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

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Direct sum of two points on an elliptic curve

Given $E:y^{2} = x^{3}+9x$ over $\mathbb{Z}_{71}$, and $A = (0,0), \: B = (1,9)$, I'm asked to find $C=A\oplus B$. I just don't know how the direct sum of two points on an elliptic curve is defined, ...
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The term “elliptic”

There are many things which are called “elliptic” in various branches of mathematics: Elliptic curves Elliptic functions Elliptic geometry Elliptic hyperboloid Elliptic integral Elliptic modulus ...
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1answer
59 views

Proving some facts about the EC $y^2 = x^3 + ax + b$ [closed]

A solution to this question would be much appreciated! If $E/F$ is the EC defined by $y^2 = x^3 + ax + b$ then prove the following: If $P = (x, y)$ is element of $E(F)$ with order 3 then $x$ is a ...
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39 views

Weil pairing of curve of genus 2

We know there is Weil pairing for elliptic curve satisfying several nice properties. So do we have Weil pairing for other curves also satisfying the nice property? Especially genus 2 curve?
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2answers
67 views
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Quotient of invariant differentials is constant

In the proof of Proposition 2.1.1 in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, he makes a comment about quotients of invariant differentials being constant, because their ...
2
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35 views

Showing $N_E/F_p^r = p^r + 1$ for a given elliptic curve [closed]

Have been studying elliptic curves, and am syuck on this problem. A detalied explanation would be much appreciated! a. Let $E/F_p^r$ be the elliptic curve $y^2 = x^3 − x$. Prove that if $p ≡ 3 (mod ...
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Proving an eliptic curve is cyclic, and determining it's order

I need a solution with an explanation for the following. Thanks! Let $E/F_q$ be an elliptic curve and let $P ∈ E(F_q)$ be a point a. if $n=ord(P)>1/2(q^{0.5}+1)^2$ prove that $E(F_q)$ is cyclic ...
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122 views

Finding zeta function of an elliptic curve

Let p=3 (mod 4) be a prime, and $E/F_{p^r}$ be the elliptic curve given by $y^2 = x^3 − x$ Find the zeta-function of $E/F_p$ and use it to determine $|E(F_{p^r} )|$ for all r>0.
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1answer
60 views

2-torsion points in a curve with genus 2

Let X be a genus 2 curve with affine equation y^2 = f(x), where f is a polynomial of degree 6. Write $P_1, ..., P_6$ for the points on X(C) with y=0. Then why every $P_i-P_j$ is a 2-torsion points in ...
2
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1answer
117 views

Finding order of a point on eliptic curve

Just started studying eliptic curves and am having trouble with this question. An explanation/solution would be much appreciated. Find the order of the point X on the elliptic curve $E/Q$ for the ...
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1answer
25 views

Is the image of $\overline{\rho_{E,p}}$ in $PGL_2$ always isomorphic to $A_5$ if $p$ does not divide the order of the image of $\rho_{E,p}$?

I have the following setting: Given an elliptic curve $E$ over $\mathbf{Q}$ and $p>5$ a prime of good ordinary reduction. Let ${G}_{k}=\text{Gal}(K(E_{p^k})/K)$ with representation ...
4
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59 views

Original proof of Ljunngren's equation

The equation $$x^2=2y^4-1$$ was studied and solved by Ljunngren, who showed that 1,1 and 293,13 are the only integer solutions.However, his proof was very difficult and L.J.Mordell thought there must ...
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1answer
76 views

How to construct isogenies between elliptic curves over finite fields for some simple cases?

From the theorem of Tate, it is known that the two elliptic curves over the same field $\mathbb{F}_p$ are isogenous iff they have the same number of points. For $p\equiv 3\mod 4$, the curve ...
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0answers
18 views

Hensel lemma and elliptic curves [closed]

What is the cardinality of this homomorphism $E(\mathbb{Z}/p^r\mathbb{Z} ) \rightarrow E( \mathbb{Z}/p\mathbb{Z} )$ where $E(\mathbb{Z}/p^r\mathbb{Z} )$, $E( \mathbb{Z}/p\mathbb{Z} ...
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23 views

Is the connection between elliptic curve and lattice unique?

If I remember correctly, elliptic curve (over C) is isomorphic to a complex lattice, and they are connected by some technical stuff(Eisenstein series, j-invariant,...) But the whole process seems so ...
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1answer
52 views

Is there another methods for counting points on the curve $x^3 + y^3 =1$ over finite fields?

