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39  Special lattices and transformations

Remark 39.1 The lattice-theoretic study of moduli of Coble and Enriques surfaces is carried out on a small stock of recurring lattices and on the group-theoretic transformations that move vectors within them. This section collects those lattices together with their discriminant data, and records the Eichler–Siegel transvections that generate the arithmetic groups acting on the associated period domains. Throughout we use the notions of the Lattice Theory section: the dual lattice \(\dualof{L}\), the discriminant group \(A_L = \dualof{L}/L\), the discriminant quadratic form \(q_L\), the twist \(L(n)\), the rank-one lattice \(\generators{n}\), the hyperbolic lattice \(U\), the negative-definite root lattices \(A_n, D_n, E_n\), and the \(2\)-elementary invariants \((r, a, \delta)\). The discriminant computations below are instances of the general exact sequence for scaled lattices, the scaled-discriminant exact sequence (Proposition 38.24).

39.1 The hyperbolic plane and its orthogonal group

Remark 39.2 (The hyperbolic plane) Recall that the hyperbolic plane \(U = \latII_{1, 1}\) is the unique even unimodular lattice of signature \((1, 1)\), realized as \(U = \ZZ e\oplus \ZZ f\) with \(e^2 = f^2 = 0\) and \(ef = 1\), and Gram matrix \[ G_U = \begin{bmatrix}0&1\\1&0\end{bmatrix} . \] Since \(U\) is unimodular, \(\dualof{U} = U\) and the discriminant group \(A_U\) is trivial. The set of primitive isotropic vectors of \(U\) is exactly \[ \theset{\pm e, \pm f} . \]

Proposition 39.3 The orthogonal group of \(U\) is a Klein four-group, \[ \Orth(U) = \theset{\pm\id,\ \pm\begin{bmatrix}0&1\\1&0\end{bmatrix}} \cong C_2\times C_2 , \] generated by the two involutions \(\sigma: (e, f)\mapsto (-e, -f)\) and \(\tau: (e, f)\mapsto (f, e)\). Over the rationals the group \(\Orth(U_\QQ)\) is much larger: it contains the two one-parameter families of isometries \[ i_a = \begin{bmatrix} a & 0 \\ 0 & \inverseof{a}\end{bmatrix}, \qquad j_b = \begin{bmatrix} 0 & b \\ \inverseof{b} & 0\end{bmatrix}, \qquad a, b\in\unitsof{\QQ} , \] written in the basis \((e, f)\).

Proof. An isometry of \(U\) permutes the four primitive isotropic vectors \(\theset{\pm e, \pm f}\) and is determined by its action on the ordered isotropic frame \((e, f)\). Preservation of \(ef = 1\) forces the image frame to be one of \((e, f), (-e, -f), (f, e), (-f, -e)\), giving the four listed matrices. Over \(\QQ\) the vectors \(ae\) and \(\inverseof{a} f\) are again isotropic with \((ae)(\inverseof{a} f) = ef = 1\), so \(i_a\in\Orth(U_\QQ)\), and likewise \(j_b(e) = \inverseof{b} f\), \(j_b(f) = be\) satisfy \((\inverseof{b} f)(be) = fe = 1\), so \(j_b\in\Orth(U_\QQ)\).

39.2 The scaled hyperbolic plane \(U(2)\)

Proposition 39.4 The scaled lattice \(U(2)\) has Gram matrix \[ G_{U(2)} = \begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix} , \] discriminant \(\disc U(2) = -4\), and dual lattice \[ \dualof{U(2)} = \tfrac{1}{2}U . \] Its discriminant group is \[ A_{U(2)} \cong U/2U \cong (\ZZ/2\ZZ)^2 , \] a \(2\)-elementary group of rank \(2\).

