38 Embeddings, overlattices, and gluing

Remark 38.1 The basic vocabulary of lattices, primitive embeddings, orthogonal complements, discriminant groups, and the unimodular lattices $\latI_{p, q}$ and $\latII_{p, q}$ was fixed in [[lattice-theory|Lattice Theory]]. We now develop the finer theory that underlies the lattice computations of this monograph: the several equivalent characterizations of a primitive sublattice, the classification of primitive embeddings up to isometry, Nikulin’s correspondence between overlattices and isotropic subgroups of the discriminant form, the splitting of a unimodular sublattice off its ambient lattice, the classification of unimodular lattices, the finiteness of the set of primitive embeddings into an even unimodular lattice, and the behaviour of the discriminant group under scaling. Throughout, $S$, $T$, and $L$ denote nondegenerate lattices in the sense of [[lattice-theory|Lattice Theory]], $\beta$ denotes the ambient bilinear form when no confusion can arise, and $A_L \definedas \dualof{L}/L$ is the discriminant group with its quadratic form $q_L$.

38.1 Primitive and saturated sublattices

Definition 38.2 (Saturation) Let $S\iscontainedin L$ be a sublattice. The saturation of $S$ in $L$ is the sublattice \[ \Sat_L(S) \definedas \theset{v\in L \mid nv\in S \text{ for some } n\in \ZZ\setminus\theset{0}} . \] Equivalently, $\Sat_L(S) = S_\QQ\intersect L$, where $S_\QQ \definedas S\tensor_\ZZ\QQ$ is regarded inside $L_\QQ$. We say $S$ is saturated in $L$ if $S = \Sat_L(S)$.

Proposition 38.3 (Characterization of primitive sublattices) Let $S\iscontainedin L$ be a sublattice. The following conditions are equivalent.

The inclusion $S\injects L$ is a primitive embedding, i.e. $\coker(S\injects L)$ is torsionfree.
$S$ is saturated in $L$: $S = \Sat_L(S)$.
$S_\QQ\intersect L = S$, where the intersection is taken in $L_\QQ$.
$S$ is a direct summand of $L$ as a $\ZZ$-module, i.e. $L\cong S\oplus T$ for some submodule $T\iscontainedin L$.
Every $\ZZ$-basis of $S$ extends to a $\ZZ$-basis of $L$.
Every $\ZZ$-linear functional $S\to\ZZ$ is the restriction of a $\ZZ$-linear functional $L\to\ZZ$.
$S = (S^{\perp L})^{\perp L}$, the double orthogonal complement of $S$ in $L$.

A sublattice satisfying these conditions is called a primitive sublattice.

Proof. We show $(1)\iff(2)$, $(2)\iff(3)$, $(2)\iff(4)\iff(5)\iff(6)$, and $(2)\iff(7)$.

$(1)\iff(2)$: By definition $\coker(S\injects L) = L/S$, and $L/S$ is torsionfree if and only if no nonzero element of $L/S$ is torsion. An element $v + S$ is torsion in $L/S$ precisely when $nv\in S$ for some nonzero $n\in\ZZ$, i.e. when $v\in\Sat_L(S)$. Thus $L/S$ is torsionfree if and only if $\Sat_L(S) = S$.

$(2)\iff(3)$: This is the second description of the saturation recorded in the saturation definition (Definition 38.2). If $v\in S_\QQ\intersect L$, then $v = \sum_i (a_i/n) s_i$ with $s_i\in S$, $a_i\in\ZZ$, and $n\in\ZZ\setminus\theset{0}$, so $nv\in S$ and $v\in\Sat_L(S)$; conversely if $nv\in S$ then $v = (nv)/n\in S_\QQ$, so $v\in S_\QQ\intersect L$. Hence $\Sat_L(S) = S_\QQ\intersect L$, and $(2)$ and $(3)$ are the same equation.

