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42  The Heegner line and the Coble arithmetic group

Remark 42.1 The Coble locus is the \((-2)\) Heegner divisor \(\mch_{-2}\) inside the Enriques period domain, and the Coble period lattice is the orthogonal complement of a \((-2)\) vector of \(\ten\). Both statements are used throughout, and both are statements about a choice of vector. This section fixes that choice explicitly, shows it is unique up to the degree-\(2\) Enriques arithmetic group, and describes the subgroup of \(\Orth(T_\Co)\) that the choice produces. The lattices are those of Chapter 48, and the arithmetic groups those of Chapter 47.

42.1 The Heegner vector inside \(\ten\)

Lemma 42.2 (An explicit $(-2)$ vector with Coble complement) Write \(\ten = U\oplus U(2)\oplus E_8(2)\) and let \(U = \ZZ u\oplus\ZZ w\) be the unimodular hyperbolic summand, so \(u^2 = w^2 = 0\) and \(u\cdot w = 1\). Set \[ \delta \definedas u - w, \qquad \zeta \definedas u + w . \] Then \(\delta^2 = -2\), \(\zeta^2 = 2\) and \(\delta\cdot\zeta = 0\), and \[ \delta^{\perp \ten} = \ZZ\zeta\oplus U(2)\oplus E_8(2) \cong \generators{2}\oplus U(2)\oplus E_8(2) = \generators{2}\oplus E_{10}(2) = T_\Co . \] The sublattice \(\ZZ\delta\oplus\delta^{\perp \ten}\) has index \(2\) in \(\ten\), the missing generator being \(u = \tfrac{1}{2}(\delta + \zeta)\), and the associated gluing class in \(A_{\delta^{\perp}}\) is \(\tfrac{1}{2}\zeta\).

Proof. The three pairings are immediate from \(u^2 = w^2 = 0\) and \(u\cdot w = 1\). A vector \(x\in \ten\) decomposes as \(x = au + bw + y\) with \(y\in U(2)\oplus E_8(2)\), and \(x\cdot\delta = a\,\bigl(u\cdot(u-w)\bigr) + b\,\bigl(w\cdot(u-w)\bigr) = -a + b\), so \(x\in\delta^{\perp}\) if and only if \(a = b\), that is \(x\in\ZZ\zeta\oplus U(2)\oplus E_8(2)\). Since \(\zeta^2 = 2\) this sublattice is \(\generators{2}\oplus E_{10}(2) = T_\Co\) by the Enriques-lattice definition (Definition 39.6).

For the index, \(\delta\) and \(\zeta\) have coordinate matrix \(\begin{bmatrix}1 & -1\\ 1 & 1\end{bmatrix}\) of determinant \(2\) in the basis \((u, w)\), so \(\ZZ\delta\oplus\ZZ\zeta\) has index \(2\) in \(U\) and hence \(\ZZ\delta\oplus\delta^{\perp}\) has index \(2\) in \(\ten\); the class of \(u = \tfrac12(\delta+\zeta)\) generates the quotient. By Nikulin’s gluing theorem (Theorem 38.9) the corresponding isotropic subgroup of \(A_{\ZZ\delta}\oplus A_{\delta^{\perp}}\) is the graph of the isometry sending \(\tfrac12\delta\) to \(\tfrac12\zeta\), whose component in \(A_{\delta^{\perp}}\) is the stated class.

Remark 42.3 The primitive embedding \(T_\Co\injects \ten\) of the primitive-embedding lemma (Lemma 48.1) sends the generator \(h\) of \(\generators{2}\) to \(\tilde e + \tilde f\), which in the notation above is \(\zeta = u + w\); the explicit Heegner-vector lemma (Lemma 42.2) therefore identifies that embedding as the inclusion of \(\delta^{\perp \ten}\) for the explicit Heegner vector \(\delta = u - w\), and supplies the gluing datum which the primitive-embedding lemma (Lemma 48.1) leaves implicit. This is the explicit form of the statement, recorded in the Period Domains section, that \(T_\Co\cong v^{\perp \ten}\) for a vector \(v\) with \(v^2 = -2\) (Dolgachev and Kondō 2013).

