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

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Suggestions for readings; Elliptic curves over function fields

I would love to know some good refercences about Elliptic curves over function fields. Especially in view with Mordell-Weil's Theorem. I am already familiar with the main proof of Mordell's theorem in ...
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0answers
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About differential 1-form associated to a ternary cubic form

I've read the following statement which I can't prove after a while, so if someone here could give me just a hint then I would be very happy! Suppose $k$ is a number field. To a ternary cubic form, ...
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Complex Atlas for Elliptic Curves over $\mathbb{C}$

I know that every elliptic curve over $\mathbb{C}$ is isomophic to a torus $\mathbb{C}/\Lambda$ in the sense of Riemann Surfaces, moreover $E(\Lambda)$ as topological subspace of ...
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1answer
61 views

Divisor on curve of genus 2

Let $C$ be a smooth, projective curve of genus 2. I want to show that there exists a non-constant rational function $f \in k(C)$ having divisor of the form $$(f) = P_1 + P_2 - P_3 - P_4 $$for points ...
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1answer
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Computing the dual of frobenius endomorphism

There is an exercise in my course Elliptic Curves and I am not sure if I am doing it right. The question is as follows: Let $E$ be the elliptic curve over $\mathbb{F}_2$ given by $Y^2+Y=X^3$. (a) ...
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0answers
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What is the right dual isogeny?

I have a question regarding dual isogenies. I read an example in Silverman's book about elliptic curves and am wondering something about this example. We have $\zeta$ as a primitive cube root of ...
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1answer
57 views

Commutativity of “extension” and “taking the radical” of ideals

Let $K$ be a field (not necessarily algebraically closed) and $\overline{K}$ its algebraic closure. By $K[\text{X}]$, I mean $K[X_1,...,X_n]$. Is it true that the operations of "extension" and ...
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1answer
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Elliptic Curves over Finite Fields as Two Cyclic Groups

Let $E$ be an elliptic curve over $\mathbb{F}_q$. I want to show $E(\mathbb{F}_q) \cong (\mathbb{Z}/m_1\mathbb{Z}) \times (\mathbb{Z}/m_2\mathbb{Z})$ where $m_1,m_2 \in \mathbb{Z}$ are such that ...
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1answer
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Categories of étale coverings of elliptic curves

Let $(E,\mathcal{O})$ be an elliptic curve over a (perfect) field $K$ and let $\textbf{Cov}(E,\mathcal{O})$ denote the category of finite pointed étale covers of $E$ from smooth varieties where the ...
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1answer
30 views

How to convert $x^3+y^3-x^2+y-1=0$ to homogenous form using the variables $X,Y,Z$

I'm to figure out how to convert algebraic curve such as $x^3+y^3-x^2+y-1=0$ to homogenous form using the variables $X,Y,Z$. Any help will be appreciated!
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1answer
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Complex Elliptic Surfaces without sections

Is there a description of smooth complex projective surfaces without sections? While working on a problem a surface $X$ showed up with the following property: it is a non-ruled surface that has an ...
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Primitive of Weierstrass $\wp$

Consider a lattice $L=\mathbb{Z}+\mathbb{Z}\tau$. Take the function $\xi(z) = \frac{-1}{z} - \sum_{w \in L\backslash \{0\}} \Big ( \frac{1}{z-w} + \frac{1}{w} + \frac{z}{w^2} \Big )$. Obviously this ...
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1answer
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Group law on elliptic curves.

Let $k$ perfect field. If we have a cubic non-singular projective curve $C(k)$ (over a field $k$), take two diferent points $P_1,P_2 \in C(k)$ and consider the line through the points, by Bezout ...
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1answer
42 views

Multiplication by $m$ isogenies of elliptic curves in characteristic $p$

I've been attempting to prove some comments I've read on MO by myself for my undergrad thesis regarding étale morphisms of elliptic curves. My definition of an étale morphism is taken from Milne's ...
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0answers
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Construction of Tate curve and formal schemes

In the notes websites.math.leidenuniv.nl/geom/tate.ps (and probably in other places), there is a construction of the Tate curve, where the steps are summarized below. 1) Take ...
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Property of Weierstrass sigma function

In theorem 1.2.3 of Schertz' Complex Multiplication says that For any $\omega \in \mathcal{L}$, a fixed lattice, we have the property: $$ \sigma(z + \omega) = \psi(\omega)e^{\eta(\omega)(z + ...
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1answer
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General theory of discriminants

I read it in the book Rational Points on Elliptic Curves by Silverman and Tate: If $f(x)$ is a polynomial with leading coefficient 1 in $Z[x]$, then the discriminant of $f(x)$ will be in the ideal ...
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2answers
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Equation of a cone

