Timeline for Supersingular elliptic curves
Current License: CC BY-SA 2.5
2 events
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| Jun 24, 2010 at 10:53 | comment | added | Boyarsky | Tate's isogeny theorem over finite $k$ is valid for all primes: $\mathbb{Z}_{\ell} \otimes {\rm{End}}_k(A)={\rm{End}}_k(A[\ell^{\infty}])$. Increase $k$ so ${\rm{End}}_k(A)={\rm{End}}_ {\overline{k}}(A)$. For $A$ a supersingular elliptic curve, ${\rm{End}}_k(A[\ell^{\infty}])$ then has rank 4 but is the Galois-invariants in the $\overline{k}$-endomorphism algebra of $A[\ell^{\infty}]$. The latter is ${\rm{M}}_2(\mathbb{Z}_{\ell})$ for all $\ell$ (verify using Dieudonne theory for $\ell=p$), so the quotient by its Galois-invariants is finite yet torsion-free! QED Conceptual, yes. Simple...? | |
| Oct 15, 2009 at 13:59 | history | answered | known google | CC BY-SA 2.5 |