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37  Discriminant forms and the genus

Remark 37.1 The Lattice Theory section introduced the dual lattice \(\dualof{L}\), the discriminant group \(A_L \definedas \dualof{L}/L\), and the discriminant quadratic form \(q_L: A_L \to \QQ/2\ZZ\) of an even lattice. We now develop the theory these constructions specialize: the general correspondence between symmetric bilinear forms and quadratic forms, its torsion-valued counterpart, the resulting structure on \(A_L\), the properties of the dual lattice, and the classification of lattices up to genus. Throughout, \((L, \beta_L)\) denotes a lattice in the sense of the Lattice Theory section, with \(L_\QQ \definedas L\tensor_\ZZ \QQ\) and \(\beta_{L_\QQ}\) the \(\QQ\)-linear extension of \(\beta_L\).

37.1 Quadratic forms and the polarization identity

Definition 37.2 A quadratic form on a \(\ZZ\)-module \(L\) is a map of sets \(q: L \to \QQ\) such that \(q(\lambda v) = \lambda^2 q(v)\) for all \(v\in L\) and all \(\lambda\in\ZZ\), and whose polar form \(\beta_q\) is a symmetric bilinear form on \(L\): \[ \begin{aligned} \beta_q: L \tensor_\ZZ L &\to \QQ \\ (v, w) &\mapsto \beta_q(v, w) \definedas q(v+w) - q(v) - q(w) . \end{aligned} \] We say \(q\) is integral if \(q(L) \iscontainedin \ZZ\), and we call the pair \((L, q)\) a quadratic \(\ZZ\)-module.

Remark 37.3 Setting \(w = v\) in the polar form and using homogeneity gives \[ \beta_q(v, v) = q(2v) - 2q(v) = 4q(v) - 2q(v) = 2q(v) , \] so that \(q(v) = \tfrac{1}{2}\beta_q(v, v)\). The polar form is thus always even, meaning \(\beta_q(v, v)\in 2\ZZ\) whenever \(q\) is integral. This identity is the source of the bijection recorded below.

37.2 The correspondence between bilinear and quadratic forms

Lemma 37.4 Every \(\QQ\)-valued symmetric bilinear module \((L, \beta)\) determines a \(\QQ\)-valued quadratic module \((L, q_\beta)\) by \[ q_\beta(v) \definedas \beta(v, v) , \] and \(q_\beta\) depends only on the symmetric part of \(\beta\). Conversely, every \(\QQ\)-valued quadratic module \((L, q)\) determines a symmetric bilinear module \((L, \beta_q)\) via its polar form from the quadratic-form definition (Definition 37.2).

Lemma 37.5 There is a bijection between even symmetric integral bilinear forms on \(L\) and integral quadratic forms on \(L\): \[ \theset{ \beta\in \Sym^2_\ZZ(\dualof{L}) \mid \beta \text{ is even} } \quad\longleftrightarrow\quad \Quad_\ZZ(L) , \] under which a bilinear form \(\beta\) is sent to \(q(v) \definedas \tfrac{1}{2}\beta(v, v)\), and a quadratic form \(q\) is sent to its polar form \(\beta_q\). Note that this forward map \(\beta\mapsto\tfrac{1}{2}\beta(v,v)\) differs by the factor \(\tfrac{1}{2}\) from the map \(\beta\mapsto q_\beta(v) = \beta(v,v)\) of the bilinear/quadratic correspondence lemma (Lemma 37.4), so this bijection is a distinct construction rather than a restriction of that lemma; only the backward (polar-form) direction is shared. Concretely, the polar form of any integral quadratic form is an even symmetric integral bilinear form, and conversely every even symmetric integral bilinear form \(\beta\) is the polar form of the integral quadratic form \(q(v) \definedas \tfrac{1}{2}\beta(v, v)\).

