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

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Constructing a homomorphism from class group to Sha.

In response to my previous question I got a wonderful answer from Prof.Emerton explaining about the similarities between $Ш$ and class group. In order to add something the comments I got from Mr. B R ...
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1answer
325 views

2-Torsion Group Scheme

Consider the elliptic curve $zy^2 + z^2y = x^3.$ I would like to explictly compute the 2-torsion group scheme, $E[2],$ over $\mathbf{Spec}(\mathbb{Z}_2),$ but I'm having a tough time writing down the ...
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1answer
477 views

How to find all rational points on the elliptic curves like $y^2=x^3-2$

Reading the book by Diophantus, one may be led to consider the curves like: $y^2=x^3+1$, $y^2=x^3-1$, $y^2=x^3-2$, the first two of which are easy (after calculating some eight curves to be solved ...
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2answers
245 views

Tamagawa numbers and Genus class numbers

I was reading the paper of Prof.Franz Lemmermeyer titled "Pell-conics" which is here, in that the author writes in page 9 that one can define Tamagawa numbers as $$ c_p = \begin{cases} 2 & \text{ ...
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1answer
387 views

An attempt of catching the where-abouts of “ Mysterious group $Ш$ ”

This question is a bit concerned with the Tate-Shaferevich group, lets start defining $C$ as $$C: X^2- \Delta Y^2=4$$ which are generally called as Pell-conics, so all in this question $K$ refers to ...
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1answer
142 views

Reference about product of elliptic curves

I am wondering if there is some accessible reference to learn about product of elliptic curves and their 'properties'. For dimension 1, there is plenty to find. I think the dimension 2 case is done as ...
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1answer
381 views

Universality of Tate-conjectures

We all know that Prof.John Tate proposed a set of conjectures(along with Prof.Emil Artin) formally spread under the name of "Tate conjectures", they have a wide range of influence on various fields of ...
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2answers
142 views

Regarding a notation related to divisors & elliptic curves

Section 5.8 of the book An Introduction to Mathematical Cryptography defines the divisor of a rational function $f(X,Y)$ defined on an elliptic curve $E: Y^2 = X^3 + AX + B$ as the formal sum: ...
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1answer
363 views

Resurrection of my Tamagawa numbers Question, to understand the Formulation of BSD

My previous question was closed very badly for asking the broad and deep things, so I now understand the consequences of asking such questions, so I refrain from asking such questions, so this is not ...
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1answer
190 views

discriminant of an étale cover of an elliptic curve

Let $\pi:X\to E$ be a finite étale morphism, where $E$ is an elliptic curve over a number field $K$. Assume $X$ to be connected, and to be of genus 1. Edit: Assume $X$ and $E$ have semi-stable ...
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1answer
2k views

Reading the mind of Prof. John Coates (motive behind his statement)

To start with the issue, I have been thinking from many days that Birch-Swinnerton-dyer conjectures should have some association with the Galois theory, but one day I got the Article of Tate called as ...
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1answer
180 views

division polynomials of elliptic curve as function on $\mathbb{C}$

I have a question about exercise 6.15 of Silverman's book AEC. Suppose that $E$ is a nonsingular elliptic curve over $\mathbb{C}$ given by the equation $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ Then we ...
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1answer
172 views

A reference and an explanation needed?

In my previous question I was asking for a method to construct a global point if we have local points with us which is here, but I got an answer, it didn't serve the entire purpose, but later on due ...
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0answers
136 views

Endomorphism of elliptic curve

Let $\alpha(x,y) = (p(x)/q(x), y\cdot s(x)/t(x))$ be an endomorphism of the elliptic curve E given by $y^2 = x^3 +Ax+B$, where $p, q, s, t$ are polymonials such that $p$ and $q$ have no common root ...
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1answer
200 views

A Generalization of Cantor's counting theory

This question may be silly to experts, but I am waiting for a response sir. My question is " Is there any existence of generalized Cantor's counting principle ( i.e the theory that decide ...
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1answer
243 views

Decomposition of Tate-Shafarevich group

We all know that Tate-Shafarevich group is defined as $$Ш(E/K)=\mathrm{Ker}(H^1(K,E)\mapsto \prod_{v}H^1(K_v,E))$$ for an abelian variety $A$ defined over a number field $K$, the non-trivial ...
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2answers
572 views

How are the Tate-Shafarevich group and class group supposed to be cognates?

