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

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The group $E(\mathbb{F}_q)/NE(\mathbb{F}_q)$

Let $E$ be an elliptic curve defined over the finite field $\mathbb{F}_q$ and let $N\geq 1$. We also make the following assumptions: $T \in E(\mathbb{F}_q)[N]$ is a point of exact order $N$, ...
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1answer
241 views

Euler Product of Modular L-series

This question is probably really easy to someone that knows the theory of modular forms well, so I apologize if this is obvious. Suppose $E_1$ and $E_2$ are elliptic curves over $\mathbb{Q}$ and the ...
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421 views

Elliptic curves with finitely many rational points

A conjecture by Goldfeld says that half of all elliptic curves have rank zero (i.e. their Mordell-Weil group has finite order.) Are there any known infinite families of elliptic curves (over ...
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1answer
348 views

Special privilege enjoyed by Elliptic Curves with Complex Multiplication

I think after reading the title one may understand the intention of me, this question is concerned about the Elliptic curves having a Complex Multiplication. I have been reading many theorems, ( ...
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1answer
69 views

Minimal degree of a field extension to obtain an elliptic curve

Let $K$ be a number field and let $X$ be a smooth projective geometrically connected curve over $K$ of genus $1$. There exists a number field $L/K$ such that $X$ has a $L$-rational point. Let $L$ be ...
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1answer
123 views

Computing the free-part

I wanted to ask about some existing algorithms for computing points over elliptic curves. Background : We know that the famous theorem of Mordell and Weil says that " Group of rational points on an ...
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130 views

What is the Hurwitz number of an elliptic curve

One can associate a Hurwitz number to any rational function $f:X\to \mathbf{P}^1$ on a compact connected Riemann surface $X$ which ramifies over precisely FOUR points. Suppose that $X$ is an elliptic ...
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1answer
62 views

Does the following define a point of the modular curve $X_1(n)$

Let $X$ be a compact connected Riemann surface of genus $1$ and suppose that there is a finite morphism $X\to \mathbf{P}^1$ of degree $n$ which ramifies totally over $0$ and $\infty$. Let $f^{-1}(0)$ ...
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576 views

Intersection of a line with an Elliptic Curve

I am trying to show that if a line given by $y = mx + b$ intersects an Elliptic Curve given by $E(\mathbb{K}): y^2 = x^3 + Ax + B$ in three points then the line is not tangent to the curve. Given ...
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305 views

Does the $p$-torsion of an elliptic curve with good reduction over a local field always determine whether the reduction is ordinary or supersingular?

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E/K$ an elliptic curve with good reduction. Does the $\mathbb{F}_p[\mathrm{Gal}(\overline{K})]$-module $E[p](\overline{K})$ determine whether the ...
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2answers
441 views

Weierstrass $\wp$ function doubly periodic

I'm working my way through Silverman and Tate's Undergraduate Introduction to Elliptic Curves. I haven't yet been able to study complex analysis, so it comes as no surprise that I'm having a tough ...
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798 views

Basic Understanding of Elliptic curve

I want to know the basic understanding about Elliptic curve. Why it is need and when it is useful. I have searched much on internet but I am not a science student hence I am not able to understand the ...
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1answer
145 views

Computing the trace of the following automorphism of the elliptic curve $y^2 = x^3+x$

Consider the elliptic curve $E$ defined by $y^2z= x^3 +xz^2$ over an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$. Consider the endomorphism $f:E\to E$ given by $(x:y:z)\mapsto ...
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1answer
146 views

How to show that the image of a certain projective embedding is an algebraic curve?

I found the following claim in a paper by Griffiths and Harris : Start with a complex torus $\mathbb{C}/\Lambda$. The vector space of meromorphic functions having period lattice $\Lambda$ and a pole ...
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2answers
685 views

Finding zeros and poles of rational functions over elliptic curve

The matter of explicitly finding the order of a rational function on an elliptic curve in the projective plane at infinity (i.e. at the point $(0, 1, 0)$) still seems unclear. For example, Silverman ...
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Local-Global Principle and the Cassels statement.

In a recent article I have read, i.e. " Lecture notes on elliptic curves ", Prof.Cassels remarks in page-110 that There is not merely a local-global principle for curves of genus-$0$, but ...
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263 views

admissible change of variables on elliptic curve

I was wondering whether an admissible change of variables of an elliptic curve given by a Weierstrass equation respects the group law. Let $E$ be defined over a field $K$ given by the equation ...
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1answer
223 views

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
329 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
487 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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389 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
144 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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387 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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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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364 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
181 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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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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137 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
201 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
585 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
478 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 ...
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1answer
703 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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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
245 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
172 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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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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200 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
680 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
213 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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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
187 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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597 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). ...