Timeline for Analytic function avoiding elements of the modular group
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 28, 2018 at 12:59 | comment | added | Venkataramana | @Eremenko: thanks for the paper. It looks interesting and I will study it. | |
| Apr 28, 2018 at 12:52 | comment | added | Alexandre Eremenko | @Venkataramana: As a result of this discussion this paper appeared: math.purdue.edu/~eremenko/dvi/picard2b.pdf. You may be interested to see it. We thank you in the end. | |
| Apr 28, 2018 at 2:55 | comment | added | Venkataramana | @Eremenko: the map $w\mapsto w^2$ from $\mathbb{C}\setminus\{0,1,-1\}$ onto $\mathbb{C}\setminus\{0,1\}$ is indeed a covering. Your $\sqrt{z}$ is in fact the function $w$. | |
| Dec 21, 2017 at 3:31 | history | edited | Alexandre Eremenko | CC BY-SA 3.0 |
added 5 characters in body
|
| Nov 15, 2014 at 5:31 | comment | added | Alexandre Eremenko | @Venkataramana: $\sqrt{z}$ lifted to the universal cover does not give a fractional-linear map. It omits perimages of $-1$ on the universal cover. $\sqrt{z}$ omits $-1$ in $C\{0,1\}$. | |
| Feb 20, 2013 at 1:10 | comment | added | Venkataramana | It is nice that you have a map $z\mapsto {\sqrt z}$ on $C= {\mathbb C}\setminus \{0,1\}$; but this is really lifting to a two sheeted cover. Therefore, its lift $f$ to the universal cover (the upper half plane) is in fact a covering map and is hence fractional linear. This seems to contradict the earlier claim that there are no fractional linear maps $f$. | |
| Feb 19, 2013 at 20:18 | history | edited | Alexandre Eremenko | CC BY-SA 3.0 |
added 286 characters in body
|
| Feb 8, 2013 at 1:37 | comment | added | Venkataramana | Very interesting story! | |
| Feb 7, 2013 at 20:40 | comment | added | Alexandre Eremenko | OK, next time we use it, let's call it the Schwarz group, and see whether this can become a standard name. (There is a funny story how Picard proved his Little Theorem. He was related to Hermit (by marriage) and once he dined with his relative. Picard told to Hermit that he was attending a lecture of Weierstrass, who asked whether an entire function can omit two values. Hermit agreed that this is an interesting problem, and explained to Picard another piece of news: how Schwarz constructed his Schwarz modular function using the reflection principle... Picard left the dinner deep in thought :-) | |
| Feb 7, 2013 at 13:04 | comment | added | Venkataramana | That bit about Schwartz is very interesting. I thought this would be a good name since it is used in the proof of "little Picard theorem" as well. I don't know if anyone else thought of it, but I would hesitate to claim credit for this! | |
| Feb 7, 2013 at 4:19 | comment | added | Alexandre Eremenko | Sounds nice. Did you invent this, or someone used this in the literature? I believe it was Schwarz who introduced it in function theory. And it is Schwarz who constructed it from tiling by triangles with zero angles. | |
| Feb 6, 2013 at 14:22 | comment | added | Venkataramana | as to naming the group $G$, how about "the little Picard group"? There is a Picard modular group (a discrete subgroup of $SU(2,1)$), and "the Picard group" in algebraic geometry, but no "little Picard group", as far as I know. | |
| Feb 4, 2013 at 4:29 | history | edited | Alexandre Eremenko | CC BY-SA 3.0 |
added 261 characters in body
|
| Feb 4, 2013 at 4:16 | history | answered | Alexandre Eremenko | CC BY-SA 3.0 |