Introduction and Context within the Section III Program

The twenty-four positive-definite even unimodular lattices of rank 24 in Euclidean space $\mathbb{R}^{24}$ form one of the most rigid finite families in arithmetic geometry . We denote their set of isometry classes by $$\mathcal{N}_{24}= \{N_1, \dots, N_{24}\}.$$ One member is the root-free Leech lattice $\Lambda_{24}$; the remaining twenty-three have root systems of ADE type whose irreducible components share a uniform Coxeter number $h$.

For a rank-24 lattice $L$, its scalar degree-$g$ Siegel theta series is defined on the Siegel upper half-space $\mathbb{H}_g = \{ \Omega \in M_g(\mathbb{C}) : \Omega = \Omega^T, \, \mathrm{Im}(\Omega) > 0 \}$ by $$\begin{equation} \Theta_L^{(g)}(\Omega) = \sum_{(x_1, \dots, x_g) \in L^g} \exp\left( \pi i \sum_{j,k=1}^g \langle x_j, x_k \rangle \Omega_{jk} \right) \in M_{12}(\mathrm{Sp}_{2g}(\mathbb{Z})). \end{equation}$$ When expanded as a Fourier series over positive semi-definite, half-integral $g \times g$ matrices $T$ ($T_{ii} \in \mathbb{Z}$ and $2T_{ij} \in \mathbb{Z}$): $$\begin{equation} \Theta_L^{(g)}(\Omega) = \sum_{T \ge 0} a(T, \Theta_L^{(g)}) \exp\left( 2\pi i \mathop{\mathrm{Tr}}(T\Omega) \right), \end{equation}$$ the Fourier coefficients count vector representations with integral Gram matrix $2T$: $$\begin{equation} a(T, \Theta_L^{(g)}) = \#\{ (v_1, \dots, v_g) \in L^g : \langle v_i, v_j \rangle = 2 T_{ij} \}. \end{equation}$$

The Section III Progression

The higher-genus theory of Niemeier lattices separates into distinct geometric stages across the papers of this series:

1. Paper III-A (Low-Genus Rigidity ): Establishes that for degrees $g \le 3$, the scalar Siegel theta series $\Theta_N^{(g)}$ depend exclusively on the Coxeter number $h$. The nineteen distinct values of $h$ partition $\mathcal{N}_{24}$ into fourteen singletons and five collision pairs sharing identical Coxeter numbers: $$\begin{align*} h = 6 &\implies \{D_4^6, \, A_5^4 D_4\}, \\ h = 10 &\implies \{D_6^4, \, A_9^2 D_6\}, \\ h = 12 &\implies \{E_6^4, \, D_7 A_{11} E_6\}, \\ h = 18 &\implies \{D_{10} E_7^2, \, A_{17} E_7\}, \\ h = 30 &\implies \{E_8^3, \, D_{16} E_8\}. \end{align*}$$

2. Paper III-B (Ambient Genus-4 Separation ): Proves that degree $g=4$ is the minimal separation genus on the ambient moduli space of principally polarized abelian varieties $\mathcal{A}_4 = \mathbb{H}_4 / \mathrm{Sp}_8(\mathbb{Z})$. Crucially, the single Fourier coefficient $a(I_4, \Theta_N^{(4)})$ counting ordered quadruplets of mutually orthogonal roots takes twenty-four distinct integer values, proving $g_{\mathrm{Siegel}}(\mathcal{N}_{24}) = 4$.

3. Paper III-C (Metric Decks and Incidence Moments ): Formulates the recovery of root systems via logarithmic moments, polynomial Vandermonde inversion, and categorical fiber products of Gram-map fibers.

4. Paper III-D (The Present Paper): Investigates the geometric restriction of the ambient genus-4 modular forms to the Torelli locus $\mathcal{J}_4 \subset \mathcal{A}_4$, defined by the image of the Torelli morphism $j: \mathcal{M}_4 \to \mathcal{A}_4$.

