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

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Elliptic Curve Group and Multiplicative Inverse of an element.

Suppose $E$ be an Elliptic Curve over a field $F_q$ and $q=p^n$ where $p=$ prime. We know that the Elliptic Curve group $E(F_q)$ under addition is an Abelian/Commutative Group of order, ...
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Prerequisites for Silverman's Arithmetic of Elliptic Curves

I would like to take a course on elliptic curves using Silverman's Arithmetic of Elliptic Curves next year. I would be taking complex analysis concurrently, but it was listed as a formal prerequisite, ...
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Topics in elliptic curves over finite fields

I have to write a paper on elliptic curves over finite fields and I was wondering if anyone had any ideas of some interesting directions to take this? Like what are some subtopics within this general ...
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Quick question: Tensoring a 2-torsion line bundle with a rank 2 nonsplit extension over a curve of genus 1

Everything is complex algebraic. Over a curve $C$ of genus $1$, let $V$ be a rank 2 vector bundle with $\deg \det(V)=1$, which is a nonsplit extension of $\mathcal{O}_C$ and $\mathcal{O}_C(p)$, where ...
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60 views

Formal Group for the Elliptic Curve $Y^2=X^3+AX$

I'm trying to solve the following problem without resorting to a direct calculation: Let $E : Y^2 = X^3 + AX$, where $A \in \mathbb{Z}$ and $A \ne 0$. Let $F(X, Y )$ be the formal group associated to ...
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The Discriminant Condition for Elliptic Curves [duplicate]

Question: Why do we need the discriminant of an elliptic curve $\Delta=-16(4a^3+27b^2)$ to be nonzero? Motivation: I am aware that when $\Delta=0$, then we obtain either a cusp (e.g. for $y^2=x^3$) ...
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Why is $E[l]\cong\mathbb Z/l\mathbb Z\times\mathbb Z/l\mathbb Z$ for an elliptic curve $E$?

René Schoof's 1995 paper contains the following statement about an elliptic curve $E$ (at the bottom of page 233): [...], we use the subgroup $E[l]$ of $l$-torsion points of $E(\overline{\mathbb ...
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189 views

Integral relations in Fricke and Klein

Can someone please explain how Fricke and Klein obtain the integral relationa stated at the top of p. 34 in this book? The entire book can be previewed on Google Books. It is an old book and I do not ...
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Elliptic curve reduction modulo $p$

While reading an introduction on elliptic curves, I stumbled upon something called reduction modulo $p$. The definition states that we want to create a group homomorphism that maps an elliptic curve ...
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142 views

Modular parametrization of elliptic curve

Let $f$ be a cusp form of weight $2$ on $\Gamma_0(N)$ and assume that $f$ is a Hecke form and a newform. Then, we easily see that $$C(\gamma)=2i\pi \int_{\tau}^{\gamma \tau}{f(\tau')d\tau'} \quad ...
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Large initial solutions to $x^3+y^3 = Nz^3$?

Let $x,y,z$ be non-zero integers. Is it true that the initial or smallest solution (in terms of absolute value) to, $$x^3+y^3 = Nz^3\tag1$$ for $N=94$ is, $$15642626656646177^3 + ...
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Galois representations and isogenies of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$. For each prime $\ell$, the action of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on $E[\ell]$ (the group of $\ell$-division points of $E$) defines a ...
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69 views

When is a modular curve defined over Q?

Let $X(N)$ and $Y(N)$ be respectively the compactified and uncompactified modular curves parametrising elliptic curves with full level $N$ structure. In other words, a point on $Y(N)$ is (essentially) ...
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The canonical height of a point on an elliptic curve

I am struggling with exercise 3.3 in Silverman-Tate Rational Points on Elliptic Curves. Here is the paraphrased problem with necessary background: Let $C:y^2 = x^3 + a x + b$ be a nonsingular cubic ...
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45 views

Birational transformation of Elliptic curves?

Let $F:V\to W$ be a birational transformation of elliptic curves; let $g$ be a generator of $V$. Is necessarily $F(g)$, a generator of $W$?
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How to show rational points of finite order on an elliptic cure are closed under addition

I would like to show that rational points of finite order on an elliptic curve are closed under addition. If $P_1$ and $P_2$ are rational (actually integral) points of finite order, say $nP_1= O$ and ...
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Relation between elliptic curves and Dirichlet L-series

I have read that an elliptic curve $E$ is modular if $a(n) = c(n)$ for all $n$, where $a(n)$ is the $n$-th coefficient in the Dirichlet series of $E$, $L(E,s)$, and $c(n)$ is the $n$-th coefficient in ...
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70 views

Points on an elliptic curve over $\mathbb F_p$

Let $E$ be an elliptic curve over $\mathbb F_p$ (the finite field of $p$ elements) defined by $y^2=x(x-n)(x-m)$ where $p\nmid nm(n-m)$. Let $N$ be the number of $\mathbb F_p$-valued points of $E$. ...
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83 views

Why does an elliptic curve have genus one?

