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.
2,280
questions
2
votes
1answer
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 ...
1
vote
0answers
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.
...
2
votes
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, ...
-2
votes
1answer
26 views
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 ...
2
votes
0answers
92 views
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 $E$ ...
2
votes
0answers
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 ...
4
votes
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 ...
2
votes
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 ...
1
vote
1answer
35 views
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 ...
0
votes
1answer
73 views
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 ...
-2
votes
0answers
20 views
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-...
3
votes
1answer
44 views
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, ...
2
votes
0answers
26 views
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-...
4
votes
2answers
86 views
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 ...
0
votes
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 ...
1
vote
1answer
46 views
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 ...
1
vote
0answers
54 views
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 ...
3
votes
0answers
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 ...
2
votes
1answer
61 views
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{-...
0
votes
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 $...
0
votes
1answer
64 views
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 $y^2=x^3+ax+b$...
1
vote
0answers
75 views
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'...
0
votes
0answers
107 views
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
$$
L_p(E,s) = \begin{cases}(1-a_pp^...
3
votes
1answer
56 views
Determinant of Frobenius action on the exterior power on cohomology
Given an elliptic curve $E$, the Frobenius action $\mathrm{Frob}$, and $\mathrm{det}(1-\mathrm{Frob}_E T | H^1(E))$, how do we find an expression for:
$$\mathrm{det}(1-\mathrm{Frob}_X T | \wedge^2 (H^...
1
vote
0answers
25 views
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 ...
3
votes
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^...
2
votes
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
$$\frac{4}{a}\int_{0}^a\sqrt{(b^2-a^2)x^2+a^4\over ({a^2-x^...
8
votes
2answers
118 views
How to determine if $n = 53467$ is prime or not using Elliptic curves?
I want to use Elliptic curves modulo $n$ to find out if $n$ is prime or not.
Now, I know that $n = 53467 = 127\times 421$, but how do I find this out using Elliptic curves?
I tried factorization ...
1
vote
2answers
43 views
Equal divisors of Weil-pairing
I try to understand the approach of Silverman to the Weil pairing.
In The Arithmetic Of Elliptic Curves he defines the Weil pairing in section III.8 with the help of two divisors:
\begin{equation*}
...
1
vote
2answers
78 views
Reference request: Sheaf for the Zariski topology
In my course notes page 62 I've read the following: let $(E,0)$ be an elliptic curve over an arbitrary scheme $S$, then $U\rightarrow\ker(\,0^*_U:\operatorname{Pic}(E_U)\rightarrow \operatorname{Pic}(...
1
vote
1answer
35 views
Is every elliptic curve isogeny over $\mathbb{F}_p$ separable?
I know that a isogeny $\varphi: E \rightarrow E^\prime$ has degree equal to its kernel if and only if the isogeny is separable.
I want to know if this always holds for elliptic curves over $\mathbb{F}...
2
votes
1answer
71 views
Elliptic curves and scheme theory
Recall that an elliptic curve over a field $k$ i.e a proper smooth connected curve of genus $1$ equipped with a distinguished $k$-rational point, i'll be really grateful for any help in understanding ...
1
vote
1answer
58 views
Lemma 4.2 Hartshorne IV
The question I have is with regards to the proof given in Hartshorne IV lemma 4.2. Let $X$ be an elliptic curve and $P,Q\in X$ be closed points. One can show that the linear system $|P+Q|$ has ...
0
votes
1answer
76 views
Transforming Elliptic Curve to Hessian Form
In the The Hessian Form of an Elliptic Curve, (NP Smart - ā2001), they began with the elliptic curve
$$E': y^2 +xy = x^3 + x^2 + b$$ defined over the $\operatorname{GF}(2^{192})$ with the polynomial ...
4
votes
0answers
33 views
good reduction for CM elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve with Complex Multiplication by the ring of integers of an imaginary quadratic field $K$. Let $p$ be an odd prime of good supersingular reduction. We know by ...
0
votes
0answers
28 views
Preimage of a point via elliptic curve isogeny
I have an elliptic curve $E$ (supersingular) defined over $\mathbb{F}_{p^2}$ with a large torsion group $E[m]$. I know a degree $m$ isogeny $\alpha:E\to E'$ over $\mathbb{F}_{p^2}$ as well as its dual ...
