Questions tagged [isogenies]

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Kato's Euler System for Isogenous Elliptic Curves

Let $E,E^\prime$ be elliptic curves over $\mathbb{Q}$ and also suppose they are $p$-isogenous. How are the Euler systems corresponding to the two isogenous elliptic curves related, if at all?
2
votes
1answer
95 views

Isogenies of degree 3 of elliptic curves with j-invariant 0

Let $E/\mathbb{Q}$ be an elliptic curve with $j(E)=0$. I.e., $E$ has Weierstrass equations $$ y^2 = x^3+ B$$ for some $B\in \mathbb{Q}$. $E$ has complex multiplication by $\mathcal{O}:= \mbox{ ring ...
2
votes
1answer
92 views

How to compute Weber polynomials efficiently?

Given $\tau\in H$ (up-half plane) and $q=e^{2\pi i \tau}$, Weber polynomail is defined as $$f(\tau)=q^{-\frac{1}{48}}\prod_{i=0}^{\infty}(1+q^{i-\frac{1}{2}}).$$ My question is: How can I compute a ...
1
vote
1answer
110 views

How to compute the Müller modular polynomials?

According to R.A.Kazmi's dissertation "Isogenies and Cryptography" (Page 22), given an isogeny degree $l$, the Müller modular polynomials are defined as $$G_l(x,y)=\sum_{r=0}^{l+1}\sum_{k=0}^{v}a_{...
1
vote
0answers
118 views

$F$-rational isogenies of CM Elliptic Curves

Let $F$ be a number field and $\mathcal{O}$ an order in an imaginary quadratic field $K$. Assume $K\subseteq F$. In Lang's Elliptic Functions, it is shown that over that there is a bijection between, ...
1
vote
1answer
280 views

Elliptic Curve, characteristic equation of Frobenius endomorphism relation to isogeny

Let E be an elliptic curve over $F_p$. Suppose that its j invarient is not supersingular and that $j\neq 0 $ or 1728. Then the modular polynomial $\Phi_l(j,T)$ has a zero $\tilde{\jmath} \in \...
7
votes
1answer
182 views

Commutation of endomorphisms of abelian varieties

Let $A$ be an abelian variety over an algebraically closed field $k$. Let $\phi:A\to A$ be an étale isogeny (over $k$). Suppose that the set $\cup_{r\geq 0}({\rm ker}\,\phi^{\circ r})(k)$ is ...
1
vote
1answer
130 views

What is the complexity of finding a distortion map on a supersingular elliptic curve?

Let $E$ be a supersingular elliptic curve which is defined over $\mathbb{F}_q$ and $P\in E$. Then there exist a distortion map with respect to $P$. I am looking for an algorithm which finds the map ...
1
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0answers
64 views

How can I find the specific endomorphism in a supersingular elliptic curve?

Let $E$ be a supersingular elliptic curve. As we know the endomorphism of $E$ is an order in a quaternion algebra. Suppose that $End(E)=\mathcal{O}$ and $a\in \mathcal{O}$. How can I find the ...
6
votes
0answers
194 views

Is there a prime degree endomorphism on supersingular elliptic curves?

Let $E$ be a supersingular elliptic curve which is defined over field $\mathbb{F}_{p^2}$ and $l$ be a prime such that $gcd(l,p)=1$. Is there an endomorphism $\phi\in End(E)$ such that $deg(\phi)=l$? ...
4
votes
1answer
602 views

What sort of ind-scheme is this?

It apparently follows from work of Velu (MathSciNet) that every isogeny between elliptic curves in (long) Weierstrass form over $k$ can be written in the form $$ \left(\frac{u(x)}{v(x)}, \frac{s_1(x)+...