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

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4
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1answer
404 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
350 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
108 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
88 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
526 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
327 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
317 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
74 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
169 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?
6
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3answers
347 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
117 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
657 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
180 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
426 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 ...
4
votes
1answer
791 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
77 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
70 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
181 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 ...
14
votes
1answer
186 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
77 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
729 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
165 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
126 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 ...
4
votes
2answers
2k 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
1k 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
247 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 ...
8
votes
2answers
219 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
92 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 $\mathcal{O}_K$ ...
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0answers
236 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
107 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 ...
3
votes
2answers
754 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
90 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
88 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
87 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
167 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
115 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
869 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 ...
6
votes
2answers
419 views

Find the rational points on $1 + 18 x + 81 x^2 + 44 x^3 = y^2$ with Sage

I'm trying to use Sage on-line,but I meet some trouble with the code of it. I want to find the rational points on an ellipse curve,such as $$1 + 18 x + 81 x^2 + 44 x^3 = y^2,\tag1$$ I know that ...
2
votes
1answer
100 views

Elliptic curve, number of elements of finite order

I've been looking at this problem for some time now and just can't seem to get the right idea. The problem is: Consider the elliptic curve $C:y^2=x^3+bx$ defined over the rational numbers with $b$ a ...
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3answers
890 views

Can you recommend some books on elliptic function?

I plan to study elliptic function. Can you recommend some books? What is the relationship between elliptic function and elliptic curve?Many thanks in advance!
23
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3answers
753 views

Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$

Let $a,b,c \in \mathbb N$ find integer in the form: $$I=\frac{a}{b+c} + \frac{b}{c+a} + \frac{c} {a+b}$$ Using Nesbitt's inequality: $I \ge \frac 32$ I am trying to prove $I \le 2$ to implies ...
1
vote
1answer
214 views

Split multiplicative reduction question

Let $E/\mathbb{Q}$ be an elliptic curve and $E_{d}$ be the quadratic twist of $E$ by a squarefree integer $d$. Let $\ell$ be a prime of multiplicative reduction for $E$. If $(d, \ell) = 1$, then ...
2
votes
1answer
125 views

Showing an elliptic curve has infinitely many points over $\mathbb{Z}_p$

I stumbled upon this question, and I can't think of how to do it, or what kind of results to use. The question is as follows: Let $$y^2=x^3+ax+b$$ be an elliptic curve ($a,b$ integers), and let $p ...
5
votes
1answer
202 views

Family of elliptic curves with trivial torsion

I'm wondering, if it is true that the torsion subgroup of $y^2=x^3+p$ (for $p$ some prime, greater than 2), is always trivial?. I was trying to prove this using Lutz-Nagell, but I can't quite get it. ...
2
votes
1answer
166 views

About fibers of an elliptic fibration.

Consider the following pencil of cubics: $\lambda C_1+ \mu C_2$ where $C_1=y^2z$ and $C_2=x(x^2+2xz+z^2)$ and the elliptic fibration $\tilde X \rightarrow \mathbb P^1$ induced by the blow-up of ...
7
votes
1answer
227 views

Surfaces ruled over elliptic curves

Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve. Suppose $E$ is an elliptic ...
5
votes
1answer
164 views

Abelian Elliptic Surfaces

By abelian surface we mean a 2-dimensional algebraic complex torus. Thus $$ S=\Bbb{C}^2/\Gamma$$ where $\Gamma$ is a rank $4$ lattice in $\Bbb{C}^2$ and such that $S$ is algebraic. It has trivial ...
2
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0answers
46 views

Question about paper on Selmer groups

Let $\textrm{Sel}_{n}E$ denote the $n$-Selmer group and $\textrm{Sel}_{p^{\infty}}E = \varinjlim_{n}\textrm{Sel}_{p^{n}}E$. Proposition 5.10 of this paper http://arxiv.org/abs/1304.3971 states that ...
2
votes
1answer
195 views

Modularity theorem and some results

Let $C$ be an elliptic curve over rationals. Then we can attach to $C$ an L-series $L(C,s)$. I read about the Modularity theorem http://en.wikipedia.org/wiki/Modularity_theorem In the section ...