Mathematics

Questions tagged [isogeny]

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

In algebraic geometry, a homomorphism is called $\phi \colon A \rightarrow B$ $\phi \colon A \rightarrow B$ of Abelian varieties $A$ and $B$ is an isogeny if $\phi$ is surjective and has a finite kernel. If there is an isogeny $\phi \colon A \rightarrow B$, the Abelian varieties $A$ and $B$ are called isogenic. In particular, isogenies are "rational" mappings between elliptic curves that respect the group law. Further reference-Wikipedia

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Isogenies of elliptic curves by kernel inclusion

In Silverman, The Arithmetic of Elliptic Curves, GTM 106, Corollary 4.11 says: If we have three elliptic curves $E_1$ , $E_2$ , $E_3$ with non-constant isogenies $f:E_1 \to E_2$ and $g:E_1 \to E_3$ and $f$ separable and $\ker f \subseteq \ker g$ ,…
Jürgen Böhm
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Isogeny $SL_2(\mathbb{Q}_p)\times SL_2(\mathbb{Q}_p) \rightarrow SO_{2,2}(\mathbb{Q}_p)$

For a field $k$ of characteristic $\neq2$ , consider the special orthogonal group $$SO_{r,s}(k):=\{g\in SL_{r+s}(k): g^TQg=Q\},\quad\text{ where }Q:=\begin{pmatrix}I_r & \\ & -I_s\end{pmatrix}.$$ As shown in Paul Garrett's notes , there exists a…
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Derivative of dual isogeny is pullback on $H^1$

Let $X$ and $Y$ be elliptic curves (over an algebraically closed field, but no assumptions on the characteristic) with Jacobians $J_X$ and $J_Y$ respectively. Suppose $f:X\to Y$ is an isogeny, with dual isogeny $\widehat{f}:J_Y\to J_X$ . How can I…
Hank Scorpio
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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 don't understand why "all supersingular elliptic curves over…
HaomengXu
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Finding supersingular elliptic curves over $\Bbb F_p$ with rational endomorphisms of degree 3 using Deuring correspondence

I found this question on this website, which gives a code that almost does what I want. Basically, I am looking for supersingular elliptic curves $E/\mathbb F_p$ that have endomorphisms of degree 3, 6 or (worst case) 9, such that those…
Clemence
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Implementation of a constructive algorithm for Deuring's correspondence

Let $E_0$ be a supersingular elliptic curve. By Deuring's correspondence, $\text{End}(E_0)\simeq \mathcal{O}_0$ is a maximal order in the quaternion algebra $B_{p,\infty}$ over $\mathbb{Q}$ ramified at $p$ and $\infty$ . When $p=17$ , $B_{p,\infty}…
Andy
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Regulator and isogeny of elliptic curve

Let $f: E \to E'$ be an isogeny of degree $n$ . Is the following correct? Regarding the height pairing of elliptic curves that appears when defining the regulator: $\langle f(a), f(b) \rangle = \deg f \langle a, b \rangle$ holds. When $f$ is…
Poitou-Tate
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How to find endomorphism ring of an isogeneous elliptic curve

I have a supersingular elliptic curve $E$ with a known endomorphism ring $\operatorname{End}(E)$ . I'd like to find an isogeny $\varphi: E \rightarrow E'$ , and by Deuring correspondence I know the corresponding ideal $I_{\varphi}$ in quaternion…
tuner007
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What does 'Isogeny is determined by its kernel up to automorphisms' mean?

This is question from Silverman's Advanced topics in the arithmetic of elliptic curves , p. 106. Let $E_1$ and $E_2$ be elliptic curves. What does isogeny $φ:E_1→E_2$ is determined by its kernel, at least up to an automorphism of $E_1$ and $E_2$…
Poitou-Tate
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About calculating isogeny between two elliptic curves

I'm trying to understand Vélu formulas for calculating isogenies. I took an elliptic curve $E: y^2 = x^3 + 3x + 5$ over $GF(7)$ . So I've got the following points on this curve: \begin{equation} \{\mathcal{O}, (1,3); (1,4); (4,2); (4,5); (6,1);…
tuner007
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Morphisms that are injective and surjective but not isomorphisms

I'm reading Silverman's Arithmetic of Elliptic Curves and read that isogenies are group homomorphisms and if $\phi:E_1\longrightarrow E_2$ is a non-zero isogeny between elliptic curves,its kernel $Ker\phi=\phi^{-1}(O)$ where $O$ is the point at…
user631874
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How to calculate an endomorphism ring for a supersingular elliptic curve

I've read a few books and papers about isogeny-based cryptography and its mathematic but didn't get the idea how to find the endomorphism ring of a supersingular elliptic curve. I know how to do it for an ordinary curve but not for a supersingular…
tuner007
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Question on isogenies of degree d.

I am trying to understand the following question (Proposition 5.12. in ABELIAN VARIETIES , Bas Edixhoven, Gerard van der Geer, and Ben Moonen) If $f: X \to Y$ is an isogeny of degree $d$ between abelian varieties then there exists an isogeny $g: Y…
Khainq
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How to compute the $j$-invariant corresponding to a given maximal order in $B_{p,\infty}$

Let $B_{p,\infty}$ is the rational quaternion algebra ramified at $p$ and $\infty$ . By Deuring's correspondence, there is a one to one correspondence between maximal orders in $B_{p, \infty}$ up to isomorphism and supersingular $j$ -invariants in…
Andy
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supersingular elliptic curve group structure

I am trying to understand the basic idea behind supersingular isogeny cryptography . With very little knowledge about the group theory and elliptic curve, I find it very hard to thoroughly understand the underlying math of this protocol. Then I read…
Alex Su
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