Complex orientations and $\text{TP}$ of complete DVRS

TL;DR

This paper investigates the structure of periodic topological cyclic homology of the ring of integers in a finite extension of p-adic numbers, revealing a formal group law influenced by the extension's Eisenstein polynomial.

Contribution

It establishes that the periodic topological cyclic homology of $\

Findings

Identifies a $p$-height one formal group law mod $p$ in the homology.

Shows dependence of the formal group law on the Eisenstein polynomial.

Connects algebraic extensions with topological cyclic homology structures.

Abstract

Let $L$ be finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_L$. I show that periodic topological cyclic homology of $\mathcal{O}_L$, over the base $\mathbb{E}_{\infty}$-ring $\mathbb{S}_{W(\mathbb{F}_q)}[z]$ carries a $p$-height one formal group law mod $p$ that depends on the Eisenstein polynomial of $L$ over $\mathbb{Q}_p$ for a choice of uniformizer $\varpi\in \mathcal{O}_L$.