Timeline for Homomorphism ring of two elliptic curves with the same supersingular reduction.
Current License: CC BY-SA 3.0
7 events
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| May 25, 2012 at 11:07 | comment | added | Dror Speiser | This is an elementary exercise. Remember that you're not calculating the norm of arbitrary elements, but of generators. My favorite solution involves the terms quadratic forms and fundamental domain. I will point out that it is crucial the discriminant is prime. If after trying a bit more you still haven't solved this problem, I suggest you post it on math.stackexchange. | |
| May 25, 2012 at 9:46 | comment | added | MPI | Thank you for the reponse. I agree with what you said, but indeed the point that is blocking me is precisely how to determine the degree of the elements of this fractional ideal. I know that this degree is going to be N$(x)$/N$(\mathfrak{a})$ for every $x\in \mathfrak{a}$ ($\mathfrak{a}$ being the Hom space.) but since I don't explicitely know $\mathfrak{a}$, I don't know how to calculate the degree of its elemtents. | |
| May 25, 2012 at 8:59 | comment | added | Dror Speiser | * When $l=3$ the class number is one and the unique j-invariant is 0, so that $P_3(X)=X$, and indeed $X=X-12^3\pmod{3}$. | |
| May 25, 2012 at 8:47 | comment | added | Dror Speiser | Finally, elements of $\mathbb{Z}[\frac{1+F}{2},I]$ which aren't in $\mathbb{Z}[I]$, quite clearly have degree at least $\frac{l+1}{4}$, the smallest element being $\frac{1+F}{2}$ with that degree. | |
| May 25, 2012 at 8:44 | comment | added | Dror Speiser | By choosing an embedding of $K_l$ into $\mathbb{C}$, then looking at the parameterising tori and their lattices, and after taking a suitable homothety, the lattices are ideals in $\mathcal{O}_l$, which cannot lie in the same class (in the class group) since they have different j-invariants. The Hom space is now a fractional ideal, that cannot be principal. That the ideal is spanned by two elements of reduced norm less than $\frac{l+1}{4}$, for $l\ge 7$ is an exercise. For $l=3$, there should be a sperate computation. | |
| May 24, 2012 at 23:18 | history | edited | MPI |
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| May 24, 2012 at 20:32 | history | asked | MPI | CC BY-SA 3.0 |