Proof. The dual form has Gram matrix \(\inverseof{G_{U(2)}} = \tfrac{1}{2}G_U\), so under the geometric identification of \(\dualof{U(2)}\) as a submodule of \(U_\QQ\) (the geometric dual-lattice identification (Theorem 37.13)) the dual lattice is \(\tfrac{1}{2}U\), that is, the module \(\tfrac{1}{2}\ZZ e\oplus\tfrac{1}{2}\ZZ f\) with the form of \(U\). Because \(U\) is unimodular of rank \(2\), the scaled-lattice sequence \(0\to U/2U\to A_{U(2)}\to A_U\to 0\) of the scaled-discriminant exact sequence (Proposition 38.24) has \(A_U = 0\), giving \(A_{U(2)}\cong U/2U\cong(\ZZ/2\ZZ)^2\). Concretely \(A_{U(2)} = (\tfrac{1}{2}U)/U\) is generated by the images of \(\tfrac{1}{2}e\) and \(\tfrac{1}{2}f\).

39.3 The lattice \(E_8(2)\)

Proposition 39.5 The scaled root lattice \(E_8(2)\), with \(E_8\) the negative-definite \(E_8\) lattice of the \(E_n\) lattice definition (Definition 36.6), satisfies \[ \disc E_8(2) = 2^8, \qquad \dualof{E_8(2)} \cong \tfrac{1}{2}E_8, \qquad A_{E_8(2)} \cong E_8/2E_8 \cong (\ZZ/2\ZZ)^8 , \] so that \(E_8(2)\) is \(2\)-elementary of rank \(8\) with \(A_{E_8(2)}\) of rank \(8\).

Proof. For any lattice \(L\) of rank \(r\) and any positive integer \(m\) one has \(\disc L(m) = m^r\disc L\), since the Gram matrix of \(L(m)\) is \(m\) times that of \(L\). Taking \(L = E_8\), which is unimodular of rank \(8\) with \(\disc E_8 = 1\), gives \(\disc E_8(2) = 2^8\). Since \(E_8\) is unimodular, \(\dualof{E_8} = E_8\), and the dual form of \(E_8(2)\) has Gram matrix \(\inverseof{G_{E_8(2)}} = \tfrac{1}{2}\inverseof{G_{E_8}}\), whence \(\dualof{E_8(2)}\cong\tfrac{1}{2}E_8\). The scaled-lattice sequence of the scaled-discriminant exact sequence (Proposition 38.24) then yields \(A_{E_8(2)}\cong E_8/2E_8\cong(\ZZ/2\ZZ)^8\).

39.4 The Enriques lattice \(E_{10}\) and its twist

Definition 39.6 (The Enriques lattice) The Enriques lattice is \[ E_{10} \definedas U\oplus E_8 \iso \latII_{1, 9} , \] the orthogonal direct sum of the hyperbolic plane and the negative-definite \(E_8\) lattice. It is even, unimodular, of signature \((1, 9)\), and is the unique even unimodular lattice of that signature up to isometry.

Remark 39.7 (Role in Enriques moduli) For an Enriques surface \(Z\) the second integral cohomology carries a torsion summand, \[ \NS(Z)\cong H^2(Z; \ZZ)\cong \ZZ^{10}\oplus\ZZ/2\ZZ , \] and the intersection pairing endows the free part \(H^2(Z; \ZZ)_f\) with the structure of an even unimodular lattice of signature \((1, 9)\). There is an isometry \(H^2(Z; \ZZ)_f\iso E_{10}\), and a marking of \(Z\) is a choice of such isometry (Cossec et al. 2025). The lattice \(E_{10}\) is thus the numerical incarnation of the Enriques surface, while its twist governs the period map.

Remark 39.8 (The twisted Enriques lattice) The relevant period lattice is the twist \[ E_{10}(2) = U(2)\oplus E_8(2) , \] which is even of signature \((1, 9)\) and, being the twist by \(2\) of a rank-\(10\) unimodular lattice, is \(2\)-elementary with \(A_{E_{10}(2)}\cong E_{10}/2E_{10}\cong(\ZZ/2\ZZ)^{10}\) by the scaled-discriminant exact sequence (Proposition 38.24). In the invariant notation of the Lattice Theory section it has \(2\)-elementary type \((r, a, \delta) = \EnriquesInvariants\). This is the lattice denoted \(\sen = E_{10}(2)\) in the lattice summary, and it recurs throughout the Coble tables as the base summand \(E_{10}(2)\) appearing alongside \(U(2)\) and the root lattices; compare [[coble-lattice-table|the Coble lattice table]].