Proof. If \(q\) is integral then \(\beta_q(v, v) = 2q(v)\in 2\ZZ\), so \(\beta_q\) is even and integral. Conversely, if \(\beta\) is even and integral then \(q(v) \definedas \tfrac{1}{2}\beta(v, v)\) takes values in \(\ZZ\), and its polar form recovers \(\beta\): \[ \beta_q(v, w) = q(v + w) - q(v) - q(w) = \tfrac{1}{2}\left( \beta(v + w, v + w) - \beta(v, v) - \beta(w, w) \right) = \beta(v, w) , \] using symmetry of \(\beta\). The two assignments are mutually inverse.

37.3 Torsion bilinear and quadratic forms

Definition 37.6 A torsion bilinear form is a pair \((G, \beta)\) where \(G\) is a finitely generated torsion \(\ZZ\)-module and \[ \beta: G \tensor_\ZZ G \to \QQ/\ZZ \] is a symmetric bilinear form. A torsion quadratic form is a pair \((G, q)\) where \(G\) is a finitely generated torsion \(\ZZ\)-module and \(q: G \to \QQ/\ZZ\) is a quadratic form, i.e. \(q(\lambda x) = \lambda^2 q(x)\) for all \(x\in G\) and \(\lambda\in\ZZ\), whose polar form is a torsion bilinear form.

Remark 37.7 The discriminant group \(A_L\) of the Lattice Theory section is a finite, hence finitely generated torsion, \(\ZZ\)-module, so its associated forms are instances of this notion. We record the discriminant data as two separate forms: the discriminant quadratic form \(q_L\colon A_L\to\QQ/2\ZZ\) (matching the \(\QQ/2\ZZ\) normalization of the Lattice Theory section) and the discriminant bilinear form \(b_L\colon A_L\times A_L\to\QQ/\ZZ\). These share the numerator \(\beta(\tilde x,\tilde y)\) read modulo different lattices; the diagonal \(q_L(\bar x) = \beta(\tilde x,\tilde x)\bmod 2\ZZ\) reduces to \(b_L(\bar x,\bar x) = \beta(\tilde x,\tilde x)\bmod\ZZ\) under the natural surjection \(\QQ/2\ZZ\surjects\QQ/\ZZ\), not under any multiplication-by-\(2\) isomorphism.

37.4 The discriminant bilinear and quadratic forms

Definition 37.8 Let \((L, \beta_L)\) be a nondegenerate even lattice with discriminant group \(A_L = \dualof{L}/L\), and let \(\beta\) also denote the \(\QQ\)-valued extension of the form to \(\dualof{L}\). The discriminant bilinear form of \(L\) is the torsion bilinear form \[ \begin{aligned} b_L: A_L \times A_L &\to \QQ/\ZZ \\ (\bar x, \bar y) &\mapsto \beta(x, y) \bmod \ZZ , \end{aligned} \] computed on any lifts \(x, y\in \dualof{L}\) of \(\bar x, \bar y\). Its associated discriminant quadratic form is \[ q_L(\bar x) \definedas \beta(x, x) \bmod 2\ZZ \in \QQ/2\ZZ , \] for any lift \(x\) of \(\bar x\); this is the \(\QQ/2\ZZ\)-valued form \(q_L\) recalled from the Lattice Theory section. Reducing modulo \(\ZZ\) recovers the diagonal of \(b_L\), i.e. \(q_L(\bar x)\bmod\ZZ = b_L(\bar x,\bar x)\), via the surjection \(\QQ/2\ZZ\surjects\QQ/\ZZ\). The orthogonal group \(\Orth(A_L)\) is the group of automorphisms of \(A_L\) preserving \(q_L\). The length \(\ell(L)\) of \(L\) is the minimal number of generators of the abelian group \(A_L\).

