Timeline for Special values of real analytic Eisenstein series
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Aug 6, 2016 at 8:15 | vote | accept | Bruce Bartlett | ||
| Aug 4, 2016 at 18:31 | answer | added | Myshkin | timeline score: 6 | |
| Aug 4, 2016 at 18:29 | history | edited | Myshkin | CC BY-SA 3.0 |
+ top level tag (nt.) + automorphic forms
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| Jul 29, 2016 at 22:27 | comment | added | Bruce Bartlett | Thanks Paul, much appreciated. But I still just want to know the answer, what is $E(\exp(\frac{2 \pi i}{3}), 0)$? I don't know much about Dedekind zeta functions and I don't know the explicit value of $\zeta_k(0)$ for the relevant $k$ here. | |
| Jul 29, 2016 at 16:32 | comment | added | paul garrett | With a cube root of unity, the outcome is still $\zeta_k(s)/\zeta(2s)$, now with $k$ the corresponding quadratic extension. | |
| Jul 29, 2016 at 14:09 | comment | added | Bruce Bartlett | Sorry, I meant $E(\exp(\frac{2 \pi i}{3}), 0)$. | |
| Jul 29, 2016 at 13:33 | comment | added | paul garrett | Probably they are not listed because they are not hard to compute whenever one wants, at least to the extent possible, for the Heegner-point cases. E.g., $E_s(1+i)=E_s(i)=\zeta_k(s)/\zeta(2s)$ for $k=\mathbb Q(i)$. | |
| Jul 29, 2016 at 13:04 | comment | added | Bruce Bartlett | Thanks, that is useful. I'd just like to know the final answer though, without having to do the linear algebra myself. Just want to know the explicit value of eg. $E(1 + i,0)$. Seems such explicit formulae are not listed anywhere. | |
| Jul 29, 2016 at 12:46 | comment | added | paul garrett | The more natural objects are suitable linear combinations (over Heegner points) giving special values of ideal-class group L-functions, from which the individual values can be recovered by linear algebra. Values at $s=0$ are residues at $s=1$, which are non-zero only for the trivial ideal-class character. For general points $\tau$, but at $s=0$ or $s=1$, over $\mathbb Q$ we have the Kronecker limit formula. | |
| Jul 29, 2016 at 7:56 | comment | added | Asaf | To give one example, the residues of the Eisenstein series appear in the computation of the fundamental domain of $G/\Gamma$ done by Langlands. | |
| Jul 29, 2016 at 7:20 | history | edited | Bruce Bartlett | CC BY-SA 3.0 |
Improved clarity.
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| Jul 29, 2016 at 7:15 | comment | added | Bruce Bartlett | Ok, that's interesting, thanks. But nevertheless, my question still isn't answered. I would like to know a list of known special values of $E(\tau, 0)$ for certain $\tau$. Or any special values whatsoever, for arbitrary $s$. You have talked about linear combinations, not values at specific points. | |
| Jul 28, 2016 at 22:52 | comment | added | paul garrett | ... and the idea of "periods of Eisenstein series" is a well-established thing, going back to Hecke and Maass, and more examples in recent years. Erez Lapid and various of his collaborators have done many examples. Rankin-Selberg integral expressions for various L-functions can be considered examples of such periods. In this vein, Deligne's Corvallis (1977, AMS 1979) piece about special values is relevant. But/and what is the question? | |
| Jul 28, 2016 at 22:33 | comment | added | paul garrett | For fixed $\tau$ in the upper half-plane, the function $s\to E_s(\tau)$ is also called an Epstein zeta function, easily google-able. The sum of these values over $\tau$ running through Heegner points for a given negative fundamental discriminant produces $\zeta_k(s)/\zeta(2s)$, where $k$ is the corresponding complex quadratic field extension of $\mathbb Q$. This was known to Hecke. Other linear combinations of special values at Heegner points produce ideal-class characters of those fields. Values at "generic" $\tau$ seem to be much less well-understood. | |
| Jul 28, 2016 at 21:44 | history | asked | Bruce Bartlett | CC BY-SA 3.0 |