PreprintarXiv licencearXiv
The contrast between real and p-adic integrability
Abstract
A fundamental question in p-adic symplectic geometry and p-adic Hamiltonian mechanics is to what extent we can learn new information about integrable systems with coefficients in the rational numbers ℚ, if we view them as p-adic systems instead of real ones. We combine geometric analytic estimates with Galois theory in order to show that at a local level, these real and p-adic viewpoints give completely different information. Then we focus on understanding this local behavior for the p-adic case by proving effective methods to compute local normal forms. Our results are motivated by a program outlined by Pelayo-Voevodsky-Warren a decade ago, to develop a p-adic theory of integrable systems and symplectic geometry.
Related
- Integrability from supersymmetric duality: a short review
- Self-similar groups and holomorphic dynamics: renormalization, integrability, and spectrum
- Work in progress: A Lean-verified conditional reduction of periodic Navier-Stokes regularity on T^3 to a single integrability hypothesis
- On Special Arithmetic Gamma Values and Thakur's Gauss Sums: From ∞-adic to v-adic
- Parallel transport for Higgs bundles over p-adic curves
BibTeX
@misc{crespo2026contrast,
title = {{The contrast between real and $p$-adic integrability}},
author = {Luis Crespo and Álvaro Pelayo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.24363},
eprint = {2606.24363},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.24363},
}
RIS
TY - UNPB
TI - The contrast between real and p-adic integrability
AU - Luis Crespo
AU - Álvaro Pelayo
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2606.24363
UR - https://arxiv.org/abs/2606.24363
AB - A fundamental question in p-adic symplectic geometry and p-adic Hamiltonian mechanics is to what extent we can learn new information about integrable systems with coefficients in the rational numbers ℚ, if we view them as p-adic systems instead of real ones. We combine geometric analytic estimates with Galois theory in order to show that at a local level, these real and p-adic viewpoints give completely different information. Then we focus on understanding this local behavior for the p-adic case by proving effective methods to compute local normal forms. Our results are motivated by a program outlined by Pelayo-Voevodsky-Warren a decade ago, to develop a p-adic theory of integrable systems and symplectic geometry.
ER -