Courses for 2009/2010:

U5530, Topics in Arithmetic Geometry, Spring Semester.

課程概述 | Modular curves as algebraic curves. Fields of definition of modular curves. Reductions of algebraic curves over global fields. Jacobians. Abelian varieties. Hecke correspondences. Eichler-Shimura relations. Drinfeld upper-half plane. Drinfeld modular curves. Rigid analytic spaces. Non-archimedean theta functions. Bruhat-Tits trees. |

課程目標 | Modular curves. Jacobians. Eichler-Shimura theory. Drinfeld upper-half plane. Drinfeld modular curves. |

課程要求 | Basic algebraic curves theory. Basic algebraic number theory, in particular basics about local fields. |

關鍵字 | Elliptic curves, modular curves, abelian varieties, Jacobians, rigid analytic geometry. |

指定閱讀 | Basic theory of elliptic curves. Algebraic curves and Riemann-Roch. Also very basic modular forms, e.g. in Chap 7 of Serre's course in arithmetic. |

參考書目 | F. Diamond and J. Shurman,
A First Course in Modular Forms, Chap-6-9, Springer GTM, 2005. M. Hindry and J. Silverman, Diophantine Geometry: an introduction, Springer GTM, 2000. S. Lang, Elliptic Functions, GTM Springer, 1987. S. Lang, Introduction to Algebraic and Abelian Functions, Chap.3, 4, 8. GTM Springer, 1982. J.-P. Serre, A course in Arithmetic, Chap 7, GTM Springer, 1978. J. Silverman, The Arithmetic of Elliptic Curves, GTM, Springer, 1986. J. Silverman, Advanced topics in the Arithmetic of Elliptic Curves, GTM Springer, 1994. G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1973. M. Van der Put, and J. Martinet, Rigid analytic geometry and its applications, Progress in Math, Birkhauser 2003. |

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