Ramified Deformation Problems
نویسنده
چکیده
0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 1. The deformation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 1.1. Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 1.2. Changing the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 2. Some applications of finite Honda systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 2.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 2.2. Initial description of ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 2.3. Translation into Honda systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 2.4. Classification of possible ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 3. Tangent space calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 3.1. Analysis of a kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 3.2. Analysis of an image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 4. Deformation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 4.1. Structure theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 4.2. Another deformation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 5. The ′-ordinary case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 5.1. Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 5.2. Honda system calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 5.3. Classification of possibilities for ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 5.4. Deformation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
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