# Questions tagged [isogenies]

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11
questions

**5**

votes

**1**answer

187 views

### 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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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**

vote

**0**answers

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**

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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

**1**answer

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)+...