42.2 Uniqueness of the Heegner line

Theorem 42.4 (The Coble Heegner line is unique up to $\gent$) The line \(\ZZ\delta\subset \ten\) of the explicit Heegner-vector lemma (Lemma 42.2) represents the unique \(\gent\)-orbit of lines spanned by a \((-2)\) vector of \(\ten\).

Proof. There is a single \(\Orth(\ten)\)-orbit of \((-2)\) vectors in \(\ten\) (Alexeev et al. 2025, sec. 2), so it suffices to show that this orbit does not split under the subgroup \(\gent\leq\Orth(\ten)\).

Write \(\mathcal A\definedas\im\bigl(\gent\to\Orth(A_{\ten}, q_{\ten})\bigr)\), so that \(\gent\iscontainedin\Orth_{\mathcal A}(\ten)\definedas \theset{\, g\in\Orth(\ten) \mid \bar g\in\mathcal A \,}\). Algorithm 2.2 of (Dawes 2022) decides \(\Orth_{\mathcal A}(L)\)-equivalence of non-isotropic vectors under three hypotheses: the vector is non-isotropic, its orthogonal complement is indefinite, and \(\Orth(L)\to\Orth(q_L)\) is surjective. All three hold here. Indeed \(\delta^2 = -2\neq 0\); the complement \(\delta^{\perp} = T_\Co\) has signature \((2,9)\); and \(\ten\) is an indefinite \(2\)-elementary lattice, so \(\Orth(\ten)\to\Orth(q_{\ten})\) is surjective by the two-elementary surjectivity theorem (Theorem 37.23). The test the algorithm applies is the discriminant-gluing condition of Nikulin’s gluing theorem (Theorem 38.9) for the rank-one sublattice \(\ZZ\delta\) and its complement: an isometry of \(\delta^{\perp}\) extends over the line \(\ZZ\delta\) exactly when it preserves the gluing class \(\tfrac12\zeta\in A_{\delta^{\perp}}\) of the explicit Heegner-vector lemma (Lemma 42.2).

That condition is vacuous here. The class \(\tfrac12\zeta\) satisfies \(q_{T_\Co}(\tfrac12\zeta) = \tfrac12\), and by the proof of the Coble discriminant-group proposition (Proposition 41.10) every isometry of \(q_{T_\Co}\) preserves the orthogonal decomposition \(A_{T_\Co} = \ZZ\tfrac12\zeta\perp A^0\) and fixes \(\tfrac12\zeta\), so \[ \Stab_{\Orth(A_{T_\Co}, q_{T_\Co})}\bigl(\tfrac12\zeta\bigr) = \Orth(A_{T_\Co}, q_{T_\Co}) . \] The double coset set controlling the split of the full \(\Orth(\ten)\)-orbit by the preimage subgroup is therefore a singleton, and the orbit does not split.

Remark 42.5 The \((-2)\) divisor \(\mch_{-2}\) is the discriminant divisor of the Enriques period space, whose points parameterize quotients of nodal K3 surfaces by an involution fixing a node, that is Coble surfaces with a \(\tfrac14(1,1)\) singularity (Alexeev et al. 2025, sec. 2). The Heegner-line uniqueness theorem (Theorem 42.4) says that this divisor stays irreducible after passing from \(\Gamma_\En = \Orth(\ten)\) to the degree-\(2\) group \(\gent\). It is a statement about the negative vector \(\delta\), and is independent of the count of \(\gent\)-orbits of primitive isotropic lines in \(\ten\), of which there are five, one for each \(0\)-cusp of \(\fentwo\) (Alexeev et al. 2025 Cor. 3.12).

42.3 The Coble arithmetic group induced from the Enriques side

Definition 42.6 (The induced Coble subgroup) For the Heegner vector \(\delta\) of the explicit Heegner-vector lemma (Lemma 42.2), define \[ \Gamma_\Co^\En(\delta) \definedas \im\Bigl( \Stab_{\gent}(\ZZ\delta) \too \Orth\bigl(\delta^{\perp \ten}\bigr) = \Orth(T_\Co) \Bigr) , \] the group of restrictions to \(\delta^{\perp}\) of those elements of \(\gent\) that preserve the line \(\ZZ\delta\). The condition is on the line, since a component of the Heegner divisor is determined by \(\ZZ\delta\) and not by a choice of sign for \(\delta\).

