Questions tagged [elliptic-curves]

Elliptic curves are objects of algebraic geometry met in somewhat advanced parts of number theory. They also appear in applications to cryptography. Use the tag, if this applies. Questions on ellipses should be tagged [conic-sections] instead.

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40 views

Elliptic curves with complex multiplication by an order $\mathcal{O}$

I am reading a paper about elliptic curves with CM, and since I'm new to this I'm having troubles understanding the essence of some things. Let $E$ be an elliptic curve with complex multiplication by ...
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15 views

Homogenous space of elliptic curve E/$\Bbb Q$

Homogenous space of elliptic curve E/$\Bbb Q$ always has $\Bbb Qp$ rational points for every prime $p$? And why? I know homegenous space has $\Bbb Q$ rational points only if it is not a trivial class. ...
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1answer
32 views

Elliptic curve: Type of reduction mod 2, how can I show the curve has a cusp?

I want to know what type of reduction the curve $E : y^2 = x^3 + 7x$ has at $p=2$. From online search, I obtain that it has additive/cuspidal reduction. But this disagrees with my own computation, ...
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1answer
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Do the all points in subgroup of an elliptic curve with prime order have the same order? [closed]

A subgroup $G$ of elliptic curve can constructed with point $P$ with order $q$ by $G=\langle P\rangle $. Now, if $q$ is prime, do the all points in subgroup $G$ (except infinity point) have same order ...
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Tate-Shafarevich group and Hasse principle

I' m looking for a proof of the fact that the Hasse local-global principle holds for an homogeneous space of elliptic curve $E $ defined over $\Bbb Q$ if and only if the Tate-Shafarevich group of $E$ ...
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55 views

Elliptic Curve over $\mathbb Z_p$ isomorphic to a subgroup of that Elliptic Curve over reals?

Is every Elliptic Curve group over finite field $\mathbb F_p$ isomorphic to some subgroup of the Elliptic Curve group over the reals having the same equation? Update: this is at last sometime the ...
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2answers
81 views

Computing the genus of $y^2=x(x^2-1)$ using 1-forms

I'm trying to compute the genus of the projective curve $C:=V(Y^2Z-X(X^2-Z^2))\subset\Bbb{P}^2_\Bbb{C}$ explicitly using differential forms. I know beforehand that this is an elliptic curve, so the ...
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1answer
34 views

Smallest solutions to show that a number is congruent

On wikipedia, under the 'smallest solutions' section, there is a table of congruent numbers, where for each congruent number $n$ they put a triple $(a, b, c)$ with $a^2 + b^2 = c^2$ and $n = ab/2$. It ...
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1answer
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Naive question about computational cost of doubling and points addition over elliptic curve

When we talk about the computational cost of doubling and points addition we usually give a result in terms of field multiplication M and field squaring S. For example, here we can read points ...
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1answer
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Group Structure of Rational Points a Supersingular Elliptic Curve over $\mathbb{F}_{p^2}$ with $p=2^a3^b-1$?

I'm looking at this Jao-De Feo-Plut paper and this expository paper by Craig Costello. Both claim that for a prime $p$ of the form $p=2^a3^b-1$, the entire $(p+1)$-torsion of a supersingular elliptic ...
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Elliptic Curve Check [closed]

I am working with a 256 bit elliptic curve. I have data which contains the following: Scalar integer, (x,y) pair. I have been able to check the scalar vs the total number of points (N) and the Y-...
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1answer
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Doubly periodic meromorphic function with prescribed poles and zeros

The field of the meromorphic functions on a complex torus $\mathbb{C} \mathbin{/} \Lambda$ is $\mathbb{C}(\wp, \wp')$, where $\wp$ is the weierstrass p-function to the lattice $\Lambda$. Furthermore, ...
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Is there an analogous of quadratic residue over elliptic curves?

I've faced this question some days ago looking at this paper Kleptography: Using Cryptography Against Cryptography In this article, the writers discuss about a kleptographic attack against Diffie-...
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2answers
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Rational functions on elliptic curves

Recall that an elliptic curve over a field $k$ i.e a proper smooth connected curve of genus $1$ equipped with a distinguished $k$-rational point, I'll be really grateful for any help in understanding ...
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1answer
30 views

Proofs of abelian group axioms for elliptic curve over finite fields + tangent-chord group law

I'm looking for the proofs of the abelian group axioms for elliptic curve over finite fields (e.g. integers mod p) with the tangent-chord group law (i.e. the "standard" group law for ...
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1answer
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Defining the action of elliptic curve $E$ on a principal homogeneous space of $E$

I recently learnt (using Milne's book about elliptic curves) about principal homogeneous space for elliptic curves. For completeness I state the definition here: Let $E$ be an elliptic curve over a ...
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Elliptic curve confusion