Remark 37.9 Both \(b_L\) and \(q_L\) are well defined: replacing a lift \(x\) by \(x + m\) with \(m\in L\) changes \(\beta(x, y)\) by \(\beta(m, y)\in\ZZ\) (so \(b_L\) is well defined modulo \(\ZZ\)) and changes \(\beta(x, x)\) by \(\beta(2x, m) + \beta(m, m)\in 2\ZZ\) (so \(q_L\) is well defined modulo \(2\ZZ\)), since \(\beta(\dualof{L}, L)\iscontainedin\ZZ\) and \(L\) is even. These forms are Nikulin’s discriminant forms (Nikulin 1980).

Proposition 37.10 The discriminant forms \(b_L\) and \(q_L\) of a nondegenerate lattice \(L\) are themselves nondegenerate, meaning that \(b_L(\bar x, \,\cdot\,) = 0\) in \(\Hom(A_L, \QQ/\ZZ)\) implies \(\bar x = 0\). For any \(\bar x, \bar y\in A_L\) the \(\QQ/\ZZ\)-valued bilinear form is recovered from the \(\QQ/2\ZZ\)-valued quadratic form by \[ b_L(\bar x, \bar y) = \tfrac{1}{2}\left( q_L(\bar x + \bar y) - q_L(\bar x) - q_L(\bar y) \right) , \] where the bracketed difference lies in \(\QQ/2\ZZ\) and equals \(2\beta(\tilde x, \tilde y)\bmod 2\ZZ\); halving this even representative yields a well-defined element \(\beta(\tilde x, \tilde y)\bmod\ZZ\) of \(\QQ/\ZZ\).

Proof. Nondegeneracy is the statement that the induced map \(A_L \to \Hom(A_L, \QQ/\ZZ)\) is an isomorphism; this holds because \(A_L\) is finite and the pairing \(b_L\) is the pairing induced by the perfect pairing \(\dualof{L}/L \times \dualof{L}/L \to \QQ/\ZZ\) coming from a nondegenerate \(\beta_L\). The polarization identity is the reduction modulo \(\ZZ\) of the identity \(\beta(x, y) = \tfrac{1}{2}(\beta(x + y, x + y) - \beta(x, x) - \beta(y, y))\) on lifts.

37.5 Properties of the dual lattice

Proposition 37.11

Let \(L\) and \(M\) be nondegenerate lattices. The dual lattice \(\dualof{L} = \Hom_\ZZ(L, \ZZ)\) satisfies the following.

  1. Duality commutes with orthogonal direct sums: \(\dualof{(L\oplus M)} = \dualof{L} \oplus \dualof{M}\).

  2. If \(L\) has Gram matrix \(G_\beta\) in a basis \(B_L\), then the dual basis is \(B_{\dualof{L}} = \inverseof{(B_L^t)}\), and the Gram matrix of the dual form is \(G_{\dualof{\beta}} = \inverseof{G_\beta}\).

  3. The discriminant of the dual satisfies \(\disc(\dualof{L}) = 1/\disc(L)\).

  4. The dual of a twist is \(\dualof{(L(m))} = \dualof{L}(1/m)\), where \(L(m)\) is the twist of \(L\) by \(m\) from the Lattice Theory section.

Remark 37.12 Property (2) is the source of (3), since \(\disc(\dualof{L}) = \det(\inverseof{G_\beta}) = 1/\det(G_\beta) = 1/\disc(L)\). Property (1) is compatible with the direct-sum decomposition \(A_{L\oplus M} = A_L\oplus A_M\) of discriminant groups recalled in the Lattice Theory section.

37.6 Geometric identification of the dual lattice

Theorem 37.13 For a nondegenerate integral lattice \((L, \beta_L)\), the dual lattice is identified with a \(\ZZ\)-submodule of \(L_\QQ = L\tensor_\ZZ\QQ\) via \[ \dualof{L} \;\cong\; \theset{ v\in L_\QQ \mid \beta_{L_\QQ}(v, L) \iscontainedin \ZZ } , \] where a functional \(\varphi\in \dualof{L}\) corresponds to the unique vector \(v_\varphi\in L_\QQ\) such that \(\varphi(w) = \beta_{L_\QQ}(v_\varphi, w)\) for all \(w\in L\). Under this identification one has the chain of inclusions \[ L \iscontainedin \dualof{L} \iscontainedin L_\QQ . \]

