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What are the best introductory texts on modular forms that are suited for a brief six week course intended for advanced undergraduates? The students will be quite sharp and as far as prerequisites go, they can be expected to have completed two courses in algebra and analysis each as well as a first course in complex analysis and a course in number theory at the level of Hardy & Wright.

I am considering Apostol's Modular Forms and Dirichlet Series in Number Theory but worry that it may not be possible to make serious progress in six weeks.

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  • $\begingroup$ Have you looked at Serge Lang's book? It's relatively elementary although you probably would need some supplementation to make it accessible to undergraduates. $\endgroup$
    – Jim Conant
    Commented Jun 6, 2014 at 4:43

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Perhaps it is a little too brisk for your purposes, but I have always liked Zagier's exposition in The 1-2-3 of Modular Forms (Springer Universitext) very much. It is about 100 pages, more than half of which are applications of various kinds and for various backgrounds; I'm sure it contains something for everyone.

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A First Course in Modular Forms, by Diamond and Shurman (D-S), is many people's favourite; not sure if it is too advanced. Also, Serre's course in arithmetic has something (chapter VII) on Modular Forms. And Shimura's Elementary Dirichlet Series and Modular Forms is of more managable size than (D-S), but assumes more familiarity with algebraic number theory than Serre.

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    $\begingroup$ Serre's chapter is pretty good, but might be a bit short... D&S's book is pretty good, but might be a bit long... $\endgroup$
    – Julien Puydt
    Commented Jun 4, 2014 at 5:17
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I would use the chapter from Koblitz's Introduction to Elliptic Curves and Modular Forms.

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gunning has a nice book. http://www.amazon.com/Lectures-Modular-Forms-Robert-Gunning/dp/0691079951

the indefatigable and always valuable James Milne has some course notes as well:

http://www.jmilne.org/math/CourseNotes/MF110.pdf

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The first one: The book---"A first course in modular forms" by F. Diamond, J. Shurman is a good book to start to study classical modular forms.

The second one: The advanced one--- "Modular forms" by Toshitsune Miyake is also a very good textbook to learn modular forms.

The more advanced books, you can consider to read the books by the Dr. Goro Shimura, who I think is the most professional mathematician in modular forms.

Good luck.

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