4
votes
1answer
40 views

Quantities $g_2$, $g_3$, $\Delta$

This question is somewhat related to this one. Let $\lambda$ be the modular lambda function. Greenhill (Elliptic Functions, p. 57) states that we may put $$g_2 = \frac{1 - \lambda + \lambda^2}{12}, ...
0
votes
2answers
90 views

defining the group law on elliptic curves in general

Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g. Any line intersects $C$ ...
1
vote
2answers
42 views

Compendium of elliptic curves?

does anyone know where I can find a collection of elliptic curves and their integral solutions? EDIT: Removed additional useless info. Thanks!
3
votes
1answer
57 views

Rank of Elliptic Curves

Recently, I have heard of some heuristics that would suggest that the rank of elliptic curves are bounded (specifically in the congruent number family). I always though that the best way to prove ...
3
votes
2answers
63 views

The torsion of an elliptic curve over a finite field

There is a result for $p$ prime, $E$ an elliptic curve over $\mathbb F_p$, then $E(\overline{\mathbb{F}_p})[m]\cong (\mathbb{Z}/m\mathbb{Z})^2$ for $m \nmid p$. The book on cryptography I am using ...
1
vote
1answer
75 views

Counting the number of elliptic curves with certain discriminant/conductor

I'm looking for some references regarding the above topic. To be more specific, references that address questions such as Given $D > 0$, how many elliptic curves over $\mathbb{Q}$ are there with ...
5
votes
1answer
249 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 ...
1
vote
0answers
138 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 $1 \pm 2q + q^2$ ...
7
votes
3answers
373 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!
1
vote
1answer
119 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 ...
5
votes
1answer
50 views

deg functions and maps

For any map $f$ between curves $C_1$ and $C_2$, one defines $\mathrm{deg}(f) = [K(C_1) : f^*K(C_2)]$ as given in "The Arithmetic of Elliptic Curves" by Silverman. For algebraic functions on elliptic ...
2
votes
1answer
191 views

Elliptic curve as an intersection of quadrics

Let $E$ be an elliptic curve. If one starts with embedding associated with invertible sheaf $\mathcal{O}(3x)$ where $x$ is some point on $E$ then one gets cubic in $\mathbb{P}^2$ and this embedding is ...
2
votes
0answers
101 views

Relations between elliptic curves and topological quantum field theory

I heard that there are relations between elliptic curves and topological quantum field theory (TQFT). I googled and found that something called "elliptic genus" might be the key word to relate these ...
25
votes
1answer
1k views

How to compute rational or integer points on elliptic curves

This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
8
votes
0answers
108 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
2
votes
1answer
140 views

Complex tori as elliptic curves

I have a question about the proof of the following theorem: A complex torus is conformally equivalent (so isomorphic as Riemann surface) to a complex elliptic curve I used the book "N.Koblitz, ...
3
votes
2answers
401 views

References for elliptic curves

I just finished reading Silverman and Tate's Rational Points on Elliptic Curves and thought it was very interesting. Could any of you point me to some more references (ex. books, articles) on ...
15
votes
2answers
403 views

Reference request in number theory for an analyst.

I am a confirmed mathochist. My background is in analysis, and fairly traditional analysis at that; mainly harmonic functions, subharmonic functions and boundary behaviour of functions, but I have for ...
2
votes
1answer
94 views

Database for size of Ш

Are there are any references that record the cardinality of Ш for elliptic curves for which Ш is known? Also their corresponding conductors. EDIT: Following the Qiaochu Yuan's comment's I should ...
3
votes
3answers
316 views

Rankin-Selberg zeta function

I was reading this paper by de Weger and in conjecture 7 he mentions "the Riemann hypothesis for the Rankin-Selberg zeta function associated to the weight 3/2 modular form associated to E (an elliptic ...
4
votes
2answers
170 views

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 ...
4
votes
3answers
268 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 ...
3
votes
3answers
181 views

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 ...
3
votes
1answer
589 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 ...
4
votes
4answers
424 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 ...
7
votes
1answer
350 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, ( ...
8
votes
3answers
806 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 ...
2
votes
2answers
246 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{ ...
2
votes
1answer
145 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 ...
0
votes
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 ...
6
votes
2answers
340 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 $E(K)$ ...
2
votes
1answer
172 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?