On Hopkins' Picard group

TL;DR

This paper computes the algebraic Picard group of the category of $K(n)$-local spectra across all heights and primes, confirming a longstanding prediction and providing explicit generators in most cases.

Contribution

It determines the structure of the Picard group for $K(n)$-local spectra at all heights and primes, using $p$-adic geometry and recent advances to explicitly identify generators.

Findings

Picard group is finitely generated over $ ext{Z}_p$ for all $n,p$.

For $n extgreater= 2$, the group has rank 2.

Explicit topological generators are provided except in the case $n=p=2$.

Abstract

We compute the algebraic Picard group of the category of $K(n)$-local spectra, for all heights $n$ and all primes $p$. In particular, we show that it is always finitely generated over $\mathbb{Z}_p$ and, whenever $n \geq 2$, is of rank $2$, thereby confirming a prediction made by Hopkins in the early 1990s. In fact, with the exception of the anomalous case $n=p=2$, we provide a full set of topological generators for these groups. Our arguments rely on recent advances in $p$-adic geometry to translate the problem to a computation on Drinfeld's symmetric space, which can then be solved using results of Colmez--Dospinescu--Niziol.