Multivariable period rings of p-adic false Tate curve extension

2025/05/29 by Yuan, Yijun · 1 citation
#11E95 #11F80 #11S15 #11S25 #14G45 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.24064

Abstract

Let p≥ 3 be a prime number and K be a finite extension of Q_p with uniformizer π_K. In this article, we introduce two multivariable period rings A_\mathfrakF,K^np and A_\mathfrakF,K^np,c for the étale (φ,Γ_\mathfrakF,K)-modules of p-adic false Tate curve extension K(π_K^{1/p^∞},ζ_p^∞). Various properties of these rings are studied and as applications, we show that (φ,Γ_\mathfrakF,K)-modules over these rings bridge (φ,Γ)-modules and (φ,τ)-modules over imperfect period rings in both classical and cohomological sense, which answers a question of Caruso. Finally, we construct the ψ operator for false Tate curve extension and discuss the possibility to calculate Iwasawa cohomology for this extension via (φ,Γ_\mathfrakF,K)-modules over these rings.

Citations

• Locally analytic vectors and decompletion in mixed characteristic
• Cohomology of (φ, τ)-modules
• Generalised power series determined by linear recurrence relations
• Explicit uniformizers for certain totally ramified extensions of the field of p-adic numbers
• Lifting the field of norms
• Perfectoid spaces
• An explicit formula for the Hilbert symbol of a formal group

Cited by

• Descent of (φ,τ)-modules in characteristic p

Related

• Relative p-adic Hodge theory, II: Imperfect period rings
• Locally analytic vectors and Z_p-extensions
• Multivariable p-adic Hodge theory for products of Galois groups
• Multivariable (φ_q,O_K^×)-modules associated to p-adic representations of Gal(K/K)
• An introduction to p-adic period rings