I read that one definition of an elliptic curve goes as follows: Let $k$ be a field. We define the elliptic curve over $k$ be a smooth projective curve $E$ over $k$, isomorphic to a closed subvariety ...
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64 views

Tate-Shafarevich groups and Hasse principle (reference)

I'm looking for a proof of the fact that the Hasse local-global principle holds for an elliptic curve $E$ defined over $Q$ if and only if the Tate-Shafarevich group of $E$ vanishes. I just need to ...
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What is known about the numbers $M_p = \left\vert C(\mathbb{F}_p )\right\vert$?

There is a question (2.4.c) marked ** (to denote "extremely difficult/currently open problem") in Silverman and Tate's Rational Points on Elliptic Curves which I found really interesting and wondered ...
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Why ins't $\mathfrak{h}$ enough to parametrize complex elliptic curves?

this a pretty idiot question and of course there is a mistake in my way of thinking. Let $E$ be a elliptic curve, $E (\mathbb{C}) \cong \mathbb{C} / \Lambda$, where $\Lambda = \langle \omega_1, ...
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How to test if a given elliptic curve has complex multiplication

Is there a general, reasonably easy to understand, algorithm for testing whether an elliptic curve has CM? For example, consider the curve $y^2=x^3+\frac{27}{1727}x+\frac{54}{1727}$ This has ...
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In cryptography, why do we reduce elliptic curves over finite fields?

What's wrong with real numbers? Is the continuous logarithm problem "easy" to solve for elliptic curves? Here's what I believe: elliptic curves over the real numbers have infinitely many points, many ...
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20 views

Multiplication by n on E(K) is surjective

What's the easiest way to see this? I can imagine a proof for $n=2^k$ since for some $P \in E(K)$ you can just move a line intersecting P round the curve till it's tangent, then that point, say $Q ...
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Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms

In Deligne's article in Séminaire Bourbaki "Formes modulaires et représentation $\ell$-adiques" ...
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(hyper) elliptic curve in characteristic two and the Jacobian criterion

Let $k$ be a field of characteristic two and let $E$ be a curve given by $$ y^2=x*(x+1)*(x^2+x+1)*(x^3+x+1)\quad\text{or}\quad y^2=f(x) $$ Now we have $dy^2/dy=2y=0$ and consider the Jacobian ...
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Unable to find any information regarding this fact (Frey, elliptic curves)

Frey states in 'Links between stable elliptic curves and certain Diophantine equations' the following "The most important fact about elliptic curves with reduction of muItipIicative type is due to ...
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Elliptic curves $\mathbb C/\Gamma , \mathbb C/\Gamma'$ are isomorphic iff $\Gamma=\lambda\Gamma'.$

Let, $\Gamma, \Gamma'$ be $lattices$ of $\mathbb C$, define $elliptic$ $curves$ by $\mathbb C/\Gamma , \mathbb C/\Gamma'$, then $\mathbb C/\Gamma , \mathbb C/\Gamma'$ are isomorphic ...
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Singularity of $V(Y^2-X^3-X)\to\mathbb{P}^1$

In "The Arithmetic of Elliptic Curves, in example I.3.7, Silverman define $\Phi:V(Y^2Z-X^3-X^2Z)\to\mathbb{P}^1$ with $\Phi(X,Y,Z)=[Y,X]$. He says that $\Phi$ is not regular at $[0,0,1]$. How to prove ...
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The mod $p$ Galois representation of the Frey curve is unramified away from $2, p$

Given a hypothetical solution to Fermat's last theorem for $p \ge 5$ $$a^p + b^p + c^p = 0$$with $a \equiv -1 \pmod 4$, $b$ even, we can write down the Frey Curve$$E: y^2 = x(x-a^p)(x+b^p)$$which has ...
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Find a point $P$ on an elliptic curve, given $2P$

Let $E$ be the Elliptic curve given by $Y^2=x^3+5x-6$ and suppose $P$ is a point on $E$ over $\mathbb F_{65537}$ with $2P=(7283,24272)$. Find $P$. I approached this question as follows. ...
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Elliptic Curve finding point of a curve backward?

Given $E: y^2=X^3+5X-6$ over $F=(65537)$ with $2P=(7283, 24272)$ how to find $P$ Can anyone provide an example in steps?
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Determine if $E(\mathbb{Q})$ is finite or infinite.

