# Tagged Questions

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

0answers
37 views

### 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, ...
2answers
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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 ...
1answer
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### 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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### 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?
2answers
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### What is the intuitive explanation of (non-singular part of) a singular elliptic curve being isomorphic to either $K^{*}$ or $K$?

Is there a rough explanation without using explicit computation?
0answers
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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 ...
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36 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 $p ≡ 3 (mod 4)$...
0answers
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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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125 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.
1answer
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### 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 ...
1answer
120 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 ...
1answer
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### 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 $E(\mathbb{Z}/p^r\mathbb{Z} )$, $E( \mathbb{Z}/p\mathbb{Z} )$...
0answers
26 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 ...
1answer
54 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 $F_p$. ...
1answer
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### (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 ...
0answers
105 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 ...
1answer
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### 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 ...
1answer
41 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 ...
0answers
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 ...
0answers
45 views

### 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 ...
1answer
38 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?
0answers
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 ...
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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2answers
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### 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 ...
0answers
38 views

### 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 ...
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 ...
0answers
55 views

### 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 ...
1answer
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### 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 ...
1answer
100 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: ...
0answers
55 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 ...
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 ...
2answers
67 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 ...
0answers
44 views

### 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 nP=\left( x-\frac{\psi_{n-1}\...
1answer
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### 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 ...
0answers
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### 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 ...
1answer
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### 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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158 views

### 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 ...
1answer
25 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 ...
2answers
132 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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44 views

1answer
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### 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 ...