Bernstein--Sato Theory for D-modules in Positive Characteristic

Abstract: In this article, we develop a positive characteristic analogue of the Bernstein--Sato theory for holonomic D-modules in the complex setting. We work with D-modules on a Noetherian regular $F$-finite $\mathbb{F}_p$-scheme $X$, and define their Bernstein--Sato roots as $p$-adic integers. When the D-module is the structure sheaf $O_X$, this recovers Bitoun's definition. When the D-module arises from a locally finitely generated unit $F<sup>e$-module and $X$ is of finite type over an $F$-finite field, we show that the roots are finite and rational, generalizing Bitoun's result. In the course of the proof, we also develop a related theory for Cartier modules.

• The paper extends Bernstein–Sato theory to positive characteristic for general holonomic D-modules, establishing finite and rational roots.
• It employs refined Cartier module techniques and pushforwards along graph embeddings to overcome obstacles inherent in characteristic p.
• Results bridge Frobenius actions with D-module theory, offering new arithmetic invariants for singularity analysis.

Bernstein--Sato Theory for D-modules in Positive Characteristic

Introduction and Motivation

This work develops a positive characteristic analog of the Bernstein–Sato theory, primarily known in characteristic zero for its deep connections to singularity theory, $D$-module theory, and algebraic geometry via the Bernstein–Sato (or $b$-) polynomial and its roots. Classically, this theory provides invariants that link the geometry of hypersurfaces to the action of differential operators and to phenomena such as monodromy and nearby cycles. The lack of an analytic theory and the failure of the naive transfer of the characteristic zero formalism to positive characteristic has long obstructed a parallel theory. This paper overcomes these obstructions by systematically extending the Bernstein–Sato root formalism to holonomic $D$-modules over regular $F$-finite schemes of characteristic $p>0$, and elucidates the structural and arithmetic nature of these invariants.

Theoretical Framework and Main Constructions

Recollection: Bernstein–Sato in Characteristic Zero

Given $X/\mathbb{C}$ smooth, the Bernstein–Sato polynomial $b_f(s)$ of a function $f$ and its generalization to a section $u$ of a holonomic $D$-module $b$0 is determined by the minimal functional equation

$b$1

for some $b$2 in the extended Weyl algebra. The roots of $b$3 contain fine topological and analytical information—e.g., they correspond to eigenvalues of local monodromy, control the existence of the $b$4-filtration, and govern the jumps in multiplier ideals.

Bernstein–Sato Theory and Its Obstacles in Characteristic $b$5

The naive approach fails in characteristic $b$6 due to inseparability and the structure of rings of differential operators. Mustaţă introduced a construction for $b$7 on regular $b$8-finite schemes using the $b$9-modules (the ring of differential operators of level $D$0), defining a tower of $D$1-functions whose collection encodes arithmetic invariants replacing the classical roots.

Bitoun refined this, replacing the scalar parameter $D$2 by a variable in the ring of locally constant functions on $D$3, utilizing the isomorphism

$D$4

with $D$5. This identification, together with the action of the ring of all differential operators, enables the definition of Bernstein–Sato roots as $D$6-adic integers, generalizing the classical roots.

Extension to $D$7-modules

The key contribution of this paper is extending Bernstein–Sato theory to general holonomic $D$8-modules in positive characteristic, following Bitoun's reformulation in terms of pushforwards along graph embeddings, but for general $D$9-modules rather than just $F$0:

• The formal object $F$1 is defined for a $F$2-module $F$3 and $F$4 using the action of $F$5.
• Bernstein–Sato roots are then declared to be those $F$6 where $F$7.
• For modules arising from locally finitely generated unit $F$8-modules (lfgu), it is shown that the roots are both finite and rational (i.e., they lie in a rational subset of $F$9), directly generalizing the rationality theorem of Kashiwara in characteristic zero.

Additionally, the paper proves these results for broader objects known as Cartier modules, refining the link between Frobenius actions and $p>0$0-module theory in positive characteristic.

Structure of the Main Results

Generalized Construction

Given a $p>0$1-module $p>0$2 and an $p>0$3-submodule $p>0$4, the paper defines for each $p>0$5 a $p>0$6-module $p>0$7, and

$p>0$8

which recovers the classical definitions in both characteristic zero and, for $p>0$9, in characteristic $X/\mathbb{C}$0.

Main Theorem Schematic

Let $X/\mathbb{C}$1 be as above, with $X/\mathbb{C}$2 an lfgu $X/\mathbb{C}$3-module and $X/\mathbb{C}$4 coherent.

