Questions tagged [isogeny]

For questions about morphisms of algebraic groups (group varieties) that are surjective and have a finite kernel.

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$\ell$-isogeny graphs over finite fields

I have a lot of questions about the following paragraph (reference), most of which are probably relatively easy to answer. What even is an $\mathbb F_q$-rational morphism? Is it a morphism over $\...
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Is every isogeny over $\mathbb Q$ separable?

I am reading a proof of a simplified version of the weak Mordell-Weil theorem, where we only consider elliptic curves over $\mathbb Q$. Now, in the proof, they mention some (non-constant) isogeny, and ...
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Proof of Theorem 6.1(b) in Silverman's AEC

I'm learning about the construction of the dual isogeny in Silverman's Arithmetic of Elliptic Curves. In particular, I'm reading the proof for Theorem 6.1 from Chapter III. There is a (probably easy) ...
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54 views

In the process of counting the number of supersinguar elliptic curve

Let $φ:E→E$ be $q$-th Frobenius map and $φ'$ be it's dual isogeny. And we define $a$ as $a=1-deg(1-φ)+deg(φ)$(what we call trace of frobenius). Then, My question is , Why $[a]=φ+φ'$ ? These questions ...
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What does it mean for an isogeny to be of “smooth degree?”

I keep seeing references to isogenies of smooth degrees, but I can't seem to find a definition. Any help would be appreciated!
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Question about the degree of the composition of two isogenies

I'm reading a paper about isogeny based cryptography and I came across a sentence that seems to imply that for separable isogenies $\phi$ and $\psi$, the following holds: deg$(\phi \circ \psi)=$ deg$(\...
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64 views

Dual isogeny of purely inseparable isogeny is not always purely inseparable

Let $φ$ be purely inseparable isogeny of elliptic curves. Then, dual isogeny of $φ$ is always purely inseparable? Background Super singular elliptic curve over a field of characteristic $p$ is defined ...
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Does the first coordinate of a morphism (isogeny) of elliptic curves only depend on the $x$-coordinate?

Let $E,E'$ be two elliptic curves over a field $k$ of characteristic $\neq 2, 3$. Assume that $E,E'$ are given by short Weierstrass equations, and let $f : E \to E'$ be a morphism given by $$f(x,y) = (...
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Isogeny between elliptic curves is always bijective?

Isogeny is morphism between elliptic curves which keeps base point. Then, is every isogeny between elliptic curves bijective? Let $E1$ and $E2$ be isogenous elliptic curves defined over Fq. Then #E1(...
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How to confirm $\phi(F_1(x,y))=F_2(\phi(x),\phi(y))$,where $F_1$ and $F_2$ are formal group law of elliptic curve $E_1$, $E_2$.

This question is from Silverman's 'the arithmetic of elliptic curves',$p134$. Let $K$ be a field of characteristic $p > 0$, let $E_1/ K$ and $ E_2/K$ be elliptic curves, and let $\phi : E_1 \to E_2$...