Upstream Inputs from Papers III-A and III-B

We take the foundational results from the preceding papers as established inputs:

For each of the five collision pairs, let $F_h = \Theta_{L_{h,1}}^{(4)} - \Theta_{L_{h,2}}^{(4)}$. Because $\Phi(F_h) = \Theta_{L_{h,1}}^{(3)} - \Theta_{L_{h,2}}^{(3)} = 0$ under the Siegel operator, each difference form is a cusp form: $$\begin{equation} F_h \in S_{12}(\mathrm{Sp}_8(\mathbb{Z})), \quad h \in \{6, 10, 12, 18, 30\}. \end{equation}$$ The differences at $I_4$ computed in Paper III-B yield the exact base factorization: $$\begin{equation} \label{eq:delta-aI4} \Delta a(I_4, F_h) = a(I_4, \Theta_{L_{h,1}}^{(4)}) - a(I_4, \Theta_{L_{h,2}}^{(4)}) = 230,400 \cdot m_h, \end{equation}$$ with base constant $230,400 = (480)^2$ and integer frame coordinates: $$\begin{equation} \label{eq:m-coords} (m_6, m_{10}, m_{12}, m_{18}, m_{30}) = (1, 9, 20, 105, 896). \end{equation}$$

Ambient Forms versus Torelli Restrictions

The Restriction Map

Let $\mathcal{M}_4$ denote the moduli space of smooth compact Riemann surfaces of genus 4. The Torelli morphism $$\begin{equation} j: \mathcal{M}_4 \longrightarrow \mathcal{A}_4 \end{equation}$$ associates to each curve $C$ its Jacobian variety $J(C)$ equipped with its canonical principal polarization. Its image is the Jacobian locus $\mathcal{J}_4 = j(\mathcal{M}_4)$. By the classical Torelli theorem, $j$ is injective on geometric points.

For each weight $k$, pullback under $j$ defines the restriction map on spaces of modular forms: $$\begin{equation} \rho_{\mathcal{J}}: M_k(\mathrm{Sp}_8(\mathbb{Z})) \longrightarrow H^0(\mathcal{M}_4, \mathcal{L}^{\otimes k}), \end{equation}$$ where $\mathcal{L}$ denotes the Hodge line bundle on $\mathcal{M}_4$. The kernel $\ker(\rho_{\mathcal{J}}) = \{ F \in M_k(\mathrm{Sp}_8(\mathbb{Z})) : \left. F \right|_{\mathcal{J}_4} \equiv 0 \}$ consists of modular forms vanishing along the Torelli locus.

Ambient separation does not imply Torelli separation: a modular form can be non-zero on $\mathcal{A}_4$ while vanishing identically on $\mathcal{J}_4$.

The Schottky Form

The moduli space $\mathcal{M}_4$ has dimension $3(4) - 3 = 9$, while $\mathcal{A}_4$ has dimension $4(5)/2 = 10$. The closure of the Jacobian locus $\overline{\mathcal{J}_4}$ is therefore an irreducible divisor in the toroidal compactification $\overline{\mathcal{A}_4}$.

By Igusa’s solution to the Schottky problem in genus 4 , the modular equation defining the divisor $\overline{\mathcal{J}_4} \subset \mathcal{A}_4$ is the weight-8 Schottky cusp form: $$\begin{equation} J_8 \in S_8(\mathrm{Sp}_8(\mathbb{Z})), \qquad \left. J_8 \right|_{\mathcal{J}_4} \equiv 0. \end{equation}$$ We adopt the classical Igusa normalization: $$\begin{equation} \label{eq:J8-normalization} J_8 := \frac{1}{3360} \left( \Theta_{E_8 \oplus E_8}^{(4)} - \Theta_{D_{16}^+}^{(4)} \right). \end{equation}$$ Evaluating at $T = I_4$, the number of mutually orthogonal 4-frames in $E_8 \oplus E_8$ is $9,064,742,400$, while in $D_{16}^+$ it is $8,858,304,000$. Under [eq:J8-normalization], the Fourier normalization of $J_8$ is fixed by: $$\begin{equation} \label{eq:J8-I4-val} a(I_4, J_8) = \frac{9,064,742,400 - 8,858,304,000}{3360} = \frac{206,438,400}{3360} = 61,440. \end{equation}$$

