Timeline for How does this calculation of Siegel make sense?
Current License: CC BY-SA 4.0
24 events
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| S Apr 4, 2020 at 0:02 | history | bounty ended | CommunityBot | ||
| S Apr 4, 2020 at 0:02 | history | notice removed | CommunityBot | ||
| Apr 3, 2020 at 4:26 | comment | added | GH from MO | In the definition on p.182 using the Kronecker symbol, $\mathfrak{p}$ and $\mathfrak{c}$ are meant to be ideals of $\mathcal{O}_k$, and their norms are meant to be $[\mathcal{O}_k:\mathfrak{p}]$ and $[\mathcal{O}_k:\mathfrak{c}]$. So I think you need to consider the product $\mathfrak{c}_k\mathcal{O_k}$, which is a fractional ideal of $\mathcal{O_k}$: write it as a quotient of two ideals of $\mathcal{O}_k$, and then apply the definition on p.182 using the Kronecker symbol. | |
| Mar 28, 2020 at 17:20 | answer | added | Franz Lemmermeyer | timeline score: 1 | |
| Mar 27, 2020 at 9:21 | comment | added | Franz Lemmermeyer | ${\mathbb Z}-modules$ in ${\mathcal O}_k$ of course - sorry. I'll go through the stuff carefully until tomorrow. | |
| S Mar 26, 2020 at 22:52 | history | bounty started | Shimrod | ||
| S Mar 26, 2020 at 22:52 | history | notice added | Shimrod | Authoritative reference needed | |
| Mar 26, 2020 at 21:12 | comment | added | Shimrod | @FranzLemmermeyer How they can be $\mathcal O_K$-modules? | |
| Mar 26, 2020 at 21:08 | comment | added | Franz Lemmermeyer | You're right. But they are ${\mathcal O}_k$-modules, and these have norm $f$. Siegel's "Ringideal" is defined as in Hecke's lecture "Analysis und Zahlentheorie" (edited by Roquette), p. 125ff. | |
| Mar 26, 2020 at 20:46 | comment | added | Shimrod | @FranzLemmermeyer Please see Update 2 above. | |
| Mar 26, 2020 at 20:39 | history | edited | Shimrod | CC BY-SA 4.0 |
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| Mar 26, 2020 at 20:11 | comment | added | Shimrod | @FranzLemmermeyer That is not true. We actually have $\mathcal O =\lbrace \alpha \in K\colon \alpha \mathfrak c_k\subset c_k\rbrace$. | |
| Mar 26, 2020 at 20:01 | comment | added | Franz Lemmermeyer | These are ideals in ${\mathcal O}_k$, and of course we are talking about the ordinary norms of these ordinary ideals. You seem to want to compute the norm of ideals in a ring to which they do not belong. | |
| Mar 26, 2020 at 19:28 | comment | added | Shimrod | @FranzLemmermeyer But $\mathfrak c_K \not \subset \mathcal O$, so what do you mean by the residue class ring in this case? One cannot use $k-\omega$ to reduce elements of $\mathcal O$, because $k-\omega \not \in \mathcal O$. I have updated the question to give a more detailed computation of the norm. | |
| Mar 26, 2020 at 19:26 | history | edited | Shimrod | CC BY-SA 4.0 |
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| Mar 26, 2020 at 14:29 | comment | added | Franz Lemmermeyer | Anyway, using the definition of the norm of an ideal as the cardinality of its residue class ring, use the element $k - \omega$ to reduce any algebraic integer to an ordinary integer; using the element $f$ of the ideal we find that the residue class ring is simply ${\mathbb Z}/f{\mathbb Z}$, so the norm if the ideal is $f$. | |
| Mar 26, 2020 at 14:21 | comment | added | Franz Lemmermeyer | I don't understand what you are doing - what references are you using? | |
| Mar 26, 2020 at 13:11 | comment | added | Shimrod | @FranzLemmermeyer Then what is wrong with my computation of the norm (see above)? | |
| Mar 26, 2020 at 13:10 | history | edited | Shimrod | CC BY-SA 4.0 |
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| Mar 26, 2020 at 7:48 | comment | added | Franz Lemmermeyer | The norm of an ideal with basis $[m, a+n\omega]$ is $mn$. | |
| Mar 25, 2020 at 17:03 | comment | added | Shimrod | @FranzLemmermeyer We have $\mathfrak c_k\subset \mathcal O_K$, but $\mathfrak c_k\not\subset \mathcal O $. Could you please show how you calculated the norm? | |
| Mar 25, 2020 at 16:56 | comment | added | Franz Lemmermeyer | Your ideals are integral ideals, and their norms seem to be $f$, not $1$. | |
| Mar 24, 2020 at 20:45 | history | edited | Shimrod | CC BY-SA 4.0 |
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| Mar 24, 2020 at 17:19 | history | asked | Shimrod | CC BY-SA 4.0 |