It is given the following algebraic curve: $$ZY^2=X^3+3XZ$$ I want to find the group of rational points of finite order $E(\mathbb{Q})_{\text{torsion}}$ and to determine if $E(\mathbb{Q})$ is finite ...
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Solving cubic equation modulo prime

I'm trying to an algorithm that can solve an elliptic curve equation for constant y: $y^2 = x^3 + ax + b \text{ mod } p$ p is 57 digits long I've tried to solve it using like a regular cubic ...
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If six points of an elliptic curve are contained in a conic, then their sum is $O$.

Let $C$ be a projective cubic without singular points and $O\in C$ an inflexion point. We consider the addition in $C$ with $O$ as neutral element. If $R_{1},...,R_{6}\in C$ are different points such ...
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Q: Deriving lambda and beta values for for an elliptical curve

You can see a little background about this on this bitcointalk post by the late Hal Finney. Beta and lambda are the values on the secp256k1 curve where: λ^3 (mod N) = 1 β^3 (mod P) = 1 In hex, N ...
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Mazur's theorem-abelian torsion group of rational points of an elliptic curve

I am looking at Mazur's theorem... $$E(\mathbb{Q})_{\text{torsion}} \cong \mathbb{Z}/n\mathbb{Z}, \text{ for } n=1,2, \dots ,10,12$$ means that the torsion group $E(\mathbb{Q})_{\text{torsion}}$ ...
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Why do modular curves parametrise elliptic curves?

Let $Y_1(N)=\Gamma_1(N)/H$, where $H$ is the upper half plane. In these lecture notes http://math.uga.edu/~pete/modularandshimura.pdf , the author makes the following statement: "$Y_1(N)$ ...
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Proof of a Proposition regarding the reduction of N-torsion groups on elliptic curves

In Diamond-Shurman A first course in Modular forms p.334 Prop. 8.4.4. It is stated, For E elliptic curve over $\bar{\mathbb{Q}}$ with good reduction at the prime ideal $\mathfrak{p}$ the reduction ...
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Mazur's theorem-abelian group of rational points of an elliptic curve

From Mazur's theorem we have the following: If $E |_{\mathbb{Q}}: y^2=x^3+ax+b, a, b \in \mathbb{Z}$ an elliptic curve, then $$E(\mathbb{Q})_{\text{torsion}} \cong \mathbb{Z}/n\mathbb{Z}, \text{ for ...
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Their product is a cubic of a rational number $x$ minus $x$

It is given the integer $6$. Analyze it into two parts such that their product is a cubic of a rational number $x$ minus $x$. $$$$ Let $y$ be the one factor. The other one is $6-y$. We have ...
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Abelian torsion group of rational points of an elliptic curve

I want to find the abelian group of rational points $E(\mathbb{Q})_{\text{torsion}}$ of the elliptic curve $y^2=x^3+8$. $$E(\mathbb{Q})_{\text{torsion}}=\{P \in E(\mathbb{Q}) | P \text{ of finite ...
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Abelian group of rational points of an elliptic curve

I want to find the abelian group of rational points $E(\mathbb{Q})_{\text{torsion}}$ of the elliptic curve $y^2=x^3-2$. $$E(\mathbb{Q})_{\text{torsion}}=\{P \in E(\mathbb{Q}) | P \text{ of finite ...
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Elliptic curve-point at infinity

In my lecture notes we have the following: $$P \oplus Q \oplus R =O \Leftrightarrow P, Q, R \text{ are collinear }$$ So $$P \oplus Q \oplus O =O \Leftrightarrow Q=-P$$ that means that $Q=-P$ ...
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Division on Elliptic Curve points

For dividing two elliptic curve points(mod some value), we use multiplication and modular inverse. Would the reason be due to division being undefined otherwise? The points of an elliptic curve, yield ...
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Equation of a non-singular cubic curve

The equation of a non-singular cubic curve in affine coordinates is $$y^2+a_1 xy+a_3 y=x^3+a_2x^2+a_4x+a_6 .$$ If $\text{ch } K \neq 2, 3$ then it is written $$y^2=x^3+ax+b .$$ Why do we write it ...
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Genus 2 Elliptic curves & their periods

The first part of my question is just a check of my knowledge on elliptic curves. I'm fairly happy with the number theory side of things (torsions, rank, whatever) but is my understanding of the more ...
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How do I generate group table for elliptic curves over finite fields

Can someone please explain how to generate a group table for an elliptic curve over a finite field? The number of solutions or points are about 16 and it is not possible to do them by adding each ...
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Group tables for elliptic curves over primes

When constructing a group table for an elliptic curve modulo a relatively large prime $p$, say 23, are adding a few points with respect to each other enough to establish symmetry and thereby deduce ...