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課程簡介 :
The purpose of this course is to introduce
the deep connection between modular forms and Galois representations,
expressed in terms of the associated L-functions. Starting with the
Eichler-Shimura relation which associates $\ell$-adic Galois
representations to weight two newforms with integral coefficients, we
proceed to Deligne's generalization to forms of higher weight. On the
reverse direction, there are the celebrated results by Wiles,
Taylor-Wiles, Breuil-Conrad-Diamond -Taylor, and Skinner-Wiles, which give
criteria for $\ell$-adic representations to be modular, that is,
associated to
automorphic forms. If time permits, we shall venture into cusp forms for
noncongruence subgroups to explore the associated representations
constructed by Scholl, their modularity, and connections to forms for
congruence subgroups. |