Theorem:

• If $X/\mathbb{C}$5 arises from an lfgu $X/\mathbb{C}$6-module and $X/\mathbb{C}$7 is coherent, the set of Bernstein–Sato roots $X/\mathbb{C}$8 is finite and rational under a mild boundedness assumption on the module structure, satisfied in practice whenever $X/\mathbb{C}$9 is of finite type over an $b_f(s)$0-finite field.
• The roots lie in the rational subset

$b_f(s)$1

and if $b_f(s)$2 is a minimal root in the sense of Cartier modules, then

$b_f(s)$3

Notably, $b_f(s)$4 can arise as a root in this generality—a phenomenon excluded in Bitoun’s original result for $b_f(s)$5—and is precisely characterized in terms of presence of constituents supported on the vanishing locus $b_f(s)$6.

Cartier Modules and $b_f(s)$7-invariants

The proof fundamentally proceeds by extending the theory of Cartier modules. For $b_f(s)$8-Cartier modules, the paper introduces "finite-level" and "infinite-level" $b_f(s)$9-invariants analogous to $f$0-jumping exponents, and shows that these invariants are also finite and rational under natural hypotheses. This further links test modules and their jumping exponents to Bernstein–Sato roots, generalizing previous results in the literature.

Technical Features and Innovations

Algebraic Instead of Analytic Formalism

A crucial innovation is the rigorous commitment to an entirely algebraic formalism, avoiding analytic tools unavailable in positive characteristic. The use of $f$1-adic analysis, $f$2-finite schemes, and the structure theory of finite-level operators replaces the analytic continuation and monodromy pictures of characteristic zero.

Extension to General $f$3-modules

While previous positive characteristic constructions either focused on the structure sheaf or special classes of test ideals and divided powers, this work demonstrates that the entire $f$4-module formalism for holonomic modules can carry Bernstein–Sato theory, provided the Frobenius actions are handled subtly via Cartier module technology.

Relationship to $f$5-filtration and Monodromy

A remarkable consequence is that the algebraic invariants constructed here admit an interpretation analogous to the $f$6-filtration and monodromy eigenvalues for regular holonomic $f$7-modules after translating via the Riemann–Hilbert correspondence (in positive characteristic, between unit $f$8-modules and (perverse) étale sheaves, cf. [BL19]). In particular, rationality of the roots again underpins the filtration structure.

Explicit Computation and Reduction Mechanisms

The paper provides explicit computation strategies for $f$9 in terms of pushforwards along graph embeddings, and shows how to reduce general statements about $u$0-modules to manageable combinatorics of Cartier module invariants and test module filtrations. Injectivity and direct sum decompositions are established via $u$1-adic expansion arguments.

Implications and Future Research Directions

The realization of a robust Bernstein–Sato theory for $u$2-modules in positive characteristic has several significant theoretical and practical implications:

• Algebraic and arithmetic invariants of singularities: These roots encode subtle information about arithmetic singularities and test module/jumping exponent behavior, extending the reach of birational representation theory in arithmetic geometry.
• Bridging $u$3-modules and étale sheaf theory in characteristic $u$4: These constructions provide an arithmetic analog of monodromy representations and nearby cycles, and, via the Riemann–Hilbert correspondence for $u$5-modules, promise applications to the theory of perverse étale sheaves and the structure of $u$6-adic cohomology.
• Foundations for $u$7-filtration and further vanishing cycle theory: The algebraic structure lays groundwork for a $u$8-filtration theory over $u$9-finite schemes, and its interaction with ramification theory, characteristic cycles, and wild ramification phenomena.
• Extension to singular schemes and other classes: The techniques here suggest avenues for extension to singular schemes (cf. [JNBQG]) and to more general (e.g., logarithmic) $D$0-modules or arithmetic $D$1-module theory.

Potential Future Directions

• Comparison with positive characteristic $D$2-filtrations: Examining the compatibility of this formalism with those defined by Stadnik and Stäbler ([Sta14], [Sta16]) may shed new light on the structure of arithmetic nearby cycles.
• Broader classes of singularities: Generalizing the theory to the non-regular setting and higher codimension subvarieties may give new perspectives on $D$3-singularities.
• Deformation and ramification theory: Interactions with the theory of test ideals and wild ramification in arithmetic and rigid geometry can be explored using these invariants.
• Explicit calculations and moduli: Constructive and computational approaches to Bernstein–Sato roots in families and moduli problems over positive characteristic fields may become feasible using this algebraic framework.

Conclusion

This paper rigorously constructs a Bernstein–Sato theory for holonomic $D$4-modules in positive characteristic, establishing both the existence and the arithmetic finiteness and rationality of Bernstein–Sato roots for a broad class of modules, and characterizing their structural, representation-theoretic, and arithmetic significance. The techniques bridge $D$5-adic, Cartier, Frobenius, and $D$6-module formalisms, laying a foundation for further advances in the algebraic analysis of singularities and arithmetic $D$7-module theory in characteristic $D$8.