39.5 Degree \(2d\) K3 lattices

Definition 39.9 (Degree $2d$ K3 lattices) For a positive integer \(d\) and an integer \(m\geq 0\), define the lattices \[ L_{\Kthree, 2d}^{(m)} \definedas \generators{-2d}\oplus U^{2}\oplus E_8^{m}, \qquad L_{\Kthree, 2d} \definedas L_{\Kthree, 2d}^{(2)} . \] The distinguished case \(m = 2\) gives a lattice of signature \((2, 19)\), which we call the degree \(2d\) K3 lattice.

Remark 39.10 The K3 lattice is \(\lkt = U^3\oplus E_8^2 = \latII_{3, 19}\), of signature \((3, 19)\). Take a primitive vector \(h = e + d f\) in one hyperbolic summand \(U = \generators{e, f}\), so that \(h^2 = 2d\); it spans a sublattice isometric to \(\generators{2d}\). Its orthogonal complement within that \(U\) is spanned by \(e - d f\), which has \((e - d f)^2 = -2d\), so \(h^{\perp U} = \generators{e - d f}\cong\generators{-2d}\). Taking the orthogonal complement of \(h\) in all of \(\lkt = U^3\oplus E_8^2\) then realizes the orthogonal complement of a degree-\(2d\) polarization as \[ h^{\perp\lkt} \cong \generators{-2d}\oplus U^{2}\oplus E_8^{2} = L_{\Kthree, 2d} . \] The lattice \(L_{\Kthree, 2d}\) therefore models the transcendental-type lattice governing the moduli space \(\ftd\) of K3 surfaces carrying a polarization of degree \(2d\); its period domain has signature \((2, 19)\) (Scattone 1987). The auxiliary lattices \(L_{\Kthree, 2d}^{(m)}\) with other values of \(m\) arise when the number of \(E_8\) summands is varied.

39.6 Nikulin’s \(2\)-adic lattices \(V_k\) and \(U_k\)

Definition 39.11 (Nikulin's lattices $V_k$ and $U_k$) Let \(\ZZ_2\) denote the ring of \(2\)-adic integers. For an integer \(k\geq 0\) define the rank-two \(\ZZ_2\)-lattices \[ V_k \definedas \left(\ZZ_2^2,\ \begin{bmatrix} 2^{k+1} & 2^{k} \\ 2^{k} & 2^{k+1} \end{bmatrix}\right), \qquad U_k \definedas \left(\ZZ_2^2,\ \begin{bmatrix} 0 & 2^{k} \\ 2^{k} & 0 \end{bmatrix}\right) . \] We abbreviate \(V \definedas V_1\). Each of \(V_k\) and \(U_k\) is even, nondegenerate, and of \(2\)-adic rank \(2\).

Remark 39.12 The lattice \(U_k\) is the \(2\)-adic hyperbolic plane scaled by \(2^k\): one has \(U_0\cong U\tensor_\ZZ\ZZ_2\) and \(U_1\cong U(2)\tensor_\ZZ\ZZ_2\). The lattice \(V_k\) has determinant \(\det\begin{bmatrix} 2^{k+1} & 2^{k} \\ 2^{k} & 2^{k+1} \end{bmatrix} = 2^{2k+2} - 2^{2k} = 3\cdot 2^{2k}\), which is positive with positive diagonal entries, so its associated real form is positive definite of signature \((2, 0)\). We caution that the symbol \(U_k\) collides with the global notation \(U\) for the hyperbolic plane over \(\ZZ\); in the \(2\)-adic setting \(U\) without a subscript is never used, and \(U_k\) always denotes the \(2\)-adic form above. These lattices are the local building blocks out of which the indecomposable factors of general even \(2\)-elementary lattices are assembled, in the sense of the decomposition below.