The Degree-12 Ambient Cusp Space and the Schottky Kernel

Dimensions and Generators

By Poor and Yuen , the spaces of degree-4, weight-12 Siegel modular forms have dimensions: $$\begin{equation} \dim M_{12}(\mathrm{Sp}_8(\mathbb{Z})) = 6, \qquad \dim S_{12}(\mathrm{Sp}_8(\mathbb{Z})) = 2. \end{equation}$$

The two-dimensional cusp space $S_{12}(\mathrm{Sp}_8(\mathbb{Z}))$ admits a geometrically natural direct-sum decomposition:

1. The Schottky Ray ($S_{\mathrm{Sch}}$): Formed by multiplying $J_8 \in S_8(\mathrm{Sp}_8(\mathbb{Z}))$ by the unique degree-4 Eisenstein series $\Theta_{E_8}^{(4)} \in M_4(\mathrm{Sp}_8(\mathbb{Z}))$: $$\begin{equation} S_{\mathrm{Sch}} := J_8 \cdot \Theta_{E_8}^{(4)} \in S_{12}(\mathrm{Sp}_8(\mathbb{Z})). \end{equation}$$ Because $J_8$ vanishes on $\mathcal{J}_4$, $S_{\mathrm{Sch}}$ vanishes identically on the Jacobian locus.

2. A Complementary Cusp Direction ($G$): Any cusp form $G \in S_{12}(\mathrm{Sp}_8(\mathbb{Z}))$ chosen to be linearly independent of $S_{\mathrm{Sch}}$ (such as the non-Schottky eigenform identified in ). The complementary form $G$ does not vanish identically on $\mathcal{J}_4$.

Hence: $$\begin{equation} S_{12}(\mathrm{Sp}_8(\mathbb{Z})) = \mathbb{C}S_{\mathrm{Sch}} \oplus \mathbb{C}G = \mathbb{C}\Big( J_8 \Theta_{E_8}^{(4)} \Big) \oplus \mathbb{C}G. \end{equation}$$

The Degree-12 Jacobian Kernel

Because the Schottky divisor $\overline{\mathcal{J}_4}$ is irreducible and defined by $J_8$, any holomorphic form in $M_k(\mathrm{Sp}_8(\mathbb{Z}))$ vanishing on $\mathcal{J}_4$ is divisible by $J_8$ in the graded ring of Siegel modular forms: $$\begin{equation} \left. F \right|_{\mathcal{J}_4} \equiv 0 \implies F = J_8 \cdot H, \quad H \in M_{k-8}(\mathrm{Sp}_8(\mathbb{Z})). \end{equation}$$ For $k=12$, the quotient $H$ belongs to $M_4(\mathrm{Sp}_8(\mathbb{Z}))$. By Freitag , $M_4(\mathrm{Sp}_8(\mathbb{Z})) = \mathbb{C}\Theta_{E_8}^{(4)}$ is strictly one-dimensional. This yields an exact characterization:

The Multi-Topology Certificate Lemma

Because $S_{12}(\mathrm{Sp}_8(\mathbb{Z}))$ is two-dimensional, the projection of any cusp form onto the complementary non-Schottky generator $G$ is uniquely detected by Fourier coefficients.