Find the equation of the cone whose vertex is at the origin and whose directing curve is given by the equations: $$\begin{cases} x^2-2z+1=0 \\ y-z+1=0\end{cases} $$ We know that an eliptic cone is ...
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2answers
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Generator for Kahler differentials of an affine elliptic curve

Consider the affine (nonsingular) elliptic curve $A = \mathbb C[x,y]/(y^2-x^3+x)$. Since the cotangent bundle is trivial, $\Omega_A^1 = A\,dx\oplus A\,dy /(2y\,dy - (3x^2-1)\,dx)$ is a free ...
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0answers
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Can I find the formula for rational point $(x,y)$ on the $y = 3x-1$? [closed]

Is it possible to find formula for rational point $(x,y)$ on the $y = 3x-1$ in term of rational parameter $t$. And also how would I write in form of $(x,y) =(f(t),g(t))$
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0answers
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Lifting a real quadratic twist of an Elliptic Curve to the modular surface

Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation $$y^2=x^3+p^2b\cdot x + p^3\cdot c$$ and parametrized by a map $$X_{0}(N\cdot {p}^{2})\rightarrow E$$ ...
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0answers
35 views

Isogeny of elliptic curves over $p$-adic field

If $K$ is a $p$-adic field, and $E_q$ and $E_{q'}$ are the corresponding Tate curves for $|q|,|q'|<1$, why does $E_q$ and $E_{q'}$ being isogenous imply that there are integers $A$ and $B$ such ...
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1answer
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Help with getting the formula for rational point $(x,y)$ on the $y^2 = x^3$

How to find formula for rational point $(x,y)$ on the $y^2 = x^3$ in term of rational parameter $t$. And also how would I write in form of $(x,y) =(f(t),g(t))$
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1answer
35 views

Finding formula for rational point ($x$,$y$) on the y = $x^2$

How to find formula for rational point ($x$,$y$) on the $y$ = $x^2$ in term of rational parameter $t$. And also how would I write in form of ($x$,$y$) =($f$($t$),$g$($t$))
2
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1answer
35 views

example of an elliptic curve without complex multiplication

What is an example of an elliptic curve $E$ without complex multiplication? This means $End(E)=\mathbb{Z}$. I know that complex elliptic curves are given by $\mathbb{C}^2/\Lambda$ for a lattice ...
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1answer
34 views

Elliptic curves over $\mathbb{Q}$, singularity points

Why is it so that $Y^2Z = X^3 + AXZ^2 + BZ^3$ is a non-singular elliptic curve if $4A^3 - 27B^2 \neq 0$? If we check the partial derivatives we get that $\frac{\partial F}{\partial Z} = Y^2 - 2AXZ - ...
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0answers
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Sum of two cubes transformed to elliptic curve

Given $x^3+y^3=N$, we can perform some substitutions to obtain an elliptic curve $u^3-432N^2=v^2$, as given here, which are $x=\frac{36N+v}{6u}$, $y=\frac{36N-v}{6u}$. Here's the details: ...
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Change of coordinate in Weistrass Normal form

From Rational Points on Elliptic Curves by Silverman and Tate To obtain Weistrass Normal form, it uses some technique of changing coordinates, but I can't understand its description, I underlined ...
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2answers
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How to prove that $\frac{1}{n}L/L\simeq (\mathbb{Z}/n\mathbb{Z})^2$?

Let $L$ be any lattice in $\mathbb{C},$ and $L'$ a lattice containing $L$ with index $n$ (i.e $n=\sharp L'/L$) I found this statement "The lattice $L'$ must be contained in $\frac{1}{n}L = ...
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1answer
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Surjectivity of morphisms of smooth projective varieties

I have a question regarding a proof of the "surjectivity of morphisms of projective varieties" (a whole mouthfull). Though there are proofs using completeness of varieties, I am interested in an ...
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1answer
41 views

Nagell-Lutz theorem

I'm a question about Nagell-Nutz theorem. For example, the point $P'=( \frac{31073}{2704},-\frac{5491823}{140608})$ on the curve $$y^2=x^3+8$$ does not meet the theorem Nagell-Nutz. So I can say what ...
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J. Silverman exercise 3.12 “The arithmetic of Elliptic curves”

I have question regarding exercise 3.12 of J. Silverman "The arithmetic of Elliptic curves". It states the following: Let $m \geq 2$ be an integer, prime to $\text{char}(K) > 0$. Prove that the ...
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1answer
36 views

Weierstrass form of elliptic curve with point with order larger than 3

L.S., Studying for my exam on elliptic curves, I tried to make exercise 8.13(a) of Silvermans "The Arithmatic of Elliptic Curves", which reads: Let $E$ be an elliptic curve defined over a field $k$, ...
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Maps between Elliptic Curves and Points at Infinity