$(2)\Rightarrow(4)$: The quotient $L/S$ is a finitely generated $\ZZ$-module, hence isomorphic to $\ZZ^k\oplus F$ with $F$ finite. The torsion subgroup $F$ is exactly $\Sat_L(S)/S$, which vanishes by $(2)$, so $L/S\cong\ZZ^k$ is free. A short exact sequence $0\to S\to L\to L/S\to 0$ with $L/S$ free splits, yielding $L\cong S\oplus T$ with $T\cong L/S$.

$(4)\Rightarrow(5)$: If $L\cong S\oplus T$, then the union of a $\ZZ$-basis of $S$ and a $\ZZ$-basis of $T$ is a $\ZZ$-basis of $L$, so any basis of $S$ extends.

$(5)\Rightarrow(6)$: Fix a basis $e_1,\ldots,e_r$ of $S$, extended to a basis $e_1,\ldots,e_r,e_{r+1},\ldots,e_n$ of $L$. Given $\varphi\colon S\to\ZZ$, define $\tilde\varphi\colon L\to\ZZ$ by $\tilde\varphi(e_i) = \varphi(e_i)$ for $i\le r$ and $\tilde\varphi(e_j) = 0$ for $j > r$; then $\tilde\varphi$ restricts to $\varphi$ on $S$.

$(6)\Rightarrow(2)$: Suppose $v\in\Sat_L(S)$ with $nv\in S$, $n\ne 0$, but $v\notin S$. Choose a $\ZZ$-basis $s_1,\ldots,s_r$ of $S$ and write $nv = \sum_i a_i s_i$ with $a_i\in\ZZ$. Since $v\notin S$, some coefficient $a_i$ is not divisible by $n$; fix such an $i$ and let $\varphi\colon S\to\ZZ$ be the coordinate functional dual to $s_i$, so $\varphi(nv) = a_i$. By $(6)$ there is $\tilde\varphi\colon L\to\ZZ$ restricting to $\varphi$, and then $n\tilde\varphi(v) = \tilde\varphi(nv) = \varphi(nv) = a_i$, forcing $n\mid a_i$, a contradiction. Hence $\Sat_L(S) = S$.

$(2)\Rightarrow(7)$: In general $S\iscontainedin (S^{\perp L})^{\perp L}$ and the right-hand side is saturated (it is cut out by the vanishing of rational linear conditions, hence equals its own $\QQ$-span intersected with $L$). Because $L$ is nondegenerate, $S$ and $(S^{\perp L})^{\perp L}$ have the same $\QQ$-span, so $(S^{\perp L})^{\perp L} = S_\QQ\intersect L$. If $S$ is saturated, this equals $S$.

$(7)\Rightarrow(2)$: If $S = (S^{\perp L})^{\perp L}$, then $S$ equals a double orthogonal complement, which as just noted equals $S_\QQ\intersect L$; by $(3)$ this is condition $(2)$.

38.2 Equivalence of embeddings

Definition 38.4 (Equivalence of primitive embeddings) Two primitive embeddings $\iota_1\colon S\injects L_1$ and $\iota_2\colon S\injects L_2$ are equivalent if there is an isometry $f\in\operatorname{Isom}(L_1, L_2)$ with $f\circ\iota_1 = \iota_2$. When $L_1 = L_2 = L$, two primitive embeddings $\iota_1,\iota_2\colon S\injects L$ are equivalent if they are related in this way by an element $f\in\Orth(L)$. The set of equivalence classes of primitive embeddings of $S$ into a fixed lattice $L$ is denoted $\Emb(S, L)$.

Remark 38.5 Since an equivalence identifies $\iota_1(S)$ with $\iota_2(S)$ as sublattices of $L$, describing $\Emb(S, L)$ amounts to classifying the $\Orth(L)$-orbits of primitive sublattices of $L$ isometric to $S$, together with the choice of isometry onto each such sublattice. This is the object controlled by the gluing theory of the next subsection and made finite, in the even unimodular case, by the embedding-finiteness proposition (Proposition 38.23).

