Timeline for Binary quadratic forms attached to supersingular elliptic curves over F_p?
Current License: CC BY-SA 2.5
6 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Sep 28, 2010 at 16:00 | comment | added | BCnrd | The tensoring method works over $\overline{\mathbf{F}}_p$ using maximal orders in the quaternion algebra (and invertible modules over such orders), and you then need an "integral" refinement of Skolem-Noether applied to the subfield $K$. The 2-to-1 will come from the fact that the CM-structure by $R$ amounts to extra data, namely a choice of isomorphism of $R$ with the endomorphism ring over %\mathbf{F}_p$, and these are two such choices. | |
| Sep 28, 2010 at 13:48 | comment | added | Tommaso Centeleghe | BCnrd, thanks for your comment. Can we explain why the number of supersingular $j$-invariants lying in the prime field is actually half as much as the number of $F_p$-isom. classes of objects of S? | |
| Sep 28, 2010 at 13:16 | comment | added | BCnrd | For #1, no natural base pt. Better formulation is not a bijection but rather a principal homogeneous space structure. For any invertible $R$-module $M$ and $E$ in $S$, the functor on $\mathbf{F}_p$-algebras defined by $A \rightsquigarrow M \otimes_R E(A)$ is represented by an elliptic curve denoted $M \otimes E$, and the natural map $M \otimes_E {\rm{Hom}}_R(E,E') \rightarrow {\rm{Hom}}_R(E,M \otimes E')$ between invertible $R$-modules is an isomorphism (pf: use Tate isogeny theorem at all primes). This shows that the tensoring operation makes $S$ a principal homogenous space for Pic($R$). | |
| Sep 28, 2010 at 11:51 | history | edited | Tommaso Centeleghe | CC BY-SA 2.5 |
edited title
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| Sep 28, 2010 at 10:45 | history | asked | Tommaso Centeleghe | CC BY-SA 2.5 |