I was trying some exercises from Silverman's book Rational Points on Elliptic Curves 2nd ed. (2015), and got stuck at this problem. 1.22 Let $C$ and $W$ be the projective curves ($b,e \ne 0$) $$ C: ...
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When is sum of squares a perfect square? [duplicate]

Recall that $$\sum_{j=1}^nj^2=\frac{n(n+1)(2n+1)}{6}.$$ When is this quantity a perfect square? It appears that the only solutions are $n=0,1,24.$ By setting $x=12n+6$, the problem reduces to finding ...
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2answers
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sheaf of relative differentials on an elliptic curve

Let $f : E\rightarrow S$ be an elliptic curve over a scheme $S$ with identity section $e : S\rightarrow E$. Why is it true that $e^*\Omega_{E/S}\cong f_*\Omega_{E/S}$? (I believe these should be ...
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1answer
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Properties of the elliptic curve $y^2 \equiv x^3 – 2 \pmod 7$

Can someone help me: 1) to list the points on the elliptic curve $E: y^2\equiv x^3 – 2\pmod 7$. 2) to find the sum $(3, 2) + (5, 5) $ on $E$.
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1answer
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Cube root of discriminant of elliptic curve

Let $E/K$ be an elliptic curve over a field $K$, with discriminant $\Delta$. Then the polynomial $x^3-\Delta$ has a root (and hence all roots since Galois) in $K(E[3])$; this can be shown laboriously ...
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1answer
37 views

Birational Equivalence of Diophantine Equations and Elliptic Curves

A while ago I saw this question Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$ which was very relevant to a undergraduate research paper I am currently working on. The answer given ...
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1answer
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Characterization of the $m$-torsion points of an elliptic curve.

Let $(E,\mathcal{O})$ be the elliptic curve of equation $$ f=Y^{2}+a_{1}XY+a_{3}Y-X^{3}-a_{2}X^{2}-a_{4}X-a_{6}, $$ $\alpha:K(E)\rightarrow K(E)$ the derivation such that $$ ...
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1answer
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Example of $n$-covering of an elliptic curve?

I've seen a couple of times the definition of $n$-covering of an elliptic curve $E/k$ but I haven't seen any explicit example of it. The definition I read is that it consists of a pair $(C,\pi)$ ...
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Elliptic curve characteristics 2 and 3

How can you show that if the characteristic of an elliptic curve $y^2 = x^3 + ax + b$ is 2 or 3 the equation fails? For characteristic 2 I know the equation must be written as $y^2 + ay = x^3 + bx^2 + ...
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1answer
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Is the sheaf of differentials on an elliptic curve over $R$ with a Weierstrass equation free?

Let $R$ be an integral domain and $E\stackrel{f}{\rightarrow}\text{Spec }R$ be an elliptic curve given by $$E := \text{Proj }R[x,y,z]/(y^2z + a_1xyz + a_3yz^2 = x^3 + a_2x^2z + a_4xz^2 + a_6z^3)$$ ...
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An implement of Constructing elliptic curves of prescribed order

In the Reinier Bröker's Phd thesis——Constructing elliptic curves of prescribed order(2006), he present a effective way to generate a elliptic curve with a given order N. And the heuristic run time of ...
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1answer
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Understanding proof by algebraic geometry, Fermat's last theorem for polynomials when $n = 3$.

This is a followup to my question here. See here. The question is as follows. How do we see that there do not exist nonconstant, relatively prime, polynomials $a(t)$, $b(t)$, and $c(t) \in ...
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Generalized elliptic curves over cusps and orbits of $\mathbb{Q}\cup\infty$

In the post http://mathoverflow.net/questions/51147/what-objects-do-the-cusps-of-modular-curve-classify, it says that the fibers over the cusps of a modular curve are n-gons. Wikipedia ...
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1answer
114 views

Finding integer solutions to $y^2=x^3+7x+9$ using WolframAlpha

I am an unconditional admirer of WolframAlpha and for this reason I want to let the people of this error (or is it really the fault of mine?). If I'm not mistaken, I would be very happy to contribute, ...
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Tate curve and cusps

I know this is a naive question, but what is the relation between the Tate curve and cusps on a modular curve? Naive googling seems to suggest that level structures on the Tate curve (up to ...
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1answer
47 views

The derivation of the Weierstrass elliptic function

I am wondering if any of you could point me to any books and/or lecture notes that explain the Weierstrass $\wp$ function for a self-studying student of elliptic curves and functions. I am interested ...
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$E[n]$ is etale locally $(\mathbb{Z}/n\mathbb{Z})^2$

I don't think we need the entire setup below (from Katz and Mazur's elliptic curve book, pages 74 and 75), but, as a beginner, I am unable to identify the assumptions I need. Let $S$ be the open ...