Proposition 42.7 (Discriminant description of $\gent$) Let \(\sen = U(2)\oplus E_8(2)\) and \(\ten\) be the invariant and coinvariant lattices of the Enriques involution on \(\lkt\), and let \[ \gamma\colon A_{\sen}\iso A_{\ten}, \qquad q_{\ten}\circ\gamma = -q_{\sen} \] be the gluing anti-isometry supplied by the embedding-gluing description (Remark 38.10) for the primitive embedding \(\sen\injects\lkt\) with complement \(\ten\). Then \[ \gent = \theset{\, g_T\in\Orth(\ten) \;\middle|\; \exists\, g_S\in\Orth(\sen) \text{ with } g_S(h) = h \textand \bar g_T\circ\gamma = \gamma\circ\bar g_S \,} , \] where \(h = e + f\in U(2) = \sdp\iscontainedin \sen\) is the degree-\(2\) polarization vector, with \(e^2 = f^2 = 0\) and \(e\cdot f = 2\), so that \(h^2 = 4\) (Alexeev et al. 2025 Def. 2.5, Def. 2.6). Equivalently, the image of \(\gent\) in \(\Orth(q_{\ten})\) is the \(\gamma\)-transport of the image of \(\Stab_{\Orth(\sen)}(h)\) in \(\Orth(q_{\sen})\).

Proof. By definition \(\gent\) is the image in \(\Orth(\ten)\) of the isometries of \(\lkt\) that commute with \(\Ien\) and fix \(h\) (Alexeev et al. 2025 Def. 2.6). An isometry of \(\lkt\) commuting with \(\Ien\) is the same as a pair \((g_S, g_T)\in\Orth(\sen)\times\Orth(\ten)\) preserving the two eigenlattices, and by Nikulin’s gluing theorem (Theorem 38.9) such a pair extends over the overlattice \(\lkt\) of \(\sen\oplus \ten\) exactly when it preserves the graph of \(\gamma\), that is when \(\bar g_T\circ\gamma = \gamma\circ\bar g_S\). The polarization \(h\) lies in \(\sen\), so the condition \(g(h) = h\) on \(\lkt\) is the condition \(g_S(h) = h\).

Proposition 42.8 (The integral stabilizer of the degree-$2$ polarization) Write \(\sen = C(2)\) with \(C = U\oplus E_8\) even unimodular of signature \((1,9)\), and let \(h\in \sen\) be the polarization vector of the discriminant description of \(\gent\) (Proposition 42.7), so that \(h = u_0 + w_0\) for a basis \(u_0, w_0\) of the \(U\) summand of \(C\) with \(u_0\cdot w_0 = 1\). Then restriction to \(h^{\perp}\) gives an isomorphism \[ \Stab_{\Orth(\sen)}(h) \;\iso\; \Orth\bigl(\generators{-2}\oplus E_8\bigr) = \theset{\pm 1}\times W(E_8) , \] of order \(2\abs{W(E_8)} = 1\,393\,459\,200\), and the element \(\id_U\oplus(-\id_{E_8})\) lies in the kernel of the reduction \(\Stab_{\Orth(\sen)}(h)\to\Orth(A_{\sen}, q_{\sen})\).

Proof. Twisting does not change the isometry group, so \(\Orth(\sen) = \Orth(C)\) and \(\Stab_{\Orth(\sen)}(h) = \Stab_{\Orth(C)}(h)\). Computing in \(C\), one has \(h^2 = 2\) and \(h^{\perp C} = \ZZ(u_0 - w_0)\oplus E_8\) with \((u_0-w_0)^2 = -2\), so \(h^{\perp C}\cong\generators{-2}\oplus E_8\). An isometry fixing \(h\) preserves \(h^{\perp}\), giving the restriction homomorphism, which is injective because \(C_\QQ = \QQ h\oplus h_\QQ^{\perp}\). It is surjective: \(A_{h^{\perp}}\cong A_{\generators{-2}}\cong\ZZ/2\ZZ\) has trivial automorphism group, so every isometry of \(h^{\perp}\) preserves the gluing class of the index-two overlattice \(C\) of \(\ZZ h\oplus h^{\perp}\) and extends over \(C\) fixing \(h\), by Nikulin’s gluing theorem (Theorem 38.9). The lattice \(\generators{-2}\oplus E_8\) is negative definite and generated by its \((-2)\) vectors, whose root system is \(A_1\oplus E_8\); no isometry mixes summands of different ranks, so its isometry group is \(\Orth(\generators{-2})\times\Orth(E_8) = \theset{\pm1}\times W(E_8)\).