Proof. Nondegeneracy of \(\beta_L\) makes the \(\QQ\)-linear extension \(L_\QQ \to \Hom_\QQ(L_\QQ, \QQ)\), \(v\mapsto \beta_{L_\QQ}(v, \cdot)\), an isomorphism, so each \(\varphi\in \dualof{L} \iscontainedin \Hom_\QQ(L_\QQ, \QQ)\) has a unique preimage \(v_\varphi\in L_\QQ\). The condition \(\varphi(L)\iscontainedin\ZZ\) translates to \(\beta_{L_\QQ}(v_\varphi, L)\iscontainedin\ZZ\), giving the stated image. The inclusion \(L\iscontainedin \dualof{L}\) is the map \(\iota\) of the Lattice Theory section, and \(\dualof{L}\iscontainedin L_\QQ\) holds because the pairing takes rational values.

37.7 The genus, class group, and class number

Definition 37.14 Two lattices \(L_1, L_2\) belong to the same genus if \(L_{1, \ZZ_p} \cong L_{2, \ZZ_p}\) for every prime \(p\), where \(L_{i, \ZZ_p} \definedas L_i\tensor_\ZZ \ZZ_p\), and \(L_{1, \RR}\cong L_{2, \RR}\). Lattices in the same genus share the same rank, signature, and determinant, but need not be isometric over \(\ZZ\). The class group \(\cl(L)\) is the set of isometry classes of lattices in the genus of \(L\), and the class number is the cardinality \(\abs{\cl(L)}\).

Remark 37.15 For indefinite even lattices \(L\) of rank \(\geq 3\) the class number is \(1\), so that the genus determines the isometry class; this is Eichler’s theorem on the spinor genus of indefinite forms (see (Conway and Sloane 2010) for a general reference). For definite lattices the situation is reversed: class number \(1\) is comparatively rare.

Proposition 37.16 If \(\rank(L) > 16 + \ell(L)\), where \(\ell(L)\) is the length of the discriminant-form definition (Definition 37.8), then the class number satisfies \(\abs{\cl(L)} \geq 2\).

Proof. This is the bound of (Scattone 1987).

37.8 Local invariants and the Jordan decomposition

Definition 37.17 (Scale, norm, and volume) Let \((L, \beta_L)\) be a lattice. Its scale is the ideal generated by all pairings and its norm the ideal generated by all squares, \[ \mathfrak{s}(L) \definedas \gcd\theset{\beta_L(x, y) \mid x, y\in L}\,\ZZ, \qquad \mathfrak{n}(L) \definedas \gcd\theset{\beta_L(x, x) \mid x\in L}\,\ZZ , \] and its volume is the index ideal \(\mathfrak{v}(L)\definedas\abs{\det G_L}\,\ZZ = \abs{A_L}\,\ZZ\). One has \(\mathfrak{n}(L)\iscontainedin\mathfrak{s}(L)\), and \(L\) is integral exactly when \(\mathfrak{s}(L)\iscontainedin\ZZ\). Under the twist of the Lattice Theory section, \(\mathfrak{s}(L(m)) = m\,\mathfrak{s}(L)\), \(\mathfrak{n}(L(m)) = m\,\mathfrak{n}(L)\) and \(\mathfrak{v}(L(m)) = m^{r}\,\mathfrak{v}(L)\) for \(r = \rank(L)\).

Definition 37.18 (Modular lattices) A lattice \(L\) is \(m\)-modular if \(m \dualof{L} = L\); equivalently, \(L\) is similar to its dual. A unimodular lattice is the case \(m = 1\), and \(L(m)\) is \(m\)-modular whenever \(L\) is unimodular.

