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Questions tagged [elliptic-curves]

For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.

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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 $$\...
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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 $m$ ...
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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 ...
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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, \...
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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 ...
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3answers
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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(\...
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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 ...
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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 ...
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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 ...
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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 $p$ ...
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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. ...
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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 ...
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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: ...
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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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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)...
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2answers
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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 ...
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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 $\...
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$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 $...
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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.
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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\...
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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 \...
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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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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 ...
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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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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 ...
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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 ...
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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 $\mathbb F_p$ ...
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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{...
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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 ...
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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 $$ F(x,y)=ax^2+bxy+cy^2+dx+ey+f=0.$$ ...
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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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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 ...
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1answer
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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? ...
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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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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 $f$ ...
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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 ...
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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}, ...
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1answer
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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 ...
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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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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 ...
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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) \...
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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 ...
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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?
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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 ...
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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 $\...