For the last claim, \(\id_U\oplus(-\id_{E_8})\) fixes \(h\in U\) and acts on \(A_{\sen}\cong C/2C\) (the twisted-unimodular discriminant proposition (Proposition 41.6)) by the identity on \(U/2U\) and by \(x\mapsto -x\equiv x\) on \(E_8/2E_8\).

Theorem 42.9 (The finite image and its isotropic orbits)

With the notation of the polarization-stabilizer proposition (Proposition 42.8) and the explicit Heegner-vector lemma (Lemma 42.2):

  1. the image of the integral stabilizer in the discriminant group, \[ G \definedas \im\bigl(\Stab_{\Orth(\sen)}(h)\to\Orth(A_{\sen}, q_{\sen})\bigr) , \] has order \(\abs{W(E_8)} = 696\,729\,600\), and is generated by the swap of the two isotropic generators of the \(U\) summand of \(C\), which fixes \(h\), together with the reflections in the simple roots of the \(E_8\) summand;

  2. the corresponding group \(G_\Co\leq\Orth(A_{T_\Co}, q_{T_\Co})\), obtained by transporting \(G\) along \(\gamma\) and restricting to the Coble Heegner complement \(\delta^{\perp} = T_\Co\), has order \(\abs{W(E_8)}\) and acts on the \(528\) isotropic classes of \(A_{T_\Co}\) with orbit lengths \[ [\,1,\ 2,\ 120,\ 135,\ 270\,] ; \]

  3. the finite stabilizer \(\Stab_{\Orth(q_{T_\Co})}\bigl(\tfrac12\tilde h_\Co\bigr)\) of the class of the transported polarization has order \(94\,755\,225\,600\), contains \(G_\Co\) with index \(136\), and acts on the same \(528\) isotropic classes with orbit lengths \([\,1,\ 255,\ 272\,]\).

Remark 42.10 (How the three orders are obtained) The generators named in (1) are the two evident families of isometries of \(C\) fixing \(h = u_0 + w_0\): the involution \(u_0\leftrightarrow w_0\), and \(W(E_8)\) acting on the second summand. By the polarization-stabilizer proposition (Proposition 42.8) these generate a group of order \(2\abs{W(E_8)}\) whose reduction has \(\id_U\oplus(-\id_{E_8})\) in its kernel, so the image has order at most \(\abs{W(E_8)}\); the stated equality, and the two orbit decompositions of (2) and (3), are exact computations in the finite quadratic space \(A_{T_\Co}\cong B/2B\) of the Coble mod-four form definition (Definition 41.7), carried out by stabilizing the four fibers of \(Q\) inside \(\GL(B/2B)\) and then computing orbits of the resulting finite group on the fiber \(\inverseof{Q}(0)\). The index \(136 = 94\,755\,225\,600 / 696\,729\,600\) in (3), and the relation \(46\,998\,591\,897\,600 = 496\cdot 94\,755\,225\,600\) between the order of the Coble discriminant-group proposition (Proposition 41.10) and that of the finite stabilizer, are consistency checks: \(\tfrac12\tilde h_\Co\) is a class with \(q_{T_\Co} = 1\), and \(496\) is the number of such classes by the Coble \(Q\)-fiber proposition (Proposition 41.8).

Remark 42.11 (Reading the two orbit decompositions) The two decompositions in the finite-image orbit theorem (Theorem 42.9) answer different questions, and their difference is the content of the theorem. The coarser one, \([1, 255, 272]\), is the decomposition under the whole finite stabilizer of the polarization class, the largest group the discriminant form alone can distinguish. The finer one, \([1, 2, 120, 135, 270]\), is the decomposition under the group that actually arises from integral isometries of \(\sen\) fixing \(h\), namely \(G_\Co\). Since the latter has index \(136\) in the former, a calculation carried out purely in \(\Orth(q_{T_\Co})\) overestimates how much of \(A_{T_\Co}\) a degree-\(2\) Coble period point can see. The four nonzero orbit lengths \(2, 120, 135, 270\) sum to \(527\), the number of nonzero isotropic classes of the Coble \(Q\)-fiber proposition (Proposition 41.8), so both decompositions refine the single nonzero orbit of the isotropic-class orbit theorem (Theorem 41.11).