Theorem 37.19 (Jordan decomposition) Let \(L\) be a nondegenerate lattice and \(p\) a prime. Then \(L_{\ZZ_p} = L\tensor_\ZZ\ZZ_p\) admits an orthogonal decomposition \[ L_{\ZZ_p} = L_1 \operatorname{\perp} L_2 \operatorname{\perp}\cdots\operatorname{\perp} L_k , \] in which each \(L_i\) is \(p^{s_i}\)-modular and \(s_1 < s_2 < \cdots < s_k\). Such a decomposition exists and is unique up to isometry, and the scales \(p^{s_i}\) and the ranks \(\rank(L_i)\) are invariants of \(L\) at \(p\).

Remark 37.20 The Jordan invariants at every prime are exactly the data compared in the genus definition (Definition 37.14): two lattices lie in the same genus precisely when they have isometric Jordan decompositions at every prime and the same signature. For a \(2\)-elementary lattice only the primes \(2\) and the archimedean place carry information, and the Jordan decomposition at \(2\) is assembled from the rank-two \(2\)-adic lattices \(V_k\) and \(U_k\) from Nikulin’s \(V_k,U_k\) definition (Definition 39.11) together with rank-one summands.

37.9 The mass formula as a class-number criterion

Definition 37.21 (The mass of a genus) Let \(L\) be a positive definite lattice and let \(L^{(1)}, \ldots, L^{(h)}\) be representatives of the isometry classes in the genus of \(L\), so that \(h = \abs{\cl(L)}\) in the notation of the genus definition (Definition 37.14). The mass of the genus is \[ m(L) \definedas \Sum_{i=1}^{h} \frac{1}{\abs{\Orth(L^{(i)})}} . \]

Remark 37.22 (The Smith--Minkowski--Siegel formula and its use) The mass is computable from local data alone: the Smith–Minkowski–Siegel mass formula expresses \(m(L)\) as a product of an archimedean factor and one \(p\)-adic factor for each prime, each factor read off from the Jordan decomposition of the Jordan-decomposition theorem (Theorem 37.19); the explicit unimodular cases are tabulated in (Conway and Sloane 2010 Ch. 16). Since each summand of the mass definition (Definition 37.21) is positive, the formula gives a criterion for class number one: \[ m(L) = \frac{1}{\abs{\Orth(L)}} \quad\Longleftrightarrow\quad \abs{\cl(L)} = 1 . \] This is how the uniqueness of \(E_8\) in rank \(8\) is certified, the mass of the genus of even unimodular lattices of rank \(8\) being \(1/\abs{\Orth(E_8)}\), and it is the definite counterpart of the indefinite criterion recorded above.

37.10 Surjectivity onto the discriminant group

Theorem 37.23 ($\Orth(H)\to\Orth(A_H, q_H)$ is surjective for indefinite $2$-elementary $H$) Let \(H\) be an indefinite even \(2\)-elementary lattice. Then the natural homomorphism \[ \Orth(H)\too\Orth(A_H, q_H) \] onto the isometry group of the discriminant form is surjective.

Proof. This is (Nikulin 1980); see (Alexeev 2022, sec. 4) for the statement in this form.

Remark 37.24 Surjectivity is what makes the finite quadratic space \(A_H\) a faithful shadow of \(\Orth(H)\), and it enters at three separate points below: it lifts an isometry of one eigenlattice of an involution to an isometry of \(\lkt\); it is the standing hypothesis of the orbit algorithms of (Dawes 2022); and it is what allows an orbit question about primitive vectors of \(H\) to be posed in \(A_H\) at all. It applies to \(S_\Co\), \(T_\Co\), \(\sen\), \(\ten\) and \(\tdp\), all of which are indefinite even \(2\)-elementary.

