Courses for 2009/2010:
U 5520 Introduction to Modular Forms, Fall semester.

 課程概述 Modular Forms occur in Number Theory, Representation Theory, Algebraic Geometry, Group Theory, and Mathematical Physics. We will present the basic theory, including Hecke operators theory, and new forms. 課程目標 looking at modular forms from various point of view. A complete account of the theory for SL(s, Z), including Hecke operators. Emphasizing explicit examples of modular forms, Eisenstein series, cusp forms, also Theta series. Modular forms for congruence subgroups, i.e. higher level theory. Hecke algebra. Euler products. Petterson inner product. Mellin transforms. Functional equations and analytic continuation. Method of Rankin-Selberg. Non-holomorphic Eisenstein series. Maass forms. Modular forms of half-integral weights. Introducing moduli, i.e. Modular curves. Introducing New forms. Connecting modular forms with Galois representations, introducing automorphic representations. Eichler-Selberg trace formula for SL2(Z). 課程要求 Background: Undergraduate algebra, and complex analysis. It would be helpful if the students are taking (or had taken) courses on algebraic number theory, compact Riemann surfaces, or algebraic curves simultaneously. 關鍵字 Modular forms, upper-half plane, modular curves, Hecke operators, Eisenstein series, cusp forms, Theta series, new forms, automorphic representations. Eichler-Selberg trace formula, Half-integral weights modular forms, Maass forms. 指定閱讀 1. Review complex analysis on the upper-half plane, including fractional linear transformations on the Riemann sphere and related geometry. 2. Read the classical problem of representing a number as sum of four squares, e.g. Chap 20 of Hardy-Wright: An Introduction to the theory of  numbers. 3. Take a good book on number theory, e.g. Hardy-Wright, or L.-K. Hua, look for arithmetic functions which are multiplicative, as many examples as possible. 參考書目 1. Bruinier-Van der Geer-Harder-Zagier: The 1-2-3 of Modular Forms. Chap.1. Springer, Universitext. 2. D. Bump : Automorphic Forms and Representations, Chap.1. Cambridge Studies in Advanced Math. 3. Diamond-Shurman: A First Course in Modular Forms, Chap.1-5. Springer GTM. 4. Kato-Kurokawa-Saito-Kurihara: Number Theory, I, II, Iwanami Shoten and AMS. 5. N. Koblitz : Introduction to elliptic Curves and Modular Forms, Chap.3, Chap.4. Springer GTM, 2nd ed. 1993. 6. S. Lang : Introduction to modular forms, Springer 1976, die Grundlehren, Band 222. 7. W. Li : Number Theory with applications, Chap. 7. World Scientific, 1996. 8. B. Schoeneberg, Elliptic modular functions, Springer, Die Grundlehren, Band 203, 1974. 9. J.-P. Serre : A course in Arithmetic, Chap 7. Springer GTM. 10. J. Silverman : Advanced topics in the arithmetic of elliptic curves, Chap.1. Springer, GTM 1994. 11. W. Stein : Modular forms, a computational approach, AMS, Graduate studies, 2007.

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