nLab Lubin-Tate space

Idea

A Lubin-Tate space is a space of deformations of a one-dimensional formal group of height $h$. One may also construct spaces $\mathcal{M}_{n}$ parametrizing deformations of formal groups with level structures. The “infinite-level” Lubin-Tate space $\mathcal{M}_{\infty}$, also known as the Lubin-Tate tower is some idea of “inverse limit” of these spaces (note that the inverse limit does not actually as a rigid analytic space, see Lecture 2 of #WeinsteinNotes for the definition of the Lubin-Tate tower on the level of points).

Application to the local Langlands correspondence

As the infinite-level Lubin-Tate space has actions of the groups appearing in the statement of the local Langlands correspondence for $\GL_{n}$, it has applications to its proof. See also Theorem 7 in #WeinsteinNotes, which is originally due to Harris-Taylor.

Generalizations

The infinite-level Lubin-Tate space is a special case of an infinite-level Rapoport-Zink space (see #ScholzeWeinstein12). These are in turn special cases of local Shimura varieties and the even more general notion of moduli of local shtukas.

Related concepts

• Lubin-Tate theory

• Lubin-Tate formal group

References

• Jared Weinstein, The geometry of Lubin-Tate spaces, pdf

• Peter Scholze, Jared Weinstein, Moduli of p-divisible groups, arXiv:1211.6357