Questions tagged [isogenies]

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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 $E_B$ ...
わくわく's user avatar
1 vote
0 answers
87 views

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 $G_{K}$-...
EAg's user avatar
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2 votes
0 answers
69 views

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 ...
matt stokes's user avatar
2 votes
1 answer
198 views

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 ...
did's user avatar
  • 595
4 votes
2 answers
400 views

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 ...
HaomengXu's user avatar
6 votes
1 answer
266 views

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 ...
Jonathan Love's user avatar
1 vote
0 answers
75 views

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};...
Manuel Bravi's user avatar
2 votes
0 answers
41 views

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 $\...
ILoveIsos's user avatar
1 vote
0 answers
74 views

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....
Dimitri Koshelev's user avatar
3 votes
0 answers
181 views

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 $A$ ...
Martina Monti's user avatar
3 votes
0 answers
154 views

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 ...
Martin Skilleter's user avatar
2 votes
0 answers
195 views

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 ...
Μάρκος Καραμέρης's user avatar
2 votes
1 answer
120 views

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 ...
José's user avatar
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1 vote
1 answer
275 views

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, ...
nam_ngn's user avatar
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2 votes
1 answer
381 views

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 \...
Watson's user avatar
  • 1,702
5 votes
1 answer
294 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?
debanjana's user avatar
  • 1,161
2 votes
1 answer
217 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 ...
Rdrr's user avatar
  • 881
2 votes
1 answer
136 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 ...
Licheng Wang's user avatar
1 vote
1 answer
137 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_{...
Licheng Wang's user avatar
1 vote
0 answers
163 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, ...
Rdrr's user avatar
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1 vote
1 answer
698 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 \...
user111264's user avatar
7 votes
1 answer
228 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 ...
Damian Rössler's user avatar
1 vote
1 answer
224 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 ...
somayeh didari's user avatar
1 vote
0 answers
86 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 ...
somayeh didari's user avatar
6 votes
0 answers
268 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 $deg(\phi)=l$? ...
somayeh didari's user avatar
4 votes
1 answer
630 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)+...
David Roberts's user avatar
  • 33.8k
18 votes
2 answers
902 views

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?
Raju's user avatar
  • 790
7 votes
1 answer
647 views

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 ...
Stefan Keil's user avatar