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

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.