2
votes
0answers
40 views
Euler Systems and Colemanās Conjecture
Iām trying to work on Colemanās conjecture for abelian extensions of imaginary quadratic fields. Iāve read most papers by Seo regarding circular distributions. However, Iām a still confused about what ...
1
vote
0answers
47 views
Cardinality of general polynomial equations over finite fields
Let $E: Q(y)=P(x)$ be an equation over a finite field $\mathbb F_p$ given any prime $p$, and any polynomials $Q(y)$ and $P(x)$ (polynomials in $y$ and $x$ respectively). Is there a general approach to ...
0
votes
0answers
22 views
Pollard kangaroo for discrete logarithm
I already made this post in the crypto community. Because nobody could answer me, I wanted to ask here: I want to understand Pollards kangaroo attack on discrete logarithms. There are two important ...
1
vote
0answers
73 views
What is the relation between Frey's Curve and Fermat's Last Theorem?
Can anyone give an overview showing how the elliptic curve $E (a, b) := y^2=x(x-a)(x-b)$ is associated with the solution of $a^n+b^n=c^n$?
In 1969 Hellegouarch performed the elliptic curves $E (a, b)$,...
18
votes
4answers
581 views
Prove: $\int_0^2 \frac{dx}{\sqrt{1+x^3}}=\frac{\Gamma\left(\frac{1}{6}\right)\Gamma\left(\frac{1}{3}\right)}{6\Gamma\left(\frac{1}{2}\right)}$
Prove:
$$
\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)}
$$
First obvious sub is $t = 1 +...
0
votes
0answers
41 views
What is an Isogeny in Complex Numbers?
Can someone give a simple explanation what it is meant to be an isogeny in complex numbers? I know that it is a homomorphism between two elliptic curves of the form $\mathbb{C}/L$ where $L$ is a ...
1
vote
1answer
51 views
transformation of singular quartics into Weierstrass form
I need a birational transformation of singular quartics into weierstrass form. the quartics are of genus one and have the following form:
$$(x^2 + c) (y^2 + i) + k = 0$$
where $c,i,k$ are given ...
1
vote
0answers
29 views
Proportion of “nice” twisted Edwards curve at given finite field size
I am trying to generate a "nice" Montgomery elliptic curve on a finite field $\mathbb{F}_q$ such that its birational equivalent twisted Edwards curve has nice properties (for cryptographic ...
1
vote
0answers
51 views
Group law of elliptic curves on finite fields
If the underlying field is $\mathbb{C}$, there is a bijective map between a given elliptic curve and $\mathbb{C} \mathbin{/} \Lambda$, where $\Lambda$ is a lattice uniquely determined by the elliptic ...
3
votes
2answers
91 views
Elliptic curve vs. Elliptic function
I am a bit unsure about the relation between elliptic curves and elliptic functions.
I believe that there is a one to one correspondence between elliptic curves and Weierstrass's elliptic functions (...
4
votes
1answer
62 views
The Weil pairing for elliptic curves over the $\mathbb{C}$
Let $E = \mathbb{C} / (\mathbb{Z} + \tau \mathbb{Z})$ be an elliptic curve over $\mathbb{C}$.
Then how can I show that $e_n(1/N, \tau/N) = \exp(2 \pi i / n)$?
If we can show it, then the fundamental ...
0
votes
2answers
70 views
Mordell equation with prime-power constant
(This question is somewhat related, but is different from this earlier question.)
I am interested in a specific case of the Mordell equation:
$$E : y^2 = x^3 + k$$
where $k=q^t$, for some prime $q \...
1
vote
0answers
33 views
Proof of Deuring's Correspondence
Let $E$ be a supersingular elliptic curve over $\overline{\mathbb{F}_q}$ where $q=p^n$ and $p$ is prime. Then $B:=\text{End}(E) \otimes \mathbb{Q}$ is a unique quaternion algebra over $\mathbb{Q}$ ...
2
votes
0answers
44 views
how to calculate rank of elliptic curve
I study about elliptic curve and I saw
Rank of the elliptic curve $y^2=x^3+px$
but i can't understand how he can calculate
$$2M^4ā2pe^4=N^2$$
$$4M^4āpe^4=N^2$$
how he revived to this formulas? what ...
