PRIMITIVE LOCAL DENSITIES FOR UNIMODULAR QUADRATIC FORMS

DONALD G. JAMES

Abstract. Local densities for the primitive representations of quadratic lattices by even unimodular lattices are determined.

1. Introduction

Let L_{p} be a unimodular ℤ_{p}-lattice on a local quadratic space V_{p}, of finite dimension n ≥ 3, with symmetric bilinear form f : V_{p} ×V_{p} → ℚ_{p}. Put Q(x) = f(x, x). The lattice L_{p} is even when Q(L_{p}) ⊆ 2ℤ_{p}. Let M_{p} be a second regular ℤ_{p}-lattice on a quadratic space with dimension m ≤ n. The lattice L_{p} represents the lattice M_{p} when there exists an injective linear map ϕ : M_{p} → L_{p} preserving the underlying quadratic forms; the representation is primitive if ϕ(M_{p}) is a direct summand of L_{p}. Let S and T be the symmetric matrices associated with L_{p} and M_{p} for some choice of bases. If L_{p} represents M_{p} there exists an n × m integral matrix X such that X^{t}SX = T; the representation induced by X is primitive when X can be expanded to a n×n unimodular matrix.

Following Siegel [14], let B_{q}(S, T) be the number of incongruent primitive integral solutions of X^{t}SX ≡ T mod q, and putting q = p^{l} with p prime, define the p-adic density of the primitive representation as

β_{p}(L_{p}, M_{p}) = lim_{l→∞} q^{m(m+1)/2−nm}B_{q}(S, T).

Let A_{q} be the local ring ℤ_{p}/qℤ_{p}, and O(L_{p}, A_{q}) the orthogonal group of L_{p} over A_{q}. Put 2α_{p}(L_{p}) = β_{p}(L_{p}, L_{p}). Then, with l ≥ 3 when p = 2,

2α_{p}(L_{p}) = |O(L_{p}, A_{q})|q^{−n(n−1)/2}.

A summary of known results can be found in Kitaoka [10], [11]; there is also the recent related work by Katsurada [6], [7], [8], [9] and Yang [15]. In §3, utilizing our methods in [1], [2], [3], we explicitly determine β_{p}(L_{p}, M_{p}) when L_{p} is an even unimodular lattice. The method can be applied to odd L_{2}, but there are now many cases to consider because of the presence of characteristic representations (see [2]); these results will be given in a later paper. Siegel’s theorem on primitive representations by an indefinite ℤ-lattice L in [14] involves the term ∏_{p} β_{p}(L_{p}, M_{p}). Our interest in these primitive densities comes from the results in [5] and [13] where the problem of classifying the conjugacy classes of the maximal Fuchsian subgroups of Bianchi groups was shown to be equivalent to a problem on primitive representations by an indefinite ℤ-lattice. Siegel’s result using primitive local densities was used in [13] to obtain a partial solution, but the methods of [3] later gave a complete answer to this problem (see [4], [5]).

Date: July, 1998.

1991 Mathematics Subject Classification. Primary 11E08; Secondary 11E12.

This research was supported by NSF Grant DMS-9500533.

2. Preliminary results

Let H^{s} = H_{1} ⊥ ⋯ ⊥ H_{s} be a hyperbolic ℤ_{p}-lattice where H_{i} = ℤ_{p}u_{i} + ℤ_{p}v_{i}, f(u_{i}, v_{i}) = 1 and u_{i}, v_{i} are isotropic. If L_{p} has even rank n = 2s, with L_{2} even when p = 2, set χ(L_{p}) = 1 when L_{p} = H^{s} is hyperbolic, and χ(L_{p}) = −1 otherwise. Set δ(2) = 1, and δ(p) = 0 when p is odd. From [10, §5.6] (note the connection between Siegel’s α and Kitaoka’s β on p. 98), we have

Theorem 2.1. Let L_{p} be a unimodular lattice. Then,

α_{p}(L_{p}) = 2^{δ(p)n}(1 + χ(L_{p})p^{−n/2})^{−1} ∏_{i=1}^{n/2}(1 − p^{−2i})

for n even with L_{2} even when p = 2, while for n ≥ 3 odd and p ≠ 2,

α_{p}(L_{p}) = ∏_{i=1}^{(n−1)/2}(1 − p^{−2i}).

