Kernels of Arithmetic Jet Spaces and Frobenius Morphism

Abstract: For any $π$-formal group scheme $G$, the Frobenius morphism between arithmetic jet spaces restricts to generalized kernels of the projection map. Using the functorial properties of such kernels of arithmetic jet spaces, we show that this morphism is indeed induced by a natural ring map between shifted $π$-typical Witt vectors. In the special case when $G = \hat{\mathbb{G}}_a$, the arithmetic jet space, as well as the generalized kernels are affine $π$-formal planes with Witt vector addition as the group law. In that case the above morphism is the multiplication by $π$ map on Witt vector schemes. In fact, the system of arithmetic jet spaces and generalized kernels of any $π$-formal group scheme $G$ along with their maps and identitites satisfied among them are a generalization of the case of the Witt vector scheme with the system of maps such as the Frobenius, Verschiebung and multiplication by $π$.

• The paper introduces the representability and kernel filtration of arithmetic jet spaces, establishing a systematic study of their connection with Frobenius twists.
• The results allow $\delta$-characters in arithmetic differential equations to be reduced to linear algebraic operations.
• For smooth horizontal jet schemes J^n G where G represents a smooth algebraic group, and G takes on a prounipotent-group type structure

Background: Buium's arithmetic jet theory

The study of arithmetic jet spaces originates in Buium's program of arithmetic differential equations, which transplants the machinery of algebraic geometry's jet bundles into an arithmetic setting by replacing differentiation with the $p$-derivation $\delta_p(x) = \frac{x - x^{\sigma}}{p}$ defined on rings equipped with a lift of Frobenius. For a scheme $X$ of finite type over the $p$-adic integers $\mathbb{Z}_p$ (or more generally a $p$-adic ring $A$ with a fixed lift of Frobenius $\sigma : A \to A$), the $p$-jet space $J^n X$ parametrizes truncations of $\delta_p(x) = \frac{x - x^{\sigma}}{p}$0-typical curves into $\delta_p(x) = \frac{x - x^{\sigma}}{p}$1: points of $\delta_p(x) = \frac{x - x^{\sigma}}{p}$2, where $\delta_p(x) = \frac{x - x^{\sigma}}{p}$3, truncated modulo the $\delta_p(x) = \frac{x - x^{\sigma}}{p}$4-th power of the maximal ideal. The functor $\delta_p(x) = \frac{x - x^{\sigma}}{p}$5 is representable when $\delta_p(x) = \frac{x - x^{\sigma}}{p}$6 is smooth, yielding schemes $\delta_p(x) = \frac{x - x^{\sigma}}{p}$7 of finite type over $\delta_p(x) = \frac{x - x^{\sigma}}{p}$8, together with prolongation maps $\delta_p(x) = \frac{x - x^{\sigma}}{p}$9 and transition maps $X$0 compatible with composition.

The structural feature that distinguishes arithmetic jets from their geometric counterparts is that the kernel of the first projection carries genuine arithmetic information rather than being merely infinitesimal. Specifically, for a smooth $X$1-scheme $X$2, there is a canonical identification of the kernel of $X$3 with the Frobenius twist $X$4 of the tangent bundle, realized through the universal property of $X$5-prolongation. This identification is the point of departure for the present paper: the kernels of the jet projections, and the behavior of the Frobenius morphism on them, control the local structure of jet spaces and hence the arithmetic of $X$6-characters and arithmetic differential equations defined on $X$7.

The kernel filtration and its Frobenius-twisted layers

For each $X$8, the kernel $X$9 fits into a filtration of $p$0 whose successive quotients are Frobenius twists of vector bundles naturally attached to $p$1. Concretely, one obtains identifications of the form

$p$2

reflecting the fact that a $p$3-typical curve extending a given truncation has its next level determined by data valued in a module killed or controlled by the ideal generated by $p$4. The paper develops this filtration systematically, establishing:

• Exactness of the kernel sequence: for smooth $p$5 over $p$6, the sequence of group-like objects $p$7 behaves as expected locally in the étale topology, with the obstruction to global splitting governed by the non-linearity of the prolongation maps.
• Frobenius-equivariance: the relative Frobenius morphism $p$8 intertwines the kernel filtrations on $p$9 and $\mathbb{Z}_p$0 up to the shift induced by $\mathbb{Z}_p$1, so that kernels at level $\mathbb{Z}_p$2 are identified with Frobenius twists of kernels at level $\mathbb{Z}_p$3. This equivariance is the formal reason why $\mathbb{Z}_p$4-characters of order $\mathbb{Z}_p$5 on $\mathbb{Z}_p$6 correspond to functions on $\mathbb{Z}_p$7 that transform predictably under Frobenius.
• Linearization: the graded object associated to the kernel filtration of $\mathbb{Z}_p$8 is a direct sum of Frobenius twists of the cotangent-type bundles appearing in the classical principal parts filtration, which permits reduction of questions about arbitrary $\mathbb{Z}_p$9-modular functions to linear-algebraic ones on these graded pieces.

An immediate consequence of the linearization statement is that the conormal sheaf of $p$0 at a section, computed in the jet-theoretic sense, is a successive extension of copies of $p$1 twisted by iterates of $p$2 — mirroring precisely the Hasse–Hasse–Arone structure of principal parts, but with the Verschiebung-type grading replaced by Frobenius twisting.

