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

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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 ...
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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 ...
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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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42 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 ...
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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?
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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 ...
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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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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
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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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22 views

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
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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
44 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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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 ...
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2answers
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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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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 ...
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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 ...
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1answer
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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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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
64 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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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( x-\frac{\psi_{n-1}\...
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1answer
60 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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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
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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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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
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 ...
2
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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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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 $E(\mathbb{Q}...
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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 $x=(u^2-a)^2/(...
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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 ...
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The Arithmetic of Elliptic Curves, Exercise 1.3

Let $V\subset \mathbb{A}^n$ be a variety given by a single equation. Prove that a point $P\in V$ is nonsingular if and only if $$\text{dim}_{\bar{K}}M_P/M_P^2=\text{dim}V.$$ For a general variety $V$,...
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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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1answer
88 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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1answer
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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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1answer
43 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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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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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 Fermat'...
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1answer
75 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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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 $\mathrm{...
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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 $E'_{\ell}$...
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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 ...