The Five Collision Differences and the $h=30$ Factorization

Definition of Difference Forms

For each Coxeter collision value $h \in \{6, 10, 12, 18, 30\}$, we define the difference cusp form: $$\begin{equation} F_h := \Theta_{L_{h,1}}^{(4)} - \Theta_{L_{h,2}}^{(4)} \in S_{12}(\mathrm{Sp}_8(\mathbb{Z})), \end{equation}$$ ordered such that $\Delta a(I_4, F_h) > 0$: $$\begin{align} F_6 &= \Theta_{D_4^6}^{(4)} - \Theta_{A_5^4 D_4}^{(4)}, \\ F_{10} &= \Theta_{D_6^4}^{(4)} - \Theta_{A_9^2 D_6}^{(4)}, \\ F_{12} &= \Theta_{E_6^4}^{(4)} - \Theta_{D_7 A_{11} E_6}^{(4)}, \\ F_{18} &= \Theta_{D_{10} E_7^2}^{(4)} - \Theta_{A_{17} E_7}^{(4)}, \\ F_{30} &= \Theta_{E_8^3}^{(4)} - \Theta_{D_{16} E_8}^{(4)}. \end{align}$$

The Exact Factorization for $h=30$

The pair with Coxeter number 30 admits an exact product factorization coming directly from orthogonal direct sums: $$\begin{equation} E_8^3 = (E_8 \oplus E_8) \oplus E_8, \qquad D_{16}E_8 = D_{16}^+ \oplus E_8. \end{equation}$$

The Schottky Ray Criterion and Multi-Topology Verification

Multi-Topology Fourier Coefficients

To evaluate representation differences across positive-definite root Gram matrices $T$, we consider five distinct connected graph topologies on four vertices (connecting directly to the incidence moments $q_X(k, c)$ analyzed in Paper III-C ):

1. $I_4$: The identity matrix ($\det = 1$), representing four mutually orthogonal roots ($\langle v_i, v_j \rangle = 0$ for $i \neq j$).

2. $T_{A_4}$: The Gram matrix of a linear 4-vertex $A_4$ Dynkin chain ($\det = 5/16$), with $\langle v_i, v_{i+1} \rangle = -1$ and other inner products zero.

3. $T_{D_4}$: The Gram matrix of a 4-vertex $D_4$ claw/star ($\det = 4/16$), with a central node connected to three mutually orthogonal leaves.

4. $T_{\mathrm{paw}}$: The Gram matrix of a paw graph (a triangle with an attached leaf, $\det = 3/16$).

5. $T_{\mathrm{dia}}$: The Gram matrix of a diamond graph (two triangles sharing an edge, $\det = 3/16$).

Representation-Theoretic Origin of the Ratio $\pm 1/40$

The rational invariant $1/40$ appearing across all five pairs is an exact consequence of the spherical 7-design property of ADE root systems and harmonic polynomials on $\mathbb{R}^{24}$.

Unconditional Proof of Schottky Collinearity

The Torelli Collapse: From 24 Ambient to 19 Jacobian Classes

The 19 Jacobian Restriction Classes

We now establish the complete classification of Niemeier theta series restricted to the moduli space of curves $\mathcal{M}_4$.

The Geometric Quotient

Theorem 10 establishes a clean geometric quotient: $$\begin{equation} \begin{tikzcd}[column sep=3.5em, row sep=2.5em] \mathcal{N}_{24}\arrow[r, "\Theta^{(4)}"] \arrow[d, equal] & M_{12}(\mathrm{Sp}_8(\mathbb{Z})) \arrow[r] \arrow[d, "\rho_{\mathcal{J}}"] & \text{24 distinct ambient classes on } \mathcal{A}_4 \arrow[d, "\text{identify 5 collision pairs}"] \\ \mathcal{N}_{24}\arrow[r, "{\left.\Theta^{(4)}\right|_{\mathcal{J}_4}}"] & H^0(\mathcal{M}_4, \mathcal{L}^{\otimes 12}) \arrow[r] & \text{19 distinct Jacobian classes on } \mathcal{M}_4 \end{tikzcd} \end{equation}$$

The Genus-5 Frontier

Separation of Pair 5 on $\mathcal{M}_5$

The Torelli collapse at genus 4 occurs because $\mathcal{J}_4$ is a hypersurface in $\mathcal{A}_4$ cut out by the single Schottky form $J_8$. At degree $g=5$, the codimension of $\mathcal{J}_5$ in $\mathcal{A}_5$ is $\dim \mathcal{A}_5 - \dim \mathcal{M}_5 = 15 - 12 = 3$.

