congruences for Fourier coefficients of modular forms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:17:23Z http://mathoverflow.net/feeds/question/103282 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103282/congruences-for-fourier-coefficients-of-modular-forms congruences for Fourier coefficients of modular forms Timo Keller 2012-07-27T08:27:31Z 2012-07-27T19:49:36Z <p>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"?</p> <p>I am looking for generalisations and <em>other explicit cases</em> than the Ramanujan $\tau$-function and the $5$ other modular forms in [Swinnerton-Dyer].</p> http://mathoverflow.net/questions/103282/congruences-for-fourier-coefficients-of-modular-forms/103290#103290 Answer by PaPiro for congruences for Fourier coefficients of modular forms PaPiro 2012-07-27T10:08:24Z 2012-07-27T10:08:24Z <p>What about K. Ono, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/002.pdf" rel="nofollow">Congruences on the Fourier coefficients of modular forms on $\Gamma_0(n)$</a>, Contemporary Mathematics, Vol. 166, 93-105, 1994.</p> http://mathoverflow.net/questions/103282/congruences-for-fourier-coefficients-of-modular-forms/103314#103314 Answer by Joël for congruences for Fourier coefficients of modular forms Joël 2012-07-27T14:33:47Z 2012-07-27T14:45:07Z <p>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.</p> <p>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.</p> <p>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... </p> http://mathoverflow.net/questions/103282/congruences-for-fourier-coefficients-of-modular-forms/103342#103342 Answer by Jon Bannon for congruences for Fourier coefficients of modular forms Jon Bannon 2012-07-27T19:49:36Z 2012-07-27T19:49:36Z <p>Also, have a look at Ken Ono's <a href="http://books.google.com/books/about/The_Web_of_Modularity.html?id=MLdRYIg6pDkC" rel="nofollow">The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and Q-Series</a>. This book is full of what you are looking for. </p>