Questions tagged [elliptic-curves]

Elliptic curves are objects of algebraic geometry met in somewhat advanced parts of number theory. They also appear in applications to cryptography. Use the tag, if this applies. Questions on ellipses should be tagged [conic-sections] instead.

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What is the idea behind the proof of the Isogeny theorem and Theorem III.7.9 (Serre) in Silverman's book?

Let $E_1$ and $E_2$ be Elliptic curves over the field $K$ and $\ell\neq\mathrm{char}(K)$ be a prime number. Let $T_\ell(E_i)$ is the Tate module of $E_i$, $i=1,2$ and $\mathrm{Hom}_K(T_\ell(E_1),T_\...
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Function Fields & Ring of regular functions

From here - https://crypto.stanford.edu/pbc/notes/elliptic/funcfield.html This leads us to define the ring of regular functions of $E$ to be $K[E] = K[X,Y]/\langle f\rangle$ Its field of fractions $...
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Field over which CM endomorphism is defined

Let $E$ be an elliptic curve with coefficients over some number field $K$. Is it true that if $E$ has complex multiplication by $\mathbb{Q}[\sqrt{-D}]$, then any endomorphism $\phi: E \rightarrow E$ ...
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Differential of Elliptic curve with bad reduction

Consider the elliptic curve defined by the Weierstrass equation $$E:y^2 = x^3+2$$ This defines a minimal Weierstrass equation for $p=2$ over $\mathbb{Q}_{2}$. It moreover has bad reduction at $p=2$. ...
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Equality of divisors under Galois action in Silverman

I am trying to understand a formula in Silverman's The Arithmetic of Elliptic Curves. Let $K$ be a field, $\overline{K}$ an algebraic closure of $K$, and $C$ be a curve defined over $K$, and $\sigma \...
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Applications of the Modularity Theorem

Besides its role in proving Fermat's Last Theorem and its well-known consequence that the $L$-function of an elliptic curve is defined at 1 (so that, in particular, the BSD Conjecture makes sense), ...
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Hilbert's Theorem 90 for polynomial rings

I have seen here: Silverman *Arithmetic of Elliptic Curves* Problem 1.12 (a) that $H^1(G_{\overline{K} / K}, \overline{K}^+) = 0$ implies that $H^1(G_{\overline{K} / K}, I(V)) = 0$, where $I(V)$ is a ...
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Proving that the first Galois cohomology group is direct limit of finite quotients

This question comes from Silverman's Arithmetic of Elliptic Curves, specifically the appendix on Galois cohomology. I am a cohomology beginner, interested (for now) in understanding just enough to get ...
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Dual isogeny of $F$ if $[n]-[m]F=0$.

I was trying to solve the following exercise of elliptic curves: Let $F$ be an isogeny such that there exists $n,m\in\mathbb{N}$ with $[n]-[m]F=[0]$, then $F=\hat{F}$, where $\hat{F}$ is the dual ...
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Elliptic curves over finite fields

We have a point $P$ on the elliptic curve $E$ over the finite field $F_q$. Consider the point on the curve $mP$ where $m$ is in $F_q$. What does multiplying $mP$ with the inverse of $m$ mean, i.e. $m^...
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Fermat curve in Weierstrass form.

In this post (Substitutions that transform Fermat Equations to Elliptic Curves) it is proved that there exists a change of variables that trasform Fermat's curve $X^3+Y^3+Z^3=0$ into $y^2=x^3-432$, ...
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Understanding “Notes on Elliptic Curves I”

Attempting to read the above paper by Birch and Swinnerton-Dyer. For reference, I’ve read to chapter 12 in Lawrence Washingtons book on elliptic curves but cannot understand Lemma 1 and Lemma 2 in the ...
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Given a fixed hash value (h) and private key (p), is it possible to find a nonce (k) that fits the following equation in ECC mathematics?

Assuming secp256k1 curve and ECDSA parameters, I'm trying to see if there's a way to solve for $k$, where: $k = {-h\over r} -p$, where $k$ is the ECDSA nonce, $p$ is the private key, $h$ is the hash ...
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Why is set of elliptic curves Zariski open in parametric space $\mathbb{P}^9$?

I'm having trouble seeing why set of elliptic curves is Zariski open in the parametric space $\mathbb{P}^9$ of all cubics. I will be grateful to receive any help on this.
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Calculating divisor of function on elliptic curve

I read Pairings for Beginners by Craig Costello. In the example 3.1.1 at 37-th page we consider $ E/F_{103} : y^2 = x^3 + 20x + 20$, with points $ P = (26, 20), Q = (63, 78), R = (59, 95), T = (77, 84)...
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Galois representation of an elliptic curve over a function field of char p.

