Grothendieck-Katz p-Curvature Conjecture

• Grothendieck-Katz p-curvature conjecture is a central concept linking the vanishing of p-curvature in differential equations to the existence of algebraic solutions and finite monodromy.
• Recent advancements extend the classical p-curvature to higher m-curvatures via global congruence conditions and effective algorithms for order one equations.
• Proven cases in rank one, Picard–Fuchs, Gauss–Manin connections, and rigid systems highlight its significance while ongoing research explores implications for higher rank and non-abelian settings.

The Grothendieck-Katz $p$-curvature conjecture is a central problem in arithmetic algebraic geometry and differential equations, connecting the arithmetic of differential equations to their monodromy and solution spaces. It posits a remarkable equivalence between the vanishing of the $p$-curvature of a flat linear differential equation modulo almost all primes $p$ and the existence of a full set of algebraic solutions or, equivalently, finite monodromy. Recent research has both strengthened the conjecture—by introducing higher “$m$-curvatures” and global congruence conditions—and related it deeply to algebraic foliations, D-modules, and the geometry of arithmetic varieties. The conjecture is proven in many important cases (e.g., rank 1, Picard–Fuchs, Gauss–Manin connections, rigid systems), and its refinements unify several finiteness properties across arithmetic geometry and D-module theory.

1. Classical Formulation and Definitions

Let $F$ be the ring of integers in a number field, and consider a linear differential system over $F(z)$: $\frac{dY}{dz} = A(z)Y,\quad A(z)\in M_n(F(z)),$ where $Y(z)$ is an $n\times n$ fundamental matrix of solutions. For almost all primes $p$ of $F$, reduction of $A(z)$ modulo $p$ (away from the poles) allows one to define the $p$-curvature endomorphism,

$\Psi_p(Y) = \frac{d^p Y}{dz^p} \bmod p,$

which acts $F_p(z)$-linearly on the space of solutions.

Grothendieck-Katz $p$-Curvature Conjecture:

The system $Y' = A(z)Y$ has a full set of algebraic solutions over $\overline{F(z)}$ if and only if for almost all primes $p$ the $p$-curvature $\Psi_p$ vanishes (Movasati, 2024).

In modern language, vanishing $p$-curvature is equivalent to the flat bundle becoming trivial over reductions modulo $p$ (i.e., admitting a full set of horizontal sections), and finite monodromy is equivalent to the global connection becoming trivial after some finite étale (Galois) cover (Bostan et al., 2023, Xu, 29 Jul 2025).

2. Higher $m$-Curvatures and Strengthened Conjecture

The concept of $m$-curvature generalizes $p$-curvature by considering all higher derivatives of the solution matrix: $${}_1 A = A, \quad {}_{n+1}A = \frac{d}{dz}({}_nA) + {}_nA,$$ so that

$\frac{d^n Y}{dz^n} = {}_nA(z) Y(z),$

and the $m$-curvature is the matrix ${}_mA(z)$. For $m=p$, ${}_pA \bmod p$ is the usual $p$-curvature. Given $k\in \mathbb{N}$ such that $\operatorname{ord}_p(m!) \geq k$, we can reduce ${}_mA(z)$ modulo $p^k$.

Movasati's Strengthened Conjecture:

There is an equivalence between the following statements for $Y'=A(z)Y$ (Movasati, 2024):

• All solutions are algebraic over $F(z)$.
• There exists $N\ge 1$ such that

$$\frac{(N\Delta(z))^m \; {}_mA(z)}{m!} \in F[z],\quad \forall m\ge 1.$$

• For all but finitely many primes $p$ and all $m,k\in\mathbb{N}$ with $\operatorname{ord}_p(m!)\geq k$,

${}_mA \equiv 0 \pmod{p^k}.$

This upgrade relates the entire tower of higher curvatures to algebraicity, not just the individual $p$-curvatures, and proves their vanishing for all but finitely many $p$ when solutions are algebraic. The converse forms a strictly stronger assertion than the classical conjecture and recovers known cases, including rank 1, Gauss–Manin, and rigid local systems (Movasati, 2024).

3. Known Cases, Effective Results, and Algorithms

The conjecture has been proven in several important settings:

• Rank One: Fully resolved by Honda and Chudnovsky; algebraicity ⇔ vanishing $p$-curvature for almost all $p$ (Bostan et al., 2023).
• Picard-Fuchs and Gauss–Manin Connections: Katz established the conjecture for de Rham cohomology with the Gauss–Manin connection; the $p$-curvature matches the Kodaira–Spencer map after taking the graded for the conjugate filtration, and its vanishing forces unitarity and thus finite monodromy (Lam et al., 12 Jan 2026, Menzies, 2019).
• Rigid Systems and Locally Symmetric Varieties: Follows via monodromy rigidity (Katz; Chudnovsky–Chudnovsky; Esnault–Groechenig) (Movasati, 2024).
• Order One Equations: An effective version is established: for $y' = u(x) y$, it suffices to check vanishing of $p$-curvature for $O(h^{12d-6}d^{12d})$ small primes (where $h$ is the coefficient height, $d$ the degree) (Fürnsinn et al., 1 Oct 2025).

