Questions tagged [isogenies]
The isogenies tag has no usage guidance.
28
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When are two elliptic curves with zero j invariant isogenous?
Consider elliptic curves of the form $E_B\colon y^2=x^3+B$ for $B\in\mathbb Q$. These are exactly the elliptic curves with zero $j$-invariant. I would like to know when are two elliptic curves $E_B$ ...
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Isogenous elliptic curves in characteristic zero and in characteristic $p$
Assume two elliptic curves (with CM), $E_{1}$ and $E_{2}$, are isogenous over a field $K$ of characteristic zero. Are the following two statements true?
(a) Their $V_{p}$ modules are $G_{K}$-...
2
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0
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Question on a certain reduced isogeny of CM elliptic curves
My question has to do with some hypotheses showing up in a Lemma of Joseph Silverman's Advanced Topics book. Here is some of the set up:
Let $K$ be an imaginary quadratic field and $E/H$ an elliptic ...
2
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1
answer
200
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Computing explicit isogenies between elliptic curves over different kinds of fields
I have some questions about isogenies of elliptic curves in two settings:
1. Elliptic curves defined over the rationals.
1.1. Given two elliptic curves $E/\mathbb{Q}$ and $E'/\mathbb{Q}$ we can decide ...
4
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2
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Why all supersingular elliptic curves over $\bar{\mathbb{F}_p}$ are isogenous?
Lemma 3.2.1 in Baker, González-Jiménez, González, Poonen, "Finiteness theorems for modular curves of genus at least 2", Amer. J. Math. 127 (2005), 1325–1387.
enter image description here
I ...
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1
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Is there a separable isogeny between any two isogenous abelian varieties?
Question: Let $k$ be an algebraically closed field, and $A,B$ abelian varieties over $k$. Suppose there exists an isogeny $A\to B$. Does this imply existence of a separable isogeny $A\to B$?
Known ...
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0
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75
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Why is the kernel cyclic if and only if the walk does not backtrack?
I'm reading Mathematics of Isogeny Based Cryptography by Luca De Feo. At some point (pg. 32), he says
"A walk of length $e_A$ in the $l_A$-isogeny graph corresponds to a kernel of size $l_A^{e_A};...
2
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0
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41
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Proportion of edges within a fraction of the diameter
Let $G = (V,E)$ be a finite, connected $k$-regular graph of diameter $D$. Fixing some $\epsilon>0$ and letting $v$ be a (random) element of $V$, what proportion of the other edges are at most $\...
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How to construct explicitly defining polynomials of an morphism between smooth irreducible curves?
Let $\phi\!: C_1 \to C_2$ be a separable morphism of smooth irreducible curves embedded as projectively normal models by invertible sheaves $\mathcal{L}_1$ and $\mathcal{L}_2$ respectively. Theorem 4....
3
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Endomorphisms ring of complex abelian variety under isogenies
I’m trying to understand if over $\mathbb{C}$ two abelian varieties have the same complex multiplication if and only if they are isogenous. Is it true?
If it is true this means that if I consider $A$ ...
3
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154
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Lifting isogeny over étale cover
I am in the situation where I need to lift a particular isogeny over an étale cover, and I am not sure how to justify the existence of such a lift. I am trying to fill in the details of the proof of ...
2
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198
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Determining existence of a $p$-isogeny from $p|E(\mathbb{F}_{\ell})$
In Siksek's notes The modular approach to Diophantine equations he uses the following result:
Let $p$ be an odd prime. For an elliptic curve $E$ over $\mathbb{Q}$ if $p|E(\mathbb{F}_{\ell})$ then for ...
2
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1
answer
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Better way to compute elliptic curves over finite fields?
I've been using modular polynomials to compute isogeny vulcanoes with prime degree $l$ over finite fields $\mathbb{F}_p$, excluding cases containing the $j$-invariants $0$ and $1728$ or $j$-invariants ...
1
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1
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276
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Find basis for the set of torsion points E[m]
In paper "On the Cost of Computing Isogenies Between Supersingular Elliptic Curves" (source) reads
Let ${P, Q}$ be a basis for $E[2^{e/2}]$. Let $R_0 = [2^{e/2}−1]P , R_1 = [2^{e/2}−1]Q, ...
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1
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383
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Degree of morphisms and isogenies
$\renewcommand{\J}{\mathrm{Jac}} \renewcommand{\F}{\mathbb{F}}$
I am reading this paper by B. Gross, and there is something I don't understand on p. 945. Here is the context: fix a prime $p \equiv 3 \...
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297
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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
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1
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218
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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
136
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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 ...
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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_{...
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$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, ...
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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 \...
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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
...
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1
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224
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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 ...
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0
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86
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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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268
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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
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1
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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)+...
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2
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902
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Infinitely many curves with isogenous Jacobians
Let $g\geq 4$. Are there infinitely many compact genus $g$ Riemann surfaces with (mutually) isogenous Jacobians?
Does the situation change in positive characteristic?
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1
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Results and conjectures on bounds on degrees of isogenies
given an isogeny between two abelian varieties $\varphi: A\rightarrow B$ (everything definied over a number field $K$), we can factor $\varphi$ through a multiplication-by-$n$-endomorphism on $A$ and ...