Timeline for How are holomorphic and real-analytic Eisenstein series related?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Feb 22, 2023 at 7:26 | comment | added | David Loeffler | Yes, this statement holds for all $s$ (except $s = 0$ and $s = 1$ when $k = 0$, in which case the series is divergent). This follows easily from analytic continuation -- the functions $E_k(\gamma z, s)$ and $j(\gamma, z)^k E_k(z, s)$ agree for $Re(s) \gg 0$, so their analytic continuations must agree everwhere they are both defined. | |
| Feb 22, 2023 at 7:12 | comment | added | Krishnarjun | I have a clarification in the last statement. When you say for a fixed value of $s$, the function $E_k(-,s)$ transforms like a modular form of weight $k$, do you require $Re(s) \gg 0$? or does this statement hold for example $s=1/2$ ? | |
| Oct 23, 2015 at 16:13 | vote | accept | Tian An | ||
| Oct 23, 2015 at 14:33 | comment | added | Pig | @paulgarrett, thanks! Somehow I have never seen this "combined" object before. | |
| Oct 23, 2015 at 14:16 | comment | added | paul garrett | @user31814, for generic $s$, they all generate the same irreducible principal series of $SL_2(\mathbb R)$, but the holomorphic/anti-holomorphic discrete series appear as proper subrepresentations of certain principal series, so the vectors inside those holo discrete series obviously cannot generate the whole thing. | |
| Oct 23, 2015 at 7:07 | comment | added | Pig | Just to clarify - the $E_-(z,s)$, for a fixed $s$, do they generate the same principal series representation for $SL_2$, except that $E_k$ corresponds to the weight $k$ part of the $SO(2)$ action in real place? | |
| Oct 23, 2015 at 6:38 | history | answered | David Loeffler | CC BY-SA 3.0 |