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

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elliptic curves and order of elements

The problem is this: Show that any elliptic curve over $\mathbb Z_{83}$ has an element of order > 30. I don't quite know which way to go on this one. We could use Hasse's thm. to show that the ...
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87 views

Adding points on an elliptic curve

I'm trying to work out a problem from a previous exam in Cryptography regarding elliptic curves. I can add points on an EC using the formulas given, but the suggested solution to this exam problem I ...
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1answer
75 views

The translation map between elliptic curves is a rational map

I want to see a reference or a prove that the following map is a rational map: Let E be an elliptic curve,$P\in E$ and $T_p$ defined as $T_p:E\rightarrow E,\text{ }T_P(Q)=P+Q$. It is important ...
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Finding number of points on elliptic curve

I'm working on a previous exam problem, and my solution does not match with the given one, and I don't know why. I have the elliptic curve $$E: Y^{2} = X^{3} + X + 46$$ over $\mathbb{F_{101}}$. We're ...
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Infinitely rational points in $y^2 = x^3 - 4$?

If the $x$-coordinate of a rational point $P$ of $y^2 = x^3 - 4$ is given by $m/n$, the $x$-coordinate of $2P$ is given by$${{(m^3 + 32n^3)m}\over{4(m^3 - 4n^3)n}}.$$Using this fact, how do I show ...
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40 views

Why is the Frob in elliptic curve not called an automorphism

Please apologize, if that's a stupid question. Why is the Frobenius Endomorphism of an elliptic curve over a finite field not regarded as an automorphism? Since it is an Isogeny, it is surjective ...
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94 views

Formal expansion of differential form on elliptic curves

First of all everything i'm asking about comes from the beginning of Katz and Mazur's book : Arithmetic moduli of elliptic curves (which you can find here). I'm considering an elliptic curve $f : E ...
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91 views

Fermat's Last Theorem

Is there any relationship between BSD conjecture and Fermat's Last Theorem? What is the importance of the analytic degree of the L-function of an elliptic curve stated in the BSD conjecture in ...
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72 views

Are elliptic curves algebraic varieties?

I got a short question. Are elliptic cubes also algebraic varietes? Say we have $E:y^2=x^3+5x=:f(x)$ Then we can $f(x)=x(x^2+5)$ So it can't be an algebraic variety.. I feel like I am totally ...
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How can we continuously deform a height 1 formal group law into a height 2 formal group law?

A Quick Review: The complex elliptic curve $\mathbb{C}/(\mathbb{Z} + \tau \mathbb{Z})$ may be rewritten using the exponential, $\text{exp(}{2 \pi i \tau}) =: q$ as $\mathbb{C}^\times/q^\mathbb{Z}$ . ...
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Rational maps between elliptic curves

I dont understand the definition of rational maps. Here is the definition: Let $E_1$ and $E_2$ be elliptic curves over a field $K$. (projectively written). A rational map $\Phi:E_1\rightarrow E_2$ ...
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33 views

Example of Constant Morphism

I am reading Basic Theory of Elliptic Curves, there I came about a statement saying : A Morphism of curves is either Surjective or Constant. While studying Isogenes I came across examples of ...
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Isogenous elliptic curve.

I'm studying elliptic curves and I have a question Take two $k$-isogenous elliptic curves defined over a number field $k$ and fix a place $v$ of good reduction. Are the reduced curves ...
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1answer
28 views

Absolute value in Hasse's theorem

The Hasse's theorem says that for an elliptic curve $E$ defined on $\mathbb{F}_p$ where $p$ is a prime number, we have: $|n-(p+1)| < 2\sqrt{p}$ with $n$ the order of $E$. I am wondering why the ...
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What will be a good source for learning elliptic curves and what viewpoints can I adopt?

I'm attending a research seminar on elliptic curves in my university where the professor is currently presenting a proof of Mordell's theorem (for elliptic curves over $\mathbb Q)$. The professor says ...
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Reduction of an elliptic curve defined over $\mathbb{Q}$

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\ell$ be a prime number such that the reduced curve $\tilde E_{\ell}$ is non singular. Assume that $\tilde E_{\ell}$ admits a subspace ...
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De Rham-Etale comparison isomorphism for elliptic curves

I can't find anywhere a proof of the following comparison isomorphishm: $$H^1_{dR}(E)\otimes \mathbb{C}=H^1_{et}(E)\otimes \mathbb{C}$$ where $E$ is an elliptic curve over $\mathbb{C}$. Any ...
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Tate curve and action of inertia group

I read the answers to this question Clarifying a comment of Serre. However I miss a passage of the second answer and since I can't comment there I have should post a new question. I don't understand ...
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Where does the terminology “fake/false elliptic curve” come from?

