arXiv subjects

Rajat Kumar Mishra

Publications and source records attributed to Rajat Kumar Mishra.

2 recordsLinked to original sources

arXiv2026

Kernels of Arithmetic Jet Spaces and Frobenius Morphism

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 $π$.

math.AG↗

arXiv2025

Differential Characters and $D$-Group Schemes

Let $K$ be a field of characteristic zero with a fixed derivation $\partial$ on it. In the case when $A$ is an abelian scheme, Buium considered the group scheme $K(A)$ which is the kernel of differential characters (also known as Manin characters) on the jet space of $A$. Then $K(A)$ naturally inherits a $D$-group scheme structure. Using the theory of universal vectorial extensions of $A$, he further showed that $K(A)$ is a finite dimensional vectorial extension of $A$. Let $G$ be a smooth connected commutative finite dimensional group scheme over $\mathrm{Spec}~ K$. In this paper, using the theory of differential characters, we show that the associated kernel group scheme $K(G)$ is a finite dimensional $D$-group scheme that is a vectorial extension of such a general $G$. Our proof relies entirely on understanding the structure of jet spaces. Our method also allows us togive a classification of the module of differential characters $\mathbf{X}_\infty(G)$ in terms of primitive characters as a $K\{\partial\}$-module.

math.AG↗