On Hopkins' Picard group

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 Zₚ and, whenever n ≥ 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.