Questions tagged [elliptic-curves]
For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.
0
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0answers
8 views
Proof of the decomposition of an isogeny into a separable part and a frobenius.
Lemma:
Let $\alpha:E_1\longrightarrow E_2 $ be an inseparable isogeny of elliptic
curves defined over a field $k$ of characteristic $p>0$. Then $\alpha$ can be
written in the form
$$\...
1
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0answers
27 views
Level Lowering obstructions
For the generalized Fermat equation:
$a^p + b^q = c^r$
with $p,q,r\ge3$ and $\gcd(a,b,c)=1$
one can construct the Frey Curve:
$y^2=x(x-a^p)(x+b^q)$
which is semi-stable for those primes $m$ ...
5
votes
0answers
80 views
Algebraic forms on an elliptic curve
On an elliptic curve defined by the equation,
$$E:y^2=x^3+a x +b$$
The algebraic form $dx/y$ is defined on the elliptic curve and it is a non-vanishing section of the (trivial) canonical bundle. From ...
1
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0answers
41 views
Certain Galois cohomology computation
Let $L/\mathbb{Q}$ be a Galois extension of degree $p$ and $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $p$ be a fixed prime (of good ordinary reduction if required). We use $L_\infty, \...
1
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0answers
25 views
Torsion of elliptic curves and abelian extensions
Let $L/K$ be an abelian $p$-extension of number fields and $E$ be an elliptic curve over $\Bbb Q$.
If $E[p](K)=0$, does it follow that $E[p](L)=0$ ? The converse is obviously true, but I don't have ...
0
votes
3answers
49 views
Sage code to check independence of rational points on elliptic curve
Suppose I have three rational points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ on certain elliptic curve. Then they are linearly independent if and only if the determinant of matrix $(<P_i,P_j>)_{i,j}$ is ...
2
votes
1answer
72 views
Proof of weak form of Mazur's theorem by Faltings' theorem
Mazur's theorem about elliptic curve claims that there are only finitely many possibilities for the torsion subgroup of a Modell-Weil group of an elliptic curve over $\mathbb{Q}$, and he also gives a ...
1
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1answer
16 views
When do Heegner points lead to a basis points?
For a rank 1 elliptic curve, a rational point can be obtained from Heegner points. When is this rational point a basis point?
If sometimes additional work is required to obtain a basis point, is ...
2
votes
1answer
37 views
How to know the j-invariant of the modular elliptic curve from the modular form?
How do people compute the $j$-invariant of an elliptic curve $E$ over $\mathbb Q$ from the associated modular form $f=\sum_{n=1}^{\infty} a_nq^n$? In other words, how to compute (at least giving some ...
1
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0answers
33 views
Generating random (torsion) point on elliptic curve efficiently
I am looking for an efficient way to generate a random point on an elliptic curve over a finite field, $E(\mathbb{K})$.
I know that you can pick a random $x$, compute e.g. in Weierstrass coordinates ...
1
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1answer
44 views
Elliptic Curve has Canonical Bundle $K_E = \mathcal{O}_E$
Let $E$ be an elliptic curve over a field $k$. I'm looking for a proof that for the canonic divisor $K_E$ we have $K_E = \mathcal{O}_E$? RR says $h^0(K_E) = h^0(\mathcal{O}_E)+deg(K_E) +g-1= h^0(\...
2
votes
2answers
64 views
Derive Group Law on Elliptic Curve with Riemann Roch
Consider $E$ be an elliptic curve and $k$ a field. I read that one way to show that $E(k)$ has an abelian group structure can be derived using Riemann Roch. Could anybody explain how it concretely ...
1
vote
0answers
43 views
Is $p$ an anomalous prime?
Let $E$ be an elliptic curve defined over the number field $K$ and $p$ be a rational prime such that for all $v\mid p$, $E$ has good ordinary reduction. If $E[p]\subseteq E(K)$, can we conclude $p$ is ...
1
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0answers
63 views
Group Law Elliptic Curve with Divisor Class Group
Let $E$ be an elliptic curve and $k$ a field. It is well know that $E(k)$ has an (additive) group structure and indeed there are a lot of sources describing what geometrically there is going on.
The ...
3
votes
1answer
52 views
Obtaining elliptic curve solution in integers from solution in quadratic field
Are there any methods or known tricks to obtain elliptic curve solutions in the integers from a solution in a quadratic field?
Starting with a Mordell curve:
$$y^2 = x^3 + k$$
Consider an integer $p$ ...
1
vote
0answers
32 views
Question about rational points on elliptics curves.
I have the following question about this lemma,
Where $A,B\in\mathbb{F}$. I tried to prove that $\mbox{Norm}_{R/F}(\alpha-\beta\theta)=f^{hom}(\alpha,\beta))$, but I don't see, why this is correct. ...
1
vote
1answer
51 views
When does a Mordell curve have non-trivial torsion?
Is there a known simple criteria for when a Mordell curve has non-trivial torsion?
A comment in this question:
Family of elliptic curves with trivial torsion
Suggests that
$$y^2 = x^3 + k$$
has ...