For the circle $(C): x^2 + y^2=1$ over finite field, we can use simple method to count the number of points. The case $p\equiv 1\mod 4$ is not difficult to find, because $-1$ is a square on $F_p$. ...
2
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1answer
48 views

(online) Tool to calculate $E(\mathbb{Q})/2E(\mathbb{Q})$ for $E: y^2 = x(x^2 + 3x + 5)$

For some exercise I need to compute the generators of $E(\mathbb{Q})/2E(\mathbb{Q})$, where $E: y^2 = x(x^2 + 3x + 5)$ I did this by the approach from Cassels book 'Lectures on Elliptic Curves' and ...
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0answers
103 views

Differentiating a period of an elliptic curve under the integral sign

Let $$g = \frac{27J}{1 - J},$$ where $J$ is the absolute invariant, and define $$\Omega = \int_{\gamma(J)} \frac{dz}{\sqrt{4z^3 + g(z + 1)}}.$$ Here, $\gamma(J)$ is a contour in the complex plane that ...
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1answer
38 views

How to pass from general weierstrass equation of elliptic curves to shorter ones

I found a page on wikipedia about weierstrass equations of elliptic curves. https://fr.wikipedia.org/wiki/%C3%89quation_de_Weierstrass The page says that one can put weierstrass equations in shorter ...
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1answer
38 views

E is an elliptic curve over the finite field Z/pZ. Let N= number of points on E. If N is divisible by p, show that either N=p or p=2

I feel like this wants to use the Hasse bound somehow since that's really the only tool we talked about with regard to counting points on a curve, but I'm not entirely sure how to get to that ...
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21 views

number of points of order 2 on an elliptic curve

Given a field $F$ with $char F \neq 2,3$ and an elliptic curve $E: y^2-(x^3+ax+b)$ I want to find the number of points of order $2$. (The given solutions say it is always exactly 3.) Let $P$ be a ...
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degree of multivariate polynomial in a quotient ring.

I'm trying to work with polynomials on Elliptic curves. Thus polynomials are elements of the ring \begin{equation*} \frac{K[X,Y]}{Y^2-X^3-aX-b} \end{equation*} (Field characteristic is supposed to be ...
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1answer
36 views

Is there a genus-one curve over $\mathbb{Q}$ with no points over any solvable extension?

Is there a (non-singular) genus-one curve $E$ over $\mathbb{Q}$ that is known to have no points over any solvable extension?
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72 views

Dual and degree of the isogeny a+b[i].

Let $[i]$ be the endomorphism such that $[i](x,y) = (-x,iy)$ with $[i]^2+[1]=[0]$. I am trying to prove that the degree of the endomorphism $[a]+[b]\circ[i]$ is equal to $a^2+b^2$. After ...
3
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1answer
37 views

height of a point on elliptic curve

I m bit confused about the definition of Weil Height over number field. In our lecture, it is defined as $|x|_p=|O_k/(p^{-ord_p(x)})|$ where $O_K$ is the rings of integers for number field $K$ and $p$ ...
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0answers
36 views

Geometric interpretation of subgroups of an elliptic curve group?

I am particularly interested in elliptic curves over finite fields of prime order, so let $\mathbb{F}_{p}$ denote the finite field of order $p$ (where $p$ is prime) and let $E$ be the elliptic curve ...
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1answer
51 views

Number of Points on an Elliptic Curve

If I have an elliptic curve $$E: y^2 = x^3 + bx + c$$, with $b, c$ integers mod some prime $p$. And $x^3 + bx + c$ has at least one root mod $p$. How can I show that the number of points on the ...
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Need clarity on calculating the y coordinate in elliptic curve cryptography

I'm just new to elliptic curve cryptography. I have been working on RSA for quite some time. Moreover I'm not from a mathematical background. The whole concept looks very complex. So tell me my ...
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1answer
31 views

If $\phi$ is an endomorphism of an elliptic curve, and $\phi = \hat{\phi}$ then $\phi = [m]$?

I heard a reference to this fact, but I cannot find a reference. (I can find the converse in Silverman, namely that $\hat{[m]} = [m]$.) Notation: $[m]$ is multiplication by $m$ in the group law, and ...
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2answers
43 views

Elliptic curve references

In the study of elliptic curves, one must have a solid ground on abstract algebra, algebraic geometry and analysis (modular forms).Would someone who is well-acquainted with the subject give me roughly ...
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Prime points and elliptic curves

Wiki in https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication given a curve, $E$, defined along some equation in a finite field (such as $E: y^2 = x^3 + ax + b$), point multiplication is ...
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2answers
42 views

Smooth curve defined over K with genus 1 is always isomorphic, over $\overline K$, to an elliptic curve over K

