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.