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Results tagged with elliptic-curves
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user 39304
An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry. Note also the tag arithmetic-geometry as well as some related tags such as rational-points, abelian-varieties, heights. Please do not use this tag for questions related to ellipses; instead use conic-sections.
4
votes
Accepted
Why can I take the quotient of a relative elliptic curve by a finite locally free subgroup?
A reference for 1. is https://stacks.math.columbia.edu/tag/07S7
We can then answer question 2. like this:
The quotient morphism $E\to E/C$ is faithfully flat (this is a part of the conclusion of the l …
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6
votes
Accepted
Lifting a splitting of an Abelian variety to characteristic 0
$\newcommand{\cA}{\mathcal{A}}\newcommand{\cB}{\mathcal{B}}\newcommand{\bZ}{\mathbb{Z}}$No, that does not imply that $\cA$ splits over $R$. In fact, if $\cA_1=\cA\times_R R/p$ is isogenous to a produc …
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12
votes
Accepted
Question on the Sato-Tate conjecture
No. If $E_p$ is a supersingular elliptic curve and $p>3$ then trace of Frobenius on $E_p$ is zero, so $\theta_E(p)=\pi/2$.
By a result of Elkies any elliptic curve over $\mathbb{Q}$ has supersingular …
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6
votes
1
answer
366
views
Lifting of Frobenius on torsors over abelian varieties
This is related to my previous question Assume that $A$ is an abelian variety over a field $k$ of characteristic $p$, $\mathcal{L}$ is a line bundle on $A$. Assume that $A$ is ordinary and $\mathcal{L …
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4
votes
1
answer
560
views
Lifting of Frobenius on semi-abelian varieties
Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume tha …
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