Maschke Octic: Geometry and Arithmetic

• Maschke Octic is a smooth algebraic surface of degree eight defined in projective space, characterized by a high-order symmetry group including a central extension of the Heisenberg group.
• It uniquely attains Miyaoka’s bound with 96 skew lines and exhibits a rich structure of invariants, lines, and cohomological decompositions linking it to modular forms.
• Its arithmetic realization is marked by an irreducible degree-8 polynomial with minimal discriminant and Galois group PSL₂(7), emphasizing its significance in number theory.

The Maschke octic refers to a distinguished smooth algebraic surface of degree eight in projective space and its associated invariants, polynomials, and arithmetic analytic structures. It occupies a central position in invariant theory, algebraic geometry, the theory of finite groups, and arithmetic geometry. The term also denotes a symmetric octic form in three variables associated with the reflection group G(4,2,2), and a number field defined by an irreducible degree-8 polynomial with Galois group PSL₂(7) and minimal discriminant.

1. Defining Equations and Core Symmetries

The canonical geometric realization of the Maschke octic is as the locus in projective space $\mathbf{P}^3(\mathbb{C})$ defined by the invariant polynomial: $$F(x_0, x_1, x_2, x_3) = x_0^8 + x_1^8 + x_2^8 + x_3^8 + 14 \sum_{0 \leq i < j \leq 3} x_i^4 x_j^4 + 168 x_0^2 x_1^2 x_2^2 x_3^2 = 0$$ This surface is non-singular over fields with characteristic other than 2, 3, 5. The polynomial is invariant under the action of a large symmetry group $G \subset \mathrm{PGL}(4,\mathbb{C})$ (order 46080), explicitly realized as a central extension containing the Heisenberg group $H \subset \mathrm{GL}(4,\mathbb{C})$ and surjecting onto $S_6$. The irreducibility of the action implies the ring of invariants is generated in degree eight with no lower-degree invariants (Bini et al., 2011, Schuett, 2011).

In three variables, the Maschke octic is the symmetric form

$$\phi(x_1, x_2, x_3) = x_1^8 + x_2^8 + x_3^8 - 2 x_1^4 x_2^4 - 2 x_1^4 x_3^4 - 2 x_2^4 x_3^4$$

which is alternatively expressed as

$$\phi(x_1, x_2, x_3) = (x_1^4 + x_2^4 + x_3^4)^2 - 4(x_1^4 x_2^4 + x_2^4 x_3^4 + x_3^4 x_1^4)$$

This form is the unique basic octic invariant of the reflection group G(4,2,2) in three variables, invariant under all coordinate permutations and certain involutions (Choudhry et al., 2022).

2. Group Theory and Invariance

The automorphism group $G$ fixing Maschke’s octic in $\mathbb{P}^3$ is constructed via the normalizer of $H$ in $\mathrm{GL}(4, \mathbb{C})$, with the exact sequence: $$1 \to \mu_4 \cdot H \longrightarrow N \longrightarrow Sp(4, \mathbb{F}_2) \cong S_6 \to 1$$ Generators include coordinate sign-changes, permutations, and Heisenberg subgroup actions. This symmetry induces a highly interconnected structure on the cohomology, especially on the Hodge decomposition and the arrangement of lines (Bini et al., 2011, Schuett, 2011).

For the polynomial form $\phi(x_1, x_2, x_3)$, S₃ acts by permutations, and additional involutive symmetries exist for the replacement $x_i \mapsto -x_i$, corresponding to the structure of binary quartic discriminants studied by Maschke (Choudhry et al., 2022).

3. Geometry: Lines, Hodge Numbers, and Néron–Severi Group

Maschke’s octic surface $\mathcal{M}$ in $\mathbb{P}^3$ contains precisely 352 lines, partitioned into two G-orbits of size 160 and 192. Of particular significance is the realization of Miyaoka’s bound: $\mathcal{M}$ contains exactly 96 pairwise disjoint (skew) lines, confirming that for degree 8, the Miyaoka bound $2d(d-2) = 96$ is optimal. Prior lower bounds for smooth octics stood at 50 skew lines; Maschke’s octic more than doubles this record (Bonnafé, 30 Dec 2025).

As a hypersurface of degree 8, Maschke’s octic has geometric genus $p_g = h^{2,0}=35$ and Hodge number $h^{1,1}=232$. The Néron–Severi group $NS(S)$ is generated by the 352 lines and has rank 202. The orthogonal complement, the transcendental lattice $T_S$, is of rank 100 (Bini et al., 2011).

