Yoshida - Non-abelian Lubin-Tate 4



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.

Excursion. Moduli Space


1. Moduli Space of Elliptic Curves


  •   Consider the category (Ell) where the objects are elliptic curves over an arbitrary base scheme
    E↓πS
    and the morphisms are the Cartesian squares
    E1a→Eπ1↓↓πS1f→S
    • The above Cartesian diagram induces an isomorphism
      E1 (α,π1) →E×SS1
    • (Ell) is called the modular stack of Deligne-Rapoport.
  • A contravariant functor P:(Ell)→(Set) is called a moduli problem for Elliptic Curves.
  • Given E/S, the set P(E/S) is called a level-P-structure.
  • The moduli problem P is called relatively representable over (Ell) if for every elliptic curve E/S, the functor (Sch/S)→(Set) defined by T↦P(ET/T) is representable. We denote such S-scheme by PE/S.
  • The moduli problem P is representable if it is representable as a functor on (Ell).  We denote the representing elliptic curve E/M(P).
    • Therefore, we have an isomorphism
      P(E/S)≅Hom(Ell)(E/S,E/M(P))
    • M(P) represents the functor (Sch)→(Set) defined by S↦
      isomorphism classes of pairs (E/S,α) with E an elliptic curve over S and α∈P(E/S) a level-P-structure on E/S
    • Converse is also true. If M(P) is representable by a scheme M(P) with universal object (E/M(P),α) and if P is rigid, then P is represented by the object E/M(P) of (Ell).

2. Universal Object

  • Let U:C→D be a functor. Let X∈D.  A initial morphism fromX to U is a pair (A,α) where A is an object of C and ϕ:X→U(A) with the following universal property: whenever Y is an object of C and f:X→U(Y) such that f,ϕ,α commute in the obvious way. (MathJAX does not support diagonal arrows.)
  • As of now, I have no idea how the two connects. I want to know whether or not this definition follows, and there exists a morphism α:X→P(E/M(P)) with some set X such that (E/M(P),α) is a universal object of P:(Ell)→(Set).

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