Formal vector spaces over a local field of positive characteristic
نویسنده
چکیده
Let OK = FqJπK be the ring of power series in one variable over a finite field, and let K be its fraction field. We introduce the notion of a “formal K-vector space”; this is a certain kind of K-vector space object in the category of formal schemes. This concept runs parallel to the established notion of a formal OK-module, but in many ways formal K-vector spaces are much simpler objects. Our main result concerns the Lubin-Tate tower, which plays a vital role in the local Langlands correspondence for GLn(K). Let Am be the complete local ring parametrizing deformations of a fixed formal OK-module over the residue field, together with Drinfeld level π structure. We show that the completion of the union of the Am has a surprisingly simple description in terms of formal K-vector spaces. This description shows that the generic fiber of the Lubin-Tate tower at infinite level carries the structure of a perfectoid space. As an application, we find a family of open affinoid neighborhoods of this perfectoid space whose special fibers are certain remarkable varieties over a finite field which we are able to make completely explicit. It is shown in joint work with Mitya Boyarchenko that the `-adic cohomology of these varieties realizes the local Langlands correspondence for a certain class of supercuspidal representations of GLn(K).
منابع مشابه
HYPERTRANSCENDENTAL FORMAL POWER SERIES OVER FIELDS OF POSITIVE CHARACTERISTIC
Let $K$ be a field of characteristic$p>0$, $K[[x]]$, the ring of formal power series over $ K$,$K((x))$, the quotient field of $ K[[x]]$, and $ K(x)$ the fieldof rational functions over $K$. We shall give somecharacterizations of an algebraic function $fin K((x))$ over $K$.Let $L$ be a field of characteristic zero. The power series $finL[[x]]$ is called differentially algebraic, if it satisfies...
متن کاملALGEBRAIC INDEPENDENCE OF CERTAIN FORMAL POWER SERIES (I)
We give a proof of the generalisation of Mendes-France and Van der Poorten's recent result over an arbitrary field of positive characteristic and then by extending a result of Carlitz, we shall introduce a class of algebraically independent series.
متن کاملLOCAL BASES WITH STRATIFIED STRUCTURE IN $I$-TOPOLOGICAL VECTOR SPACES
In this paper, the concept of {sl local base with stratifiedstructure} in $I$-topological vector spaces is introduced. Weprove that every $I$-topological vector space has a balanced localbase with stratified structure. Furthermore, a newcharacterization of $I$-topological vector spaces by means of thelocal base with stratified structure is given.
متن کاملLocal Cohomology and D-affinity in Positive Characteristic
If k is a field of positive characteristic, the local cohomology modules HjI(R) vanish for j > 2 (see Chapitre III, Proposition (4.1) in [3]). However, if k is a field of characteristic zero, H3I(R) is non-vanishing (see Proposition 2.1 of this paper or Remark 3.13 in [5]). Consider the Grassmann variety X = Gr(2, V ), where V is a 5dimensional vector space over k. Let W be a one dimensional su...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013