Timeline for Atkin-Lehner involution and class number
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10 events
| when toggle format | what | by | license | comment | |
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| Jul 2, 2011 at 14:28 | comment | added | Tommaso Centeleghe | @expmat. If you are interested in fixed points of AL involution on $X_0(p)$ then you should classify elliptic curves over C, up to isomorphism, that admit an endomorphism whose square is -p. If you think about the correspondence between elliptic curves up to isom. and lattices up to homothety then what you are doing is classifying lattices inside C that are stable under multiplication by $\sqrt{-p}$. This leads to the class number interpretation. You will have to consider the class number of all quadratic imaginary orders containing $\sqrt{-p}$. If p\equiv 3 mod 4 then there are two of them. | |
| Jul 2, 2011 at 14:00 | answer | added | David E Speyer | timeline score: 7 | |
| Jul 1, 2011 at 20:27 | answer | added | Matt Young | timeline score: 5 | |
| Jul 1, 2011 at 12:02 | comment | added | A. Pacetti | Regarding the exposition on definite quaternion algebras, you could take a look at Pizer's articles, and I suggest to look at Gross article (Heights and special values of L-series) where he first talks about the optimal embeddings. The idea is that the number of optimal embeddings is roughly speaking the number of bilateral ideals times the class number of the order you are embedding (in case there is such an ideal). So the formula written below gives you $2^t$, where $t$ is the number of prime divisors of ND (the level) which is exactly the number of bilateral ideals (if ND is square free). | |
| Jun 30, 2011 at 23:30 | comment | added | stankewicz | Dear expmat, This is exactly what I'm referring to in my answer. For $X_0(p)$, we must have $D = 1$, $N=p$, $m=p$ and so if $p\equiv 1\bmod 4$, $\mathbf{Z}[\sqrt{-p}]$ is maximal and the number of fixed points is $h(-4p)$. If $p\equiv 3\bmod 4$, an embedding of $\mathbf{Z}[\sqrt{-p}]$ must be optimal either for $\mathbf{Z}[\sqrt{-p}]$ or for $\mathbf{Z}[\dfrac{1 + \sqrt{-p}}{2}]$ so you have to count optimal embeddings of both. Hence the number of fixed points is $h(-p) + h(-4p)$. | |
| Jun 30, 2011 at 15:02 | comment | added | expmat | Hi Tommaso, this might be a silly question but why is \sum = h(-4p) ? Or why is \sum = h(-p) + h(-4p) (in the other case)? | |
| Jun 29, 2011 at 21:27 | history | edited | expmat | CC BY-SA 3.0 |
deleted 29 characters in body
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| Jun 29, 2011 at 21:23 | answer | added | stankewicz | timeline score: 11 | |
| Jun 29, 2011 at 20:43 | comment | added | Tommaso Centeleghe | This might be not very elegant on my side, but you can take a look to section 2 of: www1.iwr.uni-heidelberg.de/groups/arith-geom/centeleghe/up.pdf The idea is to interpret the Atkin-Lehner involution on $X_0(p)$ in terms of moduli of elliptic curves, and then use the relationship between fractional ideals of $Q(\sqrt{-p})$ and certain CM elliptic curves. | |
| Jun 29, 2011 at 19:41 | history | asked | expmat | CC BY-SA 3.0 |