Timeline for Isomorphic endomorphism algebras implies isogenous (for abelian varieties over finite fields)?
Current License: CC BY-SA 4.0
12 events
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| Jul 15, 2021 at 5:55 | comment | added | Watson | Update: this question and the answer below gave rise to the following preprint arxiv.org/abs/2107.06432 :-) | |
| May 25, 2021 at 13:58 | vote | accept | Watson | ||
| May 24, 2021 at 19:37 | comment | added | Watson | @HYL : thanks! In fact, Example 5.19 therein gives an example of two geometrically simple abelian varieties of dimension 3 over $\Bbb F_2$ (lmfdb.org/Variety/Abelian/Fq/3/2/a_c_c, lmfdb.org/Variety/Abelian/Fq/3/2/d_f_h), which are not isogenous, but according to the LMFDB "All geometric endomorphisms are defined over $\Bbb F_{2}$" (namely the geometric endomorphism algebra is a CM number field of degree 6, lmfdb.org/NumberField/6.0.679024.1) | |
| May 24, 2021 at 15:37 | answer | added | Yuri Zarhin | timeline score: 9 | |
| May 23, 2021 at 17:35 | history | became hot network question | |||
| May 23, 2021 at 16:29 | vote | accept | Watson | ||
| May 24, 2021 at 16:10 | |||||
| May 23, 2021 at 13:34 | answer | added | user166831 | timeline score: 5 | |
| May 23, 2021 at 13:09 | comment | added | HYL | There are examples of abelian varieties with isomorphic endomorphism algebras but different $p$-ranks (see e.g. arxiv.org/pdf/2003.05380.pdf, §5.6) and $p$-ranks are isogenous invariants. If they still have the same endomorphism algebras over $\bar{k}$, they will provide counter-examples. | |
| May 23, 2021 at 11:41 | comment | added | Watson | @Wojowu : Tate proved that any abelian variety over a finite field is a CM abelian variety (Prop. 16.27 in Moonen--Geer notes on Abelian Varieties). | |
| May 23, 2021 at 11:33 | comment | added | Wojowu | Did you mean to assume $A_i$ have CM? | |
| May 23, 2021 at 11:00 | history | edited | YCor | CC BY-SA 4.0 |
added top level tags
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| May 23, 2021 at 9:29 | history | asked | Watson | CC BY-SA 4.0 |