Timeline for Why is $(\mathbb{Z}/3\mathbb{Z})^3$ not a class group of an imaginary quadratic number field ?
Current License: CC BY-SA 3.0
10 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jun 7, 2013 at 18:29 | comment | added | Socky | This list is the "primary" source for that previous answer, which suggests that I simply counted wrong on that occasion. In particular, it looks like I omitted 54 from that list.) | |
| Jun 7, 2013 at 9:58 | comment | added | Dietrich Burde | @ABC: thank you for your nice answer. The list has now $7$ groups (at stackexchange it has $6$ groups). | |
| Jun 6, 2013 at 18:43 | history | edited | Socky | CC BY-SA 3.0 |
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| Jun 6, 2013 at 2:18 | comment | added | v08ltu | This is wrong, the hard part of Watkins's thesis wasnot the range near $e^{5077h}$, but in the reduction of the necessary sieving. The paper saves a log factor over Montgomery/Weinberger, so the sievelimit is up to $2^{52}$ typically, better by 1000x from previous approaches (Arno et.al. and Wagner). The range from $2^{162}$ to $e^{260000000}$ or something (the paper has two different numbers at places) is mechanical (5 hours) from low-lying zeros of auxiliary $L$-functions; the range from $2^{52}$ to $2^{162}$ also goes this way, though not easily. He says 100000x harder to do $h\le 1000$. | |
| Jun 6, 2013 at 2:13 | comment | added | Noam D. Elkies | OK, I get it now. Yes, even the idoneal-number conjecture (the case of $A \oplus ({\bf Z} / 2{\bf Z})^n$ where $A$ is the trivial group) is still unproved; the Goldfeld/Gross-Zagier bound is too weak. | |
| Jun 5, 2013 at 22:44 | comment | added | Socky | Dear Noam, when I said "these unconditional results are ineffective", the word "these" was referring to the claim concerning class groups of the form $(\mathbf{Z}/3 \mathbf{Z})^n$ (i.e., finding all $n$ for which these groups occur). I've edited the post for clarity. | |
| Jun 5, 2013 at 22:41 | history | edited | Socky | CC BY-SA 3.0 |
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| Jun 5, 2013 at 21:26 | comment | added | Noam D. Elkies | Actually the unconditional results are effective thanks to the Goldfeld/Gross-Zagier bound, though the effective upper bounds are exponential in the size of the class group, which is what makes it nontrivial to provably list all cases of $h \leq 100$. | |
| Jun 5, 2013 at 21:16 | history | answered | Socky | CC BY-SA 3.0 |