In particular, this theorem determines the order of the orthogonal group O(L_{p}, A_{q}).

A lattice M_{p}, with f(M_{p}, M_{p}) ⊆ ℤ_{p}, can be split as an orthogonal sum M_{p} = M_{p}^{0} ⊥ M_{p}^{1} with M_{p}^{0} unimodular and f(M_{p}^{1}, M_{p}^{1}) ⊆ pℤ_{p} (see [10, §5.3]). Put m_{p} = rank M_{p}^{1}, an invariant of M_{p}. If L_{p} primitively represents M_{p}, then m + m_{p} ≤ n (see Theorem 1 in [1]), and we may assume L_{p} = M_{p}^{0} ⊥ N with N unimodular. Define e = e(M_{2}) = 0 when M_{2} has an odd 2-modular component; otherwise, set e = e(M_{p}) = 1. When m + m_{p} = n and e(M_{p}) = 1, the lattice N must be hyperbolic (see [1]). The following is a variation of Corollary 5.6.1 in [10].

Theorem 2.2. Let M_{p} = M_{p}^{0} ⊥ M_{p}^{1}, as above, be a primitive sublattice of the even unimodular lattice L_{p} = M_{p}^{0} ⊥ N. Then,

α_{p}(L_{p}) = β_{p}(L_{p}, M_{p}^{0})α_{p}(N)

and

β_{p}(L_{p}, M_{p}) = β_{p}(L_{p}, M_{p}^{0})β_{p}(N, M_{p}^{1}).

This result holds since Witt’s extension theorem is valid (see [12]).

3. Local densities for even L_{p}

By 2.2 it suffices to study lattices M_{p} with f(M_{p}, M_{p}) ⊆ pℤ_{p}.

Theorem 3.1. Assume the hyperbolic lattice L_{p} = H^{m} primitively represents M_{p} where n = 2m and f(M_{p}, M_{p}) ⊆ pℤ_{p}. Then

β_{p}(H^{m}, M_{p}) = 2^{e+δ(p)m}(1 − p^{−m}) ∏_{i=1}^{m−1}(1 − p^{−2i}).

Proof. Let q = p^{l} with l ≥ 3, and let x_{1}, …, x_{m} be a basis for M_{p}. When e(M_{2}) = 0 choose x_{1} with Q(x_{1})/2 a unit, and Q(x_{i}) ∈ 4ℤ_{2} for i ≥ 2. By Witt’s theorem, M_{p} can be canonically embedded in the lattice H^{m} with hyperbolic basis u_{i}, v_{i}, 1 ≤ i ≤ m, as

x_{k} = ∑_{i=1}^{k−1} b_{ki}v_{i} + u_{k} + 2^{−1}b_{kk}v_{k}, 1 ≤ k ≤ m, (1)

where b_{kj} = f(x_{k}, x_{j}) ∈ pℤ_{p} (see [3, §3]). Since the group O(L_{p}, A_{q}) acts transitively on primitive representations of M_{p} in L_{p} modulo q,

B_{q}(L_{p}, M_{p}) = [O(L_{p}, A_{q}) : Stab(M_{p}, A_{q})]

where

Stab(M_{p}, A_{q}) = {ϕ ∈ O(L_{p}, A_{q}) | ϕ ≡ I mod q on M_{p}}.

We now determine s(m, q) = |Stab(M_{p}, A_{q})|. First, s(1, q) = 2^{1−e+δ(p)} since, when p = 2, the map ϕ ∈ O(L_{p}, A_{q}) with ϕ(v_{1}) = v_{1} + 2^{−1}qx_{1} always fixes x_{1} = u_{1} + 2^{−1}b_{11}v_{1}, and the symmetry about u_{1} − 2^{−1}b_{11}v_{1} also fixes x_{1} when b_{11}/2 is a unit in ℤ_{2}. Now assume m > 1.