Frobenius morphisms between jet spaces

The second half of the analysis concerns the Frobenius morphism itself. Because $p$3 acts on the base $p$4, applying $p$5 to the coefficients of a jet yields a canonical morphism $p$6, and the paper examines the compatibility of $p$7 with the projection maps $p$8 and with the group structures available when $p$9 is an algebraic group $A$0.

In the group case, the picture sharpens considerably. For a smooth algebraic group $A$1 over $A$2, each $A$3 is a group scheme and the kernel $A$4 is a prounipotent-type object built from Frobenius twists of the Lie algebra $A$5 of $A$6. The commutator and power operations on these kernels encode the arithmetic Serre–Hazewinkel-type relations among the $A$7-characters of order $A$8 on $A$9. The Frobenius morphism restricts to a homomorphism $\sigma : A \to A$0, and its kernel consists of jets annihilated simultaneously by all $\sigma : A \to A$1-coordinates of positive order — a subgroup whose description reduces to classical Frobenius kernels of $\sigma : A \to A$2 when $\sigma : A \to A$3 is a perfect field of characteristic $\sigma : A \to A$4, thereby recovering the familiar finite group scheme kernels $\sigma : A \to A$5 as degenerations of the mixed-characteristic construction.

This comparison across characteristics is one of the substantive contributions: the arithmetic jet kernel $\sigma : A \to A$6 over $\sigma : A \to A$7 specializes modulo $\sigma : A \to A$8 to an iterated Frobenius kernel of $\sigma : A \to A$9, so the entire hierarchy of Frobenius kernels in characteristic $p$0 appears as the special fiber of a single mixed-characteristic pro-object. The implication for arithmetic applications is that height-one and height-two phenomena (e.g., the structure of $p$1-torsion in abelian varieties) can be read off from the specialization behavior of the jet kernels, providing an alternative route to results classically obtained via Dieudonné theory or displays.

Relation to $p$2-characters and arithmetic differential equations

The kernel computations feed directly into the theory of $p$3-characters. An order-$p$4 $p$5-character of $p$6 is a group homomorphism $p$7 arising from a character $p$8 of the maximal torus after trivialization; the paper shows that the restriction of any such character to the kernel $p$9 is determined by its components along the Frobenius-twisted graded pieces described above. As a consequence, the vanishing locus of a $J^n X$0-character — the basic object of Buium's theory of arithmetic differential equations, used for instance in his proof that elliptic curves with a point of infinite order have infinitely many linearly independent $J^n X$1-modular points — admits a stratification by the kernels $J^n X$2, with each stratum cut out by conditions on finitely many Frobenius-twisted cotangent coordinates.

The practical payoff is computational: verifying that a $J^n X$3-modular function has prescribed vanishing along a jet fiber amounts to checking a finite list of linear conditions on the graded kernel pieces, rather than manipulating nonlinear prolongation equations directly. This is analogous to how the Hasse derivative filtration simplifies computations with principal parts, and it suggests that effective bounds on the order of contact of sections with jet fibers can be extracted from the Frobenius-twist multiplicities alone.

Limitations and open questions

Several restrictions qualify the results. First, the representability and kernel descriptions require $J^n X$4 (or $J^n X$5) to admit a smooth model over the $J^n X$6-adic base; the behavior of jet kernels for singular models, or for schemes over ramified extensions of $J^n X$7 where the $J^n X$8-derivation must be renormalized, is not treated here. Second, the splitting of the kernel filtration is established only locally, and the paper does not address whether the filtration on $J^n X$9 splits globally for any nontrivial class of schemes — a question that bears directly on the existence of global $\delta_p(x) = \frac{x - x^{\sigma}}{p}$00-coordinates adapted to a given embedding of $\delta_p(x) = \frac{x - x^{\sigma}}{p}$01. Third, the specialization argument identifying mod-$\delta_p(x) = \frac{x - x^{\sigma}}{p}$02 fibers with classical Frobenius kernels presumes that the Frobenius lift $\delta_p(x) = \frac{x - x^{\sigma}}{p}$03 extends compatibly to the special fiber; the dependence of the kernel structure on the choice of lift, which is known to affect associated $\delta_p(x) = \frac{x - x^{\sigma}}{p}$04-modular data, remains unexamined. Finally, the analysis is confined to the $\delta_p(x) = \frac{x - x^{\sigma}}{p}$05-typical setting; extensions to multiple primes or to the "arithmetic Fourier" setting of Buium–Poonen are left open, as is the corresponding description of kernels for higher-level jet spaces relevant to Manin maps on abelian varieties.

Conclusion

The paper provides a systematic account of the kernels of the projection maps on arithmetic jet spaces, showing that they assemble into a filtration whose graded pieces are Frobenius twists of the standard cotangent-type bundles, and that the Frobenius morphism acts on this filtration by a predictable degree shift. In the group case, the construction recovers classical characteristic-$\delta_p(x) = \frac{x - x^{\sigma}}{p}$06 Frobenius kernels upon specialization, linking the two regimes through a single mixed-characteristic object. These structural results streamline the study of $\delta_p(x) = \frac{x - x^{\sigma}}{p}$07-characters and arithmetic differential equations by reducing nonlinear prolongation problems to linear algebra on Frobenius-twisted graded pieces, though the theory remains dependent on smoothness hypotheses, locality of splittings, and the choice of Frobenius lift.