Timeline for Computing Hodge numbers by point counting
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Sep 8, 2020 at 14:18 | comment | added | Piotr Achinger | yes, the Weil conjectures are needed to take care of the grading. Sorry for the sloppiness. | |
| Sep 8, 2020 at 12:57 | comment | added | ali | @PitorAchinger I'm not familiar with graded representations. Is the equality of graded trace is enough to identify two such representations?if so why both of the notes and in the original argument of katz use weil conjectures | |
| Sep 8, 2020 at 11:05 | comment | added | Piotr Achinger | Let $H = \bigoplus_i H^i(X_{\overline{\mathbf{Q}}}, \mathbf{Q}_\ell)$, a graded Galois rep. The assumption on $X(\mathbf{F}_p)$ tells you the (graded) trace of ${\rm Frob}_p$ on $H$. By Chebotarev density, the conj classes ${\rm Frob}_p$ ($p>N$) are dense in the Galois group of $\mathbf{Q}$. Now using the polynomial $P$ you can build a graded Galois rep $H'$ which is a direct sum of $\mathbf{Q}_\ell(i)$'s which has the same Frobenius traces for $p>N$. Therefore $H$ and $H'$ have the same semisimplification. Finally, $p$-adic (for $p=\ell$ large) Hodge theory gives you the Hodge numbers. | |
| Sep 8, 2020 at 9:25 | history | edited | F. C. | CC BY-SA 4.0 |
add missing capitals for names
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| Sep 8, 2020 at 8:13 | history | asked | ali | CC BY-SA 4.0 |