Generators and boundary geometry of the algebra of differential Siegel modular forms DM(Γ) for genus two curves with a marked Weierstrass point

Cao–Movasati–Yau (Adv. Math. 383, 2021) constructed, for genus two curves with a marked Weierstrass point, an affine moduli space T of curves enhanced with frames of their de Rham cohomology, computed its Gauss–Manin connection and three modular vector fields R_k, identified the index-six subgroup Γ ⊂ Sp(4, ℤ) governing the functional equations, and pulled the coordinate ring O_T back to an explicit algebra of 153 meromorphic functions on the Siegel upper half space H₂ that is closed under the three canonical derivations. The subalgebra DM(Γ) ⊂ O_T of differential Siegel modular forms — elements whose pullback is holomorphic along the boundary divisor {τ₃ = 0} — is left undetermined: beyond two known elements (T₂₀ and Δ) no generators are known, the authors state that finding them "would be essential" as the genus-two generalization of Igusa's classical generator theorem, and the required geometric input is open: explain why the invariant combinations A = χ₁₂/χ₁₀ and E = χ₃₅/χ₁₀² of Igusa's generators pull back with pole orders exactly 2 and 4 along τ₃ = 0, and clarify why the R_k are meromorphic along the boundary although the abelian-variety framework predicts holomorphic modular vector fields. The problem remains open.

Background

The classical setting. Let H₂ denote the Siegel upper half space of genus two: the set of symmetric complex 2×2 matrices τ = [[τ₁, τ₃], [τ₃, τ₂]] whose imaginary part is positive definite. The integral symplectic group Sp(4, ℤ) acts on H₂ by τ ↦ (aτ+b)(cτ+d)⁻¹, and a Siegel modular form of weight w for a subgroup Γ ⊂ Sp(4, ℤ) is a holomorphic function f on H₂ satisfying f((aτ+b)(cτ+d)⁻¹) = det(cτ+d)^w·f(τ) for all elements of Γ, together with holomorphic behavior at the cusps. Via the period map (which parametrizes a complex torus by integrals of differential forms over cycles), the quotient Sp(4, ℤ)\H₂ identifies with the moduli space A₂ of principally polarized abelian surfaces; the moduli M₂ of smooth genus two curves sits inside it as an open subset whose complement is the locus of products of two elliptic curves, which corresponds to the divisor {τ₃ = 0}. Igusa (1967) computed the ring of genus two Siegel modular forms for the full group: it is generated by five explicit forms E₄, E₆, χ₁₀, χ₁₂, χ₃₅ (subscripts are weights), obtained from the invariant theory of binary sextics. Genus two curves are hyperelliptic (double covers of the projective line), and after marking a Weierstrass point — a fixed point of the hyperelliptic involution, of which a genus two curve has six — every such curve can be written as y² = x⁵ + t₂x³ + t₃x² + t₄x + t₅; the moduli M₂(w) of marked curves is then a weighted projective space P(2,3,4,5) with a discriminant and t₅ removed, and the forgetful map M₂(w) → M₂ is six-to-one.

The differential side. The three canonical derivations ∂/∂τ_k, k = 1, 2, 3, act on meromorphic functions on H₂, and one may ask for differential equations among Siegel modular forms. Resnikoff (1970) found the first such equations for genus two Eisenstein series, and Bertrand–Zudilin (2003) proved that the field generated by genus g Siegel modular forms and their derivatives has transcendence degree 2g² + g. The prototype is Ramanujan's system of differential equations for the elliptic Eisenstein series E₂, E₄, E₆: the smallest algebra of functions on the elliptic upper half plane that contains modular forms and is closed under d/dτ is the algebra of quasi-modular forms, and modular forms are exactly the elements with controlled growth at the cusp.

The Gauss–Manin connection in disguise. Movasati and collaborators reinterpreted such differential equations geometrically. On a moduli space of varieties enhanced with a frame of their de Rham cohomology, the Gauss–Manin connection — the flat connection describing how the cohomology basis varies as the variety moves in a family — produces distinguished vector fields, called modular vector fields, which under the period map become the derivations ∂/∂τ_k. Cao–Movasati–Yau carried this out for genus two curves with a marked Weierstrass point: they constructed the affine moduli space T (curve plus marked point plus basis of H¹_dR compatible with the cup product), computed the Gauss–Manin connection explicitly, produced the three modular vector fields R₁, R₂, R₃, identified the resulting functional-equation group — an explicit index-six subgroup Γ ⊂ Sp(4, ℤ), with Γ\H₂ containing M₂(w) as an open subset — and defined the t-map t: H₂ − {τ₃ = 0} → T, assigning to τ the unique enhanced curve whose period matrix has the form [[τ, −I], [I, 0]]. Pulling O_T back along t yields 153 explicit meromorphic functions X₁, …, X₁₅₃ on H₂ with poles only along τ₃ = 0; five of them, T₄, T₈, T₁₂, T₁₆, T₂₀, are Siegel modular forms for Γ of weights 1, 2, 3, 4, 5 (the grading is a quarter of the canonical weight because the enhancement carries an extra $\mathbb{G}_m$-symmetry, $\mathbb{G}_m$ being the multiplicative group), with the relation $T_{16}^2 = T_{20}T_{12}$ and T₂₀ = t₅² vanishing exactly along τ₃ = 0; the ring ℚ[X]/I is an open subset of a weighted projective space, and each derivative ∂X_i/∂τ_k becomes a polynomial in the X's after multiplication by X₅ = T₂₀ (Theorem 1.1 of the source).

