Perfectoid $C_i$ transfer

TL;DR

The paper develops a comprehensive model-theoretic and perfectoid framework to transfer $C_i$-type properties across tilts and untilts of perfectoid fields. By combining tame-field theory, tilt/tilt-into-residue embeddings, and ultraproduct techniques, it constructs finite totally ramified extensions $E/ obreak olinebreak null null null t p$ ensuring that every untilt containing $E$ is $C_2(d)$, and it extends these ideas to rationally connected and rationally simply connected settings via Starr–Xu results for global function fields. Conditional results relying on Kollár–Colliot-Thélène conjectures yield a geometric $C_1$ transfer from tilt to untilt for perfectoid fields, implying RC/RC-solvable phenomena persist under tilting, and giving a pathway to Lang–Manin type statements in the perfectoid realm. The paper thus connects Ax–Kochen type transfer, perfectoid tilting, and RC/RSC geometry to produce robust transfer principles, with concrete consequences for hypersurfaces and rational points over perfectoid extensions of ${f Q}_p^{ur}$ and related fields.

Abstract

We prove a perfectoid analogue of the Ax-Kochen theorem on zeros of $p$-adic forms: Given $d\in \mathbb{N}$, there is a finite totally ramified extension $E/\mathbb{Q}_p$ such that every untilt of $\mathbb{F}_p(\!(t^{1/p^{\infty}})\!)$ containing $E$ is $C_2(d)$. We also prove a similar result for the existence of rational points in rationally connected varieties over perfectoid field extensions of $\mathbb{Q}_p^{ur}$.

Theorems & Definitions (63)

• Theorem : Ax-Kochen
• Theorem : Duesler-Knecht
• Theorem : Pieropan
• Theorem 1.1
• Theorem 1.2
• Conjecture A
• Theorem 1.3
• Definition 2.1
• Proposition 2.4
• proof