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

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Prove that the only negative real zeroes are at the integers

Let $$L(C,s)=\prod_{p\mid\Delta}(1-a_{p}p^{-s})^{-1}\cdot\prod_{p\nmid\Delta}(1-a_{p}p^{-s}+p^{1-2s})^{-1}=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse-Weil $L$-function of ...
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1answer
2k 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 ...
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1answer
74 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 ...
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214 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?
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355 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?
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54 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 ...
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1answer
79 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 ...
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1answer
392 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 ...
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66 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 ...
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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 ...
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90 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.
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369 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 ...
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319 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 ...
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1answer
105 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 ...
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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 ...
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1answer
432 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 ...
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294 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 ...
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261 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 $1 \pm 2q + q^2$ ...
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1answer
69 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 $P$ ...
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156 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?
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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 ...
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113 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. ...
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2answers
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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 ...
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1answer
178 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?
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366 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 ...
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1answer
612 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 ...
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2answers
75 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 ...
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67 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 ...
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161 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 ...
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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 ...
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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 ...
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701 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 ...
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141 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 $y^2=x^3+ax+b$?
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124 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 ...
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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 ...
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1answer
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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 ...
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220 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 ...
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196 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 ...
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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 ...
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90 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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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= ...
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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 ...
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615 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 ...
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87 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 ...
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1answer
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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 ...
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1answer
86 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?
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162 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 $\mathbb C[[u,v]]/(uv)$, ...
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111 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. ...
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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 ...
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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 ...