For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.
16
votes
1answer
513 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 ...
7
votes
0answers
78 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
86 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, ...
0
votes
0answers
69 views
Meaning of having a rational $m$-torsion point
Suppose I have an elliptic curve $E/\mathbb{Q}$. What does it mean when one says $E$ has a rational $m$-torsion point over $\mathbb{Q}$? What does this mean for the torsion subgroup, ...
2
votes
2answers
86 views
Elliptic curve condition on coefficients
I am working something where a picture like this one appeared :
Say the curve is written in the form
$$
y^2 = x^3 + ax^2 + bx + c
$$
(if this is the wrong form of coefficients, feel free to correct ...
3
votes
0answers
110 views
when do two rational elliptic curves have identical size when reduced mod $p$ for all primes $p$?
If $E_1$ and $E_2$ are two elliptic curves over $\mathbb{Q}$ such that $|E_1(\mathbb{F}_p)|=|E_2(\mathbb{F}_p)|$ for all primes $p$, what does this tell us about the relationship between $E_1$ and ...
5
votes
1answer
139 views
Is the pushforward of the sheaf of differentials on an elliptic curve over a scheme necessarily trivial?
If $f:E\rightarrow S$ is an elliptic curve over a scheme $S$ (so $f$ is proper and smooth of relative dimension one with geometrically connected fibers of genus one, equipped with a section ...
2
votes
0answers
53 views
Galois action on CM elliptic curves
Let $E$ be an elliptic curve $E$ defined over a number field $K$ such that $E$ has complex multiplication by the maximal order in the ring of integers of an imaginary quadratic field $F$. Let ...
4
votes
2answers
193 views
Why do we define the group law on elliptic curves only for Weierstrass forms and $O$ an inflexion point?
In almost all texts concerning the group law on an elliptic curve it is first proven that any nonsingular cubic can be given by a Weierstrass equation and then the group law using the point $O$ at ...
2
votes
1answer
67 views
Tate models and subgroups of type$(m,m)$ - re Silverman, Hindry 88
I need help in understanding a passage in a paper by Hindry and Silverman, "The Canonical Height and Integral Points on Elliptic Curves". (re. page 439)
Let $ E(K) $ be an elliptic curve with ...
2
votes
2answers
163 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 you point me to some more references (ex. books, articles) on elliptic ...
8
votes
2answers
186 views
Concrete Example of the Birch and Swinnerton-Dyer Conjecture
The Setup
Consider an elliptic curve $E$ in Weierstrass form
$y^2=x^3+ax+b$
with $a,b \in \mathbb{Z}$. As usual, we let $\Delta_E$ be the discriminant of the polynomial, and we set
$N_p := $ ...
2
votes
1answer
283 views
What is the Birch and Swinnerton-Dyer Conjecture?
This is probably a really silly question, but I was wondering if someone could explain the Birch and Swinnerton-Dyer conjecture to me in a simple way. I've read a lot about it, but cannot understand ...
4
votes
1answer
485 views
Proving Fermat's Last Theorem (easily) using “assumed” conjectures
It can easily be proven assuming Szpiro's conjecture that Fermat's Last Theorem is true for sufficiently large $n$. The proof consists of extremely straightforward computations. My question is, is ...
1
vote
1answer
110 views
Drawing elliptic curve
Consider an elliptic complex curve in $\mathbb{C}^2$ given by equation $w^2 = (z-a)(z-b)(z-c)$ where $a,b,c$ are complex mutually distinct constants. It is a $2$-dimensional surface in $4$-dimensional ...
2
votes
2answers
227 views
On the relationship between Fermats Last Theorem and Elliptic Curves
I have to give a presentation on elliptic curves in general. It does not have to be very in depth. I have a very basic understanding of elliptic curves (The most I understand is the concept of ranks). ...
4
votes
2answers
129 views
Subvariety of Product of Elliptic Curves
This is almost certainly known (and maybe written down somewhere?). Is there an example of two elliptic curves $C, E/k$ that are not isomorphic, yet there is an embedding $C\hookrightarrow E\times E$ ...
3
votes
3answers
159 views
Show that the curve $y^2 = x^3 + 2x^2$ has a double point, and find all rational points
Show that the curve $y^2 = x^3 + 2x^2$ has a double point. Find all rational points on this curve.
By implicit differentiation of $x$, $-3x^2 - 4x$ vanishes iff $x = -4/3$ and $0$.
By implicit ...
2
votes
3answers
68 views
Show that if the curve $y^2 = p(x)$ has a double point, then it must be of the form $(r,0)$ where $r$ is a double root of $p(x)$.
Let $p(x) = ax^3 + bx^2 + cx + d$ where $a,b,c,d \in\mathbb{R}$. Show that if the curve $y^2 = p(x)$ has a double point, then it must be of the form $(r,0)$ where $r$ is a double root of $p(x)$.
...
7
votes
2answers
124 views
Clarifying a comment of Serre
Let $\rho_{\ell}$ be the "mod $\ell$" Galois representation associated to an elliptic curve $E/K$ (i.e., corresponding to the action of Galois on the $\ell$-torsion points). Serre proved that in the ...
3
votes
1answer
77 views
Reduction of endomorphism ring of elliptic curve
Let $E$ be an elliptic curve defined over a number field without complex multiplication and with ordinary reduction at a prime $p\in\mathbb{N}$. When is the reduction mod $p$ map a surjection on the ...