Both are statements in the finite quadratic space \(A_{T_\Co}\). Neither is yet a classification of \(\Gamma_\Co^\En(\delta)\)-orbits of primitive isotropic vectors of \(T_\Co\), for the reason recorded in the primitive-isotropic-class proposition (Proposition 41.12): an integral parabolic stabilizer may have proper image in the finite one, so a finite orbit can split.

42.4 The Coble folding involution

Proposition 42.12 (The sign involution of the Coble primitive embedding) Let \(S_\Co\injects\lkt\) be the primitive embedding of the Coble invariant-lattice proposition (Proposition 32.4), with complement \(T_\Co = S_\Co^{\perp\lkt}\), and let \[ \gamma_\Co\colon A_{S_\Co}\iso A_{T_\Co}, \qquad q_{T_\Co}\circ\gamma_\Co = -q_{S_\Co} \] be the associated gluing anti-isometry. Then the pair \((-\id_{S_\Co}, \id_{T_\Co})\) preserves the graph of \(\gamma_\Co\) and therefore defines an isometry \[ \theta_\Co\in\Orth(\lkt), \qquad \restrictionof{\theta_\Co}{S_\Co} = -\id, \qquad \restrictionof{\theta_\Co}{T_\Co} = \id , \] whose \((+1)\) and \((-1)\) eigenlattices are \(T_\Co\) and \(S_\Co\). An isometry of \(\lkt\) commutes with \(\theta_\Co\) if and only if it preserves both eigenlattices.

Proof. Both \(A_{S_\Co}\) and \(A_{T_\Co}\) are \(2\)-elementary, so \(-\id\) and \(\id\) agree on each of them; the pair therefore acts as the identity on the graph of \(\gamma_\Co\) and by Nikulin’s gluing theorem (Theorem 38.9) extends over the overlattice \(\lkt\) of \(S_\Co\oplus T_\Co\). The eigenlattice description is the definition of the extension. For the last claim, an isometry commuting with an involution preserves its eigenspaces, which by the involution-eigenspace proposition (Proposition 40.3) are the rational spans of \(T_\Co\) and \(S_\Co\); conversely an isometry preserving both eigenlattices commutes with \(\theta_\Co\) on each of them and hence on \(\lkt\).

Remark 42.13 The folding-involution proposition (Proposition 42.12) settles the existence of the folding involution \(\theta\) as a lattice isometry, and does so before any \(22\times 22\) matrix is written: the involution is determined by the primitive embedding and the gluing anti-isometry. What a matrix realization additionally supplies is a basis in which \(\theta_\Co\), the polarization class, and the roots of the Coxeter diagram can be compared with one another.

Combining the folding-involution proposition (Proposition 42.12) with the discriminant description of \(\gent\) (Proposition 42.7) gives the Coble-side analogue of the discriminant description of \(\gent\): the restriction to \(T_\Co\) of the centralizer of \(\theta_\Co\) inside the stabilizer of a chosen polarization class \(\tilde h_\Co\in S_\Co\) consists of those \(g_T\in\Orth(T_\Co)\) for which some \(g_S\in\Orth(S_\Co)\) satisfies \(g_S(\tilde h_\Co) = \tilde h_\Co\) and \(\bar g_T\circ\gamma_\Co = \gamma_\Co\circ\bar g_S\). Identifying that group with \(\Gamma_\Co^\En(\delta)\) of the induced Coble subgroup definition (Definition 42.6) requires an isometry between the Coble primitive embedding and the Enriques Heegner complement carrying \(\tilde h_\Co\) to the class \(h = e+f\) of the discriminant description of \(\gent\) (Proposition 42.7). The images of the two groups in \(\Orth(q_{T_\Co})\) do agree, by the finite-image orbit theorem (Theorem 42.9), but agreement of finite images is weaker than an isomorphism of the lattice subgroups.