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By Thomas Lehmkuhl

Listed here the writer describes intimately a compactification of the moduli schemes representing Drinfeld modules of rank 2 endowed with a few point constitution. The boundary is a union of copies of moduli schemes for Drinfeld modules of rank I, and its issues are interpreted as Tate facts. the writer additionally reports infinitesimal deformations of Drinfeld modules with point constitution

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As an intermediate step towards compactification we develop the general concept of a formal boundary. Then we show that the failure of MId being proper can be described by means of Tate data. The construction of a formal boundary and of the compactification will be carried out in the next chapter. 1. Formal schemes In this section we recall some facts on formal schemes. We only need noetherian adic formal schemes. For details and a more general concept the reader is referred to [EGA I], chap. 10; see also [Ha], II, 9.

Let x1 , . . , xd be a base. Assume that Λ ⊗A k∞ → is not injective. Then we can find an unbounded sequence d xi ⊗ λni yn = i=1 in the kernel of that map. As k∞ /A is compact, we can find C > 0 and elements ani ∈ A such that |λni − ani |∞ < C holds for all n, i. Therefore, if D > 0 is such that |xi | < D for i = 1, . . , d, we have d d ani xi (λni + (ani − λni ))xi = i=1 i=1 d ≤ |ani − λni ||xi | < d · C · D i=1 Therefore { d ani xi | n = 1, 2, . } is an infinite set lying in i=1 Λ ∩ B(0, d · C · D).

By definition, isomorphisms of deformations specialize to the identity. 1. Let (E0 , ι0 ) be a Drinfeld module of rank d > 0 and of level I. Then the functor Def(E0 , ι0 ) is pro-represented by a regular algebra RI in CˆO of dimension d. For J ⊆ I the canonical homomorphism RI → RJ is finite and flat. 3. DEFORMATIONS OF LEVEL STRUCTURES 43 Proof. We proceed in several steps. Step 1. The case p = 0 and I = p. This is the crucial case. Let E pp be the universal deformation of E0 , let E → E/E[p] be the projection, let L be a splitting field of pp over Q(R0 ), and let R1 ⊆ L be the R0 algebra generated by all roots of pp.

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