Timeline for isomorphism of abelian varieties
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2010 at 4:30 | comment | added | Fei YE | @Bjorn: In the remark (3) in your paper "The Grothendieck ring of varieties is not a domain", you mentioned elliptic curve over $\mathbb{C}$. Does that mean that there exist elliptic curves $A\not\cong B$ over $\mathbb{C}$ such that $A\times A \cong B\times B$? | |
| Apr 11, 2010 at 18:39 | comment | added | Torsten Ekedahl | @Bjorn: You are right of course, I misremembered. (In my defence, an example of non-cancellation also gives examples of zero-divisors in the Grothendieck ring, the tricky thing is to get an example over $\mathbb Q$.) | |
| Apr 10, 2010 at 1:02 | comment | added | Bjorn Poonen | @Torsten: I think the example in my paper was of something slightly different, namely A x A = B x B with A and B not isomorphic. | |
| Apr 9, 2010 at 4:16 | vote | accept | Tuan | ||
| Apr 9, 2010 at 4:04 | comment | added | Torsten Ekedahl | It is false also in characteristic $0$ though there it is true for elliptic curves (see Angelo's reply). It is a question of finding an example of non-cancellation for projective modules over a suitable ring and then mirror it for abelian varieties. See Poonen, "The Grothendieck ring of varieties is not a domain" for an example. (Bjorn's example is somewhat involved as he wants an example over $\mathbb Q$. An example over $\mathbb C$ is easier to construct.) | |
| Apr 9, 2010 at 3:44 | answer | added | Angelo | timeline score: 20 | |
| Apr 9, 2010 at 3:23 | comment | added | Tuan | I meant the complete reducibility theorem. Thanks. | |
| Apr 9, 2010 at 0:21 | comment | added | Pete L. Clark | I think that rather than "irreducibility" you mean "complete reducibility" or "Poincare's complete reducibility theorem". As for the question itself: I seem to recall that it is famously false for supersingular abelian varieties, i.e., in positive characteristic. Over C, I might guess that it's true, but that's just a guess. | |
| Apr 8, 2010 at 21:56 | history | asked | Tuan | CC BY-SA 2.5 |