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

S. Tchuiaga, C. Dor Kewir

TL;DR

The paper develops a systematic theory of Weil bundles $M^A$ over $p$-adic analytic manifolds to encode infinitesimal deformations in a non-archimedean setting. It constructs $M^A$ as a $p$-adic analytic space, proves lifting theorems for analytic functions, vector fields, forms, and connections, and shows a Galois-equivariant structure when $M$ is defined over a number field. A cohomological comparison is established, linking $H^k_{ ext{dR}}(M^A/ obreak\mathbb{Q}_p)$ with crystalline cohomology via $H^k_{ ext{crys}}(M/ obreak\mathbb{Z}_p) ens A$, thereby connecting infinitesimal geometry to $p$-adic Hodge theory. The framework yields applications to $p$-adic modular forms, Diophantine geometry, and infinitesimal solutions on elliptic curves, offering a geometric lens for deformation theory in the arithmetic context.

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.

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

TL;DR

The paper develops a systematic theory of Weil bundles over -adic analytic manifolds to encode infinitesimal deformations in a non-archimedean setting. It constructs as a -adic analytic space, proves lifting theorems for analytic functions, vector fields, forms, and connections, and shows a Galois-equivariant structure when is defined over a number field. A cohomological comparison is established, linking with crystalline cohomology via , thereby connecting infinitesimal geometry to -adic Hodge theory. The framework yields applications to -adic modular forms, Diophantine geometry, and infinitesimal solutions on elliptic curves, offering a geometric lens for deformation theory in the arithmetic context.

Abstract

We introduce a systematic theory of Weil bundles over -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 -adic setting, we establish that Weil bundles associated with a -adic manifold and a Weil algebra 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 to . 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 and the crystalline cohomology of , unifying infinitesimal and crystalline perspectives. Applications to Diophantine geometry and -adic Hodge theory are central to this work. We show that spaces of sections of Hodge bundles on parametrize -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 -adic deformations.

Paper Structure

This paper contains 13 sections, 17 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The field of $p$-Adic numbers $\mathbb{Q}_p$, Fon94
  4. $p$-Adic manifolds
  5. Weil Bundles over $p$-adic manifolds
  6. Infinitely near points
  7. Transition maps on $M^A$
  8. Lifting results from $M$ to $M^A$
  9. Arithmetic aspects of Weil bundles
  10. Cohomology of Weil bundles
  11. Arithmetic Subgroups and Invariant Sections
  12. Diophantine applications
  13. Additional examples and applications

Key Result

Theorem 2.5

Ber02 Let $f: \mathbb{Z}_p \to \mathbb{Q}_p$ be a continuous function. Then $f$ can be uniquely represented by the Mahler expansion: $f(x) = \sum_{n=0}^{\infty} a_n \binom{x}{n},$ where the Mahler coefficients $a_n$ are given by $a_n = \sum_{k=0}^n (-1)^{n-k} \binom{n}{k} f(k),$ and $\binom{x}{n}$ d Furthermore, $f$ is continuous if and only if $|a_n|_p \to 0$ as $n \to \infty$.

Theorems & Definitions (54)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5: Mahler's Theorem
  • Remark 2.6
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Definition 4.1
  • ...and 44 more