39.7 Eichler–Siegel transvections

Definition 39.13 (Eichler--Siegel transvection) Let \(L\) be a lattice, let \(e\in L_\QQ\) be an isotropic vector, and let \(a\in \generators{e}^{\perp L_\QQ}\) be a vector orthogonal to \(e\), i.e. \(ae = 0\). The Eichler–Siegel transvection associated to the pair \((e, a)\) is the isometry \(E_{e, a}\in\Orth(L_\QQ)\) defined by \[ E_{e, a}(v) = v + (ev)\,a - \left( av + \tfrac{1}{2}(aa)(ev) \right) e . \] If \(a\) is itself isotropic this simplifies to \[ E_{e, a}(v) = v + (ev)\,a - (av)\,e . \] The construction is due to Eichler (Eichler 1974).

Definition 39.14 (The Eichler transvection group) The Eichler transvection group \(E(L)\leq\Orth(L)\) is generated by all transvections \(E_{e, a}\) for which \(e\in L\) is a primitive isotropic vector of divisibility \(\div_L(e) = 1\) and \(a\in\generators{e}^{\perp L}\). For lattices containing two orthogonal hyperbolic planes it acts transitively on primitive vectors of fixed norm and divisibility, and coincides with the stable orthogonal group up to finite index; see (Gritsenko et al. 2008).

Theorem 39.15 (The Eichler criterion) Let \(L\) be a lattice containing an orthogonal direct summand isometric to \(U^{\oplus 2}\), and let \(v_1, v_2\in L\) be primitive vectors. Then there exists \(g\in\widetilde{\SO}^+(L)\) with \(g v_1 = v_2\) if and only if

  1. \(v_1^2 = v_2^2\), and

  2. \(v_1^{*}\equiv v_2^{*}\) in \(A_L\), where \(v_i^{*}\definedas v_i/\div_L(v_i)\in \dualof{L}\).

Thus for such \(L\) the \(\widetilde{\SO}^+(L)\)-orbit of a primitive vector is determined by the pair \(\bigl(v^2,\ v^{*}\bmod L\bigr)\), and the classification of orbits becomes a finite computation in the discriminant group.

Proof. The criterion is due to Eichler (Eichler 1974); see (Gritsenko et al. 2008) for the formulation used here, where the transvections of the Eichler transvection-group definition (Definition 39.14) supplied by the two hyperbolic planes are what move a primitive vector into normal form.

Remark 39.16 (The hypothesis is genuine) The hypothesis \(U^{\oplus 2}\iscontainedin L\) cannot be dropped. A lattice all of whose pairings are even contains no copy of \(U\) at all, since \(U\) requires two vectors pairing to \(1\); this is the case for every twist \(M(2)\), and in particular for \(\sen = E_{10}(2)\), for \(S_\Co = \generators{-2}\oplus E_{10}(2)\) and for \(T_\Co = \generators{2}\oplus E_{10}(2)\). Among the lattices of Chapter 48 only \(\ten\) and \(\tdp\) contain a unimodular hyperbolic summand, and each contains exactly one. Orbit statements for the Coble lattices therefore have to come from elsewhere; the alternatives used below are the transitivity results for split maximal lattices and the algorithms of (Dawes 2022) for a subgroup prescribed by its image in \(\Orth(q_L)\).

