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Theta Functions and Weighted Theta Functions of Euclidean Lattices, with Some Applications

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Theta Functions and Weighted Theta Functions of Euclidean Lattices, with Some Applications Theta functions and weighted theta functions of Euclidean lattices, with some applications Noam D. Elkies1 March, 2009 0. Introduction and overview [...] 1. Lattices in Rn: basic terminology, notations, and examples By “Euclidean space” of dimension n we mean a real vector space of dimension n, equipped with a positive-definite inner product , . We usually call such a space “Rn” even when there is no distinguished choiceh· of·i coordinates. A lattice in Rn is a discrete co-compact subgroup L Rn, that is, a discrete sub- ⊂ group such that the quotient Rn/L is compact (and thus necessarily homeomorphic with the n-torus (R/Z)n). As an abstract group L is thus isomorphic with the free abelian group Zn of rank n. Therefore L is determined by the images, call them v ,...,v , of the standard generators of Zn under a group isomorphism Zn ∼ L. 1 n → We say the vi generate, or are generators of, L: each vector in L can be written as n a v for some unique integers a ,...,a . Vectors v ,...,v Rn generate i=1 i i 1 n 1 n ∈ a lattice if and only if they constitute an R-linear basis for Rn, and then L is the ZP-span of this basis. For instance, the Z-span of the standard orthonormal basis n n e1,...,en of R is the lattice Z . This more concrete definition is better suited for explicit computation, but less canonical because most lattices have no canonical choice of generators even up to isometries of Rn. An equivalent approach defines L as a free abelian group of rank n with a positive- definite symmetric pairing; the Euclidean space is then recovered as L Z R, ⊗ which inherits a symmetric bilinear pairing from L. With this approach we must be careful about the definition of a “positive-definite pairing” so that the extension to L Z R remains positive-definite. For most of the lattices we consider, the pairing takes⊗ values in Q, and then the usual definition for inner products suffices: v L, v = 0 = v, v > 0. (1) ∈ 6 ⇒ h i 1Department of Mathematics, Harvard University, Cambridge, MA 02138 USA; e-mail: ([email protected]). Supported in part by NSF grant DMS-0501029. DRAFT1 But for a general R-valued pairing, (1) guarantees only that the pairing on L Z R ⊗ is positive semidefinite, not necessarily positive definite: a standard counterexam- ple is L = Z2 with v, w =(v tv )(w tw ) h i 1 − 2 1 − 2 for some irrational t, when the nonzero vector x = (t, 1) L Z R satisfies ∈ ⊗ x,x = 0. For this general case we say the pairing , is “positive-definite” if h i h· ·i it has a positive-definite Gram matrix A. The Gram matrix of , with respect to h· ·i generators v ,...,v of L is the symmetric matrix with (i,j) entry A = v , v . 1 n ij h i ji The Gram matrix depends on the choice of generators, but if w1,...,wn are any n other generators then each wk = i=1 bikvi for some integers bik forming a matrix B of determinant 1, and the Gram matrix with respect to w ,...,w is BTAB, ± P 1 n which is positive-definite if and only if A is. If v = m v + + m v for some 1 1 ··· n n m =(m ,...,m ) Zn then v, v is the value at m of the quadratic form 1 n ∈ h i n n n n (m,AmT) = A m m = v , v m m , (2) ij i j h i ji i j Xi=1 Xj=1 Xi=1 Xj=1 a homogeneous polynomial of degree 2 in n variables. We next recall some further invariants2 of L and the corresponding properties of the Gram matrix. The discriminant of L is disc L = (Vol(Rn/L))2 = det A; (3) in particular, det A is independent of the choice of generators, which we can also verify directly: if det B = 1 then det A = det BTAB. The volume √disc L of ± the torus Rn/L is known as the covolume of L. A lattice of discriminant 1 is said to be unimodular. The dual lattice L∗ is defined by n L∗ = v∗ R v L, v, v′ Z . (4) { ∈ | ∀ ∈ h i ∈ } If L is the Z-span of v1,...,vn with Gram matrix A, then L∗ is the Z-span of the T 1 dual basis