Let $E/K$ be a non-isotrivial elliptic curve over a function field of characteristic $p$. Let $$\rho_{E,l}:Gal(K^{sep}/K)\to GL_2(\mathbf{Z}_l)$$ For $l\neq p$, the image of $\rho_{E,l}$ is a $l$-adic ...
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Prove 3-torsion in characteristic 2 field is isomorphic to $\mathbb{Z}_3 \times \mathbb{Z}_3$ [duplicate]

From washington's book exercise 3.2. I have to show that $E[3] = \mathbb{Z}_3 \times \mathbb{Z}_3$ where $E$ is defined over a characteristic 2 field. Given the generalized Weierstrauss equation $y^2 +...
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Characteristic polynomial of endomorphism of the Tate module of an elliptic curve.

In Milne's book Elliptic Curves, he states (Corollary 3.23) that for any endomorphism $\alpha$ of $E$, we have the following facts about the induced endomorphism $\alpha$ of the Tate module $T_\ell(E)$...
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Question on elliptic curve of Weierstrass form $y^2 = x^3+ax+b$: Any class there?

I want to present a brief question. I'm curious whether there is any class of Weierstrass form $y^2 = x^3+ax+b$ that we can assign them as rank $0$ by some particular property. In other words, is ...
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Show that $\int_0^{\pi/2} \frac{dt}{\sqrt{\sin t}}=\int_0^1 \frac{dx}{\sqrt{x-x^3}}$

This is intended to be an example of an elliptic integral but I'm not sure how to go about showing it. I'm not sure which identities to start with, any tips would be greatly appreciated!
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Elliptic curves are one dimensional abelian varieties

I am trying to prove that the group law on an elliptic curve induces abelian variety structure. Now I am aware of the following group structure of an elliptic curve, $$X(k)\longleftrightarrow \text{...
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Are all algebraic sets generated by a single irreducible element varieties?

I am reading Silverman's Arithmetic of Elliptic Curves, and his early examples of (affine algebraic) varieties are confusing me. This is because he defines algebraic varieties as algebraic sets $V$ ...
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If $E/\mathbb{Q}$ is an elliptic curve and $j(E)=0$ then $y^2=x^3+D$ for some $6$-th power free integer $D$.

I am self studying Silverman's arithmetic of elliptic curves book and I am trying to solve exercise 10.19. It has four parts, and I was able to do everything but the first one. Here is the problem: ...
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Lang's proof concerning ray class fields of imaginary quadratic number fields

In Lang's book Elliptic Functions, he shows how to generate the ray class field with conductor $N$ of an imaginary quadratic number field $k$ using the $j$-invariant of an elliptic curve $A/\mathbb{C}$...
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Is the Weierstrass $\wp$-function compatible with automorphisms of $\mathbb{C}$?

Let $\Lambda$ be a lattice in $\mathbb{C}$ and $\sigma\in\mathrm{Aut}(\mathbb{C})$. Is it true that \begin{equation} \tag{$*$} \sigma(\wp(z; \Lambda))=\wp(\sigma(z); \sigma(\Lambda))? \end{equation} ...
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$4$-torsion points of elliptic curve in field extension of odd degree

Let $E$ be an elliptic curve over some field $k$ of characteristic not $2$. Let $m$ be the maximal number of $4$-torsion points of $E(L)$ where $L$ ranges over all finite field extensions of $k$ of ...
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Birationally equivalent elliptic curves and singularities

I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3 \alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma -\beta ^2$ for known ...
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On the definition of an elliptic divisibility sequence

I have been learning about elliptic divisibility sequences, I started reading this article on Wikipedia (https://en.wikipedia.org/wiki/Elliptic_divisibility_sequence#:~:text=In%20mathematics%2C%20an%...
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Mordell-Weil rank growth in Iwasawa tower

This is more of a reference request in case anyone can direct me to the right literature. If you have an elliptic curve $E/\mathbb Q$, and you consider the $\mathbb Z_p$ extension, $\mathbb Q_{\infty}$...
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How to understand the statement "two (very) general elliptic curves are not isogenous"

Given two elliptic curves $E_1$ and $E_2$ over $\mathbb{C}$. How to understand the statement "two (very) general elliptic curves are not isogenous"? Question 1: Should it be "general&...
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Reductions of elliptic curves over number fields (implementation on Sage)

I have found two elliptic curves over a number field whose reductions are isomorphic to supersingular elliptic curves (used Deuring Lifting Theorem). How can I check which reduction is isomorphic to ...
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Calculate the divisor of the differential $dx/y$ on $C$ and use the result to show that $C$ has genus $g$.

