Introduction ................................................... ix
Chapter 1. Line Bundles ......................................... 1
1. Basic notions ............................................. 1
2. Homomorphisms of line bundles ............................. 3
3. Quotients ................................................. 7
4. The convergence lemma .................................... 10
Chapter 2. Drinfeld Modules .................................... 15
1. Analytical definition of Drinfeld modules ................ 15
2. The category of Drinfeld modules ......................... 19
3. Drinfeld modules over fields ............................. 22
4. Level structures ......................................... 25
5. Modular manifolds ........................................ 28
6. Pseudo-Drinfeld modules .................................. 30
Chapter 3. Deformation Theory .................................. 33
1. Deformations of Drinfeld modules ......................... 33
2. Deformations of isogenics ................................ 40
3. Deformations of level structures ......................... 42
4. Smoothness of the moduli spaces .......................... 46
5. Group action on the moduli space ......................... 47
Chapter 4. Tate Uniformization ................................. 51
1. Formal schemes ........................................... 51
2. Good and stable reduction ................................ 54
3. Lattices and Tate data ................................... 57
4. Group action ............................................. 64
Chapter 5. Compactification of the Modular Surfaces ............ 67
1. Formal represent ability of Tate data .................... 67
2. The universal Drinfeld module with bad reduction ......... 73
3. Algebraization ........................................... 78
Appendix ....................................................... 85
A. Induced schemes .......................................... 85
В. Construction of coherent sheaves ......................... 86
Bibliography ................................................... 89
Glossary of Notations .......................................... 91
Index .......................................................... 93
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