Definition 38.6 (Isomorphic embeddings versus isomorphic primitive sublattices) Two primitive embeddings $\iota_1\colon S\injects M_1$ and $\iota_2\colon S\injects M_2$ can be compared in two inequivalent ways.

They are isomorphic as embeddings if there is an isometry $\varphi\colon M_1\iso M_2$ with $\varphi\circ\iota_1 = \iota_2$, so that $\varphi$ restricts to the identity of $S$. This is the embedding-equivalence definition (Definition 38.4).
They define isomorphic primitive sublattices if there is an isometry $\varphi\colon M_1\iso M_2$ with $\varphi(\iota_1(S)) = \iota_2(S)$, so that $\varphi$ restricts to some automorphism of $S$, not necessarily the identity.

The first relation is finer than the second, and the two classifications differ by the action of $\Orth(S)$: the classes of the second kind are the orbits of the classes of the first kind under $\Orth(S)$ acting by precomposition.

Remark 38.7 The distinction matters whenever $\Orth(S)$ is large, and it is the source of two different answers to the question of how many primitive embeddings $S\injects L$ there are. Nikulin’s classification of primitive embeddings (Nikulin 1980 Prop. 1.15.1) is stated for the first relation, in terms of the gluing data $(H_S, H_T, \gamma)$ of the embedding-gluing description (Remark 38.10); passing to the second relation divides out by the action of the image of $\Orth(S)$ in $\Orth(q_S)$ on those data, which is the extra quotient appearing in the uniqueness statement of Nikulin’s gluing theorem (Theorem 38.9).

38.3 Overlattices and gluing

Definition 38.8 (Overlattice) An overlattice of a lattice $S$ is a lattice $L$ containing $S$ as a finite-index sublattice, with $\restrictionof{\beta_L}{S} = \beta_S$. Equivalently, $L$ is a lattice with $S\iscontainedin L\iscontainedin \dualof{S}$, where the inclusions use the canonical map $S\injects \dualof{S}$ of [[lattice-theory|Lattice Theory]] and its dual; the finite quotient $L/S$ is then a subgroup of $A_S = \dualof{S}/S$.

Theorem 38.9 (Nikulin's gluing correspondence) Let $S$ be an even lattice. There is a bijection \[ \theset{\text{even overlattices } L \text{ of } S} \;\longleftrightarrow\; \theset{\text{isotropic subgroups } H\le A_S} \] between the set of even overlattices of $S$ and the set of subgroups $H\le A_S$ on which the discriminant quadratic form $q_S$ vanishes identically. The bijection sends an overlattice to the isotropic subgroup it cuts out, and an isotropic subgroup to the overlattice it glues: \[ \begin{aligned} L &\longmapsto H_L \definedas L/S \iscontainedin A_S, \\ H &\longmapsto \inverseof{\eta}(H)\iscontainedin \dualof{S}, \end{aligned} \] where $\eta\colon \dualof{S}\to A_S$ is the quotient map. Under this correspondence, \[ [L : S] = \abs{H}, \qquad A_L \cong H^{\perp}/H, \qquad \disc L = \frac{\disc S}{\abs{H}^2}, \] where $H^{\perp}\le A_S$ is the orthogonal complement of $H$ with respect to $q_S$. Two even overlattices $L$, $L'$ of $S$ are isometric by an isometry restricting to an element of $\Orth(S)$ if and only if $H_L$ and $H_{L'}$ are conjugate under the image of $\Orth(S)$ in $\Orth(q_S)$.

