Leaf schemes and Hodge loci

This is a collection of articles, written as sections, on arithmetic properties of differential equations, holomorphic foliations, Gauss-Manin connections and Hodge loci. Each section is independent from the others and it has its own abstract and introduction and the reader might get an insight to the text by reading the introduction of each section. The main connection between them is through comments in footnotes. Our major aim is to develop a theory of leaf schemes over finitely generated subrings of complex numbers, such that the leaves are also equipped with a scheme structure. We also aim to formulate a local-global conjecture for leaf schemes.

💡 Research Summary

The manuscript “Leaf Schemes and Hodge Loci” is a collection of six largely independent chapters that together develop a new framework for studying arithmetic aspects of differential equations, holomorphic foliations, Gauss‑Manin connections, and Hodge loci. The central notion introduced is that of a leaf scheme, a scheme‑theoretic enhancement of a leaf of a foliation, which may be non‑reduced and of varying codimension. The author’s overarching goal is to formulate a local‑global principle for leaf schemes and to apply it to the algebraicity of Hodge loci.

Chapter 1 revisits the Grothendieck‑Katz p‑curvature conjecture. The author proves that if every solution of a linear differential equation is algebraic, then after multiplying the n‑th derivative by a suitable factorial factor, all but finitely many primes disappear from the denominators of the resulting matrix entries (Theorems 1.1 and 1.3). This motivates the definition of an mp,k‑curvature, a refinement of the classical p‑curvature (mp,1 coincides with the usual p‑curvature). The converse statement—if the mp,k‑curvature vanishes for all k then the solutions must be algebraic—is posed as Conjecture 1.1, suggesting that the original Grothendieck‑Katz conjecture might be false.

Chapter 2 focuses on Lamé equations, providing explicit examples where the p‑curvature vanishes for a single prime but the higher mp,k‑curvature does not. This leads to the introduction of a p‑curvature density for a differential equation, a notion not previously studied. The author shows that for certain Lamé families the set of primes with vanishing p‑curvature can have positive density, yet the equations are not globally algebraic.

Chapter 3 is the conceptual core. A leaf scheme is defined as a (possibly non‑saturated) module of differential 1‑forms, allowing the underlying analytic set to lie inside the singular locus of the foliation. Hodge loci are realized as leaf schemes of specific foliations. The author proves that, after reduction modulo a prime, only finitely many primes appear in the denominators of the defining ideal of a leaf scheme (Theorem 3.5). This finiteness mirrors the algebraicity of Hodge loci proved by Cattani‑Deligne‑Kaplan, but now it is expressed in the language of leaf schemes. The central Local‑Global Conjecture for Leaf Schemes (Conjecture 3.2) asserts that integrality of the coefficients of a leaf scheme over a finitely generated ring forces the leaf scheme to be algebraic, thereby generalizing the Grothendieck‑Katz conjecture.

Chapter 4 departs from the abstract theory to study a concrete Ramanujan vector field on the moduli space of elliptic curves (denoted “ibip oranga”). The author identifies loci where the vector field is invariant under the Frobenius at a fixed prime p (vₚ = v) and shows that these loci control the integrality of Fourier coefficients of modular forms. Theorem 4.1 and 4.8 give a new characterization of CM elliptic curves via the Ramanujan field, providing a bridge to the integrality phenomena needed later for Calabi‑Yau modular forms.

Chapter 5 gathers computational evidence for three conjectures (0.1–0.3) concerning Hodge loci of cubic hypersurfaces in ℙ⁷ and ℙ⁹, and of degree‑8 surfaces in ℙ³. Using high‑precision Taylor expansions of period integrals (coefficients in ℚ(ζ₂ᵈ)), the author verifies the conjectures up to 14‑th, 6‑th, and 5‑th order respectively. The calculations reveal that denominators involve only a slowly growing set of primes, supporting the belief that the leaf‑scheme integrality condition holds in practice.

Chapter 6 treats vector fields themselves as differential equations. Theorem 6.1 establishes existence and uniqueness for integral curves; Theorem 6.3 shows that if a vector field admits an algebraic integral curve, then after multiplying by n! the coefficients become integral away from finitely many primes. Conjectures 6.2 and 6.3 formulate a local‑global principle for vector fields, again mirroring Conjecture 3.2.

Overall, the paper proposes a unifying perspective: leaf schemes provide a scheme‑theoretic setting where arithmetic properties (integrality, reduction modulo primes) and geometric properties (algebraicity of Hodge loci, foliation leaves) interact. The author supplies both theoretical results (integrality theorems, finiteness of bad primes) and extensive computational data (Lamé equations, period integrals) to support the conjectural framework. While the exposition is fragmented—each chapter repeats definitions and the overall narrative can be hard to follow—the underlying ideas are original and could stimulate further research on the arithmetic of foliations, the refinement of p‑curvature concepts, and the arithmetic geometry of Hodge loci.