Timeline for What is the meaning of the ideal class group?
Current License: CC BY-SA 4.0
17 events
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| Jan 29, 2022 at 21:10 | audit | First answers | |||
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| Jan 26, 2022 at 21:13 | audit | First answers | |||
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| Jan 19, 2022 at 10:01 | audit | First answers | |||
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| Jan 10, 2022 at 1:30 | vote | accept | Adithya Chakravarthy | ||
| Jan 8, 2022 at 14:26 | audit | First answers | |||
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| Jan 6, 2022 at 17:19 | audit | First answers | |||
| Jan 6, 2022 at 18:02 | |||||
| Jan 5, 2022 at 18:35 | comment | added | Bill Dubuque | Yes, I agree that is well worth emphasis. My only quibble was that the first paragraph might mislead readers into thinking no interesting work has been done on the topics raised by the OP. | |
| Jan 5, 2022 at 18:30 | comment | added | KCd | @BillDubuque That's why I had included the long paragraph near the end related to $S$-integers. From the way the class group of $\mathcal O_{K,S}$ is a quotient group of the class group of $\mathcal O_K$ we can enlarge $\mathcal O_K$ in a finite way (i.e., using a finite set of places $S$) to some $\mathcal O_{K,S}$ that is a UFD by forcing the class group of $\mathcal O_{K,S}$ to be trivial. That is really useful in arithmetic geometry. | |
| Jan 5, 2022 at 18:22 | comment | added | Bill Dubuque | No doubt "measure the failure" is a subjective term, but I don't think what you have in mind universally agrees with "what others have in mind" by such, and I see no good reason to restrict its denotation so narrowly. Indeed, the OP's questions go beyond that, as does the research I linked to. | |
| Jan 5, 2022 at 18:18 | comment | added | KCd | @BillDubuque I edited my first paragraph, but I don't think what I wrote is highly misleading because the line of research you describe is not what people have in mind when they use the phrase "measure the failure". They're just thinking about $h = 1$ vs. $h > 1$. When people say a class group measures the failure or unique factorization or the Tate-Shafarevich group or Brauer-Manin obstruction measures the failure of the local-global principle, the only common thread of all this "measure the failure" behavior is distinguishing when the group or obstruction is trivial and nontrivial. | |
| Jan 5, 2022 at 18:05 | history | edited | KCd | CC BY-SA 4.0 |
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| Jan 5, 2022 at 13:03 | comment | added | Bill Dubuque | The claim in the first paragraph is highly misleading given the line of research linked to in my comment on the question. | |
| Jan 4, 2022 at 9:54 | audit | First answers | |||
| Jan 4, 2022 at 9:57 | |||||
| Jan 4, 2022 at 4:25 | history | edited | KCd | CC BY-SA 4.0 |
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| Jan 4, 2022 at 1:29 | history | edited | KCd | CC BY-SA 4.0 |
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| Jan 4, 2022 at 1:23 | history | edited | KCd | CC BY-SA 4.0 |
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| Jan 4, 2022 at 1:18 | history | answered | KCd | CC BY-SA 4.0 |