Timeline for Values of cusp forms at q = 1 ?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 6, 2010 at 11:15 | vote | accept | Franz Lemmermeyer | ||
| Mar 27, 2010 at 4:02 | comment | added | KConrad | Oh, and the convergence of $L(s)$ for $s$ with real part greater than 5/6 uses the elliptic modularity theorem. | |
| Mar 27, 2010 at 4:01 | comment | added | KConrad | Voloch is right: this $\sqrt{2}$ is related to partial Euler product behavior as $s$ tends to 1, not to the Dirichlet series representation for $L(s)$, which in fact converges for $s$ with real part greater than 5/6 (so at $s=1$). If you replace elliptic curve $L$-functions with Hecke $L$-functions for a quadratic character, you can turn the unexpected $\sqrt{2}$ into $1/\sqrt{2}$ with the Euler product as $s$ tends to $1/2$. Oh, and all such Euler product convergence implies GRH so you can numerically test this stuff and see it happen but you'll never unconditionally prove it. | |
| Mar 26, 2010 at 17:52 | comment | added | Felipe Voloch | @Kevin, you mean Goldfeld, not Szpiro. What he proved was to do with $\prod N_p/p$, the quantity originally computed by Birch and Swinnerton-Dyer. See: ams.org/mathscinet/search/… I don't think this is related with $\sum a_n/n$ which does seem to converge conditionally to $L(1)$. | |
| Mar 26, 2010 at 16:15 | comment | added | Marty | I don't think I've got the analytic expertise for that one. But I saw it, and I'm curious about any answers from others. | |
| Mar 26, 2010 at 15:53 | comment | added | Kevin Buzzard | Based on your comments Marty, you might want to take a stab at mathoverflow.net/questions/19410/… which may be talking about a similar issue? | |
| Mar 26, 2010 at 15:52 | comment | added | Kevin Buzzard | Sounds like I have indeed got the wrong end of the stick then :-) Apologies. | |
| Mar 26, 2010 at 15:23 | comment | added | Marty | But here I'm looking at $f(z)$ as $z$ approaches $1$, via certain paths in the upper half-plane. These values have almost nothing in common with the values of the Dirichlet series as $s$ approaches zero. For cusp forms, the limits I'm talking about are in the same spirit as the convergent integral representation of the L-function. | |
| Mar 26, 2010 at 15:17 | comment | added | Kevin Buzzard | Somehow it would surprise me if $f(q)$ really did tend to $L(E,0)$ in any reasonable sense, because isn't it a result of Szpiro that if $\sum_n a_n.n^{-s}$ converges as $s$ tends to 1, then it won't converge to $L(E,1)$ but rather to something like $\sqrt{2}$ times this? I forgot what he actually proved unfortunately. Somehow if you can't get it right at 1 with 'naive' methods, what chance have you of getting it right at 0? Or do you think I've got the wrong end of the stick? | |
| Mar 26, 2010 at 15:13 | history | answered | Marty | CC BY-SA 2.5 |