Tagged Questions

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

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10
votes
1answer
992 views

Explicit Derivation of Weierstrass Normal Form for Cubic Curve

In page 22-23 of Rational Points on Elliptic Curves by Silverman and Tate, authors explain why is it possible to put every cubic curve into Weierstrass Normal Form. Here are relevant pages: (My ...
0
votes
1answer
66 views

This proof is completely unclear for me

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. I find the following proof in an old Russian book: I want to ...
4
votes
1answer
196 views

Rational Points on Elliptic Curves

I have this homework problem: Can there be an elliptic curve, view as a projective curve, with no rational points with at least one 0 as a coordinate?
0
votes
2answers
228 views

Subtraction two points on elliptic curve.

Suppose Q, T and S are three points on an elliptic curve, such that Q+T = S. With knowing Q and S, can we compute T? In other word whether exists subtraction operation on elliptic curve, or not?
0
votes
0answers
52 views

Addition, Subtraction points on an elliptic curve with integer values

I see in some papers that their authers claimed that with operation such as XOR, Addition and subtraction, hide an integer value by a point on an elliptic curve. to clarify, suppose Q is a point in ...
5
votes
1answer
53 views

How to relate the valuation of x/y (For a minimal Weierstrass equation)

I'm reading an article about elliptic curves, but since I'm not very experienced on this subject, I ended up getting stuck. The problem starts as: "Let $K/\mathbb{Q}$ be a number field and $E/K$ an ...
1
vote
1answer
65 views

About the quotient group of degree zero divisors on $C$ by the principal divisors on $C$

Let $C$ be an elliptic curve with distinguished point $O$. My question is about a mathematical desription of this set denoted by $Pic(C)$ which is the quotient group of degree zero divisors on $C$ by ...
5
votes
1answer
278 views

On Bachet's Duplication Formula and the number $-432$

While reading "Rational Points on Elliptic Curves" by Silverman and Tate, I came across this interesting passage about Bachet's duplication formula: I know how to derive Bachet's duplication ...
0
votes
0answers
61 views

Addition a point on an elliptic curve with an integer value

Suppose $Q$ is a point in an elliptic curve such that $Q=dP$ and $d$ is an integer value, and $P$ is base point of that elliptic curve. Note $Q = dP$ means that $P+\cdots+P$ for $d$ times** and since ...
7
votes
0answers
75 views

Ex. $2.30$ in Silverman Adv. Topics

I would like to refer you to Exercise $2.30(c)$ in Silverman's Advanced Topics in Elliptic Curves. Question: Let $E/L$ be an EC with CM by $K$. Assume that $K\nsubseteq L$, and let $L'=LK$ and let ...
0
votes
2answers
69 views

Can one explain to me this Theorem

Can one explain to me Theorem 2. (page 3) in this link: http://www.math.leidenuniv.nl/~evertse/siksek-modular.pdf I am but confused about the nature the bijection defined in that result.
4
votes
1answer
322 views

Square root in Characteristic 2 Field

Let $K$ be a field of characteristic 2. For each $a\in K$, can we always find some $x$ such that $x^2=a$? I came upon this question while reading "Arithmetic of Elliptic Curves". The original ...
9
votes
2answers
229 views

Making an elliptic curve out of a cubic polynomial made a cube, or $ax^3+bx^2+cx+d = y^3$

What is the transformation such that a general cubic polynomial to be made a cube, $$ax^3+bx^2+cx+d = y^3\tag{1}$$ can be transformed to Weierstrass form, $$x^3+Ax+B = t^2\tag{2}$$ (The special ...
1
vote
1answer
89 views

Diffie-Hellman key exchange for three user.

Assume that there are three users that have their own secret key $d_i$ and corresponding public key $Q_i = d_i G$ such that $Q_i$ is a point in an elliptic curve. Now I'm looking for a solution to ...
3
votes
1answer
86 views

Why the author choose $s$ real?

My question is: Why the author of this book (http://wstein.org/books/bsd/bsd.pdf) page 8 on Sec 1.4 choose $s$ real in despite that the variable is complex in the entire chapter. I am very confused ...
0
votes
1answer
264 views

Point Division in Elliptic Curve Cryptography?

I want to implement a crypto protocol using Elliptic Curve Cryptography. However, it requires a division which I cannot handle. In multiplicative notation, it requires: Let $\mathbb{G}=\left ...
6
votes
1answer
234 views

What do the involutions of an elliptic curve look like?

