Timeline for The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2019 at 18:40 | comment | added | David E Speyer | It took me a bit of time to check but, yes, you are using the terms abelian function and Theta function correctly. | |
| Mar 21, 2019 at 20:03 | vote | accept | MCS | ||
| Mar 21, 2019 at 20:03 | |||||
| Mar 18, 2019 at 23:01 | history | bounty ended | CommunityBot | ||
| Mar 15, 2019 at 19:29 | comment | added | MCS | I'm still waiting on that explanation of the classical and straightforward solution to (1) (the problem of computing a basis of holomorphic differentials for a given curve). :) | |
| Mar 13, 2019 at 1:52 | comment | added | MCS | Hmm... is the local inverse of your map $\Psi:X^{g}\rightarrow\mathbb{C}^{g}$ an abelian function (viz. encyclopediaofmath.org/index.php/Abelian_function)—in which case, could they be used to piece together the global (non-injective!) inverse of $\Psi$? Doing some searching, it seems that Riemann Theta Functions are involved in this (encyclopediaofmath.org/index.php/Riemann_theta-function). Finally, if you wouldn't mind, that "classical and straightforward" solution to (1) would be most appreciated. :) | |
| Mar 13, 2019 at 1:33 | comment | added | David E Speyer | For any $x$, there are only finitely many $y$ with $F(x,y)=0$, so $(\partial \wp)/(\partial z_j)$ can only take finitely many values. The easiest possibility is that $(\partial \wp)/(\partial z_1) = \cdots = (\partial \wp)/(\partial z_n)$, but then $\wp$ is just a function of $\sum z_j$. I haven't carefully thought through the more complicated combinatorial options, but I don't think it would help. For example, if our equation is $y^2 = p(x)$, then we would have $\pm (\partial \wp)/(\partial z_1) = \cdots = \pm (\partial \wp)/(\partial z_n)$ and $\wp$ would be a function of $\sum \pm z_j$. | |
| Mar 13, 2019 at 1:30 | comment | added | David E Speyer | I don't think it makes sense to add more variables in that way. And I think I have already said what there is to be said about parametrizing a curve as $(\phi, \phi')$: It works locally, but globally there are all sorts of branching issues with $\phi$. I'll explain why I don't think adding variables in this way is useful: | |
| Mar 13, 2019 at 1:20 | comment | added | David E Speyer | That doesn't match my understanding of Jacobi inversion? Here is what I think Jacobi inversion is (see your link, near eqn (1)). Let $\omega_1$, ..., $\omega_g$ be a basis of holomorphic $1$-forms on $X$. Fix a base point $x_0$. Locally, we have a map $X^g \to \mathbb{C}^g$ sending $(x_1, \ldots, x_g)$ to $\left( \sum_j \int_{x_0}^{x_j} \omega_1, \ldots, \sum_j \int_{x_0}^{x_j} \omega_g \right)$. I understood Jacobi inversion as locally inverting this map. I don't know a way to make that look like the equations you give. Maybe someone else does. | |
| Mar 13, 2019 at 0:30 | comment | added | MCS | I want to know what happens when $N\geq2$. Looking at the linked article, my guess would be that instead of “parameterizing” $F$'s vanishing set, we obtain a system: $$F\left(\wp,\frac{\partial\wp}{\partial z_{1}}\right)=F\left(\wp,\frac{\partial\wp}{\partial z_{2}}\right)=\ldots=F\left(\wp,\frac{\partial\wp}{\partial z_{N}}\right)=0$$ or something like that. Moreover, I want to know how to construct and compute $\wp$ for a given $F$. | |
| Mar 13, 2019 at 0:30 | comment | added | MCS | This is an excellent answer, though it's a bit over my head. To the extent that I understand the content of this article (encyclopediaofmath.org/index.php/Jacobi_inversion_problem), given a formula $F\left(x,y\right)$ there is a complex manifold $S$ of complex dimension $N$ and a complex-valued function $\wp\left(\mathbf{z}\right)$ (where $\mathbf{z}=\left(z_{1},\ldots,z_{N}\right)$) defined on $S$ that, at least in the case where $N=1$, parameterizes the vanishing set of $F$ via $F\left(\wp\left(z\right),\wp^{\prime}\left(z\right)\right)=0$ for all $z$. | |
| Mar 12, 2019 at 19:30 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 12 characters in body
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| Mar 12, 2019 at 19:25 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 42 characters in body
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| Mar 12, 2019 at 18:57 | history | answered | David E Speyer | CC BY-SA 4.0 |