The gap. The t-pullback of a general element of O_T is only meromorphic along τ₃ = 0, so the genuinely modular part is, by definition,

$\mathrm{DM}(\Gamma) := \{\, f \in \mathcal{O}_T \mid f \circ \mathsf{t} \ \text{is holomorphic along}\ \tau_3 = 0 \,\}$

the algebra of differential Siegel modular forms. It plays the role modular forms play inside quasi-modular forms, and it carries the differential structure induced by the R_k. The source proves it is nontrivial (T₂₀ and a weight-10 form Δ belong to it) but does not determine it. The authors state that, to generalize Igusa's result, "it would be essential … to find generators for the DM(Γ)", and identify the prerequisite: a geometric explanation of why the invariant-theoretic generators A = χ₁₂/χ₁₀ and E = χ₃₅/χ₁₀² pull back with pole orders exactly 2 and 4 along τ₃ = 0 (χ₁₀ vanishes to order two there); the source studies this through the behavior of periods near the nodal degeneration (§7.3). A related structural mismatch: the general theory for principally polarized abelian varieties (Fonseca's higher Ramanujan equations; Movasati's book framework) predicts modular vector fields holomorphic on the whole moduli, yet the R_k have a pole of order one along t₅ = 0 — equivalently, along τ₃ = 0 after pullback — so the differential structure acting on DM(Γ) is at present only meromorphic.

Problem Statement

Determine the algebra of differential Siegel modular forms DM(Γ) for the index-six subgroup Γ ⊂ Sp(4, ℤ) attached to genus two curves with a marked Weierstrass point, as constructed by Cao–Movasati–Yau. With T the affine moduli of enhanced marked genus two curves, t: H₂ − {τ₃ = 0} → T the period-enhancement map, and

$\mathrm{DM}(\Gamma) := \{\, f \in \mathcal{O}_T \mid f \circ \mathsf{t} \ \text{is holomorphic along}\ \tau_3 = 0 \,\},$

the problem has three linked components that together give the explicit structure theory of this differential algebra:

(i) Generators and relations. Produce an explicit finite set of generators of DM(Γ) as a ℂ-algebra, the ideal of polynomial relations among them, and their weights under the Γ-action — the generalization of Igusa's theorem that the ring of genus two Siegel modular forms for Sp(4, ℤ) is generated by E₄, E₆, χ₁₀, χ₁₂, χ₃₅. The answer must be compatible with the ambient data of the source's Theorem 1.1 (the 153 functions X_i, the ring ℚ[X]/I as an open subset of a weighted projective space, and the polynomiality of the derivations after multiplication by T₂₀).

(ii) Boundary geometry. Give a geometric characterization of which elements of O_T pull back holomorphically along the boundary divisor {τ₃ = 0} (the nodal, product-of-elliptic-curves degeneration); in particular, explain why the combinations A = χ₁₂/χ₁₀ and E = χ₃₅/χ₁₀² of Igusa's invariants acquire pole orders exactly 2 and 4, and which algebraic functions on T are holomorphic versus meromorphic after pullback.

(iii) Holomorphicity of the modular vector fields. Clarify the status of the vector fields R₁, R₂, R₃: the framework for principally polarized abelian varieties predicts holomorphic modular vector fields, but the R_k have pole order one along t₅ = 0 and thus act meromorphically along τ₃ = 0. Determine whether this meromorphicity is intrinsic to the marked-Weierstrass-point enhancement or can be removed.

The three components form one scientific objective: (ii) is explicitly presented in the source as the prerequisite for (i), and (iii) concerns the derivations that make DM(Γ) a differential algebra; an answer that resolved (i) without the geometric input of (ii)–(iii), or vice versa, would not complete the structure theory. Scope note: the source additionally lists four further directions — writing the constructed forms as Poincaré series (series obtained by averaging a seed function over the group action, the standard route to the Fourier, i.e. q-, expansions of modular forms) together with explicit recursions for their coefficients; proving geometrically that Resnikoff's differential operator maps $\mathcal{O}_S$ — the algebra of regular functions on the source's intermediate moduli space $S$ of marked genus two curves, $S = \mathrm{Spec}\,\mathbb{C}[T_4, T_8, T_{12}, T_{16}, T_{20}, (T_{20}\Delta)^{-1}]/\langle T_{16}^2 - T_{12}T_{20}\rangle$, through which Igusa's generators are expressed as polynomials in the five Γ-modular functions $T_i$ — into its cusp forms (modular forms vanishing at the cusps); deriving explicit equations for Humbert surfaces (the loci in the moduli of principally polarized abelian surfaces parametrizing those admitting an extra symmetric endomorphism of a given positive discriminant) from the foliations induced by the vector fields $R_k$ on $\mathbb{H}_2$; and extending the whole construction to the Clingher–Doran family of lattice-polarized K3 surfaces (K3 surfaces — simply connected complex surfaces with trivial canonical bundle — whose Néron–Severi lattice contains a specified fixed lattice). These are adjacent open directions of the same research program, not parts of this problem's completion boundary, since they concern other deliverables or other moduli spaces.

