Timeline for Correspondence between binary quadratic representations and proper ideals of quadratic number fields
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Jan 29, 2021 at 18:43 | vote | accept | Melanka | ||
| Jan 29, 2021 at 18:43 | vote | accept | Melanka | ||
| Jan 29, 2021 at 18:43 | |||||
| Jan 29, 2021 at 17:02 | vote | accept | Melanka | ||
| Jan 29, 2021 at 18:43 | |||||
| Jan 28, 2021 at 23:35 | comment | added | GH from MO | In my answer below, I gave a natural one-to-one correspondence between $R(d,n)$ and $I(d,n)$. It is independent of my previous comment, which was based on Siegel's mass formula. | |
| Jan 28, 2021 at 23:34 | answer | added | GH from MO | timeline score: 2 | |
| Jan 28, 2021 at 22:36 | comment | added | GH from MO | By Siegel's mass formula, $|R(d,n)|$ is a product of local densities given explicitly in Siegel's original paper (Annals of Mathematics, 1935). It equals $\sum_{m\mid n}\chi_d(m)=|I(d,n)|$. About an explicit correspondence: define an ordering on both $R(d,n)$ and $I(d,n)$, and map the $k$-th element of $R(d,n)$ to the $k$-th element of $I(d,n)$. | |
| Jan 28, 2021 at 22:30 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title, formatting
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| Jan 28, 2021 at 22:30 | answer | added | Matt E | timeline score: 3 | |
| Jan 28, 2021 at 22:23 | history | edited | Melanka | CC BY-SA 4.0 |
added 21 characters in body
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| Jan 28, 2021 at 22:22 | comment | added | Melanka | @AllenHatcher Yes. Sorry I have missed it in the post. I'll edit it. Thank you for pointing it out. | |
| Jan 28, 2021 at 22:07 | comment | added | Melanka | @GHfromMO My question is what the actual correspondence is? Can we define an explicit map between $R(d, n)$ and $I(d, n)$? Also, I have a second question, whether this relationship of cardinalities holds in the real quadratic case also? | |
| Jan 28, 2021 at 20:35 | comment | added | GH from MO | By definition, two sets have the same cardinality if and only if there is a one-to-one correspondence between them. So it is not clear what your question is. | |
| Jan 28, 2021 at 19:57 | comment | added | Will Jagy | There is a recent book that follows binary quadratic forms and quadratic number fields all through the book, Lehman bookstore.ams.org/dol-52 There is a more concentrated discussion in Cohen, A Course in Computational Algebraic Number Theory, especially section 5.2 in pages 225-230. In brief, a simple mapping from forms to ideals is one-to-one for positive forms, also indefinite forms when the principal form integrally represents $-1,$ and two-to-one otherwise (so that forms $f$ and $-f$ are distinct). | |
| Jan 28, 2021 at 19:10 | comment | added | Melanka | @PeterHumphries Sorry if I have missed it. I went through the book, especially chapter 7. But I didn't find what I was looking for. The exposition is about the correspondence between the narrow class group and the equivalence classes of forms - which is well known. Not quite what I was looking for. I am looking for the correspondence between ideals of a fixed norm $n > 0$ and the integral representations $(x, y)$ of $n$ by the forms. | |
| Jan 28, 2021 at 17:42 | comment | added | Peter Humphries | Yes, this is indeed the case and is well known. A nice exposition is written up in chapter 7 of "Algebraic Theory of Quadratic Numbers" by Mak Trifkovic. | |
| Jan 28, 2021 at 17:06 | history | asked | Melanka | CC BY-SA 4.0 |