Timeline for Families of abelian varieties on the line (or more generally simply connected varieties)
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Apr 29, 2016 at 20:18 | comment | added | Alex Youcis | Thanks so much! I'm going to accept Donu's answer because it answers all three of my questions, but I greatly appreciate your help. | |
| Apr 29, 2016 at 20:15 | comment | added | ACL | The argument in your second comment is more or less the (simple) one given by Donu Arapura. The complex definition of $\mathcal A_{g,n}$, as a quotient of a bounded symmetric domain by the free action of a discrete group, shows that it is hyperbolic. Consequently, it does not receive non-constant holomorphic maps from the affine line. | |
| Apr 29, 2016 at 20:13 | comment | added | ACL | I think it's true, but the litterature on abelian schemes is scarce. There are notes from a 1967-68 seminar in Orsay, math.u-psud.fr/~biblio/numerisation/docs/04_SEMINAIRE/pdf/…. | |
| Apr 29, 2016 at 16:58 | comment | added | Alex Youcis | Also, in this paper it's not clear to me whether the following is true. In the case of elliptic schemes the following works: choose a trivilization $\alpha:\mathscr{E}[n]\xrightarrow{\approx}(\mathbb{Z}/N\mathbb{Z})^2$. This then defines a map $\mathbb{A}^1_\mathbb{C}\to Y(N)$ which extends to a map $\mathbb{P}^1_\mathbb{C}\to X(N)$. For $N\gg 0$ the genus of $X(N)$ is positive and so this map must factor through a point—thus the family is isotrivial. Do you know if there is a way to proceed using the geometry of $\mathcal{A}_{g,1}$ (or $\mathcal{A}_{g,n}$, or its spaces with level structure)? | |
| Apr 29, 2016 at 16:55 | comment | added | Alex Youcis | Hey ACL, thanks for the information! Do you have a belief that $\mathscr{A}$ should be isogenous to a Jacobian (as in the case of fields)? And is your statement 'I don't know whether there is a reference in the literature' mean "I think it's true" or "It is true—no one's bothered to type it up yet"? Also, do you have any opinion about the 'proof'I described about above? | |
| Apr 29, 2016 at 16:52 | history | answered | ACL | CC BY-SA 3.0 |