How can one consider the Tate-Shafarevich group and class group of a field to be analogues? I have heard many authors and even many expository papers saying so, class group as far as I know is ...
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2answers
444 views

Geometric reason why elliptic curve group law is associative

The question title says it all. I am looking for a geometric proof for the fact that the group law defined on elliptic curves is associative. I've heard somewhere about something on the internet about ...
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1answer
699 views

A Hunt for a Mathematical Machine That Gives Points

The central question is : Is there any method for Producing the global Points on the curve (any cubic curve, or at least a Degree-2 curve ) , if we have local Part with us ? Explanation: ...
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1answer
235 views

Nice formulas for the lambda invariant of an elliptic curve

Where can I find some nice formulas for the lambda invariant of an elliptic curve? I vaguely recall there's a nice product formula in terms of $q$, but a google search didn't give me much. Also, are ...
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1answer
99 views

question about j-invariant

In Hartshorne, there's a formula for the j-invariant in terms of $\lambda$. It says that $$ j = 2^8 \frac{(\lambda^2-\lambda+1)^3}{\lambda^2(\lambda-1)^2}.$$ Can one reverse this formula? That is, can ...
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1answer
243 views

Fencing the Group size,and its implication to Finiteness of Tate-Shafarevich Group

This question is an interesting one,not like my previous one. Can we judge the size of a Quotient Group by seeing the size of its constituents ? To add something ,Suppose consider a group ...
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1answer
181 views

$j$-invariant for a singular curve

Say I have a plane cubic $f(x,y,z) \subset \mathbb{C}^3$ and I identify it with an elliptic curve by setting $z=1$ and end up with (perhaps after a change of variables) something of the form ...
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1answer
116 views

Real elliptic curves in the fundamental domain of $\Gamma(2)$

An elliptic curve (over $\mathbf{C}$) is real if its j-invariant is real. The set of real elliptic curves in the standard fundamental domain of $\mathrm{SL}_2(\mathbf{Z})$ can be explicitly ...
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1answer
170 views

What is the “$\tau$” of this elliptic curve

For any $n\geq 1$, let $E_n $ be the elliptic curve given by the equation $y^2 = x(x-1)(x-\zeta_{15^n})$. Here $\zeta_{m} = \exp(2\pi i /m)$ for any positive integer $m$. There is a unique element ...
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1answer
180 views

Ring on an Elliptic Curve

I know that for a given elliptic curve $E$ we can define a group $G$ with the points on this curve. However, can we define a ring on it? That is, can we define a multiplication on the curve, where we ...
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0answers
197 views

How can I check I have a birational transformation

Updated this question, just focusing on the relevant part $$f_2(u,v) = \tfrac{27}{64} u^3 - \tfrac{81}{64} u^2 v + \tfrac{189}{20} u^2 + \tfrac{81}{64} u v^2 - \tfrac{189}{10} u v + \tfrac{1764}{25} ...
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1answer
81 views

Violating assertion in Cohen's instructions for Weierstrass reduction

I am trying to follow case 2 of the procedure given in Cohen: for the cubic $f(x,y,z) = x^3 + 3 y^3 - 11 z^3$ using the rational point $P_0 = (2 : 1 : 1)$. The tangent at this point is $y = - ...
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1answer
669 views

Birational equivalence of cubic with a Weierstrass form

I want to convert the cubic $x^3 + 3y^3 - 11z^3 = 0$ to Weierstrass form (to find its rank) so I tried to follow the suggestion from Timothy: I found three points ...
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1answer
210 views

Divisor of a function on a curve

Let $k$ be an algebraic closed field with character not equal to 2, $a,b,c\in k$ be distinct numbers, and consider the curve $C: y^2=(x-a)(x-b)(x-c)$. Let $P=(a,0),P_{\infty}$ for the point at ...
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2answers
2k views

Is the Birch and Swinnerton-Dyer conjecture solved?

