Timeline for Primes and $x^2+2y^2+4z^2$
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Nov 12, 2012 at 14:33 | vote | accept | Joël | ||
| Oct 10, 2012 at 6:00 | answer | added | paul Monsky | timeline score: 5 | |
| Jun 27, 2012 at 22:55 | comment | added | Will Jagy | So when you suggested that you and Paul talk, you meant actual talking. It makes sense now. | |
| Jun 27, 2012 at 22:07 | comment | added | Joël | Will, I am Joël Bellaïche, from Brandeis University. I happen to know Paul very well, as he is an emeritus at the same university. | |
| Jun 27, 2012 at 19:08 | comment | added | Will Jagy | Joel, I had no trouble finding an email address for Paul at ams.org/cml and emailed him a year or two ago. You could do the same. Paul may not know who you are or how to contact you; I certainly do not. | |
| Jun 27, 2012 at 11:55 | vote | accept | Joël | ||
| Nov 12, 2012 at 14:33 | |||||
| Jun 27, 2012 at 11:54 | comment | added | Joël | There is a natural Galois representation $G_{\Q,2} \rightarrow \Gl_2(A)$, and by describing it (its image has cardinal 32), one can prove the proposition by Cebotarev. There is indeed an involution appearing in the computation, and I wonder if it has anything to do with yours. | |
| Jun 27, 2012 at 11:50 | comment | added | Joël | Dear Paul, we have to talk! What lead me to that question is that I was studying the action of the Hecke operators on the forms $\Delta^7$ in characteristic $2$. As you know, its $n$-th coefficient is $1$ iff $n$ is represented exactly as above, i.e. by $x^2+2y^2+4z^2$ an odd number of times (with $x,y,z$ odd positive). The form $\Delta^7$ is not stable by Hecke (even in char 2), but the space $\Delta,\delta^3,\Delta^5,\Delta^7$ is, and the Hecke algebra on this space is isomorphic to $A=\mathbb{F}_2[x,y]/(x^2,y^2)$ (where $x=T_3$, and $y=T_5$). ... | |
| Jun 27, 2012 at 10:56 | comment | added | paul Monsky | if he had any idea what the involution should be,and whether it could be constructed in an elementary fashion, and he gave me the answer I quoted to you in my previous comment. He also gave me a reference to your question which, chased back, led to the Hasse article I mentioned to Will. I eventually discovered what the involution should be, but was only able to describe it using composition of binary quadratic forms and Gauss's theorem. (See my answer mentioned by Will). | |
| Jun 27, 2012 at 10:45 | comment | added | paul Monsky | I also was led to your question in a context that I think won't surprise you. Suppose that p is 7 mod 8 and prime. Let S be the set of unordered triples of squares summing to 2p. It's easy to see that the number whose parity you're asking for is just the cardinality of S. In studying identities involving characteristic 2 theta series I was led to guess the existence of a certain involution on S having 1 or no fixed points according as p was 7 or 15 mod 16. Ira Gessel made extensive calculations supporting my guess. At that point I asked Franz Lemmermeyer (to be continued) | |
| Jun 27, 2012 at 3:21 | comment | added | paul Monsky | Joel: According to Lemmermeyer all known proofs use Gauss's result relating the number of representations of 2p as a sum of 3 squares to a class number. So an elementary proof is perhaps unlikely. | |
| Jun 26, 2012 at 19:15 | answer | added | Will Jagy | timeline score: 13 | |
| Jun 26, 2012 at 17:12 | history | asked | Joël | CC BY-SA 3.0 |