Here, the point is the smooth curve defined over K with genus 1 may not have rational point. But to be an elliptic curve defined over K, the base point must be a rational point. I tried to use ...
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Automorphism of m-torsion subgroup of an elliptic curve determines the automorphism of the entire elliptic curve

For $m\ne2$ I want to show that if two automorphisms coincide on $E(m)$, which is the $m$-torsion subgroup of the elliptic curve $E$, then these automorphisms are the same. The statement is very ...
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1answer
45 views

The rank of elliptic curves of the form $y^2=x^3+ax$

I am looking for references of the following two questions: 1) For with class of primes the rank of the elliptic curves $y^2=x^3+px$ is exactly $0,1$ or $2$. It was quite easy to show that the rank ...
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1answer
78 views

The degree of a principal divisor

I've become extremely confused (due to having no experience with varieties) over a remark (Remark 3.7) Silverman makes in his book The Arithmetic of Elliptic Curves. Here's the relevant background: ...
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52 views

Can I choose the base point of an elliptic curve arbitrarily?

If I define an elliptic curve as a smooth curve with genus one and with base point $\mathbb O$, it seems that I can choose this base point arbitrarily. When I go through the proof that establish ...
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1answer
20 views

Reflect point in Group law on elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve. When we add two points on an elliptic curve, we take the line joining them, take the third intersection point and then reflect the point and use that as the ...
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2answers
61 views

Why should the fibers of a surjective morphism of curves be finite?

Let $\phi : C_1 \rightarrow C_2$ be a nonconstant (and therefore surjective) morphism of smooth curves, and let $\phi^* : K(C_2)\rightarrow K(C_1)$ be the pullback homomorphism it induces on function ...
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0answers
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how to prove that multiplication of points of an elliptic curve can be done with division polynomials

I'm trying to solve an exercise in the book The Arithmetic of Elliptic Curves by Joseph H. Silverman on the page 106. The exercise asks to prove that \begin{equation} nP=\left( ...
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1answer
58 views

Meromorphic function written as Weierstrass Elliptic Function [closed]

Let $\Lambda$ be a lattice in the complex plane. And Weierstrass Elliptic Function $$\wp(z)=\frac{1}{z^2}+\sum_{\omega \in \Lambda - \{0\}}\frac{1}{(z-\omega)^2}-\frac{1}{\omega ^2}$$ How can I ...
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24 views

advice for curve fitting

I have numerically obtained some curves, corresponding with it I have also obtained some roots. I strongly believed these curves can be fitted with some (elliptic) functions taken the roots as ...
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1answer
50 views

Interpretation of enhanced elliptic curves

In "A first course in modular forms" (Diamond-Shurman) the author defines something called an 'enhanced elliptic curve' for the congruence subgroups $\Gamma_0(N), \Gamma_1(N)$ and $\Gamma(N)$. For ...
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Galois cohomologies of an elliptic curve

I am studying basic theory of elliptic curves. I encountered Galois cohomology. But two introductory textbooks I read used only $H^0$ and $H^1$. I am curious why higher cohomologies did not appear. I ...
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1answer
23 views

Elliptic Curves and Mod P

I'm trying to figure out why the number of points (Np) equals any Prime (P) when: P (is congruent) 2 (mod 3) To the Elliptic Curve y^2=x^3+17 Does anyone know why this is?
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Why are supersingular elliptic curves useful for cryptography?

I don't know very much about cryptography and would like to learn more. I know the basics of RSA alogrithm and how elliptic curves over finite fields can be used to do something similar. But I would ...
2
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2answers
127 views

rational number solutions to $\frac{a}{a^2+1} + \frac{b}{b^2+1} = \frac{c}{c^2+1}$ with $abc\ne 0$

This question concerns the equation $$\frac{a}{a^2+1} + \frac{b}{b^2+1} = \frac{c}{c^2+1}$$ and the possibility of rational number solutions with $abc \ne 0$. In comments arising from: Using ...
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Surjectivity of good reduction maps of elliptic curves

For simplicity, Let $E/\mathbb{Q}$ be an elliptic curve with good reduction, call it $E'$, at $p$. We know that the reduction map $E(\mathbb{Q}_p)\to E'(\mathbb F_p)$ is surjective, but ...
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32 views

Elliptic curve of points order 4

So how do you find the points on the eliptic curve $y^2=x^3+ax$ of order 4, where $4\mid a$ but $4^n$ does not divide $a$ for $n>1$. We proved that for $(x,y)=2(u,v)$, we must have ...
2
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1answer
94 views

Using Modularity Theorem and Ribet's Theorem to disprove existence of rational solutions

This is likely overly optimistic, but can one take the results from the Modularity theorem and Ribet's theorem, and distill these down to an undergrad math level of a way to check if certain rational ...