Algorithmic Table for Order One Equations (Fürnsinn et al., 1 Oct 2025):

Step	Description	Complexity
Compute resultant $R(w)$	For $u(x)=a(x)/b(x)$ compute splitting polynomial	$\tilde O(d^2 \log H)$
Search for roots	Check splitting mod $p$ for $p < \sigma$	Arithmetic in $p$, $d$, $H$
$p$-curvature computation	Via finite field reduction, rational function manipulation	$\tilde O(H^{12d-6}d^{12d})$

4. Geometric, Topological, and Foliation-Theoretic Perspectives

Recent studies embed the conjecture within broader geometric frameworks:

• Foliations and Algebraic Integrability: Ekedahl–Shepherd-Barron–Taylor’s Conjecture F states: An algebraic foliation $\mathcal{F} \subset T_X$ is algebraically integrable iff its reduction is $p$-integrable (closed under $p$-th powers) for almost all $p$ (Xu, 29 Jul 2025). Xu proves this is equivalent to the $p$-curvature conjecture by associating to any connection a horizontal (flat) foliation; finite monodromy corresponds to algebraic integrability, and vanishing $p$-curvature to $p$-integrability.
• Topological Reformulations: The monodromy representation’s finite image can be characterized topologically, e.g., via isomonodromic deformations and mapping class group orbits (Lawrence–Litt) (Lawrence et al., 2019). Universally finite orbits correspond to the vanishing of $p$-curvature across finite covers, providing a group-theoretic perspective.
• Non-Abelian Generalizations: For the stack of flat connections or "non-abelian Gauss–Manin connection," if the $p$-curvature of the isomonodromy foliation vanishes for infinitely many $p$, then the monodromy action on integral characters is finite (Lam et al., 12 Jan 2026, Menzies, 2019).

5. Known Partial Results, Refinements, and Open Problems

Partial Results include:

• Complete proofs for rank 1, Picard-Fuchs, Gauss–Manin, and rigid systems.
• For generic curves, proven for rank 2 bundles (Patel–Shankar–Whang; Shankar) (Patel et al., 2018, Shankar, 2016).
• For $q$-difference equations, a tight $q$-analogue is established, substituting $q_v$-curvature for $p$-curvature and cyclotomic places for primes (Vizio et al., 2012).
• Effective bounds and decision procedures for order 1 (Fürnsinn et al., 1 Oct 2025).

Refinements and Strengthenings:

• Movasati’s higher $m$-curvature criterion gives a strictly sharper necessary and sufficient condition (Movasati, 2024).
• Integrality criteria: algebraicity of formal power series solutions correlates with integrality of Taylor coefficients (and thus, finiteness of primes dividing their denominators), yielding a more refined diagnostic for algebraicity than pure $p$-curvature checks (Lam et al., 22 Jan 2025).

Open Directions:

• Proving the full converse in Movasati’s “higher curvature” hypothesis for all $(m,k)$.
• Constructing general effective algorithms for higher rank and arbitrary order, and establishing Chebotarev-type density results for differential Galois groups (Bostan et al., 2023).
• Extending the theory to families with regular singularities along divisors, higher-dimensional varieties, and higher $D$-modules (Movasati, 2024, Esnault et al., 2016).
• Non-abelian generalizations and links to the algebraicity of leaves of isomonodromy foliations (Bost/Ekedahl–Shepherd-Barron–Taylor) (Lam et al., 12 Jan 2026).

6. Key Research Milestones and Evidence Table

Result / Case	Proof status	Techniques / Reference
Rank 1 (arbitrary curves)	Proven	Reduction, Chudnovsky, Cartier theory
Order 1 (arbitrary coefficients)	Proven, effective algorithm	Rothstein–Trager, Honda, Hermite–Padé
Picard–Fuchs, Gauss–Manin	Proven	Kodaira–Spencer, Hodge theory, Katz
"Generic" rank 2 bundles	Proven	Deformation, topological rigidity
Rigid systems, hypergeometric	Proven	Monodromy rigidity, explicit calculations
$q$-difference analogues	Proven	Cyclotomic reduction, functional methods
Higher $m$-curvature	Algebraic $\Rightarrow$ vanishing, converse is conjecture	Divided powers, Eisenstein, factorials

7. Broader Impact and Theoretical Connections

The $p$-curvature conjecture and its refinements establish a deep arithmetic local-to-global principle for differential equations, analogous to the Hasse principle for quadratics. The conjecture unifies aspects of algebraic geometry, number theory, D-module theory, arithmetic jet spaces, algebraic/analytic foliations, and non-abelian Hodge theory. It also relates to the algebraicity of formal power series, Galois-theoretic classifications, and the arithmetic properties of periods and motives. Recent work (e.g., on $p$-integrability, higher curvatures, and $q$-difference analogues) indicates that resolution of the full conjecture would not only settle a central problem of Grothendieck’s anabelian program but also harmonize several finiteness conjectures across algebraic geometry, differential algebra, and arithmetic dynamics (Movasati, 2024, Xu, 29 Jul 2025, Lam et al., 12 Jan 2026).