# congruences for Fourier coefficients of modular forms

Are there other good articles on congruences for Fourier coefficients of modular forms beside Swinnerton-Dyer's article in "Modular Functions of One Variable III"?

I am looking for generalisations and other explicit cases than the Ramanujan $\tau$-function and the $5$ other modular forms in [Swinnerton-Dyer].

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Could you be a bit more specific regarding what you are looking for. If not, I am tempted to vote to close this as not a real question. – user9072 Jul 27 '12 at 10:19
I have edited my question accordingly. – Timo Keller Jul 27 '12 at 14:41
Dear Timo, The theory of congruences of modular forms is a massive and central topic in contemporary number theory, lying at the basis of the proof of Mazur's theorem on torsion in elliptic curves, Fermat's Last Theorem, and Sato--Tate, for example. There are dozens and dozens of articles on this topic written over the last several decades. (Actually, I just saw Joel's answer, where he puts the number in the thousands!) Some of the key names are Gross, Hida, Katz, Mazur, Ribet, Serre, Taylor, and Wiles. Regards, – Emerton Jul 28 '12 at 3:08

I understand and share quid's bewilderment. There are easily more than 5000 articles whose main theme can be considered, without stretching too much, as "congruencies between Fourier coefficients of modular forms - and generalizations", including some of the most famous of the last four decades, like the one proving Fermat's last theorem.

The theory of congruences between modular forms has blossomed in many directions. You may want to look at work of Serre, Katz, Hida, Mazur, Ribet, Wiles (and others) in the 70's and 80's for the beginning of the story.

Edit: There are so many directions of generalizations that it's hard to decide where to begin with. Swinnerton-Dyer works with congruences between two modular forms of the same level and the same weights. Those congruences are in some sense "accidental". Several papers by Hida (like this one "congruences between cusp forms and special values of their Zeta functions") hace generalized this line of thought. Besides, one can consider congruences between forms of the same level and various weights -- and you get the theory of p-adic modular forms, Hida families, eigencurve, etc. Or congruences between forms of the same weights and different level -- you get to the level-raising and level-lowering result of Ribet's and other. Ribet's ICM talk is a good introduction to this. Then you get all the generalization to other automorphic forms...

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Thank you very much for your response. – Timo Keller Jul 27 '12 at 14:49
Here is a nice survey article on families of modular forms: math.uchicago.edu/~emerton/pdffiles/padic.pdf. Note that the existence of a family implies a great number of congruences between the various members. – Kevin Ventullo Jul 27 '12 at 18:47

What about K. Ono, Congruences on the Fourier coefficients of modular forms on $\Gamma_0(n)$, Contemporary Mathematics, Vol. 166, 93-105, 1994.

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Thank you. This is what I was looking for. – Timo Keller Jul 27 '12 at 10:40