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

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Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
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Prove that $y^2 = x(x-1)(x- \lambda)$ is irreducible for all $\lambda \in k$

I wish to prove that $y^2 = x(x-1)(x- \lambda)$ is irreducible for all $\lambda \in k$. It seems like this follows from the fact that $x(x-1)(x- \lambda)$ cannot be written as the square of any ...
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How to relate the valuation of x/y (For a minimal Weierstrass equation)

I'm reading an article about elliptic curves, but since I'm not very experienced on this subject, I ended up getting stuck. The problem starts as: "Let $K/\mathbb{Q}$ be a number field and $E/K$ an ...
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241 views

On Bachet's Duplication Formula and the number $-432$

While reading "Rational Points on Elliptic Curves" by Silverman and Tate, I came across this interesting passage about Bachet's duplication formula: I know how to derive Bachet's duplication ...
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116 views

Show that an ideal is unramified

See Advanced Topics in elliptic curves for the full question(see also errata: http://www.math.brown.edu/~jhs/ATAEC/ATAECErrata.pdf): 2.30 (pg 184) Given $E/L$ an elliptic curve with complex ...
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Abelian Elliptic Surfaces

By abelian surface we mean a 2-dimensional algebraic complex torus. Thus $$ S=\Bbb{C}^2/\Gamma$$ where $\Gamma$ is a rank $4$ lattice in $\Bbb{C}^2$ and such that $S$ is algebraic. It has trivial ...
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Attacking Elliptic Curve Cryptography Problem with a Bad Reduction $\pmod p$

I'm working on a crypto problem as a puzzle and unfortunately my math isn't at the level I need it to be to answer the question. I have been given a prime $p$, a curve $E$ defined over $F(p)$, a ...
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What is the complex *algebraic* moduli of elliptic curves?

It's well-known that $SL_2(\mathbb{Z}) \backslash \mathfrak{h}$ is a coarse moduli space for complex elliptic curves. Thus, I would expect this to be related to the pullback of $\mathcal{M}_{ell} ...
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Why is this a characterization of isogenies of elliptic curves? (From Silverman)

In the proof of Theorem III.6.2 (c) in Silverman's The Arithmetic Of Elliptic Curves it says: Let $x_1, y_1 \in K(E_1)$ and $x_2, y_2 \in K(E_2)$ be Weierstrass coordinates. We start by looking at ...
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Group Law for an Elliptic curve

I was reading this book "Rational points on Elliptic curves" by J.Silverman, and J.Tate, 2 prominent figures in Number theory and was very intrigued after reading the first couple of pages. The ...
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Integral points on an elliptic curve

Let's start with an elliptic curve in the form $$E : y^2 = x^3 + Ax + B, \qquad A, B \in \mathbb{Z}.$$ I am wondering about integral points. I know that Siegel proved that $E$ has only finitely many ...
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Subvariety of Product of Elliptic Curves

This is almost certainly known (and maybe written down somewhere?). Is there an example of two elliptic curves $C, E/k$ that are not isomorphic, yet there is an embedding $C\hookrightarrow E\times E$ ...
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563 views

Intuition and Stumbling blocks in proving the finiteness of WC group

After reading many articles about the Tate-Shafarevich Group ,i understood that "in naive perspective the group is nothing but the measure of the failure of Hasse principle, and coming to its ...
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267 views

The elliptic curve $y^2 = 23328x^3-890273x^2+14755570x-7^7$

The elliptic curve, $$y^2 = 23328x^3-890273x^2+14755570x-7^7 \tag{1}$$ has the small solution $x = 58$. I know how to find other rational points, but the number of digits in the denominator gets ...
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Isomorphic Elliptic Curves

I want to solve the following exercise: Show that the two elliptic curves $E/ \mathbb{Q}$ and $E'/ \mathbb{Q}$ are isomorphic. $E: y^2 = x^3+x-2$ and $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. ...
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Elliptic Curves Without Geometry

Unfortunately geometry terrifies me, so I was hoping to understand the basic theory of elliptic curves algebraically (via their function fields). Let F be a transcendence degree 1 extension of ...
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Elliptic curves with finitely many rational points

A conjecture by Goldfeld says that half of all elliptic curves have rank zero (i.e. their Mordell-Weil group has finite order.) Are there any known infinite families of elliptic curves (over ...
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295 views

Square root in Characteristic 2 Field

Let $K$ be a field of characteristic 2. For each $a\in K$, can we always find some $x$ such that $x^2=a$? I came upon this question while reading "Arithmetic of Elliptic Curves". The original ...
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333 views

Why do we define the group law on elliptic curves only for Weierstrass forms and $O$ an inflexion point?

