Arithmetic Aspects of Weil Bundles over $p$-Adic Manifolds

Abstract: We introduce a systematic theory of Weil bundles over ( p )-adic analytic manifolds, forging new connections between differential calculus over non-archimedean fields and arithmetic geometry. By developing a framework for infinitesimal structures in the ( p )-adic setting, we establish that Weil bundles ( M^A ) associated with a ( p )-adic manifold ( M ) and a Weil algebra ( A ) inherit a canonical analytic structure. Key results include: \text{Lifting theorems :} for analytic functions, vector fields, and connections, enabling the transfer of geometric data from ( M ) to ( M^A ). A \text{Galois-equivariant structure :} on Weil bundles defined over number fields, linking their geometry to arithmetic symmetries. A \text{cohomological comparison isomorphism:} between the Weil bundle ( M^A ) and the crystalline cohomology of ( M ), unifying infinitesimal and crystalline perspectives. Applications to Diophantine geometry and ( p )-adic Hodge theory are central to this work. We show that spaces of sections of Hodge bundles on ( M^A ) parametrize ( p )-adic modular forms, offering a geometric interpretation of deformation-theoretic objects. Furthermore, Weil bundles are used to study infinitesimal solutions of equations on elliptic curves, revealing new structural insights into ( p )-adic deformations.