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

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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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42 views

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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31 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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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
30 views

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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61 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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57 views

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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51 views

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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47 views

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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81 views

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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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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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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39 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
35 views

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
36 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 ($x$,$y$) =($f$($t$),$g$($t$))
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1answer
39 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
38 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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45 views

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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51 views

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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69 views

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
50 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
45 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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44 views

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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55 views

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
86 views

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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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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71 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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150 views

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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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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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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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
125 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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61 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 ...
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The Picard group of an Elliptic Curve

Let $(E,O)$ be an elliptic curve. Let $\operatorname{Pic}^0(E)$ stand for the divisors that have degree $0$ where : $$D = \sum_{p\in E}n_p(P) \text{ and } \deg D = \sum_{p\in E}n_p.$$ I understand ...
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Canonical sheaf of bielliptic surface

This is an example of bielliptic surface on page 84 from Beauville's book "Complex algebraic surfaces". Let $\rho^3=1$, $\rho\neq1$ and $F_\rho=\mathbb{C}/(\mathbb{Z}+\rho\mathbb{Z})$, ...
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1answer
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How can I find $E(\mathbb F_{17})$ for the elliptic curve $E:$ $y^2=x^3+c$ where $c$ is any element in $\mathbb F_{17}^*$?

This was left as an exercise in a seminar in my college. I tried to figure it out myself, but haven't been able to make any progress thus far. I don't think it should need any non-trivial result (or ...
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Elliptic curve over algebraically closed field of characteristic $0$ has a non-torsion point

Let $E/k$ be an elliptic curve over an algebraically closed field $k$ of characteristic $0$. Can one prove that the abelian group $E(k)$ is non-torsion? Better yet, can one prove that $E(k) ...
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On $p^2 + nq^2 = z^2,\;p^2 - nq^2 = t^2$ and the “congruent number problem”

(Much revised for brevity.) An integer $n$ is a congruent number if there are rationals $a,b,c$ such that, $$a^2+b^2 = c^2\\ \tfrac{1}{2}ab = n$$ or, alternatively, the elliptic curve, $$x^3-n^2x = ...
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Understanding the group structure of quotient group derived from elliptic curve group

I am working through some content in L.C. Washington's Elliptic Curves, Number Theory, and Cryptography and I am unsure about what the group structure of a certain group looks like. Some background: ...
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Is the coordinate ring of an elliptic curve principal?

Let $K$ a field, $E$ an elliptic curve. I would like to know if the coordinate ring of $K[E]=K[X,Y]/(E)$ is principal. I think the answer is no. I tried to prove that the ideal $J=\langle ...
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1answer
104 views

find the structure of an elliptic curve over a finite field

For the elliptic curves E1,E2,E3, and E4 defined below, determine the structure of the groups Ek(F13) by using the information given below together with a minimal amount of extra (hand) ...
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72 views

An elliptic curve has a point of order $n$ iff $E[n]\cong\mathbb{Z}/n\times\mu_n$?

Let $E$ be an elliptic curve over some field $k$, and let $n$ be coprime to the characteristic of $k$. Then I claim that $E(k)$ has a point of order $n$ if and only if ...