I am confused by some lecture I heard. It was stated that the elliptic curve over a finite field $\mathbb{F}_{q^k}$ would be a subset of the elliptic curve over the finite field $\mathbb{F}_{q}$. But ...
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97 views

Elliptic curve over finite non-prime fields

I have a problem understanding elliptic curves over finite non-prime fields, i.e., $E / {\Bbb F}_{p^k}$, where $p$ is prime and $k > 1$. Let's say we have a defining Weierstrass equation of the ...
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1answer
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Question about supersingular elliptic curves

It is a fact that given $D>0$ there exist only finitely many isomorphism classes of elliptic curves over $\overline{\mathbb{Q}}$ with complex multiplication by $O_D=\mathbb{Z}[\frac{1}{2}(D+\sqrt{-...
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1answer
40 views

Degree of a isogeny

I have this exercise: Let be $\phi:(E_1,O_1)\rightarrow (E_2,O_2)$ a isogeny. Let be $\phi=(r_1(x),r_2(x)y)$ its standard affine representation and write $r_1(x)=\frac{p(x)}{q(x)}$ with polynomials $...
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Understanding the compactification of elliptic curves

I'm having trouble understanding the following example from my lecture notes: Example: Let $E$ be an elliptic curve over a field $k$ given by the compactification of the affine equation $y^2=x^3+ax+b$...
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Function field for projective variety

Let $V$ be a projective variety over a field $k$. For any affine patch we can define the function field of $V$ to be $K(V)=K(V\cap\mathbb{A}^n)$, and these are all canonically isomorphic. In Silverman'...
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Definition of $L$-function of elliptic curves

$L-$function of elliptic curves is Dirichlet series and defined to be $$ L(E,s) = \sum_{n\ge 1}\frac{a_n}{n^s} = \prod_p L_p(E,s), $$where the Euler factor at $p$ is $$ L_p(E,s) = \begin{cases}(1-a_pp^...
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1answer
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Determinant of Frobenius action on the exterior power on cohomology

Given an elliptic curve $E$, the Frobenius action $\mathrm{Frob}$, and $\mathrm{det}(1-\mathrm{Frob}_E T | H^1(E))$, how do we find an expression for: $$\mathrm{det}(1-\mathrm{Frob}_X T | \wedge^2 (H^...
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Confusion about the definition of regular maps

So I'm reading Milne's book on elliptic curves, which is freely available here. On page 38, he defines a regular map: Let $k$ be a perfect field. A regular map $\varphi : C_{g_1} \to C_{g_2}$ of ...
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2answers
68 views

Right triangle with rational sides and area = 1 equivalent to n = 3 case of Fermat's Last Theorem

I watched a talk by Andrew Wiles in which he spoke about the proof of Fermat's Last Theorem, and he said something that has puzzled me. He mentioned that the $n=3$ case of FLT (i.e. proving $a^3+b^3=c^...
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1answer
64 views

Elliptic integrals: do they actually exist?

On my very first question on this website (Perimeter of an ellipse solution and elliptic integrals.) I ended up asking for an expression for $$\frac{4}{a}\int_{0}^a\sqrt{(b^2-a^2)x^2+a^4\over ({a^2-x^...
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2answers
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How to determine if $n = 53467$ is prime or not using Elliptic curves?

I want to use Elliptic curves modulo $n$ to find out if $n$ is prime or not. Now, I know that $n = 53467 = 127\times 421$, but how do I find this out using Elliptic curves? I tried factorization ...
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2answers
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Equal divisors of Weil-pairing

I try to understand the approach of Silverman to the Weil pairing. In The Arithmetic Of Elliptic Curves he defines the Weil pairing in section III.8 with the help of two divisors: \begin{equation*} ...
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2answers
78 views

Reference request: Sheaf for the Zariski topology

In my course notes page 62 I've read the following: let $(E,0)$ be an elliptic curve over an arbitrary scheme $S$, then $U\rightarrow\ker(\,0^*_U:\operatorname{Pic}(E_U)\rightarrow \operatorname{Pic}(...
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1answer
35 views

Is every elliptic curve isogeny over $\mathbb{F}_p$ separable?

I know that a isogeny $\varphi: E \rightarrow E^\prime$ has degree equal to its kernel if and only if the isogeny is separable. I want to know if this always holds for elliptic curves over $\mathbb{F}...
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1answer
71 views

Elliptic curves and scheme theory

Recall that an elliptic curve over a field $k$ i.e a proper smooth connected curve of genus $1$ equipped with a distinguished $k$-rational point, i'll be really grateful for any help in understanding ...
1
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1answer
58 views

Lemma 4.2 Hartshorne IV

The question I have is with regards to the proof given in Hartshorne IV lemma 4.2. Let $X$ be an elliptic curve and $P,Q\in X$ be closed points. One can show that the linear system $|P+Q|$ has ...
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1answer
76 views

