Yoshida - Non-abelian Lubin-Tate 3
This will be a series of personal notes on the paper of Yoshida, "On Non-abelian Lubin-Tate Theory via Vanishing Cycles". The general format would be to have outlines in the first part and (stupid) questions in the second part. Also, in the middle, I will put background materials. Hopefully, most of the questions will be answered by the end of the month.
Current Reference: Drinfeld, "Elliptic Modules".
1. Formal Modules
- The definition of a formal group and a homomorphism coincide with the one with Yoshida.
- There is a canonical homomorphism D:End(F)→B where F is a formal group over a ring B. If ϕ is an endomorphism of F, then D(ϕ)=ϕ′(0).
- This is a homomorphism as D(ϕ∘ψ)=ϕ′(ψ(0))⋅ψ′(0), but as ψ∈(X)⊂B[[X]], we conclude that D(ϕ∘ψ)=ϕ′(0)ψ′(0).
- (Example) F(X,Y)=X+Y is a formal group over any ring. This group is called additive.
- We consider an endomorphism ϕ of F over a ring B with characteristic p. This means that
ϕ(F(X,Y))=F(ϕ(X),ϕ(Y))⇒ϕ(X+Y)=ϕ(X)+ϕ(Y) - Write ϕ(X)=∑aiXi, then ϕ(X+Y)=∑ai(X+Y)i and ϕ(X)+ϕ(Y)=∑ai(Xi+Yi). If p|i or i=1, then (X+Y)i=Xi+Yi, so ai survives, but when p⧸|i, ''intermediate coefficient" in (X+Y)i is nonzero, hence creates an addition term aiXkYl with k+l=i. Therefore, ai dies.
As a conclusion, ϕ(X)=∑i=0aiXpi. - Elements in B is identified with the endomorphism bX. We also have the Frobenius endomorphism Xp denoted. Hence the additive group consists of the ''series'' ∑biτp with the rule that
τb=bpτ
We denote the ring as B{{τ}}. - Let B be an O-algebra. A Formal O-module over B is the pair (F,f) where F is a formal group over B and f:O→End(F) such that D∘f=γ where γ is the structure map of B as a O-module.
- As D is exactly looking at its linear term and O acts on B exactly by γ, D(f(a))=γ(a) is equivalent to saying that f(a)≡aX (mod X2), i.e., coincide with the definition given in Yoshida's paper.
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