References

1. Jin Cao, Hossein Movasati, Shing-Tung Yau, "Gauss-Manin connection in disguise: Genus two curves", Advances in Mathematics 383 (2021) 107684. DOI: 10.1016/j.aim.2021.107684; arXiv:1910.07624; https://arxiv.org/abs/1910.07624
2. Jun-ichi Igusa, "Modular forms and projective invariants", American Journal of Mathematics 89 (1967) 817–855. DOI: 10.2307/2373243; https://doi.org/10.2307/2373243
3. H. L. Resnikoff, "A differential equation for an Eisenstein series of genus two", Proceedings of the National Academy of Sciences USA 65 (1970) 495–496. DOI: 10.1073/pnas.65.3.495; https://doi.org/10.1073/pnas.65.3.495
4. D. Bertrand, W. Zudilin, "On the transcendence degree of the differential field generated by Siegel modular forms", Journal für die reine und angewandte Mathematik 554 (2003) 47–68. DOI: 10.1515/crll.2003.008; https://doi.org/10.1515/crll.2003.008
5. Tiago J. Fonseca, "Higher Ramanujan equations and periods of abelian varieties", Memoirs of the American Mathematical Society 281 (2022), no. 1391. DOI: 10.1090/memo/1391; arXiv:1807.11044; https://arxiv.org/abs/1807.11044
6. Charles Doran, Andrew Harder, Hossein Movasati, Ursula Whitcher, "Humbert surfaces and the moduli of lattice polarized K3 surfaces", Proceedings of Symposia in Pure Mathematics 93 (2016) 109–140. DOI: 10.1090/pspum/093/01559; https://doi.org/10.1090/pspum/093/01559
7. Walter Páez Gaviria, "Modular Vector Fields for Lattice Polarized K3", Journal of Algebra 673 (2025) 410–454. DOI: 10.1016/j.jalgebra.2025.03.002; arXiv:2404.06662; https://arxiv.org/abs/2404.06662
8. Atsuhira Nagano, "Picard-Fuchs system for family of Kummer surfaces as subsystem of GKZ hypergeometric system", arXiv:2408.14271 (2024). https://arxiv.org/abs/2408.14271
9. Murad Alim, Vadym Kurylenko, Martin Vogrin, "The algebra of derivations of quasi-modular forms from mirror symmetry", Pure and Applied Mathematics Quarterly 18 (2022), no. 3. DOI: 10.4310/pamq.2022.v18.n3.a6; arXiv:2008.06523; https://arxiv.org/abs/2008.06523
10. Samuel Grushevsky, Tomoyoshi Ibukiyama, Gabriele Mondello, Riccardo Salvati Manni, "Differentiating Siegel modular forms, and the moving slope of A_g", International Mathematics Research Notices 2024 (2024), no. 4, 3442–3486. DOI: 10.1093/imrn/rnad203; arXiv:2207.04139; https://arxiv.org/abs/2207.04139
11. Jin Cao, Hossein Movasati, Roberto Villaflor Loyola, "Gauss-Manin connection in disguise: Quasi Jacobi forms of index zero", International Mathematics Research Notices 2024 (2024), no. 8, 6680–6709. DOI: 10.1093/imrn/rnad260; arXiv:2109.00587; https://arxiv.org/abs/2109.00587
12. Felipe Espreafico, "Gauss-Manin Connection in Disguise: Open Gromov-Witten Invariants", Communications in Mathematical Physics 407 (2026), no. 3, article 61. DOI: 10.1007/s00220-025-05504-1; arXiv:2205.08302; https://arxiv.org/abs/2205.08302
13. Hossein Movasati, "Ramanujan vector field", arXiv:2406.13432 (2024). https://arxiv.org/abs/2406.13432
14. T. Shaska, "Degrees of Genus-Two Split-Jacobian Loci and Humbert-Form Reconstruction", arXiv:2608.16426 (2026). https://arxiv.org/abs/2608.16426
15. Adrian Clingher, Charles F. Doran, "Lattice polarized K3 surfaces and Siegel modular forms", Advances in Mathematics 231 (2012) 172–212. DOI: 10.1016/j.aim.2012.05.001; https://doi.org/10.1016/j.aim.2012.05.001
16. Albrecht Klemm, Maximilian Poretschkin, Thorsten Schimannek, Martin Westerholt-Raum, "On direct integration for mirror curves of genus two and an almost meromorphic Siegel modular form", Communications in Number Theory and Physics 10 (2016), no. 4, 587–701. DOI: 10.4310/cntp.2016.v10.n4.a1; arXiv:1502.00557; https://arxiv.org/abs/1502.00557