Proof. Any lattice $L$ intermediate between $S$ and $\dualof{S}$ contains $S$ with finite index, and the pairing $\beta_S$ extends to $L$ with integer values precisely when the image $H_L = L/S\iscontainedin A_S$ is isotropic for $q_S$: for $x + S, y + S\in H_L$ one has $\beta_{S_\QQ}(x, y)\in\ZZ$ if and only if the associated bilinear form on $A_S$ vanishes on $H_L$, and evenness of $L$ requires in addition $q_S(x + S) = 0$ for all $x + S\in H_L$. Conversely, given an isotropic $H\le A_S$, the preimage $\inverseof{\eta}(H)\iscontainedin \dualof{S}$ is an even overlattice of $S$ with $\inverseof{\eta}(H)/S = H$; the two constructions are mutually inverse.

For the numerical statements, $[L:S] = \abs{L/S} = \abs{H}$. Since $\disc$ scales by the square of the index under passage to a finite-index sublattice, $\disc S = [L:S]^2\disc L = \abs{H}^2\disc L$, giving the displayed formula. Finally, the discriminant form of $L$ is computed on $\dualof{L}/L$; one has $\dualof{L} = \inverseof{\eta}(H^{\perp})$ inside $S_\QQ$, whence $A_L = \dualof{L}/L \cong H^{\perp}/H$.

The orbit statement is immediate from the definitions: an isometry of $S$ extends to an isometry of $\dualof{S}$ and hence acts on $A_S$ through the natural map $\Orth(S)\to\Orth(q_S)$, carrying the overlattice attached to $H$ to the one attached to its image.

Remark 38.10 The correspondence of Nikulin’s gluing theorem (Theorem 38.9) is the engine behind the classification of primitive embeddings. A primitive embedding $S\injects L$ with orthogonal complement $T\definedas S^{\perp L}$ realizes $L$ as an even overlattice of the orthogonal direct sum $S\oplus T$, whose discriminant group is $A_S\oplus A_T$ by the additivity recorded in [[lattice-theory|Lattice Theory]]. The corresponding isotropic subgroup $H\le A_S\oplus A_T$ is the graph of an isometry $\gamma\colon H_S\isoto H_T$ between subgroups $H_S\le A_S$ and $H_T\le A_T$, anti-isometric for the two discriminant forms; the embedding is thus determined by the gluing data $(H_S, H_T, \gamma)$. Comparing discriminants across the overlattice $L$ of $S\oplus T$ gives the discriminant formula \[ \abs{\disc T} = \frac{\abs{\disc L}\cdot\abs{H}^2}{\abs{\disc S}}, \qquad H\definedas H_L, \] recovering, when $L$ is unimodular, the statement that $\abs{\disc T} = \abs{\disc S}$ and that $H_S = A_S$, $H_T = A_T$ so that $\gamma\colon A_S\isoto A_T$ is an isometry onto $A_T$ equipped with the negated form; see the embedding-finiteness proposition (Proposition 38.23). The construction and these formulas are due to Nikulin (Nikulin 1980, secs. 1.4–1.5). The gluing subgroup $H$ is also what decides whether a pair of isometries of $S$ and of $T$ assembles into an isometry of $L$; that question is settled in the automorphism-lifting criterion (Theorem 43.5).

38.4 Splitting of unimodular sublattices

Definition 38.11 (Splitting) Let $S\injects L$ be a primitive embedding with orthogonal complement $T\definedas S^{\perp L}$. We say $S$ splits $L$ if $L = S\oplus T$; equivalently, if $L$ is the trivial index-$1$ overlattice of $S\oplus T$, i.e. the isotropic subgroup $H_L\le A_{S\oplus T}$ from Nikulin’s gluing theorem (Theorem 38.9) is the zero group.

Proposition 38.12 (Unimodular sublattices split) Let $S$ be a unimodular lattice admitting a primitive embedding into a nondegenerate lattice $L$, and let $T\definedas S^{\perp L}$ be its orthogonal complement. Then $S$ splits $L$: \[ L \cong S\oplus T . \] Moreover, if $L$ is unimodular then $T$ is unimodular as well.