37.11 Invariants that do not classify

Theorem 37.25 (Milgram's formula) Let \(L\) be a nondegenerate even lattice with discriminant form \(q_L\colon A_L\to\QQ/2\ZZ\) and signature \((n_+, n_-)\). Then \[ \Sum_{\lambda\in A_L} e\!\left(\tfrac{1}{2}q_L(\lambda)\right) = \sqrt{\abs{A_L}}\; e\!\left(\frac{n_+ - n_-}{8}\right), \qquad e(z)\definedas e^{2\pi i z} . \] In particular the discriminant form determines the index \(n_+ - n_-\) modulo \(8\).

Proof. This is the Gauss-sum formula of (Milnor and Husemöller 1973 Appendix 4).

Remark 37.26 (What each invariant determines) Milgram’s formula is a constraint linking the two halves of the classifying data of the genus definition (Definition 37.14); it is an invariant of the discriminant form alone. It is an invariant of the discriminant form, so two even lattices with the same discriminant form give the same Gauss sum whether or not they are isometric, and it recovers the index only modulo \(8\), whereas the pair \((n_+, n_-)\) is what the genus needs.

Two weaker invariants are sometimes offered in place of the discriminant form and do not classify. The Arf invariant is defined for a quadratic form valued in \(\bF_2\); the reduction \(q_L\bmod\ZZ\) of a discriminant form loses the \(\QQ/2\ZZ\)-valued information that distinguishes, for example, \(\generators{2}\) from \(\generators{-2}\), so lattices agreeing in Arf invariant need not be isometric. The Brown invariant, valued in \(\ZZ/8\ZZ\), is the bordism-theoretic shadow of the index, and it records exactly what Milgram’s formula supplies, namely \(n_+ - n_-\) modulo \(8\). For an even lattice the classifying data are the signature \((n_+, n_-)\in\ZZ^2\) together with the discriminant form \((A_L, q_L)\), which determine the genus (Nikulin 1980), and for the indefinite lattices of rank at least three considered here the genus determines the isometry class.

Remark 37.27 (The consequence for complementary lattices) If \(S\iscontainedin\Lambda\) is primitive in an even unimodular \(\Lambda\) with complement \(T = S^{\perp\Lambda}\), then \(q_T\cong -q_S\) by the embedding-gluing description (Remark 38.10), and applying Milgram’s formula (Theorem 37.25) to both sides gives the congruence \[ \sign(S) + \sign(T)\equiv 0 \pmod 8 . \] This congruence follows from \(q_T\cong -q_S\): verifying the isometry of discriminant forms is a normal-form comparison of the pairs \((A_S, q_S)\) and \((A_T, -q_T)\), and a matching Gauss sum is one numerical shadow of that comparison.

37.12 Finiteness of orbits of vectors of fixed norm

Theorem 37.28 (Finitely many orbits in each norm) Let \(L\) be an integral lattice and \(n\in\ZZ\). The set \[ S_n \definedas \theset{ v\in L \mid v^2 = n } \] of representations of \(n\) by \(L\) decomposes into finitely many \(\Orth(L)\)-orbits.

Remark 37.29 (Why the statement is needed and where it comes from) For a definite lattice the statement is trivial, \(S_n\) itself being finite. The content is the indefinite case, where \(S_n\) is typically infinite: it is the assertion that the integral points of the affine quadric \(\theset{v^2 = n}\) fall into finitely many orbits under the arithmetic group \(\Orth(L)\), and it belongs to the classical theory of representations of an integer by an indefinite quadratic form, where the mass-formula count of representations by the classes of a genus is replaced by a count of orbits.

Finiteness is what makes the enumeration of cusps a well-posed finite problem: it guarantees that the \(0\)-cusps and \(1\)-cusps of a Baily–Borel compactification are finite in number before any of them is exhibited, and that an orbit computation can terminate. Under the hypothesis \(U^{\oplus 2}\iscontainedin L\) the Eichler criterion (the Eichler criterion (Theorem 39.15)) makes it effective, bounding the number of \(\widetilde{\SO}^+(L)\)-orbits in \(S_n\) by \(\abs{A_L}\); without that hypothesis finiteness still holds but a bound has to come from elsewhere.