For ϕ ∈ Stab(M_{p}, A_{q}), put

ϕ(v_{m}) = ∑_{i=1}^{m}(a_{i}u_{i} + c_{i}v_{i}),

and let n(v_{m}, q) be the number of possible ϕ(v_{m}) modulo q. Since f(x_{k}, v_{m}) = 0 for k < m, each c_{k} is uniquely determined modulo q in terms of C = {a_{k}, b_{kj} | k < m}. Likewise, f(x_{m}, v_{m}) = 1 and Q(v_{m}) = 0 determines c_{m} ≡ 1 mod p uniquely modulo q in terms of the elements in C and b_{mj}, with two possibilities for a_{m} when p = 2 (after eliminating the unit c_{m}, use a Hensel’s lemma type argument on the equation for a_{m} to determine it uniquely modulo q/2). Thus we are free to choose each a_{i}, for i < m, so that n(v_{m}, q) = 2^{δ(p)}q^{m−1}. Splitting off the unimodular sublattice spanned by x_{m} and ϕ(v_{m}), and restating the problem on the hyperbolic orthogonal complement, gives s(m, q) = 2^{δ(p)}q^{m−1}s(m − 1, q). Hence

s(m, q) = 2^{1−e+δ(p)m}q^{m(m−1)/2},

and the result now follows from 2.1.

Let B denote the binary anisotropic unimodular ℤ_{2}-lattice with f-

matrix (\begin{matrix}2 & 1 \\ 1 & 2 \end{matrix}). Note that B ⊥ B = H^{2}.

Corollary 3.2. Assume the lattice L_{2} = B ⊥ H^{m−1} primitively represents M_{2} where n = 2m, f(M_{2}, M_{2}) = 2ℤ_{2}, and e(M_{2}) = 0. Then

β_{2}(L_{2}, M_{2}) = 2^{−m}α_{2}(L_{2}).

Proof. First B ⊥ L_{2} = H^{m+1} and, since there exists x ∈ M_{2} with Q(x)/2 a unit, N = B ⊥ M_{2} = H ⊥ M′_{2} where f(M′_{2}, M′_{2}) = 2ℤ_{2}. By 2.2,

β_{2}(H^{m+1}, N) = β_{2}(H^{m+1}, B)β_{2}(L_{2}, M_{2})
= α_{2}(H^{m+1})α_{2}(L_{2})^{−1}β_{2}(L_{2}, M_{2}).

Also, by 2.1, 2.2 and 3.1, since e(M′_{2}) = 0,

β_{2}(H^{m+1}, N) = β_{2}(H^{m+1}, H)β_{2}(H^{m}, M′_{2}) = 2^{−m}α_{2}(H^{m+1})

and the result follows.

Lemma 3.3. Let D, E be symmetric even r × r matrices over ℤ_{p} with D unimodular. Then

det D ≡ det(D + pE) mod 4p

and the associated even unimodular lattices are isometric.

Proof. The result is trivial when p is odd so assume p = 2 and hence r is even. Changing the basis associated with D, we may assume all the antidiagonal elements of D are 1, the main diagonal elements are in 2ℤ_{2}, and the remaining elements are all 0. Let d_{ij} be an element in D + 2E not on either the antidiagonal or the main diagonal, and consider a product term t in the expansion of det(D + 2E) containing d_{ij}. Then 8|t unless r − 2 of the other factors in t are units along the antidiagonal, and then t ≡ d_{ij}d_{kl} mod 8 where d_{kl} is the mirror image of d_{ij} in the antidiagonal so that i + l = j + k = r + 1. But the mirror pair, of this pair, in the main diagonal then contributes the term t′ ≡ d_{ji}d_{lk} mod 8, and these two terms cancel modulo 8 by symmetry. Thus det(D + 2E) mod 8 is determined solely from elements on the main diagonal and antidiagonal. The result now easily follows since the main diagonal terms of 2E are in 4ℤ_{2}. In fact,

det D ≡ (−1)^{r/2} ∏_{i+j=r+1}(1 − d_{ii}d_{jj}) ≡ (−1)^{r/2} + ∑_{i+j=r+1} d_{ii}d_{jj} mod 8

where i < j in the product and sum over i + j = r + 1.