0
votes
0answers
20 views
how to find the generator of elliptic curve using matlab
my question is that
my Matlab program for elliptic curve generated all points which satisfy the elliptic curve equation
now how to find the generator which generates all the points
example:
...
0
votes
1answer
44 views
Non-singular elliptic curve parametrization
It is known that some singular elliptic curves can be expressed with parametric equations.
For example :
$y^2=x^3$ can be parametrized with $x=t^2$ and $y=t^3$
$y^2=x^3+x^2$ can be parametrized with $...
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0answers
35 views
Singular points of a Weierstrass equation over a perfect field are rational
Let $k$ be a perfect field, $E$ a curve defined by $f = y^2 + a_1 xy + a_3 y - x^3 - a_2 x^2 - a_4 x - a_6$ over $k$, and let $P = (x, y) \in E(\overline{k})$ be a singular point.
Then does $P \in E(k)...
5
votes
2answers
78 views
Solving $y^2 = 4x^3 - p$, with prime $p \equiv 7 (\text{mod } 8)$
I'm trying to find integer solutions to equations of the form
$$y^2 = 4x^3 - p \tag{1}$$
where $p$ is a prime and $p \equiv 7 (\text{mod } 8)$.
1) Is there a simple way to check if solutions do not ...
0
votes
0answers
23 views
Divisor over ellitptic curves
I struggle to prove the following theorem :
Let $E$ be an elliptic curve over a field $K$. Let $D=\sum n_p P$ be a divisor on $E$. Then $D \sim 0$ if and only if $\sum [n_p]P=\mathcal{O}$ where $\...
6
votes
2answers
96 views
$p$-adic values of rational points on elliptic curves
The following question came up naturally whilst studying diophantine equations: given an elliptic curve $E$ of the form $Y^2 + aY = X^3 +bX^2 + cX + d$ defined over $\mathbb{Q}$, consider the subset $...
4
votes
2answers
44 views
What are some sources to learn projective geometry?
I am looking for notes, books etc. for learning projective geometry for getting started with elliptic curves.
1
vote
1answer
86 views
Finding a kernel generator of the dual isogeny
Let's say we have an isogeny $\phi:E\to E/\ker\phi$ between two elliptic curves over some finite field. Let's also assume we know $\ker\phi$ explicitly, or at least a generator of it, e.g. $\langle A\...
5
votes
0answers
152 views
What are the positive integer solutions to $x^2-x+1 = y^3$?
The only solutions that I know of till now are $(x,y) = (1,1) \space , (19,7)$. We can note that:
$$x^2-x+1 = y^3 \implies (2x-1)^2 = 4y^3-3$$
Thus, if odd prime $p \mid y$, then $(2x-1)^2 \equiv -3 \...
2
votes
1answer
54 views
Weierstrass $\wp$-function defines a map from the torus to an elliptic curve. Why is it injective?
For $L$ a lattice in $\mathbb C$, the Weierstrass $\wp$-function is the meromorphic function
$$\wp(z) = \frac{1}{z^2} + \sum\limits_{0 \neq \lambda \in L}\frac{1}{(z-\lambda)^2} - \frac{1}{\lambda^2}$...
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0answers
78 views
Is there a substitution which transforms every Fermat curve into an elliptic curve?
A Fermat Curve of degree $n$ is the set of solutions to $x^n+y^n=z^n$, $x,y,z\in \mathbb R$. In this question, the OP provides a substitution which relates a Fermat Curve of degree $n=3,4$ to two ...
2
votes
0answers
66 views
Are there counterexamples of isogeny elliptic curves with non-isomorphic integral Tate modules?
Let $K$ be a field and $G_K$ be its absolute Galois group. Let $E_1,E_2$ be two elliptic curves over $K$. Assume that there exists an isogeny $f:E_1\rightarrow E_2$. Let $p$ be a prime number. Then $f$...
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0answers
23 views
Have these Generalised Fermat Curves been studied before?
While trying to solve a problem in number theory, I recently came across the concept of Fermat curves. This is the set of points in the complex projective plane, defined by
$X^n+Y^n=Z^n$. In the ...
0
votes
1answer
58 views
Scalar Point Multiplication for Elliptic Curve Diffie-Hellman Key Exchange
I am trying to understand elliptic curve Diffie-Hellman key exchange and here is a book example which I don't understand. Given values of $G=(2,2)$ and I should calculate $203(2,2)$, which actually ...
4
votes
2answers
59 views
Finding elliptic curves achieving the upper and lower bounds of Hasse's Interval
I always thought that Hasse's bound is sharp (at least for elliptic curves). In other words I always thought that given a prime number $p$, I can find two elliptic curves $E_1,E_2$ over $\mathbb F_p$ ...
0
votes
1answer
52 views
Elliptic Curve scalar multiplication on $\mathbb{R}$
I have an elliptic curve $ y^2=x^3+109x^2+224x$ and a point $P(-100;260)$ on it. And I need to find point $2P$. I took a formulas $$x_2=\left(\frac{ax_1-b}{y_1}\right)^2 -a+x_1$$ and $$y_2=-y_1+\frac{...
