Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2503.05567 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866929747494174720 |
|---|---|
| author | Tchuiaga, S. Kewir, C. Dor |
| author_facet | Tchuiaga, S. Kewir, C. Dor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05567 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic Aspects of Weil Bundles over $p$-Adic Manifolds Tchuiaga, S. Kewir, C. Dor Number Theory Differential Geometry 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. |
| title | Arithmetic Aspects of Weil Bundles over $p$-Adic Manifolds |
| topic | Number Theory Differential Geometry |
| url | https://arxiv.org/abs/2503.05567 |