v1∗,...,vn∗ with Gram matrix (A )− ; in particular 1 disc L∗ = (disc L)− . (5) We say L is integral if v, v Z for all v, v L; equivalently, if L L . In this h ′i ∈ ′ ∈ ⊆ ∗ case L∗/L is a finite group with #(L∗/L) = disc(L). In particular, L = L∗ if and only if L is integral and unimodular; we naturally say such a lattice is self-dual. 2These are “invariant” in the sense that they do not depend on a choice of generators, nor on other extrinsic features such as an embedding in Rn. The Gram matrix depends on the choice of generators, and is thus not an invariant: different Gram matrices may give rise to the same lattice. DRAFT2 The basic example of a lattice is Z with the pairing x, y = xy; this lattice is self- h i dual, and is the unique unimodular lattice in the 1-dimensional Euclidean space R. More generally, for every real D > 0 there is a unique lattice of discriminant D in R, namely D1/2Z, or equivalently Z with the pairing x, y = Dxy instead; h i this lattice is integral if and only if D Z. ∈ We next give some constructions of new lattices from old. A subgroup L′ of finite n n index in a lattice L in R is itself a lattice in R whose dual contains L∗. Com- paring covolumes, we see that [L′∗ : L∗]=[L : L′]; in fact much more is true: the n inner product on R induces a perfect pairing (L′∗/L∗) (L/L′) Q/Z on the quotient subgroups, so in particular these subgroups are× isomorphic,→ albeit not in general canonically isomorphic. If v ,...,v generate L and w ,...,w generate L , then [L : L ] = det B , 1 n 1 n ′ ′ | | where B is the integer matrix formed by the coefficients bik of the expansions n wk = i=1 bikvi. If A is the Gram matrix of L with respect to the vi, then as before det BTAB is the Gram matrix of L with respect to the w . Therefore P ′ k 2 disc L′ =[L : L′] disc L, (6) an identity that can also be obtained from the first equality in (3) because n n Vol(R /L′)=[L : L′] Vol(R /L). If L is integral then so is L′. In the other direction, if L′ is integral, let G be any set of generators of L/L′, and G˜ an arbitrary lift of G to a subset of L; then L is integral if and only if L L and g,˜ g˜ Z for all g,˜ g˜ G˜. ⊂ ′∗ h ′i ∈ ′ ∈ n1 n2 If n = n1 + n2 and L1 and L2 are lattices in R and R respectively, then the direct sum L = L L := (v , v ) v L , v L (7) 1 ⊕ 2 { 1 2 | 1 ∈ 1 2 ∈ 2} n is a lattice in R of discriminant disc(L1) disc(L2); if Ai (i = 1, 2) is a Gram matrix for Li then L has a block-diagonal Gram matrix with blocks A1, A2. The dual of L L is L L . The direct sum L is integral if and only if L and 1 ⊕ 2 1∗ ⊕ 2∗ 1 L2 are integral, and self-dual if and only if L1 and L2 are self-dual. For example, Z2 = Z Z R2 is self-dual; iterating the construction yields the self-dual lattice ⊕ ⊂ Zn Rn for each n =1, 2, 3,.... ⊂ Of course once n > 1 this self-dual lattice is no longer unique, because we can obtain uncountably many others by applying an orthogonal linear transformation to Rn; but the resulting lattices are isomorphic. It is known that for n 7 every ≤ self-dual lattice is isomorphic with Zn (see Proposition 7 below for one approach to this result), and for every n there are only finitely many isomorphism classes. DRAFT3 2 But for large n the number of isomorphism classes grows rapidly, exceeding (cn)n for some positive c. We conclude this section by constructing the self-dual lattice E R8 and showing that E = Z8. 8 ⊂ 8 6∼ For n =1, 2, 3,..., define a lattice D Rn by n ∈ n D = (x ,...,x ) Zn : x 0 mod 2 , (8) n 1 n ∈ i ≡ i=1 X 3 n a sublattice of Z of index 2. An explicit Z-basis consists of ei + ei+1 for 0 < n i < n together with 2e1. The dual lattice Dn∗ is the union of Z and the translate of Zn by the half-lattice vector n 1 h := (1, 1,..., 1)/2 = e . (9) 2 i Xi=1 The 4-element quotient group Dn∗ /Dn is cyclic if n is odd, and of exponent 2 when n is even. In the latter case, 2h D , so ∈ n D+ := D (D + h) (10) n n ∪ n + n is a lattice, which is unimodular because [Dn : Dn]=2=[Z : Dn]. An explicit basis consists of e + e for 0 <i<n 1 together with 2e and h.
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