Timeline for Elliptic curves over proper variety over $\mathbf{F}_q$ isotrivial
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Jul 2, 2010 at 14:57 | answer | added | Holger Partsch | timeline score: 4 | |
| Jul 2, 2010 at 6:37 | answer | added | Boyarsky | timeline score: 6 | |
| Jul 1, 2010 at 20:21 | comment | added | Boyarsky | The fiberwise definition of isotrivial in Scott's comment is not the one in the question, and is too weak; the definition in the revised question is too strong. For a smooth proper geometrically connected curve $X$ over a field $k$ so that there is a geometrically connected etale double cover $X' \rightarrow X$, the corresponding quadratic twist $\mathcal{E}$ of a constant elliptic curve $E \times X$ is a new elliptic curve over $X$ which is not isotrivial in the sense of the OP, but it is isotrivial in the right sense: becomes constant over an etale cover of the base (namely $X'$!). | |
| Jul 1, 2010 at 19:57 | comment | added | S. Carnahan♦ | The coarse moduli space is a canonical map from $\mathcal{M}_{Ell}$ to the $j$-line, and the properties of your base imply its image in the $j$-line is a point. Since the $j$-invariant classifies elliptic curves up to isomorphism, all $\overline{\mathbb{F}_q}$-fibers of the family are isomorphic. | |
| Jul 1, 2010 at 19:34 | history | edited | user6960 | CC BY-SA 2.5 |
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| Jul 1, 2010 at 19:22 | comment | added | Boyarsky | It's false when the base is non-reduced (e.g., dual numbers over a field). You want the base to be smooth and geometrically connected, not just proper. It would be good if you indicated in the question precisely how you are defining isotriviality (in case your problem is caused by using a definition which is too restrictive). | |
| Jul 1, 2010 at 19:19 | history | edited | user6960 | CC BY-SA 2.5 |
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| Jul 1, 2010 at 18:58 | history | edited | Charles Rezk | CC BY-SA 2.5 |
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| Jul 1, 2010 at 18:44 | history | asked | user6960 | CC BY-SA 2.5 |