(Silverman's The Arithmetic of Elliptic Curves Exercise 2.14) For this exercise we assume that char K $\neq 2$. Let $f(x)\in K[x]$ be a polynomial of degree $d\geq 1$ with nonzero discriminant, let $...
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Give me one non-isotrivial elliptic curve over $\mathbb{F}_2(t)$ with supersingular reduction at some place

I would like the equation of a non-isotrivial elliptic curve over the rational function field $\mathbb{F}_2(t)$ with exactly one place of supersingular reduction and I would like to know which place. ...
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Confusion in showing [2] is a rational map between elliptic curves

Let $E:y^2=x^3+ax+b$ be an elliptic curve over $\mathbb{C}$ with the identity $\mathcal{O}=(0:1:0)$. It is well known that if $P=(x,y)$ (affine coordinate), then $2P=(f(x,y),g(x,y))$ where $$ f(x,y) ...
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prime index coefficients of $L$-functions of Elliptic Curves over $\mathbb{Q}$ and analytical rank [closed]

I have searched without success if there was any known relationships or conjectures between the rank and the prime index coefficients of the $L$-functions of a modular elliptic curves. Anyone has a ...
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How to change coordinates so that $\pi|a_3,a_4,a_6$ in the second step of Tate's algorithm.

I am trying to apply Tate's algorithm for the elliptic curve $$y^2-2(7T+3)xy-36(7T+3)y-x^3-2(16T^2-7)x^2+324x+648(16T^2-7)$$ over the rational function field $\mathbb{Q}(T)$ at the primes $\pi=T+1,T-1,...
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Is elliptic curves an group object in some category?

Is elliptic curves an group object in some category? Group like object, like lie group,formal group, algebraic group,topological group, is sometimes define as group object in some category. What ...
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Clarification in Silverman's proof of the descent theorem

In the book by Silverman called The Arithmetic of Elliptic Curves, there is the Descent Theorem (Theorem 3.1). He proves the theorem and allong the way he writes $$h(P_{n}) \leq \left( \frac{2}{m}\...
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How to compute the $j$-invariant corresponding to a given maximal order in $B_{p,\infty}$

Let $B_{p,\infty}$ is the rational quaternion algebra ramified at $p$ and $\infty$. By Deuring's correspondence, there is a one to one correspondence between maximal orders in $B_{p, \infty}$ up to ...
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Why is 2-descent called "descent"?

My understanding is 2-descent today means calculating $E(\mathbb Q)/2E(\mathbb Q)$ by computing the Selmer group and trying to figure out which curves of the Selmer group actually have a K-point. (See ...
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supersingular elliptic curve group structure

I am trying to understand the basic idea behind supersingular isogeny cryptography. With very little knowledge about the group theory and elliptic curve, I find it very hard to thoroughly understand ...
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Bringing an Elliptic curve in homogenous form to Weierstrass form

I have a family of curves given by $F(U,V,W)= U^3 +V^3 + W^3- 3\lambda UVW$ in $\mathbb{P^2C}$, with an origin $O = [1,-1,0]$. I am struggling to bring this to the form $y^2 = x^3 - ax +b$ as I can't ...
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What is the proof of - $E[2]$ torsion group is isomophic to $Z_2 \oplus Z_2$?

This is from the book "Elliptic Curves" by Lawrence Washington. $E$ is an elliptic curve over $K$ $E[n] = \{P \in E(\bar K) \mid nP = \infty\}$ where $\bar K$ is the algebraic closure of $K$ ...
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Elliptic point of surfaces

Why does a point $p$ in a surface $S \subset \mathbb{R}^3$ such that the Gaussian curvature is positive is called elliptic point?
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Second Galois cohomology of elliptic curves over number fields

Let $E$ be an elliptic curve over a number field $K$. Do I understand correctly that the second Galois cohomology of $E$ is a product of $\mathbb{Z}/2\mathbb{Z}$ where each factor corresponds to a ...
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$(x_1, y_1) = \infty \implies x = -y$ with elliptic curve of a cubic equation

In Lawrence Washington's book on Elliptic Curves section 2.5.2 on Cubic Equations, starting from $x^3 + y^3 + z^3 = 0$ such that $xyz \neq 0$, he derives $$\frac{x}{z} = u + v, \qquad \frac{y}{z} = u -...
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Why would you have an Elliptic Curve where coordinates of points lie on a different group as opposed to the field over which the curve is defined?

Normally, when you define an Elliptic Curve over a field $F$ - it means that the coefficients of the Curve equation lie in field $F$. However, in most cases, I have seen that even the $x$ & $y$ ...
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Where does this theorem of Ogg appear?

The webpage http://www-personal.umich.edu/~asnowden/teaching/2013/679/L20.html claims that Ogg proved that the order of $[0] - [\infty]$ on $J_0(N)$ is $(N-1)/\gcd(N-1, 12).$ I cannot find any paper ...
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Kernel of the local-global map for first galois cohomology of $m$-torsion group

Let $E$ be an elliptic curve over a number field $K$. Are there examples when the local-global map $$H^1(K,E[m])\to \prod_\nu H^1(K_\nu,E_{K_\nu}[m])$$ has a nontrivial kernel? I know that if we ...
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A question on the $2$-torsion points of a elliptic curve

I am reading Saito's Fermat's Last Theorem: Basic tools and on page 15, it was claimed that Proposition 1.4. Let $K$ be a field with char($K$) $\neq 2$, and let E be an elliptic curve over $K$. Then ...
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