Proof. First observe that $S\intersect T = \theset{0}$: any $x\in S\intersect T$ satisfies $\beta_L(x, S) = 0$, and since $S$ is nondegenerate this forces $x = 0$. Thus the sum $S + T$ inside $L$ is direct, giving an inclusion $S\oplus T\iscontainedin L$.

We show this inclusion is an equality. Let $x\in L$ be arbitrary and consider the functional $\restrictionof{\beta_L(x,\,\cdot\,)}{S}\colon S\to\ZZ$. Since $S$ is unimodular, the canonical map $S\to \dualof{S}$ is an isomorphism, so there exists $s\in S$ with $\beta_L(x, y) = \beta_S(s, y)$ for all $y\in S$; that is, $\beta_L(x - s, S) = 0$, so $x - s\in T$. Hence $x = s + (x - s)\in S\oplus T$, and $L = S\oplus T$, which is the first claim.

For the second claim, $L = S\oplus T$ gives $\disc L = \disc S\cdot\disc T$. When $L$ is unimodular, $\disc L = \pm 1$, and since $S$ is unimodular with $\disc S = \pm 1$ we obtain $\disc T = \pm 1$, so $T$ is unimodular.

Lemma 38.13 (Divisibility of isotropic vectors in unimodular lattices) Let $L$ be a nondegenerate unimodular lattice and let $v\in L$ be a primitive isotropic vector. Then there exists $w\in L$ with $\beta_L(v, w) = 1$. In particular $\div_L(v) = 1$ for every primitive vector $v\in L$.

Proof. Since $L$ is unimodular, the canonical map $L\to \dualof{L}$ is an isomorphism, so every functional in $\dualof{L} = \Hom_\ZZ(L, \ZZ)$ has the form $\beta_L(u,\,\cdot\,)$ for a unique $u\in L$. Because $v$ is primitive, by the primitive-sublattice characterization (Proposition 38.3) it extends to a $\ZZ$-basis $e_1 = v, e_2,\ldots,e_n$ of $L$. Let $\varphi\colon L\to\ZZ$ be the dual-basis functional with $\varphi(v) = 1$ and $\varphi(e_j) = 0$ for $j > 1$. Writing $\varphi = \beta_L(w,\,\cdot\,)$ for the corresponding $w\in L$ and using symmetry gives $\beta_L(v, w) = \beta_L(w, v) = \varphi(v) = 1$. For the divisibility statement, the image $\beta_L(v, L) = \div_L(v)\ZZ$ contains $\beta_L(v, w) = 1$, so $\div_L(v) = 1$.

Corollary 38.14 (Hyperbolic splitting of unimodular lattices) Let $L$ be a nondegenerate unimodular lattice containing a primitive isotropic vector. Then $L$ splits off a rank-$2$ unimodular hyperbolic plane: \[ L \cong P\oplus P^{\perp L}, \qquad P \cong \begin{cases} U, & L \text{ even},\\ \latI_{1, 1}, & L \text{ odd}. \end{cases} \]

Proof. Let $e\in L$ be a primitive isotropic vector. By the unimodular-divisibility lemma (Lemma 38.13) choose $w\in L$ with $\beta_L(e, w) = 1$, and set $k\definedas w^2$. The sublattice $P\definedas\generators{e, w}$ has Gram matrix $\begin{bmatrix}0 & 1\\ 1 & k\end{bmatrix}$ of determinant $-1$, hence $P$ is a rank-$2$ unimodular sublattice, and it is primitive since a unimodular sublattice is saturated by the primitive-sublattice characterization (Proposition 38.3). The isometry type of $P$ is governed by the parity of $k = w^2$: the Gram matrix $\begin{bmatrix}0 & 1\\ 1 & k\end{bmatrix}$ gives $P\cong U$ when $k$ is even and $P\cong\latI_{1, 1}$ (the odd rank-$2$ unimodular hyperbolic lattice) when $k$ is odd. We adjust $w$ within its coset to realize the parity dictated by $L$; note that adding to $w$ any vector of $\generators{e}^{\perp L}$ preserves $\beta_L(e, w) = 1$.

