Timeline for Why does the definition of modularity demand weight 2?
Current License: CC BY-SA 4.0
17 events
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| Nov 19, 2021 at 16:13 | history | edited | Timothy Chow | CC BY-SA 4.0 |
Fixed typo in LaTeX formula
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Aug 26, 2011 at 13:53 | vote | accept | Barinder Banwait | ||
| Aug 26, 2011 at 13:53 | comment | added | Barinder Banwait | I'd like to thank everybody who contributed to this thread. I have learned a huge amount from the answers and comments. | |
| Aug 24, 2011 at 17:55 | comment | added | Rob Harron | One should note that Taniyama's statement was in 1955, at least 10 years before Serre conjectured the existence of ℓ-adic representations attached to higher weight forms (which, afaik, he did in order to study congruences such as Ramanujan's for $\tau(n)$). | |
| Aug 24, 2011 at 17:49 | comment | added | Rob Harron | According to footnote 2 in Lang's article Some history of the Shimura–Taniyama conjecture (ams.org/notices/199511/forum.pdf), Shimura's rationale for the conjecture was (Hasse's) conjectural functional equation of the Zeta function of the elliptic curve. And if you go back to Taniyama's original "Problem 12" (available in English in Shimura's article (dx.doi.org/10.1112/blms/21.2.186)), Taniyama seems to use the same reasoning (so, of course, they both use the $s$ to $k-s$ normalization for the $L$-function). | |
| Aug 16, 2011 at 19:59 | comment | added | GH from MO | @Dror: Yes, but he talks about $L$-function associated to a Galois representation, while I talk about the $L$-function associated to a cusp form. I am emphasizing that if you use the correct "God given" normalization (actually Riemann's), i.e. the one in which $s$ is related to $1-s$, you can read off the weight from the gamma factor. If you use your favorite normalization depending on the weight (e.g. the one in which $s$ is related to $k-s$), then it is no surprise that you can read it off from the gamma factors. | |
| Aug 16, 2011 at 10:55 | comment | added | Dror Speiser | @GH: note that "the gamma factors determine the weight" is also in Kevin's comments. | |
| Aug 16, 2011 at 10:36 | comment | added | GH from MO | @Barinder: My response below should clarify the role of the weight. | |
| Aug 16, 2011 at 10:15 | answer | added | GH from MO | timeline score: 4 | |
| Aug 15, 2011 at 17:10 | answer | added | Kevin Buzzard | timeline score: 18 | |
| Aug 15, 2011 at 0:15 | answer | added | Dror Speiser | timeline score: 28 | |
| Aug 15, 2011 at 0:13 | answer | added | Rob Harron | timeline score: 3 | |
| Aug 14, 2011 at 21:58 | answer | added | Álvaro Lozano-Robledo | timeline score: 12 | |
| Aug 14, 2011 at 21:55 | comment | added | Barinder Banwait | @James: Maybe you are referring to Theorem 11.4 in Ch 1; which says that the weight must be two to ensure the L-function of E satisfies the right functional equation, i.e, s goes to 2-s? | |
| Aug 14, 2011 at 21:42 | comment | added | James D. Taylor | I'm sure someone will answer more fully, but in the meantime: it is explain very pleasantly and with the insight you require in Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves". | |
| Aug 14, 2011 at 20:52 | history | asked | Barinder Banwait | CC BY-SA 3.0 |