Rigid local systems, Hilbert modular forms, and Fermat’s last theorem
نویسنده
چکیده
1. Frey representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 1.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 1.2. Classification: The rigidity method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 1.3. Construction: Hypergeometric abelian varieties . . . . . . . . . . . . . . . . . . . . . . 419 2. Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 2.1. Hilbert modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 2.2. Modularity of hypergeometric abelian varieties . . . . . . . . . . . . . . . . . . . . . . 430 3. Lowering the level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 3.1. Ribet’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 3.2. Application to xp+yp = z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 4. Torsion points on abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
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