If $L$ is even then $k = w^2$ is even; replacing $w$ by $w - \tfrac{k}{2}e$ (and $e\in\generators{e}^{\perp L}$ since $e^2 = 0$) leaves $\beta_L(e, w) = 1$ unchanged and makes $w^2 = 0$, so $P\cong U$.

If $L$ is odd we arrange $w^2$ to be odd, so that $P\cong\latI_{1, 1}$. Every $x\in L$ satisfies $x - \beta_L(e, x)\,w\in\generators{e}^{\perp L}$, so $L = \generators{e}^{\perp L} + \ZZ w$. Were $\generators{e}^{\perp L}$ to consist entirely of even-norm vectors and $w^2$ even, every $x = y + mw$ ($y\in\generators{e}^{\perp L}$) would have $x^2 = y^2 + 2m\,\beta_L(y, w) + m^2 w^2$ even, forcing $L$ even — contrary to hypothesis. Hence either $w^2$ is already odd, or $\generators{e}^{\perp L}$ contains a vector $u$ of odd norm; in the latter case replace $w$ by $w + u$, which preserves $\beta_L(e, w) = 1$ and gives $(w + u)^2 = w^2 + 2\beta_L(w, u) + u^2$ odd. Either way $k = w^2$ is odd and $P\cong\latI_{1, 1}$.

Since $P$ is unimodular, the unimodular-splitting proposition (Proposition 38.12) gives $L\cong P\oplus P^{\perp L}$.

38.5 Maximal and split maximal lattices

Definition 38.15 (Maximal lattices) An even lattice $L'$ is maximal if its discriminant group contains no nonzero isotropic subgroup, that is if $q_{L'}$ vanishes on no nonzero subgroup of $A_{L'}$. An inclusion $L\iscontainedin L'$ with $L'$ maximal and $L'$ an overlattice of $L$ in the sense of the overlattice definition (Definition 38.8) is a maximal overlattice of $L$.

Lemma 38.16 Every even lattice admits a maximal overlattice. A unimodular lattice is maximal.

Proof. By Nikulin’s gluing theorem (Theorem 38.9) the even overlattices of $L$ correspond to the isotropic subgroups of $A_L$, and the discriminant form of the overlattice attached to $H$ is carried by $H^{\perp}/H$. Starting from $H = 0$ and repeatedly enlarging $H$ by an isotropic element of $H^{\perp}$ terminates, since $A_L$ is finite, at a subgroup for which $H^{\perp}/H$ has no nonzero isotropic subgroup; the corresponding overlattice is maximal. A unimodular lattice has $A_L = 0$, so the condition is vacuous.

Definition 38.17 (Split maximal lattices) A maximal lattice $L'$ of signature $(2, n)$ is split if it decomposes as \[ L' \cong U\oplus L_1, \qquad L_1 \cong U\oplus L_0 , \] for some lattice $L_0$; equivalently, if $L'$ contains $U^{\oplus 2}$ as an orthogonal direct summand.

Theorem 38.18 (Maximal lattices of signature $(2, n)$ split for $n\geq 5$) Every maximal lattice of signature $(2, n)$ with $n\geq 5$ is split.

Proof. This is recorded in (Dawes 2022, sec. 3.1), where it is attributed to Attwell-Duval; split maximal lattices are also common for $n < 5$, by the criterion of (Nikulin 1980 Cor. 1.13.5).

Theorem 38.19 (Transitivity on isotropic sublattices of a split maximal lattice)

Let $L'$ be a split maximal lattice of signature $(2, n)$.

