I am a graduate student majoring in number theory. Recently I have to give a report to graduate students studying mathematics. I am interested in this field, but I know little about it. Can you give me some advice? Welcome to providing some references! Thanks!
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5$\begingroup$ This is pretty vague. You really don't have any idea what you'd like to say in an introductory talk on a subject that you say are studying? $\endgroup$– Todd TrimbleCommented Oct 24, 2013 at 16:49
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$\begingroup$ Thanks! I have no idea about the attractive part of the report. So the first thing I should do is to find the attractive aspects, such as the application in solving Fermat's last theorem, but I can't understand it well. So I hope you can give me some advice. What book/paper should I refer to? $\endgroup$– chluoCommented Oct 25, 2013 at 11:14
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$\begingroup$ The book Elliptic and Modular Functions from Gauss to Dedekind to Hecke by Ranjan Roy might be relevant. $\endgroup$– WatsonCommented May 30, 2018 at 21:16
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$\begingroup$ I wonder why Poincaré's work on Automorphic Functions is not developed in Roy's book $\endgroup$– PonceCommented May 3, 2023 at 17:27
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3 Answers
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The 1-2-3 of modular forms, by J. H. Bruinier, G. van der Geer, G. Harder, and D. Zagier (2009)
An introduction to the roles modular forms have played throughout history, emphasizing both classical and contemporary problems and applications.
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"Modular forms, a computational approach", by W.A.Stein (2007) explains many simple things about modular forms (and how to compute with them) ; it should give you a firm grasp of the basics before diving into "The 1-2-3 of modular forms" by Bruinier, van der Geer, Harder and Zagier (as pointed out by Carlo Beenakker in his answer).
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Two other references which you may like and which in particular emphasize the historical aspects:
"Elliptic Curves, Function Theory, Geometry , Arithmetic" by Henry McKean and Victor Moll.
"Linear Differential Equations and Group Theory, from Riemann to Poincaré" by Jeremy Grey.