Component	Count/Rank	Structural Role
Lines on surface	352 (160+192 orbits)	Generate NS(S), rank 202
Skew lines on surface	96	Maximal for degree 8 (sharp Miyaoka bound)
Transcendental lattice	100	Modular components, cohomological splits

4. Arithmetic Realizations and Number Fields

In arithmetic geometry, the Maschke octic is represented by the degree-8 irreducible polynomial: $$f(x) = x^8 - 4x^7 + 7x^6 - 7x^5 + 7x^4 - 7x^3 + 7x^2 + 5x + 1$$ This defines a number field of minimal discriminant $|\Delta(f)| = 3^8 \cdot 7^8 = (21)^8$ among octic fields with Galois group $\mathrm{PSL}_2(7)$. The normal closure realizes the group as the transitive subgroup 8T37 of $S_8$, acting on the eight roots via projective linear transformations of $\mathbb{P}^1(\mathbb{F}_7)$. The field is the unique octic subfield inside the splitting field of the Trinks septic $x^7 - 7x + 3$ (Jones et al., 2016).

5. Associated Diophantine Equations and Elliptic Curves

The octic form $\phi(x_1, x_2, x_3)$ enables construction of integer and rational solutions to $\phi(x_1,x_2,x_3) = \phi(y_1, y_2, y_3)$, yielding parametric families of triples $(x_i)$ and $(y_i)$ such that triangles with sides $x_i^2$ and $y_i^2$ have equal areas (via Heron's formula, with $16A^2 = \phi(x_1, x_2, x_3)$) (Choudhry et al., 2022).

Two explicit families—one degree-8, one degree-5 in the parameter $t$—generate infinitely many such pairs. Each family leads to a one-parameter family of elliptic curves over $\mathbb{Q}(t)$ of rank five. For the degree-5 family, the associated Weierstrass form is

$E_t : V^2 = U^3 + A(t)U$

with $A(t) = 36\prod_{i=1}^6 h_i(t)$ given explicitly. Five rational points $P_1(t),...,P_5(t)$ are shown to generate the Mordell–Weil group modulo torsion by specialization and regulator computations.

6. Calabi–Yau Threefolds and Cohomological Decomposition

By taking the double cover branched along the Maschke octic, one obtains the Calabi–Yau threefold $X$, given in weighted projective space by

$$w^2 = x_0^8 + x_1^8 + x_2^8 + x_3^8 + 14\sum x_i^4 x_j^4 + 168 x_0^2 x_1^2 x_2^2 x_3^2$$

$X$ has Hodge numbers $h^{3,0}(X) = 1$, $h^{2,1}(X) = 149$, and $h^{1,1}(X) = 2$. The action of $G$ on $X$ induces a decomposition of $H^3(X)$ into 150 two-dimensional Hodge substructures, each conjecturally or provably modular and associated to classical modular forms (Bini et al., 2011).

Further, the quotient by the Heisenberg subgroup $H \cong (\mathbb{Z}/2)^3$ yields another Calabi–Yau threefold $Y = X/H$, with $b_3(Y) = 30$ and cohomology explicitly described in terms of modular representations (Schuett, 2011).

7. Modularity and Galois Representations

Arithmetic aspects of Maschke’s octic relate its cohomology to modular forms. The Néron–Severi group decomposes into one-dimensional Artin characters, while the transcendental lattice splits into modules proven or conjectured to correspond to newforms of specific level and weight:

• $W_3$: 2-dimensional, attached to the unique weight-3 newform of level 15 with Nebentypus character $\chi_{-15}$.
• $W_7$: 3-dimensional module, conjecturally attached to a singular K3 surface.

This modularity is proven via geometric isogenies between K3 surfaces and point-counting over finite fields using the Lefschetz trace formula and representation-theoretic decompositions. Explicit formulas relate Frobenius traces to Fourier coefficients of these newforms, and confirm the compatibility of Galois action with the predicted arithmetic structure (Schuett, 2011).

8. Research Context and Open Problems

The extremal properties of the Maschke octic—maximal skew line arrangements, high symmetry, modular cohomological components—demonstrate the utility of complex reflection groups and invariant theory in constructing algebraic surfaces with significant geometric and arithmetic invariants. For octic surfaces, the bound of 96 skew lines is only attained by Maschke’s octic, and for degrees other than 2, 3, 4, 8, sharpness of Miyaoka’s bound remains unresolved.

Interplay between surface geometry and arithmetic structure—visible in the splitting of cohomology and explicit modularity results for associated Calabi–Yau varieties—positions Maschke’s octic as a central object in the contemporary study of algebraic surfaces, automorphism groups, diophantine equations, and the modularity of Galois representations (Bonnafé, 30 Dec 2025, Bini et al., 2011, Schuett, 2011, Choudhry et al., 2022, Jones et al., 2016).