For any primitive isotropic $x, y\in L'$ there exists $\tau(x, y)\in\Orth^+(L')$ with $\tau(x, y)\,x = y$; in particular $\Orth^+(L')$ acts transitively on primitive isotropic vectors.
Two primitive totally isotropic sublattices $E_1, E_2\iscontainedin L'$ lie in the same $\Orth^+(L')$-orbit if and only if $E_1^{\perp}/E_1\cong E_2^{\perp}/E_2$.

Proof. Both statements are in (Dawes 2022, sec. 3.1): (1) is Algorithm 3.3, which constructs $\tau(x,y)$ explicitly from the two hyperbolic summands of the split-maximal definition (Definition 38.17), and (2) is the orbit criterion of Attwell-Duval recorded there, together with the canonical $\ZZ$-basis $S(E)$ that a totally isotropic sublattice of a maximal lattice determines.

38.6 Classification of unimodular lattices

Theorem 38.20 (Classification of indefinite unimodular lattices) Any indefinite unimodular lattice is determined up to isometry by its rank, index, and parity. Explicitly, an indefinite unimodular lattice of signature $(p, q)$ is isometric to $\latI_{p, q} = \generators{1}^{\oplus p}\oplus\generators{-1}^{\oplus q}$ if it is odd, and to $\latII_{p, q}$ (which requires $p - q\equiv 0\pmod 8$) if it is even. The same uniqueness by rank, index, and parity holds for definite unimodular lattices of rank at most $8$.

Theorem 38.21 (Classification of small unimodular lattices)

Let $L$ be any unimodular lattice, definite or indefinite, with $\rank_\ZZ L\le 4$. Then either

$L$ is odd and $L\cong\latI_{p, q}$ for some $(p, q)$ with $p + q = \rank_\ZZ L$, or
$L$ is even and $L\cong U$ or $L\cong U^{\oplus 2}$.

Remark 38.22 The indefinite classification is Serre’s theorem on integral quadratic forms (Serre 1973 Ch. V); the uniqueness statement for definite unimodular lattices of small rank, and the $\rank\le 4$ enumeration, are classical and may be found in Milnor–Husem"oller (Milnor and Husemöller 1973 Ch. II) and in Conway–Sloane (Conway and Sloane 2010 Ch. 15). The even case is especially rigid: in the range $\rank_\ZZ L\le 4$ the only even unimodular lattices are $U$ (rank $2$) and $U^{\oplus 2}$ (rank $4$), there being no even unimodular lattice of odd rank and none of rank $0$ other than the zero lattice. The constraint $p - q\equiv 0\pmod 8$ for $\latII_{p, q}$ reflects the fact that the signature of an even unimodular lattice is divisible by $8$.

38.7 Finiteness of embeddings

Proposition 38.23 (Finiteness of embeddings into even unimodular lattices) If $S$ and $L$ are even lattices and $L$ is unimodular, then $\Emb(S, L)$ is a finite set.

Proof. By the unimodular-splitting proposition (Proposition 38.12) and the embedding-gluing description (Remark 38.10), a primitive embedding $S\injects L$ into an even unimodular $L$ is equivalent to the data of its orthogonal complement $T\definedas S^{\perp L}$ together with an isometry \[ \gamma\colon A_S\xrightarrow{\;\sim\;} A_T(-1), \] because unimodularity of $L$ forces the gluing subgroup to be the whole graph of $\gamma$ (equivalently $H_S = A_S$, $H_T = A_T$). Two such embeddings are equivalent in $\Emb(S, L)$ if and only if the corresponding $\gamma$ differ by the action of $\Orth(q_T)$; the equivalence class therefore depends only on the isometry class of $T$ and on a $\Orth(q_T)$-orbit of isometries $A_S\isoto A_T(-1)$.