In describing, say, the moduli of Shimura curves, people often refer to "fake" or "false elliptic curves" ("les fausses courbes elliptiques"), which are abelian surfaces whose endomorphism ring is an ...
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element of an $\ell$-adic Galois representation with certain eigenvalues.

Let $\mathscr{G}=Gal(\bar{\mathbb{Q}} / \mathbb{Q})$, $E$ an elliptic curve over $\mathbb{Q}$, and consider the $\ell$-adic representation $$ \varphi_{\ell}: \mathscr{G} \longrightarrow ...
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Showing $P=(o,m) \not\in 2E_m(\mathbb{Q})$ for an elliptic curve $E_m$

I have been having trouble with this question. Let $m\in \mathbb{Z}$ with $m > 0$ and define $E_m : y^2 = x^3 −x+m^2$ Then $E_m$ is an elliptic curve Determine the group sturcture of ...
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intersection of plane elliptic curve with coordinate hyperplanes

Let $E: y^2z = x^3 - Axz^2 - Bz^3$ be a plane elliptic curve. I want to calculate the intersection of $E$ with the coordinate hyperplanes $H_i = \{x_i = 0\}$, $i=1,2,3$. I write $H_x = \{x=0\}, H_y = ...
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Trisectible Angle

How do we prove that a triangle with sides $(one, x, y)$, where $x$ is any constructible length from one to three at the elliptic curve $$y^2 = x^3 -x^2 -x +1$$then the triangle possess at least ...
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Rational solutions to $x^4+y^4=cz^2$

Suppose $c\neq 1$ is a squarefree number, and consider the curve $x^4+y^4=cz^2$. How can I find rational points on this curve? What I really want to know is how to transform this into an elliptic ...
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What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like?

What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like? And also what would the ring $\mathbb{Z}[x,y]/(x^3-x-y^2)$ look like? These are two sorts of rings I have been curious about.
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The sum of an isogeny and its dual for the Frobenius homeomorphism

This is from page 150 of Silverman's "The Arithmetic of Elliptic Curves". Any my only questions is: How you can conclude that $[a]=\phi+\hat{\phi}$? I tried to use the formula on page 85 which ...
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119 views

Galois invariants of the Tate module of an elliptic curve over a number field

Let $K$ be a number field, $E$ be an elliptic curve over $K$, $l \neq p$ be two different prime numbers and $v$ be a place of $K$ above $l$. I am trying to understand the proof of proposition I.6.7 ...
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Isogenies between elliptic curves with specified torsion groups

For each of the $15$ possible torsion groups of an elliptic curve defined over $\mathbb{Q}$ we have an infinite family of curves with that torsion group. This sometimes goes under the name of Kubert ...
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23 views

Order's size (in bits) of an elliptic curve

I am trying to prove that, given an Elliptic Curve defined on $\mathbb{F}_p$ with $p$ a prime number, the order $q$ verifies: $|p| \le |q| \leq |p|+1$ where $|x|$ denotes the length in bits of $x$. ...
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Polynomial representation of elliptic curve points (Frobenius Endomorphism)

I'm trying to understand the Schoof algorithm for counting the number of points on elliptic curves in finite fields. I.e. the most basic algorithm to efficiently determine $\#E(F_p)$. For literature, ...
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Elliptic curve is self dual.

How to prove $E[p^\infty] \cong Hom ( T_E, \mathbb{Q}_p/\mathbb{Z}_p(1)) $ where $T_E$ denotes the Tate module of $E$ ?
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Proving $x^2 - x = y^5 - y$ is a hyperelliptic curve

Greetings to one an all! How can we prove the curve "$x^2 -x = y^5-y$" is a hyperelliptic curve? Is a hyperelliptic curve the same as a hyperbolic elliptic curve or are there any differences?
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The form of the zeta function of an elliptic curve over a finite field

I seek a (very) elementary proof that the zeta function of an elliptic curve $E$ over $\mathbb{F}_q$ has the form $$Z(T)=\frac{1-aT+qT^2}{(1-T)(1-qT)}.$$ Something tedious and computational making use ...
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Addition of x-coordinate on elliptic curve given by Möbius Transformation

Consider the elliptic curve $y^2=(x-\alpha)(x^2+ax+b)=x^3+(a-\alpha)x^2+(b-a\alpha)x-\alpha b$ over the field $K$ with $\text{char}\ K\not= 2$. The questions I am doing asks for a formula for the ...
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101 views

