Timeline for Reference Request: CM Motives over Function Fields
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Apr 11, 2021 at 21:32 | comment | added | Angus McAndrew | Ah ok, that makes more sense. Thanks for the reference! | |
| Apr 11, 2021 at 21:30 | comment | added | Will Sawin | See Proposition 2.6 of Motives over Finite Fields by J.S. Milne jmilne.org/math/articles/1994aP.pdf | |
| Apr 11, 2021 at 21:23 | comment | added | Will Sawin | It's not all Galois representations, but the ones arising from algebraic varieties, which necessarily have eigenvalues Weil numbers. I learned a long time ago on MO that you can use the fact that Weil numbers lie in CM fields to write them as products of weight 1 Weil numbers that look like the Frobenius eigenvalues of abelian varieties, and use Honda's theorem to conclude that these actually come from abelian varieties. | |
| Apr 11, 2021 at 21:12 | comment | added | Angus McAndrew | Thanks for the insight! I wasn't aware of the fact that all Galois representations from $\mathbb{F}_q$ were tensor products of Galois representations of abelian varieties. How does one see this? | |
| Apr 11, 2021 at 20:19 | review | First posts | |||
| Apr 12, 2021 at 5:24 | |||||
| Apr 11, 2021 at 20:17 | comment | added | Will Sawin | Rank one Galois representations over $\mathbb F_q(T)$ are classified - they are all products of finite order characters with characters factoring through the Galois group of $\mathbb F_q$. Galois representations factoring through the Galois group of $\mathbb F_q$ all are tensor products of Galois representations of abelian varieties over $\mathbb F_q$. So we might guess that such motives are tensor products of one-dimensional Artin motives with motives arising from abelian varieties over $\mathbb F_q$. But I don't know in what cases we can prove this. | |
| Apr 11, 2021 at 20:10 | history | asked | Angus McAndrew | CC BY-SA 4.0 |