Each ingredient is finite. The discriminant groups $A_S$ and $A_T$ are finite abelian groups, so the set $\operatorname{Isom}(A_S, A_T(-1))$ of discriminant-form isometries and the group $\Orth(q_T)$ are finite. The isometry class of $T$ is constrained to a fixed genus in the sense of the genus definition (Definition 37.14) (its signature is $\sign L - \sign S$ and its discriminant form is $-q_S$), and a genus of lattices contains only finitely many isometry classes; equivalently, the class group $\cl(T)$ from the genus definition (Definition 37.14) is finite. A finite union of finite sets of $\Orth(q_T)$-orbits is finite, so $\Emb(S, L)$ is finite.

38.8 The exact sequence for scaled lattices

Proposition 38.24 (Discriminant group of a scaled lattice) Let $L$ be a lattice and $m$ a positive integer, and let $L(m)$ be the twist of $L$ by $m$ ([[lattice-theory|Lattice Theory]]). There is a short exact sequence of finite abelian groups \[ 0 \to L/mL \to A_{L(m)} \to A_L \to 0 . \] In particular, if $L$ is unimodular then $A_L = 0$ and $A_{L(m)}\cong L/mL\cong(\ZZ/m\ZZ)^{\rank L}$.

Proof. The underlying module of $L(m)$ is $L$, with form $\beta_{L(m)} = m\beta_L$. Consequently the canonical map $\iota_{L(m)}\colon L(m)\to \dualof{L(m)}$ is $m$ times the canonical map $\iota_L\colon L\to \dualof{L}$ after the identification $\dualof{L(m)}\cong \dualof{L}$ of underlying modules. Thus $\dualof{L(m)}\cong \dualof{L}$ and the image of $L(m)$ inside $\dualof{L(m)}$ is $m\iota_L(L)$. The discriminant group is \[ A_{L(m)} = \dualof{L(m)}/L(m) \cong \dualof{L}/mL . \] The inclusion $mL\iscontainedin L\iscontainedin \dualof{L}$ then yields the short exact sequence \[ 0 \to L/mL \to \dualof{L}/mL \to \dualof{L}/L \to 0, \] which is the claimed sequence $0\to L/mL\to A_{L(m)}\to A_L\to 0$. If $L$ is unimodular then $\iota_L$ is an isomorphism, so $A_L = 0$ and $A_{L(m)}\cong L/mL\cong(\ZZ/m\ZZ)^{\rank L}$.

Proposition 38.25 (The orthogonal group exact sequence) Let $L$ be a lattice. Every isometry $f\in\Orth(L)$ extends by functoriality to an isometry of $\dualof{L}$ and hence induces an automorphism of the discriminant form $A_L$, defining a group homomorphism \[ \psi\colon\Orth(L)\to\Orth(q_L) . \] Writing $\tilde\Orth(L)\definedas\ker\psi$ for the stable orthogonal group of $L$ ([[lattice-theory|Lattice Theory]]) and $\Orth^*(q_L)\definedas\coker\psi$, there is an exact sequence \[ 0 \to \tilde\Orth(L) \to \Orth(L) \xrightarrow{\;\psi\;} \Orth(q_L) \to \Orth^*(q_L) \to 0 . \]

Remark 38.26 The cokernel $\Orth^*(q_L)$ measures the obstruction to lifting an automorphism of the discriminant form $q_L$ to an isometry of $L$. By Nikulin’s surjectivity criterion, $\psi$ is surjective, i.e. $\Orth^*(q_L) = 0$, whenever $L$ is indefinite and \[ \ell(A_L) + 2 \le \rank L, \] where $\ell(A_L)$ is the length from the discriminant-form definition (Definition 37.8), i.e. the minimal number of generators of $A_L$ (Nikulin 1980 Cor. 1.5.2, Thm. 1.14.2). For a unimodular lattice such as $U$ or $E_8$ the discriminant group is trivial, so $\Orth(q_L) = 0$ and $\tilde\Orth(L) = \Orth(L)$; the same triviality underlies the surjectivity statement invoked for $\lkt$ in the embedding-sequence lemma (Lemma 48.2).