The aim of this book is to provide a detailed proof of a theorem of Gerd Faltings [F], asserting the existence of an isomorphism between Lubin-Tate towers and Drinfeld towers.
If F is a local field with ring of integers O, the Lubin-Tate and the Drinfeld spaces are defined as deformation spaces of formal O -modules with additional structures. They are actually the local analogue of the Shimura varieties (in mixed characteristics) and of Drinfeld’s Shtukas (equal characteristics).
Deformations of formal groups, and more generally of formal modules, occur frequently in arithmetic geometry. An illustration of this can be seen in the work of M. Harris and R. Taylor on the local Langlands conjecture [HT]. Also, an extensive use of the theory has been made by H. Hida in his proof of the vanishing of the μ-invariant of CM fields. An excellent introduction to the deformation theory of formal groups can be found in Katz’s paper [K].
As already mentioned, a beautiful application of the deformation theory of formal groups is the local Langlands correspondence, which is a nonabelian generalization of the reciprocity law of local class field theory. It gives a link between certain representations of Galois groups and certain infinite-dimensional representations of reductive algebraic groups. Its proof represents a milestone in algebraic number theory. Loosely speaking, it turns out that the ‘right’ Galois representations in the Langlands correspondence can be built inside the (l-adic or crystalline) cohomology of Shimura varieties (higher dimensional generalizations of modular curves). The local properties (ramification, Hodge-Tate numbers, etc.) of such Galois representations are then reflected by the local structure of the Shimura variety, which is in some sense coded in the Lubin-Tate towers.
The book is divided into two parts. The first, written by Fargues, deals with the case of a local field of mixed characteristic, whereas the second explains an alternative construction which is applicable only to fields of equal characteristics. These two parts are completely independent.
Needless to say, this is a very technically demanding reading, addressed to researchers in the field.
[F] A relation between two moduli spaces studied by V.G. Drinfeld, G. Faltings, vol. 300 of Contemp. Math., 2002
[HT] The geometry and cohomology of some simple Shimura varieties, M.Harris and R.Taylor, Annals of Math. Studies 151, PUP 2001.
[H] Non-vanishing modulo p of Hecke L-values and application, H. Hida , London Mathematical Society Lecture Note Series 320 (2007) 207-269
[K] Serre-Tate local moduli. N. Katz, Algebraic surfaces (Orsay, 1976--78), pp. 138--202, Lecture Notes in Math., 868, Springer, Berlin-New York, 1981.
Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at email@example.com.