Every automorphism $\varphi \in \mathrm{Aut}(E)$ of an elliptic curve $E$ (with base point $O$ over a field $k$) can be written $\varphi = \tau_Q\phi$ where $\phi \in \mathrm{Aut}(E,O)$ is an isogeny ...
1
vote
0answers
151 views

Twist of elliptic curve

It is continuation of this question: explict form of the equation of elliptic curve Let $p$ is prime and $p = 3 ($mod $4)$. $q = p^n$. It is easy to see that $E: y^2 = x^3 + x$ has $1 \pm 2q + q^2$ ...
1
vote
1answer
59 views

Ireland-Rosen Hecke Character for $y^2=x^3-Dx$

I would like to refer you to page $310$ of Ireland-Rosen: A Classical Intro to Modern Number Theory. Firstly, to construct the Hecke character, it is enough to specify $\chi(P)$ for prime ideals $P$ ...
6
votes
1answer
137 views

explict form of the equation of elliptic curve

Let $E(\mathbb{F}_{q^2})$ is elliptic curve with #$E(\mathbb{F}_{q^2}) =q^2 + q + 1$. Can we write equation of this curve in the explicit form?
5
votes
3answers
202 views

Purpose of cusps

In the theory of of modular forms, there is the set of of cusps defined by $\mathbb{P}^1 (\mathbb{Q})= \mathbb{Q} \cup \{\infty\}$. For an subgroup $\Gamma < \text{SL}_2(\mathbb{Z})$ of finite ...
3
votes
1answer
102 views

$q$-expansion of Modular forms

I am trying to compute the $q$-expansion of $g\theta_2$ and $g\theta_4$, the $q$-expansion of modular forms of weight $3/2$ and level $128$ and trivial character and character $\chi_8$ respectively. ...
2
votes
2answers
419 views

The curve $y^2 = f(x)$ where $f$ has degree $d$ and no repeated roots has genus $[(d-1)/2]$?

Let $f$ be a polynomial in $x$ of degree $d$ (over $\mathbb{C}$, say) without repeated roots. I've heard that the curve $y^2 = f(x)$ has genus $[(d-1)/2]$, but I can't find a proof. To be more ...
1
vote
1answer
174 views

What is the amount of abstract algebra needed to study elliptic curves?

To be more specific, how much is needed to understand the book 'Rational points on elliptic curves' by Silverman?
3
votes
1answer
257 views

Real Period of an Elliptic Curve

Trying to work out what the real period of an elliptic curve is as seen in the Birch Swinnerton-Dyer conjecture. From what I've read, given an elliptic curve E over the rationals, one can associate ...
3
votes
1answer
371 views

Adding points of an elliptic curve over a finite field

I'm a bit confused with how fractions are handled with adding points of elliptic curves over finite fields. Below is an example from the text which I am trying to understand: The part that ...
4
votes
2answers
70 views

Other names for $E_{p+1}$ $\pmod{p}$?

If I want to know properties of $E_{p+1}$ modulo $p$, do you know a name for this modular form, so that it is easier to search via the internet? So far, what I know is that $E_{p-1}$ is the Hasse ...
3
votes
0answers
61 views

Weil operator of elliptic curve

Let $V$ be a $1$-dimensional $\mathbb{C}$-vector space and $\Lambda \subset V$ be an elliptic curve (=lattice). Let $C : V \rightarrow V$ be the multiplication by $i$. Consider the two following ...
3
votes
1answer
133 views

Division polynomials of elliptic curves

This is exercise 3.7 from Silvermans AEC (2nd edition). Let $E$ be a nonsingular elliptic curve over $\mathbb{C}$ given by $$ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ The $n^{th}$ division polynomls ...
0
votes
0answers
175 views

Analogy between Picard group and Ideal class group

Can you give a reference where the conformity between Picard group and Ideal class group is explained? What is analogy of Picard group of elliptic curve over finite field?
12
votes
1answer
152 views

Galois Properties of the Values of Modular Forms

Let $f \in S_0(N)$ be a normalized Hecke eigenform. It is well known that its coefficients are algebraic integers, and $f^\sigma$ lies in $S_0(N)$ for $\sigma \in G_{\mathbb{Q}}$. At CM points $z \in ...
4
votes
1answer
69 views

Hecke $L$-series exercise in Silverman's Advanced Topics in Arithmetic of EC

I would like to refer you to 2.30 & 2.32 in Silverman's book Advanced Topics in the Arithmetic of Elliptic Curves. 2.30(b)[(c) in errata]: Suppose $\mathfrak{P}$ remains inert in $L'$, say ...
5
votes
1answer
637 views

How do I show that this curve has a nonsingular model of genus 1?

