of Dortmund, 44221 Dortmund, Germany

E-mail address: rudolf.scharlau@mathematik.uni-dortmund.de

Department of Mathematics, Ohio State University, Columbus, Ohio 43210 Current address: Department of Mathematics, University of Florida, Gainseville, Florida 32611

E-mail address: tiep@math.ufl.edu