Timeline for What is the "reason" for modularity results?
Current License: CC BY-SA 3.0
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| Nov 14, 2011 at 21:30 | history | edited | Joël | CC BY-SA 3.0 |
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| Sep 14, 2011 at 2:07 | comment | added | B R | James, automorphic $L$-functions for cusp forms on $GL_2$ are essentially Mellin transforms of cusp forms, so you could recover the cusp form by taking the inverse Mellin transform of the $L$-function (unless the cusp form is unramified everywhere you'd also need to worry about twists of the $L$-function). See Proposition 1.5.1 and Theorem 1.5.1 of Bump's book, or Theorem 7.3 of Iwaniec's Topics in Classical etc. For $GL_n$, inverting the integral requires the notion of an automorphic form on $GL_{n-1}$, so you could bootstrap it . . . | |
| Sep 13, 2011 at 19:58 | comment | added | James D. Taylor | Joel, you said: "Actually, what Weil's result proves is that those nice behavior is essentially the same thing as being modular." Would it be fair to say that one can deduce the definition of a cuspidal representation by having its L-function have analytic continuation and a functional equation is some way that I can't make precise but others (you?) can? | |
| Sep 13, 2011 at 19:43 | comment | added | user9072 | Regarding historic developpment the following question is somewhat related and perhaps of interest mathoverflow.net/questions/61959/taniyama-original-conjecture | |
| Sep 13, 2011 at 19:22 | history | answered | Joël | CC BY-SA 3.0 |