Let $C$ be the projective closure of $Z(f) \subset \mathbf{A}^2$ where $f$ is an irreducible polynomial of degree 4 in $x$ and degree 2 in $y$, so $C = Z(f^*) \subset \mathbf{P}^2$ where $f^*$ is the ...
6
votes
1answer
99 views

transform into weierstrass-form

How can I transform the elliptic curve $E/\mathbb{C}$ of the form $$y^2=4(x-e_1)(x-e_2)(x-e_3)$$ with $e_1>e_2>e_3\in\mathbb{R}$ roots of $E$ into a Weierstrass-Form $y^2=x^3+ax+b$?
5
votes
1answer
118 views

Show that an ideal is unramified

See Advanced Topics in elliptic curves for the full question(see also errata: http://www.math.brown.edu/~jhs/ATAEC/ATAECErrata.pdf): 2.30 (pg 184) Given $E/L$ an elliptic curve with complex ...
3
votes
2answers
838 views

Point addition on an elliptic curve

I have an elliptic curve $y^2 = x^3 + 2x + 2$ over $Z_{17}$. It has order $19$. I've been given the equation $6\cdot(5, 1) + 6\cdot(0,6)$ and the answer as $(7, 11)$ and I'm unsure how to derive ...
1
vote
1answer
613 views

Doubling a point on an elliptic curve

I've a programming background and am just about to get into a project where Elliptic Curve Cryptography (ECC) is used. Although our libraries deal with the details I still like to do background ...
9
votes
0answers
168 views

Hecke Characters

My question today concerns Hecke characters or Größencharaktere. I'm doing a project on complex multiplication of elliptic curves and need some help understanding Hecke characters. My main ...
7
votes
1answer
142 views

Completion along zero section of an elliptic curve.

I am trying to understand the intuition that I should have about the formal group of an elliptic curve. Say that I have an elliptic curve $E\to \text{Spec} R$ for some ring $R$, with section $0\colon ...
3
votes
1answer
79 views

How do infinite series contain “local” information?

I would like to know why we consider infinite series (Dirichlet series, zeta function, elliptic curve $L$-series) or their Euler product. How is the local information "stored/contained" in the ...
4
votes
2answers
84 views

Silverman Adv. Topics example

I would like to refer you to Silverman's Advanced Topics in the Arithmetic of Elliptic Curves example 10.6: Let $D$ be a nonzero integer, $E:y^2=x^3+D$ with complex multiplication by $\mathcal{O}_K$ ...
1
vote
0answers
233 views

Solving equation

Assume that $a$, $b$, $c$ and $d$ are known value integers , and $P$, $Q$, $R$ and $G$ are known value points on an elliptic curve with these equations: $$a = by+x, \\ c=\frac{1}{y}d +z , \\ P= ...
1
vote
0answers
65 views

egg curve estimation

Let $p_{1...3}$ be three points on an ellipse, and $t_{1...3}$ be their tangent lines. For $i={1..2}$, let $M_i$ be the point of intersection of $t_i$ and $t_{(i+1)\%2}$, and $K_i$ be the midpoint of ...
2
votes
2answers
417 views

Why is this curve nonsingular?

Let $C$ be the projective closure of $Z(f) \subset \mathbf{A}^2$ where $f$ is an irreducible polynomial of degree 4 in $x$ and degree 2 in $y$, so $C = Z(f^*) \subset \mathbf{P}^2$ where $f^*$ is the ...
0
votes
0answers
82 views

Is Hash(bG) equal to b(Hash(G))?

Assume b is an integer, G is a basepoint in an elliptic curve, and Hash is a one-way hash function. Is Hash(bG) equal to b(Hash(G)) ? or not? Note: A hash function is any algorithm or subroutine ...
6
votes
1answer
80 views

Reduction of kernel of isogenies in the CM case

Let $F$ be a number field and $E/F$ an elliptic curve with CM by an order $\mathcal{O}$ in a quadratic imaginary field $K$. Let us suppose that $K\subseteq F$. Let $p$ be a prime that splits in ...
0
votes
1answer
82 views

Addition on elliptic curves

assume $a$, $b$ are two integer numbers, and $G$ is a basepoint in an elliptic curve. Is $(a+b)G$ equal to $aG+bG$ or not?
5
votes
1answer
136 views

Finding the completion of a coordinate ring

Consider $A=\mathbb C[x,y]/(y^2-x(x+1))$, and consider the $\mathfrak m$-adic completion, where $\mathfrak m =(x,y)$. I want to show that this completion is isomorphic to $\mathbb C[[u,v]]/(uv)$, ...
3
votes
1answer
98 views

Write an elliptic curve with coefficients only depending on its j-invariant

Let $$E:y^2 = 4x^3-g_2 x - g_3$$ be an elliptic curve and $$j=\frac{g_2^3}{g_2^3-27 g_3^2}$$ denote to its $j$-invariant. I want to transform $E$ to find $f$ and $g$ s.t. ...
11
votes
2answers
769 views

Is there a more elementary proof of this special case of Riemann-Roch?

I'm looking for an elementary proof of the fact that $\ell(nP) = \dim L(nP) = n$, where $L(nP)$ is the linear (Riemann-Roch) space of certain rational functions associated to the divisor $nP$, where ...