Theorem 39.17 (Eichler transvections on $U\oplus U$) There is an isometry \[ \phi\colon U\oplus U \iso \left(\Mat_{2\times 2}(\ZZ),\ q = \det\right) , \] identifying the rank-four even lattice \(U\oplus U\) with the lattice of integer \(2\times 2\) matrices equipped with the determinant as its quadratic form (so that the associated bilinear form is \(2\det\)). Under \(\phi\) the group generated by the Eichler–Siegel transvections of \(U\oplus U\) corresponds to the image of \(\SL_2(\ZZ)\times\SL_2(\ZZ)\) acting on a matrix \(X\in\Mat_{2\times 2}(\ZZ)\) by \[ (A, B)\cdot X \definedas A X \inverseof{B} . \]

Remark 39.18 The quadratic form \(q(X) = \det X\) on \(\Mat_{2\times 2}(\ZZ)\) is integer-valued, so the lattice is even, and its associated bilinear form is \(b(X, X) = 2\det X\); it has signature \((2, 2)\), matching that of \(U\oplus U\), and indeed \((\Mat_{2\times 2}(\ZZ),\det)\cong U\oplus U\) is even unimodular. This realizes the exceptional isogeny between \(\Orth(U\oplus U)\) and a quotient of \(\SL_2\times\SL_2\). Since \(\det(AX\inverseof{B}) = \det X\) for \(A, B\in\SL_2\), each pair \((A, B)\) acts by an isometry, and the transvections \(E_{e, a}\) generate exactly the image of \(\SL_2(\ZZ)\times\SL_2(\ZZ)\); a proof is given in (Gritsenko et al. 2008).

39.8 Decomposability of even \(2\)-elementary lattices

Theorem 39.19 (Decomposition of $2$-elementary discriminant forms) Let \(L\) be an even \(2\)-elementary lattice. Then the discriminant quadratic form on \(A_L\) is an orthogonal direct sum of the four elementary discriminant forms \(p\), \(q\), \(u\), \(v\), subject only to the relations \[ \begin{aligned} u^{\oplus 2} &= v^{\oplus 2}, & p^{\oplus 4} &= q^{\oplus 4}, \\ u\oplus p &= (p\oplus q)\oplus p, & u\oplus q &= (p\oplus q)\oplus q, \\ v\oplus p &= q^{\oplus 3}, & v\oplus q &= p^{\oplus 3} . \end{aligned} \] Here \(p\) and \(q\) are the two rank-one forms on \(\ZZ/2\ZZ\), realized as the discriminant forms of \(\generators{2}\) and \(\generators{-2}\), while \(u\) and \(v\) are the two rank-two forms on \((\ZZ/2\ZZ)^2\), realized as the discriminant forms of \(U(2)\) and of \(V = V_1\) respectively.

Remark 39.20 The relations record the coincidences among finite \(2\)-adic quadratic forms established by Nikulin (Nikulin 1980): for instance \(u^{\oplus 2} = v^{\oplus 2}\) expresses that two orthogonal copies of the even rank-two form agree with two copies of the odd one, and \(v\oplus p = q^{\oplus 3}\) resolves the odd rank-two form against the rank-one forms. Together they present the semigroup of \(2\)-elementary discriminant forms under orthogonal sum, so that every \(A_L\) has a normal form in the generators \(p, q, u, v\).

Theorem 39.21 (Building blocks of K3-embeddable $2$-elementary lattices) The even \(2\)-elementary lattices admitting a primitive embedding into the K3 lattice \(\lkt\) decompose as orthogonal direct sums of basic lattices, such as \[ A_1,\quad D_4,\quad D_6,\quad D_8,\quad E_7,\quad E_8,\quad E_8(2),\quad \generators{2},\quad U,\quad U(2) . \] This standard building-block list is not asserted to be exhaustive.

Remark 39.22 This building-block list is drawn from Nikulin’s study of the even \(2\)-elementary hyperbolic lattices embedding into \(\lkt\) (Nikulin 1980), the same theory that underlies the invariant triple \((r, a, \delta)\) of the Lattice Theory section. The Coble lattices of [[coble-lattice-table|that table]] are exactly of this form: each \(2\)-elementary lattice \(M\) appearing there is assembled from these blocks — for example \(M = E_8(2)\oplus U\oplus A_1^{\oplus 2}\) at \(n = 2\) and \(M = E_8\oplus D_8\oplus U(2)\) at \(n = 8\) — so the table is a concrete census of the decomposition in the range relevant to Coble surfaces.