In almost all texts concerning the group law on an elliptic curve it is first proven that any nonsingular cubic can be given by a Weierstrass equation and then the group law using the point $O$ at ...
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What is a primitive point on an elliptic curve?

While working with elliptic curves for cryptography reasons, I found the notion of a primitive point, but no definition. For example, $P(0,6)$ is a primitive point on the elliptic curve $y^2\equiv ...
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How could I calculate the rank of this elliptic curve?

The birational change of variables $(u,v) = (\frac{36+y}{6x},\frac{36-y}{6x})$ maps $u^3+v^3=1$ to $y^2 = x^3 - 432$ which has discriminant $-2^{12}\cdot 3^9$. Using pari/gp we can compute the ...
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Example of an elliptic curve with trivial torsion subgroup and rank 0

What is an example of an elliptic curve over $\mathbb{Q}$ with trivial torsion subgroup and rank 0?
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The rational points on the curve: $y^2=ax^4+bx^2+c$.

I wonder how to find the rational points on the curve: $y^2=ax^4+bx^2+c$. Is there infinite rational points on this curve? For example:$y^2=x^4+3x^2+1.$If we set $y=x^2+k$,then $2kx^2+k^2=3x^2+1$, ...
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Is it possible to do elliptic curve cryptography over $\mathbb{Q}$ instead of a finite field?

Whenever I read about elliptic curve cryptography (ECC), the writer always works over a finite field. But as I understand it there is no group-theoretic reason not to use $\mathbb{Q}$ as the ...
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168 views

Exposition on Modular Curves

I was recently reading this paper by Weston, whereby he talks about the modular curves $X_0(11)$ and $X_1(11)$. I was wondering if anyone can recommend a more general exposition of modular curves ...
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Bound for number of points on surface over $\mathbb{F}_p$

I know of the bound for the number of points on an elliptic curve over a finite field: $$|\# E(\mathbb{F}_q) - q - 1| < 2\sqrt{q}$$ where this includes the point at infinity. I have been told that ...
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Fundamental period of the WeierstrassP elliptic function?

Consider the WeierstrassP elliptic function $\wp(z, g_2, g_3)$ with the invariants $g_2\in\mathbb{R}$ and $g_3\in\mathbb{R}$: $$\wp'(z)^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3$$ According to Wikipedia when ...
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Silverman Adv. Topics example

I would like to refer you to Silverman's Advanced Topics in the Arithmetic of Elliptic Curves example 10.6: Let $D$ be a nonzero integer, $E:y^2=x^3+D$ with complex multiplication by $\mathcal{O}_K$ ...
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Elliptic curve question

Let $P$ be a point on an elliptic curve over $\mathbb{R}$. Give a geometric condition that is equivalent to P being a point of order (a) $2$ , (b) $3 $ , (c) $ 4$ . Could someone explain this to ...
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How does $\text{Gal}(L/K)$ act on the automorphism group of an elliptic curve?

Let $L/K$ be a finite Galois extension of number fields; I'm interested mainly in the case $K = \Bbb{Q}$ and $L= \Bbb{Q}(\sqrt{d})$. Let $X$ be an elliptic curve over $K$ and $\text{Aut}(X_L)$ the ...
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Why are mathematicians more interested in elliptic curves than other algebraic curves?

Why are mathematicians more interested in elliptic curves than other algebraic curves? There must be some reason that motivates mathematicians to research elliptic curves specifically.
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There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am ...
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Cohomology group and elliptic curve

Let $E$ be an elliptic curve with a 3-torsion point $P$ and $G = \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Let $X = \{O, P, -P\}$ where $O$ is the point at infinity and $X$ is a ...
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Proving Fermat's Last Theorem (easily) using “assumed” conjectures

It can easily be proven assuming Szpiro's conjecture that Fermat's Last Theorem is true for sufficiently large $n$. The proof consists of extremely straightforward computations. My question is, is ...
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division polynomials of elliptic curve as function on $\mathbb{C}$

I have a question about exercise 6.15 of Silverman's book AEC. Suppose that $E$ is a nonsingular elliptic curve over $\mathbb{C}$ given by the equation $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ Then we ...
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Cokernel of morphism of Tate module of elliptic curves

