Timeline for Existence of a holomorphic map between Riemann surfaces
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 4, 2023 at 12:23 | vote | accept | Alexandre Eremenko | ||
| Aug 3, 2023 at 22:02 | comment | added | Alexandre Eremenko | @Christian Remling: we are discussing Picard's paper of 1887; it was published long before the general uniformization theorem was proved, and before the universal cover was defined. | |
| Aug 3, 2023 at 19:08 | comment | added | abx | @Christian Remling: yes, of course. But the question is to understand how Picard (or Nevanlinna?) wanted to proceed. | |
| Aug 3, 2023 at 18:19 | comment | added | Christian Remling | If we have a map $\mathbb C\to X$, we can also just lift it to the universal cover of $X$ (=$D$ if $g(X)\ge 2$) to obtain a bounded entire function. | |
| Aug 3, 2023 at 17:12 | comment | added | abx | Oh I see, sorry I didn't pay attention to this. Well, suppose you have a nontrivial map $\mathbb{C}\rightarrow X$; using the notation in my previous comment, since $\tilde{\rho} $ is étale, this map lifts to $\tilde{X} $, then by projection to $C$ gives a nontrivial map $\mathbb{C}\rightarrow C$. | |
| Aug 3, 2023 at 16:06 | comment | added | Alexandre Eremenko | Correct. But Picard used this construction to PROVE this fact, that there is no map from C to a curve of genus >2. How exactly did he reduce the proof of this theorem to hyperelliptic case? | |
| Aug 3, 2023 at 12:42 | comment | added | abx | I am sorry I don't understand your question. $X$ has genus $\geq 2$, there is no non-constant holomorphic map from the complex plane to $X$. | |
| Aug 3, 2023 at 12:27 | comment | added | Alexandre Eremenko | Could you please extend your explanation a bit? Suppose $X$ receives a non-constant holomorphic map from the complex pane. Why $\tilde{X}$ does? | |
| Aug 3, 2023 at 7:13 | history | answered | abx | CC BY-SA 4.0 |