Differential Characters and $D$-Group Schemes
Rajat Kumar Mishra, Arnab Saha
TL;DR
This work generalizes Buium’s differential-characters framework from abelian schemes to arbitrary smooth connected commutative group schemes G over a differential field K. It proves that the kernel K(G) of differential characters on the jet space J^∞G forms a finite-dimensional D-group scheme, and that K(G) is a vectorial extension of G; it also characterizes the module of differential characters X_∞(G) as a finitely generated K{∂}-module of rank g, generated by g primitive characters with orders bounded in terms of r = dim_K Ext(G, G_a) and m_u. The authors provide a coordinate-based analysis of differential characters, define primitive generators via splitting numbers, and construct K(G) explicitly as the kernel of a canonical map arising from a primitive basis. They also discuss connections to the universal vectorial extension in the abelian case and suggest a pushout perspective and avenues for future work in differential-algebraic geometry and Diophantine geometry.
Abstract
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.