I read today that in 2010 Manjul Bhargava with Arul Shankar proved the conjecture basing upon the work of Kolyvagin. Is it right? Does it satisfy for all elliptic curves, or is it limited to some ...
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1answer
186 views

Tamagawa number conjecture

I heard somewhere that the above formulation of conjecture is for predicting the exact leading term of a L-function at an integer. But i didnt find any reference about how it is stated, anyone please ...
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2answers
591 views

Zeros and poles of rational functions over elliptic curve

On this page author states: It turns out this definition can be extended to points of order 2, and also the point O (when we homogenize the functions and work over the projective plane). ...
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421 views

Doubt on class group

I started reading Class group after some one's advice ,so I got the following doubts,I would be happy if someone clarify the doubts, I understood that the class group measures the failure of the ...
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1answer
241 views

Fixing Hasse principle

As everyone know that Hasse principle (I am referring to Hasse Local-Global Principle) doesn't work for cubics, but today my question is concerned about: Is there any method or any known theorem, ...
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1answer
82 views

Isogenies between curves

Is it always possible to find an isogeny from a hyperelliptic curve of genus 4, to a 'normal' elliptic curve (genus 1), or a product of elliptic curves? Are such isogenies easy to compute?
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117 views

Hyperelliptic curve order

How to compute order of a hyperelliptic curve ($y^2=f(x)$, $deg(f)=2 \cdot g+1$, $g=4$), over $F_p$ for small $p$ ($p$ prime)? Are there any efficient algorithms to do so? Is it possible with ...
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1answer
420 views

Elliptic curves, inflection points and divisors

I'm studying basics of elliptic curves. I'm reading An Elementary Introduction to Elliptic Curves by Leonard Charlap and David Robbins. It is stated there that the divisor of a line (i.e. a polynomial ...
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1answer
165 views

Groups where discrete logarithm is hard

What are examples of groups, where DLP (discrete logarithm problem) is hard? Two obvious ones are: integers modulo $p$ ($p$ being prime) and elliptic curves over finite fields. What are the others?
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1answer
82 views

Bound of point's order on elliptic curve

For a given elliptic curve over a finite field and a point $P$ on that curve, how can we bound its order (integer $k$, such that $k*P=O$).
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1answer
96 views

Determining elliptic curve's parameters from addition procedure

Given a procedure that adds two points on an unknown elliptic curve, is it possible to determine curve's parameters, treating this procedure as a black box? We are given two points on this curve $P$ ...
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1answer
306 views

Elliptic curve point addition

Are there any elliptic curves, that require computing GCD for point addition? I've an algorithm, that apparently adds two points on an elliptic curve, but it uses GCD, which is strange, because I ...
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1answer
563 views

Intuition and Stumbling blocks in proving the finiteness of WC group

After reading many articles about the Tate-Shafarevich Group ,i understood that "in naive perspective the group is nothing but the measure of the failure of Hasse principle, and coming to its ...
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2answers
330 views

The Néron-Tate canonical height on elliptic curves

I have been trying to understand the Néron-Tate global canonical height of algebraic points on elliptic curves. Let $K$ be a number field, $E$ an elliptic curve (over $\mathbb{Q}$, say), and $E(K)$ ...
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1answer
210 views

Finding a pencil of elliptic curves parametrized by a given modular surface

The following is an attempt to formulate a couple of questions which have been lurking in the back of my mind for a while. I'm sorry if this is long, or if my terminology is not correct, or if my ...
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241 views

About How to Elliptic Curve Equation and Discriminant

I am study public key cryptology and interested in elliptic curve cryplogical algorithms. I have some problem about elliptic curve equation. First I can't find inter process and transformation steps ...
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1answer
162 views

How to nicely extend finite field?

I'm working on an implementation of Miller's algorithm that computes the Weil pairing (elliptic curves, cryptography). In order to do that, I have to implement finite fields. So far I have managed to ...
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0answers
71 views

For an elliptic curve E, does there exist a cofinite Fuchsian group without elliptic elements with quotient E minus a finite subset

Let $E$ be a compact Riemann surface of genus 1, i.e., an elliptic curve. Let $P$ be the identity element of $E$. Question 1. Does there exist a cofinite Fuchsian group (or a Fuchsian group of the ...
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1answer
171 views

Poincaré Residue Theorem

Can anyone point me to a reference which talks about periods of elliptic curves and the Poincaré Residue Theorem, hopefully one which uses this residue theorem to explicitly write out the period?
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178 views

Weierstrass Equation and K3 Surfaces

Let $a_{i}(t) \in \mathbb{Z}[t]$. We shall denote these by $a_{i}$. The equation $y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}$ is the affine equation for the Weierstrass form of a ...