Timeline for Which abelian varieties over a local field can be globalized?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| 8 hours ago | comment | added | Laurent Moret-Bailly | @ChrisWuthrich Agreed, thanks! | |
| 16 hours ago | history | became hot network question | |||
| 19 hours ago | vote | accept | curious math guy | ||
| 20 hours ago | comment | added | Chris Wuthrich | @LaurentMoret-Bailly If $j(\mathcal{A})$ is in $\mathbb{Q}$ there is an elliptic curve $E/\mathbb{Q}$ such that $E \times \mathbb{Q}_p$ is a twist of $\mathcal{A}$. If $j\not \in \{0,1728\}$, there is a $D\in\mathbb{Q}_p^{\times}$ such that $\mathcal{A}$ is the quadratic twist $E^D$. Now approximate $D$ by a rational $D'$ such that $\mathbb{Q}(\sqrt{D}) = \mathbb{Q}(\sqrt{D'})$. Then $A=E^{D'}$ will do. Similar with sextic, cubic and quartic twists. | |
| 23 hours ago | answer | added | Daniel Loughran | timeline score: 9 | |
| 23 hours ago | comment | added | Vik78 | You might be able to say something about Brauer-Manin obstructions for various moduli spaces of abelian varieties, especially modular curves. See math.mit.edu/~poonen/papers/heuristic.pdf | |
| yesterday | comment | added | Laurent Moret-Bailly | @ChrisWuthrich: This condition is necessary, but if $j(\mathcal{A})\in\mathbb{Q}$ I can only conclude, a priori, that there is $A$ over $\mathbb{Q}$ that becomes isomorphic to $\mathcal{A}$ over $\overline{\mathbb{Q}_p}$. | |
| yesterday | comment | added | Chris Wuthrich | For an elliptic it is simply the question if $j(\mathcal{A})\in \mathbb{Q}$. | |
| yesterday | history | asked | curious math guy | CC BY-SA 4.0 |