Transforming Elliptic Curve to Hessian Form

In the The Hessian Form of an Elliptic Curve, (NP Smart - ‎2001), they began with the elliptic curve $$E': y^2 +xy = x^3 + x^2 + b$$ defined over the $\operatorname{GF}(2^{192})$ with the polynomial ...
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good reduction for CM elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve with Complex Multiplication by the ring of integers of an imaginary quadratic field $K$. Let $p$ be an odd prime of good supersingular reduction. We know by ...
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28 views

Preimage of a point via elliptic curve isogeny

I have an elliptic curve $E$ (supersingular) defined over $\mathbb{F}_{p^2}$ with a large torsion group $E[m]$. I know a degree $m$ isogeny $\alpha:E\to E'$ over $\mathbb{F}_{p^2}$ as well as its dual ...
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Euler Systems and Coleman’s Conjecture

I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what ...
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Cardinality of general polynomial equations over finite fields

Let $E: Q(y)=P(x)$ be an equation over a finite field $\mathbb F_p$ given any prime $p$, and any polynomials $Q(y)$ and $P(x)$ (polynomials in $y$ and $x$ respectively). Is there a general approach to ...
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22 views

Pollard kangaroo for discrete logarithm

I already made this post in the crypto community. Because nobody could answer me, I wanted to ask here: I want to understand Pollards kangaroo attack on discrete logarithms. There are two important ...
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What is the relation between Frey's Curve and Fermat's Last Theorem?

Can anyone give an overview showing how the elliptic curve $E (a, b) := y^2=x(x-a)(x-b)$ is associated with the solution of $a^n+b^n=c^n$? In 1969 Hellegouarch performed the elliptic curves $E (a, b)$,...
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Prove: $\int_0^2 \frac{dx}{\sqrt{1+x^3}}=\frac{\Gamma\left(\frac{1}{6}\right)\Gamma\left(\frac{1}{3}\right)}{6\Gamma\left(\frac{1}{2}\right)}$

Prove: $$ \int_{0}^{2}\frac{\mathrm{d}x}{\,\sqrt{\,{1 + x^{3}}\,}\,} = \frac{\Gamma\left(\,{1/6}\,\right) \Gamma\left(\,{1/3}\,\right)}{6\,\Gamma\left(\,{1/2}\,\right)} $$ First obvious sub is $t = 1 +...
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What is an Isogeny in Complex Numbers?

Can someone give a simple explanation what it is meant to be an isogeny in complex numbers? I know that it is a homomorphism between two elliptic curves of the form $\mathbb{C}/L$ where $L$ is a ...
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1answer
51 views

transformation of singular quartics into Weierstrass form

I need a birational transformation of singular quartics into weierstrass form. the quartics are of genus one and have the following form: $$(x^2 + c) (y^2 + i) + k = 0$$ where $c,i,k$ are given ...
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0answers
29 views

Proportion of “nice” twisted Edwards curve at given finite field size

I am trying to generate a "nice" Montgomery elliptic curve on a finite field $\mathbb{F}_q$ such that its birational equivalent twisted Edwards curve has nice properties (for cryptographic ...
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0answers
51 views

Group law of elliptic curves on finite fields

If the underlying field is $\mathbb{C}$, there is a bijective map between a given elliptic curve and $\mathbb{C} \mathbin{/} \Lambda$, where $\Lambda$ is a lattice uniquely determined by the elliptic ...
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2answers
91 views

Elliptic curve vs. Elliptic function

I am a bit unsure about the relation between elliptic curves and elliptic functions. I believe that there is a one to one correspondence between elliptic curves and Weierstrass's elliptic functions (...
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1answer
62 views

The Weil pairing for elliptic curves over the $\mathbb{C}$

Let $E = \mathbb{C} / (\mathbb{Z} + \tau \mathbb{Z})$ be an elliptic curve over $\mathbb{C}$. Then how can I show that $e_n(1/N, \tau/N) = \exp(2 \pi i / n)$? If we can show it, then the fundamental ...
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2answers
70 views

Mordell equation with prime-power constant

(This question is somewhat related, but is different from this earlier question.) I am interested in a specific case of the Mordell equation: $$E : y^2 = x^3 + k$$ where $k=q^t$, for some prime $q \...
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0answers
33 views

Proof of Deuring's Correspondence

Let $E$ be a supersingular elliptic curve over $\overline{\mathbb{F}_q}$ where $q=p^n$ and $p$ is prime. Then $B:=\text{End}(E) \otimes \mathbb{Q}$ is a unique quaternion algebra over $\mathbb{Q}$ ...
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0answers
44 views

how to calculate rank of elliptic curve

I study about elliptic curve and I saw Rank of the elliptic curve $y^2=x^3+px$ but i can't understand how he can calculate $$2M^4−2pe^4=N^2$$ $$4M^4−pe^4=N^2$$ how he revived to this formulas? what ...

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