Theorem 3.4. Let L_{p} = H^{m} ⊥ N be an even unimodular lattice, and let M_{p} be a lattice with Q(M_{p}) ⊆ 2pℤ_{p} and n − 2m = r > 0. Then, for n odd,

β_{p}(L_{p}, M_{p}) = ∏_{i=(r+1)/2}^{(n−1)/2}(1 − p^{−2i}),

and for n even, including p = 2,

β_{p}(L_{p}, M_{p}) = 2^{δ(p)m}(1 + χ(L_{p})p^{−n/2})^{−1}(1 + χ(L_{p})p^{−r/2}) ∏_{i=r/2+1}^{n/2}(1 − p^{−2i}).

Proof. We may assume that M_{p} = ℤ_{p}x_{1} + ⋯ + ℤ_{p}x_{m} ⊂ H^{m}. Put M′_{p} = M_{p} ⊥ N. Then, from 2.2,

B_{q}(L_{p}, M′_{p}) = B_{q}(L_{p}, N)B_{q}(H^{m}, M_{p}).

Let ϕ : M_{p} → H^{m} be the canonical embedding with ϕ(x_{k}) = x_{k} as above in (1). By Witt’s theorem, any embedding is equivalent to a canonical embedding, so that also

B_{q}(L_{p}, M′_{p}) = n(q)B_{q}(L_{p}, M_{p}),

where n(q) is the number of distinct modulo q extensions of ϕ to M′_{p}. Let N = ℤ_{p}y_{1} + ⋯ + ℤ_{p}y_{r} and consider an extension of the embedding ϕ to N with ϕ(y_{k}) = w_{k} + z_{k}, 1 ≤ k ≤ r, where

w_{k} = ∑_{i=1}^{m}(a_{ki}u_{i} + c_{ki}v_{i})

and z_{k} ∈ N. Then f(ϕ(x_{i}), ϕ(y_{k})) = f(x_{i}, w_{k}) = 0 determines

c_{ki} = −2^{−1}b_{ii}a_{ki} − ∑_{j=1}^{i−1} b_{ij}a_{kj}

with c_{ki} ∈ pℤ_{p} since 2p|Q(x_{i}). Therefore

f(y_{j}, y_{k}) = f(w_{j}, w_{k}) + f(z_{j}, z_{k}) ≡ f(z_{j}, z_{k}) mod p

with Q(w_{k}) = ∑_{i=1}^{m} 2a_{ki}c_{ki} ∈ 2pℤ_{p} and f(w_{j}, w_{k}) ∈ pℤ_{p}. Hence the matrix (f(z_{j}, z_{k})) is unimodular and the z_{k} ∈ N are primitive and linearly independent. Now count the number of distinct modulo q extensions of ϕ to N. There are at most q^{mr} possibilities for the a_{ki}, and at most B_{q}(N, N) = |O(N, A_{q})| possibilities for the z_{k}. Hence the number of extensions of ϕ to M′_{p} is at most q^{mr}B_{q}(N, N). Conversely, all these possibilities occur. For, given any choice of the a_{ki}, the c_{ki} and w_{k} are determined as above, and we can then find z_{k} ∈ N such that the matrix (f(z_{j}, z_{k})) is isometric to

R = (f(y_{j}, y_{k})) − (f(w_{j}, w_{k})) = D + pE

(since det R ≡ det N mod 4p by 3.3). This constructs one ϕ, and the action of O(N, A_{q}) then completes the set. Hence n(q) = q^{mr}B_{q}(N, N). Therefore, using 2.2 and 3.1,

β_{p}(L_{p}, M_{p}) = 2^{−δ(p)m}α_{p}(L_{p})α_{p}(N)^{−1}

and the result follows from 2.1.

The above argument fails when M_{2} contains an odd 2-modular Jordan component, since then we cannot be sure that the z_{k} ∈ N exist to extend ϕ for all choices of a_{ki}. We must determine the fraction of choices of a_{ki} where R and N are isometric.

Theorem 3.5. Let L_{2} = H^{m} ⊥ N be an even unimodular lattice, and M_{2} a lattice with f(M_{2}, M_{2}) = 2ℤ_{2}, e(M_{2}) = 0 and n − 2m = r > 0. Then

β_{2}(L_{2}, M_{2}) = 2^{m}(1 − χ(L_{2})2^{−n/2}) ∏_{i=r/2+1}^{n/2−1}(1 − p^{−2i}).

