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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 information. the writer additionally reviews infinitesimal deformations of Drinfeld modules with point constitution

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Now we have to compute Ext1A⊗A (A, tE0 ) = tE0 ⊗ ω. Since ( ⊗ A) ⊗A⊗A A ∼ = and ω is invertible, we obtain Ext1A⊗A (A, tE0 ) ∼ = tE0 ⊗ ⊗A . This is d by the following lemma. Hence dim tD = d − 1. 5), the proposition is proved. 4. The ⊗ A-module tE0 is projective of rank d. Proof. 3), and the module tE0 is torsion free. In fact a cyclic torsion submodule would have finite dimension over , so the degree of its elements would be bounded. This is not possible. Now fix any a ∈ A with |a|∞ > 1, and let N = d · logq |a|∞ .

Let ¯ be an algebraic closure of . Since H( ¯) is finite and A-invariant, it is annihilated by some b = 0. Hence H is contained in r·s the kernel of ea with a = b if the characteristic is generic, and a = bq otherwise. 1. Finally, from the equalities ϕ ◦ ψ ◦ ϕ = ϕ ◦ ea = fa ◦ ϕ , it follows that ϕ ◦ ψ = fa . 5. Let E and F be Drinfeld modules of rank d over . Then (i) Hom(E, F) is a finitely generated projective A-module of rank ≤ d2 . (ii) Let q ∈ Spec A be away from the characteristic of . Then the canonical homomorphism Hom(E, F) ⊗A Aˆq −→ HomAˆq (E( ¯)q−tors , F( ¯)q−tors ) is injective.

Cit. The proof of the proposition is complete. As an application we will prove that the equality of divisors in the definition of a level I structure holds for all ideals containing I. 3. Let (0) = I be an ideal in A, and let (E, ι) be a Drinfeld module of rank d and level I over an A-scheme S. Then we have ι(x) . E[I] = x∈(I −1 /A)d Proof. 5, it will be sufficient to prove this for the universal Drinfeld module over RI as discussed above. 3, (1). 4. Smoothness of the moduli spaces Now we are prepared to prove the main result of this chapter.

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