The Frobenius and Monodromy Operators for Curves and Abelian Varieties
نویسندگان
چکیده
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part I. Definitions of the operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Definitions of N and F for curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. The monodromy operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2. The Frobenius operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. N and F for Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1. The monodromy operator for Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . 7 2.2. The Frobenius operator for Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . 8 3. Equality of the monodromy operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4. Equality of the Frobenius operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1. Review of Fontaine’s rings and Colmez’s integration . . . . . . . . . . . . . . . . . . . . . . 17 2. The universal covering space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3. Integration of differential forms of the second kind along paths . . . . . . . . . . . . 24 4. Explicit description of Fontaine’s monodromy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5. Explicit description of Fontaine’s Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6. The p-adic integration map commutes with the monodromies . . . . . . . . . . . . . 33 7. The integration map commutes with the Frobenii . . . . . . . . . . . . . . . . . . . . . . . . . 35
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