For Pair 5 ($h=30$), multiplicativity under orthogonal direct sums extends to degree 5: $$\begin{equation} F_{30}^{(5)} = \Theta_{E_8^3}^{(5)} - \Theta_{D_{16}E_8}^{(5)} = \left( \Theta_{E_8 \oplus E_8}^{(5)} - \Theta_{D_{16}^+}^{(5)} \right) \cdot \Theta_{E_8}^{(5)}. \end{equation}$$

The Open Problem for Pairs 1 to 4

For the remaining four collision pairs ($h \in \{6, 10, 12, 18\}$), the lattices are not orthogonal direct sums of 16-dimensional unimodular sublattices. Establishing that the global Jacobian separation genus is strictly $$\begin{equation} g_{\mathrm{Jac}}(\mathcal{N}_{24}) = 5 \end{equation}$$ requires proving that the difference forms $F_h^{(5)}$ do not vanish identically on $\mathcal{J}_5$. This constitutes the primary open problem for the subsequent paper in this series (Paper III-E).

Summary of the Geometric Hierarchy

The complete geometric progression across the Niemeier classification is summarized below in a two-tier sequence to maintain visual clarity: $$\begin{equation} \begin{aligned} \boxed{ \begin{array}{c} \mathbf{g = 1, 2, 3 \text{ on } \mathcal{A}_g} \\ \midrule \text{19 classes on } \mathcal{A}_g \\ (\text{equal } h \text{ rigid}) \end{array} } &\quad \xlongrightarrow{\quad\text{Paper III-B}\quad} \quad \boxed{ \begin{array}{c} \mathbf{g = 4 \text{ on } \mathcal{A}_4} \\ \midrule \textbf{24 classes} \\ (a(I_4) \text{ injective}) \end{array} } \\[14pt] &\qquad\quad\Bigg\downarrow \text{Torelli Restriction } \rho_{\mathcal{J}} \text{ (Schottky Collapse)} \\[10pt] \boxed{ \begin{array}{c} \mathbf{g = 4 \text{ on } \mathcal{M}_4} \\ \midrule \textbf{19 classes} \\ (\text{Schottky ray vanishes}) \end{array} } &\quad \xlongrightarrow{\quad\text{Paper III-E}\quad} \quad \boxed{ \begin{array}{c} \mathbf{g = 5 \text{ on } \mathcal{M}_5} \\ \midrule \text{Pair 5 separates outside } \mathcal{T}_5 \\ (\text{Pairs 1--4 open}) \end{array} } \end{aligned} \end{equation}$$

Multi-Topology Fourier Verification and Implementation Notes

All entries in Proposition 7 were obtained by exact combinatorial enumeration of root quadruplets across the ADE root systems, matching the incidence moments $q_X(k, c)$ established in Paper III-C .

For an irreducible simply-laced component $X$:

• The linear chain $T_{A_4}$ corresponds to ordered 4-tuples $(v_1, v_2, v_3, v_4)$ with Gram matrix $\langle v_i, v_j \rangle = 2\delta_{ij} - \delta_{|i-j|, 1}$. Since the graph is connected, all four roots must reside in the same irreducible component.

• The claw $T_{D_4}$ corresponds to a central root connected to three mutually orthogonal leaves.

The exact values for $A_n, D_n, E_6, E_7, E_8$ are recorded in Table 2 of Paper III-C . The constant ratio $\Delta a(T) / \Delta a(I_4) = \pm 1/40$ reflects the underlying spherical design properties of the ADE root shells.