Proof. Continue the notation in 3.4. Since e(M_{2}) = 0 we may take b_{11} = Q(x_{1}) ≡ 2 mod 4, and also assume 4 divides b_{ii} for i ≥ 2. Then

Q(w_{k}) = ∑_{i=1}^{m} 2a_{ki}c_{ki} ≡ ∑_{i=1}^{m} a_{ki}^{2}b_{ii} ≡ 2a_{k1} mod 4.

Since N is an even unimodular lattice we can choose y_{k} such that f(y_{j}, y_{k}) = d_{jk} with d_{jk} = 1 when j+k = r+1, d_{11} = d_{rr} = 2d ∈ {0, 2}, and all the remaining d_{jk} = 0. Then L_{2} and N are hyperbolic precisely when d = 0. As in 3.3, det N ≡ (−1)^{r/2} + 4d^{2} mod 8 and

det R ≡ (−1)^{r/2} + ∑_{i+j=r+1}(d_{ii} − 2a_{i1})(d_{jj} − 2a_{j1}) mod 8

where it is again understood that i < j in the sum over i + j = r + 1. Hence det N ≡ det R mod 8 if and only if

d(a_{11} + a_{r1}) ≡ ∑_{i+j=r+1} a_{i1}a_{j1} mod 2.

Let A(r) = a(r)2^{−r} where a(r) is the number of distinct sets of solutions to this congruence for the elements a_{i1}. When d = 0 we find that a(2) = 3 and inductively

a(r + 2) = 3a(r) + (2^{r} − a(r)) = 2a(r) + 2^{r}.

The case d = 1 is similar, but now a(2) = 1. It follows that a(r + 2) = 2^{r+1} + χ(L_{2})2^{r/2} and 2A(r) = 1 + χ(L_{2})2^{−r/2}. The rest of the proof is similar to 3.4, but with n(q) = q^{mr}B_{q}(N, N)A(r).

References

[1] D.G. James, Primitive representations by unimodular quadratic forms, J. Number Theory 44 (1993), 356-366.

[2] D.G. James, Representations by unimodular quadratic ℤ-lattices, Math. Z. 215 (1994), 465-475.

[3] D.G. James, The number of embeddings of quadratic S-lattices, in “Ktheory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras,” Proc. Symp. Pure Math. 58.2 (1995), 265-274.

[4] D.G. James, Bianchi groups and integral quadratic forms, in “Proceedings of the Beijing International Symposium on Group Theory” (to appear).

[5] D.G. James and C. Maclachlan, Fuchsian subgroups of Bianchi groups, Trans. Amer. Math. Soc. 348 (1996), 1989-2002.

[6] H. Katsurada, A certain formal power series of several variables attached to local densities of quadratic forms I, J. Number Theory 51 (1995), 169-209.

[7] H. Katsurada, A certain formal power series attached to local densities of quadratic forms II, Proc. Japan Acad. 70(A) (1994), 208-211.

[8] H. Katsurada, An explicit formula for Siegel series, Amer. J. Math. 121 (1999), 415–452.

[9] H. Katsurada and M. Hisasue, A recursion formula for local densities, J. Number Theory 64 (1997), 183–210

[10] Y. Kitaoka, Arithmetic of Quadratic Forms, Cambridge University Press (1993).

[11] Y. Kitaoka, Representations of positive definite quadratic forms and analytic number theory, Sugaku Expositions 7 (1994), 1-17.

[12] M. Kneser, Witts Satz für quadratische Formen über lokalen Ringen, Nachr. Akad. Wiss. Gött. Math.-Phys II 9 (1972), 195-203.

[13] C. Maclachlan and A.W. Reid, Parametrizing Fuchsian subgroups of the Bianchi groups, Canadian J. Math. 43 (1991), 158-181.

[14] C.L. Siegel, On the theory of indefinite quadratic forms, Ann. of Math. 45 (1944), 577-622.

[15] T. Yang, An explicit formula for local densities of quadratic forms, J. Number Theory 72 (1998), 309–356.

Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

  E-mail address: james@@math.psu.edu