1
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0answers
34 views
Finitely many zeros and poles for a function in a function field of a smooth curve
Let $\bar{K}$ be a perfect field and let $f \in \bar{K}(C)$ be a nonzero function in the function field of $C$, a smooth curve (projective variety of dimension 1). I'm trying to understand why (even ...
1
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0answers
17 views
Am I right in this proof of a criterion for the nonsingularity of a conic curve?
$\newcommand{\C}{\mathcal{C}}$
This is an exercise in Silverman and Tate's Rational Points on Elliptic Curves:
Let $\C$ be the conic given by the equation
$$ F(x,y)=ax^2+bxy+cy^2+dx+ey+f=0.$$
...
1
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0answers
34 views
Existence of rational parametrization of elliptic curves
I read somewhere that it is not possible to have rational parametrization for elliptic curves. So there is possibility of the existence of rational parametrization for a 'part' of elliptic curve or ...
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0answers
42 views
What are some prerequisites for studying Elliptic Curves over $\mathbb Q$?
Suppose you had an undergraduate friend who's looking to take an introductory course on Elliptic Curves over $\mathbb Q$, in the context of Number Theory. The difficulty is around the level of ...
2
votes
1answer
26 views
Distribution of torsion subgroups of elliptic curve
By Mazur's theorem, we know the list of possible torsion subgroups of elliptic curves over $\mathbb{Q}$. Now, if we order them by height, can we compute the distribution of each possible groups? ...
0
votes
1answer
27 views
elliptic curve over GF(2^3)
This is $f(x) = x^3+x+3$ over $GF\left(2^3\right)$
How to know numbers of points on this equation?
How to find those points?
Is it an irreducible polynomial?
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1answer
20 views
Modular parametrization from equality of $L$-functions
In the literature, an elliptic curve $E/\mathbb{Q}$ is defined to be modular in two different ways
1) if there exists a nonconstant morphism $X_0(N) \to E$,
2) if there exists a modular form $f$ ...
0
votes
0answers
53 views
Trivial solutions of a diophantine equation
Let $K$ be an odd degree number field. Consider the Diophantine equation:
$$
X^4 + bY^4 =Z^2
$$
where $b\neq 0$.
Say we know that the above equation has only trivial roots in $K$ (for some fixed ...
2
votes
1answer
52 views
No elliptic curves over $\mathbb{Q}$ with everywhere good reduction
I'm trying to prove that there aren't any elliptic curves $E$ over $\mathbb{Q}$ with everywhere good reduction. I first suppose that $\Delta = \pm 1$ and am trying to reduce the quantities for $c_{4}, ...
0
votes
1answer
14 views
Elliptic curves: twists and homogeneous spaces
I have a slight confusion with these 2 concepts. It is my understanding that twists of an elliptic curve $E/k$ are elliptic curves $E'/k$ with $j(E)=j(E')$.
Then in Chapter X of The Arithmetic of ...
3
votes
1answer
59 views
What is p-adic logarithmic map of an elliptic curve? How to compute it?
I was reading about elliptic curves in this pdf. Page 55 of the pdf states that if number of points on elliptic curve #$E(F_p) = p$, then there exists a p-adic logarithmic map that homomorphically ...
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0answers
15 views
Singular points on a variety. $V: 4x^2y^2=(x^2+y^2)^3$
So I was studying some stuff about projective varieties from the book "The Arithmetic of Elliptic Curves" from Silverman, and there is this exercise at the very beginning, about determine the singular ...
0
votes
1answer
36 views
Use of frobenius map of an elliptic curve
I was reading about elliptic curves from https://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf. Page No. 44 defines Frobenius map. It defines the frobenius map as $f(x,y)=(x^p,y^p) \...
2
votes
1answer
61 views
Independence of points on Elliptic curve
Let $P_1(x_1,y_1),P_2(x_2,y_2)...P_n(x_n,y_n)$ be $n$ rational points on given Elliptic curve. How do we prove they are independent? Are there any theorems/results/algorithms/softwares to prove their ...
1
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0answers
69 views
Derivation of the Frey curve for $a^2+b^6=c^p$
At the bottom of page $5$ in Bennet Chen's paper, they give the Frey curve for $a^2+b^6=c^p$ as:
$Y^2=X^3−3(5b^3+ 4ai)bX+ 2(11b^6+ 14ib^3a−2a^2)$
Can anyone tell me how they derived this Frey curve?
1
vote
0answers
36 views
Calculus of order in an elliptic curve
This is my first time in this website.
I'm a student in master 2 ans I'm following a course about elliptic curves.
And I'm didn't really understand how to calculer the order of a function at a point P ...
2
votes
0answers
62 views
Over Silverman’s differential forms on curves
I’m reading Silverman’s Arithmetic of Ellipctic curves. In II.4 he gives an definition of differential form as the set of $dx$ for $x$ in $\overline{K}(C)$. Taking the complex circle $x^2+y^2=1$ in $\...