2
votes
2answers
101 views
Embedding elliptic curves into the general linear group
Is it possible to embedd an elliptic curve $E:\;\; y^2=x^3+ax+b$, defined over an algebraically closed field $k$, into some $GL_n(k)$ ?
2
votes
1answer
130 views
Congruence subgroups and modular curves of type (M,N)
I would like to study the "modular curve" $Y(M,N)$, parametrizing an elliptic curve $E$ together with $p \in E[M]$ and $q \in E[N]$ (here and in the following $M$ divides $N$).
Let $\Gamma(M,N)$ be ...
7
votes
1answer
104 views
Line Bundle on subvarieties
I've been having problem actually restricting a Line bundle $L$ defined on some projective space $\mathbb C \mathbb P^{N-1}$ to a subvariety $X$.
I know how to do this on an abstract level, but ...
7
votes
1answer
194 views
splitting of quaternion algebras
A rational (definite) quaternion algebra is an algebra of the form
$$ \mathcal{K} = \mathbb{Q} + \mathbb{Q}\alpha + \mathbb{Q}\beta + \mathbb{Q}\alpha \beta $$
with $\alpha^2,\beta^2 \in ...
4
votes
0answers
78 views
Approach to elliptic curve $y^2=x^3+1/4+p/a^2$
While taking a brute-force look at this question I discovered that it seems that almost every prime (I'll conjecture every prime larger than 20627) can be written as $p=w^2+wc+d$ for $w,c,d\in ...
15
votes
2answers
303 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 ...
5
votes
2answers
101 views
Computing rank using $3$-Descent
For an elliptic curve $E$ over $\Bbb{Q}$, we know from the proof of the Mordell-Weil theorem that the weak Mordell-Weil group of $E$ is $E(\Bbb{Q})/2E(\Bbb{Q})$. It is well known that
$$
0 \rightarrow ...
9
votes
3answers
385 views
History of elliptic curves
In one sense elliptic curves are a rather modern object as some of its properties have been studied only in the last century or so. But in another sense there are a very classical object for studying ...
2
votes
2answers
77 views
sum of torsion of an elliptic curve
It is clear from the isomorphism between elliptic curves over $\mathbb{C}$ and complex tori that the sum of the $m$-torsion points is the identity in the group law of the elliptic curve. How generally ...
2
votes
1answer
97 views
Elliptic Curves over Noncommutative rings
It is known that we can define elliptic curves over commutative rings. However can we define an elliptic curve over a noncommutative ring?
This question is considered to some extent in this thesis ...
3
votes
2answers
257 views
Elliptic curves over a finite field $\mathbb{F}_p$ where $p$ is prime.
Let $Y^2=f(X)$ be an Elliptic curve over a finite field $\mathbb{F}_p$ where $f(X)=X^3+aX+b$
In an undergraduate coursebook on an Applied Algebra course it states that "It is plausible to suggest ...
5
votes
1answer
161 views
A question about modular curves and base change
Let $X$ be a smooth projective geometrically connected curve over a number field $K$.
Suppose that the curve $X\times_{K,\sigma} \mathbf{C}$ is a modular curve for some $\sigma:K\to \mathbf{C}$.
Can ...
1
vote
1answer
79 views
Silverman's Lefschetz Principle
Let K be a field of characteristic 0, E/K an elliptic curve. The "Lefschetz principle" implies that $E[m] \simeq \mathbb{Z}/m \times \mathbb{Z}/m$, but for this to follow from the result for complex ...
1
vote
1answer
87 views
Conductor of $ABC$, Frey-Hellegouarch curves, and twists
In page 109 of de Weger's paper, he says that for coprime $A, B, C$ the conductor $N$ of the Frey-Hellegouarch curve
$$
E: y^2 = x(x - A)(x + B)
$$
equals $N(A,B,C)$ (product of primes dividing $ABC$ ...
2
votes
1answer
80 views
Database for size of Ш
Are there are any references that record the cardinality of Ш for elliptic curves for which Ш is known? Also their corresponding conductors.
EDIT: Following the Qiaochu Yuan's comment's I should ...
2
votes
2answers
120 views
Epsilon conjecture analog
Recently this question caught my eye. Is there a relation to the modularity problem of elliptic curves over $\mathbb{Q}(\zeta_m)$ and this problem? Namely, if all elliptic curves over ...
3
votes
3answers
185 views
Rankin-Selberg zeta function
I was reading this paper by de Weger and in conjecture 7 he mentions "the Riemann hypothesis for the Rankin-Selberg zeta function associated to the weight 3/2 modular form associated to E (an elliptic ...
1
vote
1answer
155 views
Consequences of Szpiro's conjecture
Let $E/\mathbb{Q}$ be an elliptic curve. Recall that Szpiro's conjecture says that for every $\epsilon > 0$, there exists $C_\epsilon$ such that
$$
|\Delta_E| \leq C_\epsilon(N_E)^{6 + \epsilon},
...
2
votes
1answer
43 views
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 \; ...
4
votes
2answers
130 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 ...
2
votes
1answer
72 views
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 ...
4
votes
1answer
115 views
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 ...
1
vote
2answers
472 views
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 ...
2
votes
1answer
131 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 ...
4
votes
1answer
121 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), ...
2
votes
0answers
152 views
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) ...
4
votes
3answers
221 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
140 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 ...
2
votes
1answer
258 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 ...
