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

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Consequences of Szpiro's conjecture

Let $E/\mathbb{Q}$ be an elliptic curve. Recall that Szpiro's conjecture says that for every $\epsilon > 0$, there exists $C_\epsilon$ such that $$ |\Delta_E| \leq C_\epsilon(N_E)^{6 + \epsilon}, ...
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Bounding the product of exponents

I was reading de Weger's paper on bounding the cardinality of the Tate-Shafarevich group and in lemma 1 (pg 111), he claims that for any $n \in \mathbb{N}$, we have that $$ c(n) << N^{((log \; ...
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Exposition on Modular Curves

I was recently reading this paper by Weston, whereby he talks about the modular curves $X_0(11)$ and $X_1(11)$. I was wondering if anyone can recommend a more general exposition of modular curves ...
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The 2-primary Part of Ш

I was reading Silverman's Arithmetic of Elliptic Curves I have a question on computing the Mordell-Weil group of an elliptic curve over $E(\mathbb{Q})$. Adapting the argument given in Silverman we ...
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UPDATE: How to find the order of elliptic curve over finite field extension

I want to find the order of elliptic curve over the finite field extension $\mathbb{F}_{p^2}$, where $E(\mathbb{F}_{p^2}):y^2=x^3+ax+b $ I am using the method illustrated by John J. McGee in his ...
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How to find the order of elliptic curve over finite field extension

I want to find the order of elliptic curve over the finite field $\mathbb{F}_{5^2}$, where $E(\mathbb{F}_{5^2}):y^2=x^3+10x+17$. I am using the method illustrated by John J. McGee in his thesis ...
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236 views

Point addition on an elliptic curve over $\mathbb{F}_{5^2}$

I have the elliptic curve equation $E(\mathbb F_{5^2}): y^2=x^3+10x+17$, and I have that the points $(3,7)$ and $(8,3)$ belong to $E$. According to the addition law, the slope ...
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205 views

Global minimal model of elliptic curve over $\mathbb{Q}$

I am basically trying to solve the cannonball problem using elliptic curves. In other words I have to show that the only integer points on the "elliptic curve" $6y^2 = 2x^3 + 3x^2 + x$ are $(0,0), ...
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Elliptic curves, 2-torsion and branch points.

I'm currently reading through Ravi Vakil's notes on Algebraic Geometry. I've been having trouble grasping some things conceptually though and I hope that you can help me. For an elliptic curve (E,p) ...
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271 views

The elliptic curve $y^2 = 23328x^3-890273x^2+14755570x-7^7$

The elliptic curve, $$y^2 = 23328x^3-890273x^2+14755570x-7^7 \tag{1}$$ has the small solution $x = 58$. I know how to find other rational points, but the number of digits in the denominator gets ...
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Reference: Elliptic curves as complex tori

I'm looking for books which contain a more or less self-contained description of how elliptic curves over $\mathbb{C}$ - that is, nonsingular plane cubic curves - can be realized as a quotient of the ...
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597 views

Reference request for “Weierstrass equation” and “Weierstrass normal form”

I would like to know more about the history of the widely used terms "Weierstrass equation" and "Weierstrass normal form", as they appear in the theory of elliptic curves. When were these terms first ...
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130 views

Are elliptic curves also Galois covers of degree 3

Let E be an elliptic curve with equation $y^2=x^3+Ax+B$. The projection onto the $x$-coordinate is a Galois morphism of degree $2$. But what about the projection onto the $y$-coordinate? Is it ...
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How can I determine in practice whether two elliptic curves over $\mathbb{Q}$ have isomorphic $p$-torsion?

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ with good, ordinary reduction at an odd prime $p$. I'm wondering how to determine whether $E_1[p]$ and $E_2[p]$ are isomorphic ...
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129 views

Can an algebraic group only have trivial elements over $k$

Let $G$ be an algebraic group over $k$ such that $G(k) = \{e\}$ is the trivial group. Does this imply that $G_{\overline{k}}$ is trivial? I think the answer is no. I think you can just take an ...
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Complex elliptic curve for the “conjugate” lattice

Let $\Lambda$ be a lattice in $\mathbb{C}$, and $E=\mathbb{C}/\Lambda$ the corresponding complex elliptic curve. Let $\bar{\Lambda}$ be the "conjugate" lattice, i.e. the one obtained by conjugating ...
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Form of minimal integral Weierstrass equation for elliptic curve over $Q$ with good reduction at $2$ and $3$.

If $E$ is an elliptic curve over $\mathbb{Q}$ which has good reduction at $2$ and $3$, is it always possible to find a minimal integral Weierstrass equation for $E$ of the form $y^2 = x^3 + Ax + B$ ...
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320 views

Nontrivial Rational solutions to $y^2=4 x^n + 1$

Are there any nontrivial rational solutions to the following equation: $$y^2=4 x^n + 1,$$ where $n>2$?
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How to know whether an elliptic curve has a low-degree isogeny?

Given an elliptic curve with a Weierstrass equation, is there any easy way to see whether it has got an isogeny of low degree?
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Does there exist a finite morphism of algebraic curves such that…

Let $K\subset L$ be a finite field extension. Let $X$ and $Y$ be (smooth projective geometrically connected) curves over $L$. Let $f:X\to Y$ be a finite morphism of curves over $L$. Assume that ...
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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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246 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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427 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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353 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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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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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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136 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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63 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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606 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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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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484 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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822 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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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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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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730 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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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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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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333 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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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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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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391 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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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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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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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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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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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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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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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 ...