Let $K$ be a field, and $\phi: E_1\to E_2$ be an isogeny of elliptic curves over $K$. Given a prime $\ell$ different from the characteristic of $K$, $\phi$ induces an injection $T_\ell \phi: T_\ell ...
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Can one check by hand whether the Tate module of an elliptic curve is semi-simple

Let $E$ be an elliptic curve over $\mathbb Q$, and $\ell$ a prime number. Then, the $\ell$-adic Tate module $V_\ell(E)$ of $E$ is semi-simple as a $\mathbb Q_\ell$-representation of ...
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Group law for an elliptic curve using schemes

I was trying to understand better the definition of the group law for an elliptic curve given in Katz and Mazur's book ...
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Confusion with computing kernel of an isogeny between two elliptic curves

Consider the two elliptic curves $$E_3: y^2+y=x^3+x^2+x \enspace [Cremona:19A3]$$ and $$E_1: y^2+y=x^3+x^2−9x−15 \enspace [Cremona:19A1]$$ Let $\varphi$ be the $3$-isogeny from $E_3$ to $E_1$. I want ...
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Computing cohomology groups of elliptic curves

I'm skimming through Silverman's text to recall some theory of elliptic curves that I've learned in undergrad. In practice however, I'm having trouble actually computing the cohomology groups. For ...
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Rank of the elliptic curve $y^2=x^3+px$

I need to prove that the rank of the curve $y^2=x^3+px$ is $0$, if $p\equiv 7 \pmod {16}$ is a prime. Using the standard technique, we need to show that none of the following two equations admits an ...
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Computing the divisors of a meromorphic function defined by a hyperelliptic curve.

Let $X$ be a hyperelliptic curve defined by $y^2=h(x).$ Let $\pi : X\to \mathbb{P}^1$ be the double covering map sending $(x,y)$ to $x$. Let $\omega=\pi^{*}(dx/h(x)).$ Compute div$(\omega)$. I ...
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Discriminant of isogenous elliptic curves

Let $E$ be an elliptic curve with a rational $p$-torsion point $P$. Then $E$ is isogenous to the elliptic curve $E' := E/\langle P \rangle$ via the mod $P$ map. I know that the conductor of $E$ and ...
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191 views

Global minimal model of elliptic curve over $\mathbb{Q}$

I am basically trying to solve the cannonball problem using elliptic curves. In other words I have to show that the only integer points on the "elliptic curve" $6y^2 = 2x^3 + 3x^2 + x$ are $(0,0), ...
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How to show that the image of a certain projective embedding is an algebraic curve?

I found the following claim in a paper by Griffiths and Harris : Start with a complex torus $\mathbb{C}/\Lambda$. The vector space of meromorphic functions having period lattice $\Lambda$ and a pole ...
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seeing the differential dx/y on an elliptic curve as an element of the sheaf of differentials

$\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\Spec}{\text{Spec }}$ It's a well known fact that every elliptic curve (say, over a field $k$) has a global holomorphic nowhere vanishing differential. If ...
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Where does a CM elliptic curve have bad reduction?

Let $d>1$ be square-free, and $K=\mathbf Q(\sqrt{-d})$. Choose an embedding of $K$ in $\mathbf C$, and let $E = \mathbf C/\mathcal O_K$. It is known that $E$ admits a model over the Hilbert class ...
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104 views

Prym variety associated to an étale cover of degree 2 of an hyperelliptic curve.

In view of this question, I have an additional question. The situation is as follows. Let $C$ be the hyperelliptic curve over $\mathbb{C}$, which is given on an affine by the equation $y^2 = x^5 +1 ...
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Computing the kernel of an isogeny between two elliptic curves

Consider the two rational elliptic curves - $ E_{1}: y^{2}+y=x^{3}+x^{2}-131x-650 $ $ [\text{Cremona}:35a2] $ $ E_{2}: y^{2}+y=x^{3}+x^{2}-x $ $ [\text{Cremona}:35a3] $ We know that ...
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Part of verifying that the Weil pairing $e_m$ is well-defined.

As part of a homework problem, I need to show that the value of $e_m(P,Q)$ is independent of the choice of a point $S \in E[m] \setminus \{\mathcal{O},P,-Q,P-Q\}$, where $E[m]$ is the collection of ...