Timeline for Varieties with the same number of $\mathbb{F}_p$-points for all but finitely many primes
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 20, 2021 at 6:40 | comment | added | alpoge | whatcha mean? i was just thinking that the zeta function of $X$ mod $p$ is rational and an alternating product of the $L$-functions of the $H^i$’s, and you can pick out the $i$-th guy by taking the polynomial in the factorization (there’s no cancellation by purity) whose roots are all the roots/poles of the zeta function of $X$ mod $p$ of absolute value $p^{i/2}$. | |
| Sep 20, 2021 at 6:36 | comment | added | lkx | @alpoge actually even if everything is smooth and proper can't the eigenvalues conspire to satisfy a polynomial equation? The trace on $H^0$ and $H^{2n}$ is predetermined so we need to consider at least surfaces. | |
| Sep 20, 2021 at 6:17 | comment | added | lkx | Thank you, corrected | |
| Sep 20, 2021 at 6:16 | history | edited | lkx | CC BY-SA 4.0 |
added 22 characters in body
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| Sep 20, 2021 at 6:13 | comment | added | alpoge | yeah exactly! that’s why i did that in the first comment above —- i was just correcting your summary | |
| Sep 20, 2021 at 6:09 | comment | added | lkx | @alpoge if we look at the trace on the alternating sum of $H^i_c$ then we don't have to pick them apart right? | |
| Sep 20, 2021 at 5:55 | comment | added | alpoge | i agree with everything except: in the answer to the first question because you don't assume the varieties are smooth and proper i dunno how to extract the trace on just the $H^i_c$'s from only the information of all point counts (when they're smooth proper you can use purity) --- but im not an expert. anyway otherwise i agree with the summary! | |
| Sep 20, 2021 at 5:38 | history | answered | lkx | CC BY-SA 4.0 |