The cardinality of elliptic curves over finite field

Given an elliptic curve over $\mathbb Q$ as $y^2=f(x)$ where $f(x)$ is a cubic polynomial. In some places I read that if $p$ is a prime of good reduction then we have that $E(\mathbb F_p)=p+1$. Is ...
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Algebraic independence of `Riemann-Roch' elements

First of all, I'm not too sure on what terminology should be used in the title: the question deals with the vector spaces $$\mathcal{L(D)}=\{f\colon E\to\mathbb{C} \mid f\text{ is meromorphic}, ...
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Is it normal surface of general type to have infinitely many positive rank elliptic curves?

I am not good at algebraic geometry and almost surely am misunderstanding something. Got an alleged argument against Bombieri-Lang conjecture and would like to know what the mistake is. One of the ...
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Problem with Elliptic Curve in Montgomery form

I am trying to understand how points are added in Elliptic Curves in Montgomery form. I am working with the curve $$3y^2 = x^3 + 5x^2 + x \mod 65537$$ Adding the point $(3,5)$ with itself gives (or ...
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Why do we assume the ring to be torsion free when dealing with formal logarithms in the context of formal group laws?

Let $F$ be a formal group over a ring $R$. Why do we require that $R$ has no additive torsion before we discuss formal logarithms?
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Calculating Elliptic Curve cofactor h

An Elliptic Curve in short Weierstrass form over a finite field $F_p$ is given by the equation: $$y^2 = x^3 + ax + b \mod p$$ To use this curve for cryptographic purposes, in the domain parameters ...
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1answer
48 views

Galois Representation with $D_{10}$ image

I want to construct an explicit elliptic curve $E$ over a number field $K$ such that $Gal(K(E[l])/K) \cong D_{10}$ where $D_{10}$ is the dihedral group of order 10 and $l$ is a prime number. ...
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height of formal group of an elliptic curves

I have an elliptic curve $E$ defined over a complete discrete-valued field $K$ of characteristc $0$. the residue field $k$ is of positive characteristic $p$. Then $E[p]=\mathbb{Z}/p\mathbb{Z} \times ...
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Show that Weierstrass function is elliptic function.

Prove that Weierstrass function is periodic with respect to lattice $L (L\subset \mathbb{C})$ .i-e $f(z+w,L)=f(z,L)$ ($w\in L$). $f(z,L)=\frac{1}{z^2}+\sum_{0\ne w\in ...
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Point conversion between Twisted Edwards and Montgomery curves

With the great help of Birational Equvalence of Twisted Edwards and Montgomery curves I know how to convert twisted Edwards curves into their birationally equivalent Montgomery counterparts where I ...
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149 views

Why does the elliptic curve for $a+b+c = abc = 6$ involve a solvable nonic?

The curve discussed in this OP's post, $$\color{brown}{-24a+36a^2-12a^3+a^4}=z^2\tag1$$ is birationally equivalent to an elliptic curve. Following E. Delanoy's post, let $G$ be the set of rational ...
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95 views

can someone explain Nagell-Lutz theorem

(elliptic curve $y^2 = x^3 + ax^2 + bx + c$) Nagell-Lutz theorem: If $p(x, y)$ is finite order on a given integer coefficient elliptic curve satisfy: (1) x and y are integer (2) y = 0 or y | D (D ...
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Birational Equvalence of Twisted Edwards and Montgomery curves

I'm trying to understand the birational equivalence between Twisted Edwards and Montgomery curves and try to calculate some examples. In particular, as an example, I'm looking at the Ed25519 Twisted ...
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Reduction map on torsion points of an elliptic curve and their valuation

Let $K$ be a field of characteristic zero, complete with respect a discrete valuation $v$. Assume that the residue field $k$ is of positive characteristic $p$. Now take an elliptic curve $E$ defined ...
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Structure of first-coordinate-projection of set of solutions of “elliptic” diophantine equation $xy(6-(x+y))=6$

Say that a rational number $a$ is good iff there is a rational number $b$ with $ab(6-a-b)=6$, or equivalently iff $a^4 - 12a^3 + 36a^2 - 24a$ is the square of a rational number. Denote by $G$ the set ...
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Group operations on Montgomery Curves in affine representation

I'm trying to understand group operations on elliptic Montgomery curves in affine representation. Let's say the curve I use is Curve25519, i.e.: $$y^2 = x^3 + A\,x^2 + x\quad\text{where}\quad ...