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(ϕψ)=ϕ(ψ(0))ψ(0), but as ψ(X)B[[X]], we conclude that D(ϕψ)=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:OEnd(F) such that Df=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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