Questions tagged [elliptic-curves]
An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry. Note also the tag arithmetic-geometry as well as some related tags such as rational-points, abelian-varieties, heights. Please do not use this tag for questions related to ellipses; instead use conic-sections.
173
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Elliptic curves with the same mod $p$ representation
What is the largest prime number $p$ for which one knows examples of nonisogenous elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$ with isomorphic mod $p$ Galois representations: $E_1[p] \cong E_2[p]$...
- 2,623
9
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5
answers
2k
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The significance of modularity for all Galois representations
On pg. 1 of the slides of a talk, Henri Darmon wrote:
Question: What is an interesting Diophantine equation?
A “working definition”. A Diophantine equation is interesting
if it reveals or ...
- 6,942
9
votes
1
answer
799
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Parametric families for large torsion subgroups of elliptic curves
The following are two facts about $\mathbb{Z}/9\mathbb{Z}$, $\mathbb{Z}/10\mathbb{Z}$, $\mathbb{Z}/12\mathbb{Z}$,
$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$.
(a) According to Andrej ...
- 276
8
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1
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446
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Existence of newforms which are non-ordinary at a given prime
Let $f$ be a newform of weight $k \geq 2$ and level $N \geq 1$ without complex multiplication. A prime $p$ is said to be ordinary for $f$ if the $p$-th Fourier coefficient $a_p(f)$ is a $p$-adic unit (...
- 20.5k
8
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2
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2k
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Can someone suggest a path to study Mordell-Weil theorem for someone studying on their own?
So, I want to read the proof of Mordell-Weil theorem and so, I picked up the book 'Arithmetic of Elliptic Curves' by J. Silverman and J.S. Milne's Elliptic Curves book. But after going through both ...
- 391
8
votes
2
answers
714
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An elliptic curve for Ramanujan-type cubic identities?
Given the roots $x_i$ of the depressed cubic,
$$x^3+px+q=0$$
with rational coefficients. It can be shown that, in general, one can find rational $u,v$ such that,
$$(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-...
- 12.1k
8
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1
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Proof of Prop 1.1 in Wiles' "Modular elliptic curves and Fermat's last theorem"
On p. 459 of "Modular elliptic curves and Fermat's last theorem", proof of Prop. 1.1, where it says "Since $H^2(G,\mu_{p^r}) \rightarrow H^2(G,\mu_{p^s})$ is injective for $r \leq s$...", is there any ...
- 2,005
8
votes
3
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1k
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Ranks of elliptic curves depend only on the field?
Let $K/\mathbb{Q}$ be an algebraic extension, and let $E_1,E_2/\mathbb{Q}$ be elliptic curves. Is it possible that the Mordell-Weil rank of $E_1(K)$ is finite while that of $E_2(K)$ is infinite?
- 11.2k
7
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528
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A constructive proof of the theorem of the cube
Do you know a constructive proof of the theorem of the cube ? More precisely, let $X$, $Y$, $Z$ be projective varieties (e.g., over an algebraically closed field $k$) with points $x$, $y$, $z$ ...
- 1,801
7
votes
0
answers
504
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$a^5+b^5=c^5+d^5$ and polynomial identities
No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known.
(1) has infinitely many solutions in an extension of $\mathbb{Z}$
(root of $9-15x+37x^2 $ ) resulting
from a genus 0 ...
- 24.2k
7
votes
1
answer
1k
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Modular polynomials for elliptic curves point counting
The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" ...
- 637
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398
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Algebraic equations for modular parameterizations
I was wondering if there some place where for some small $N$ I can find explicit modular parameterizations in an algebraic way.
One type of model for $X_0(N)$ is just given by a single algebraic ...
- 2,185
7
votes
1
answer
652
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How do you compute modular symbols?
In John Cremona's book, he defines the modular symbol of an elliptic curve in the following way.
Let $E/\mathbf{Q}$ be an elliptic curve and let $f_E$ be the modular form associated to $E$. The ...
- 1,869
7
votes
1
answer
898
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Modular curve X(2)
Let $\mathfrak{M}(2)$ be the algberaic stack over $\mathbb{Z}[1/2]$ which classifies the elliptic curves with the two level structure and let $X(2)$ be the coarse moduli space of $\mathfrak{M}(2)$ ($X(...
- 1,046
7
votes
1
answer
475
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Is an elliptic curve that is isomorphic to its Frobenius conjugate defined over $\mathbb{F}_p$?
Let $p$ be prime and $q = p^n$. Let $E$ be an elliptic curve over $\mathbb{F}_q$, and let $E^{(p)}$ be the pullback of $E$ by the $p$-power Frobenius of $\mathbb{F}_q$. If $E$ is isomorphic (over $\...
- 2,623
7
votes
1
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473
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Discriminant of elliptic curve (Frey-Hellegouarch), j-invariant and positive definite kernels, similarities?
Consider the Frey-Hellegouarch curve given $a,b$ natural numbers:
$$y^3= x(x-\frac{a}{\gcd(a,b)})(x+\frac{b}{\gcd(a,b)})$$
Then the discriminant is given by $\Delta = \Delta(a,b) = 16 \left(\frac{ab(a+...
user6671
7
votes
3
answers
571
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Uniform bounds on the number of integer points on a family of elliptic curves
Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the ...
- 5,129
7
votes
1
answer
386
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Why are some solutions of these diophantine equations off the usual patterns?
This is inspired by a recent question about complete multipartite integral graphs. I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It is easy to see ...
- 13.2k
7
votes
3
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891
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Do there exist elliptic curves over schemes which have all primes as residue characteristics?
It's well known that there are no elliptic curves over Spec $\mathbb{Z}$, but it's unclear (to me at least) if the proof generalizes.
My question is: If $S$ is a connected scheme such that has every ...
- 10k
6
votes
1
answer
358
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Lifting of Frobenius on torsors over abelian varieties
This is related to my previous question Assume that $A$ is an abelian variety over a field $k$ of characteristic $p$, $\mathcal{L}$ is a line bundle on $A$. Assume that $A$ is ordinary and $\mathcal{L}...
- 7,027
6
votes
0
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921
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Can you get Siegel's theorem "for free" from modularity and Mazur's Eisenstein Ideal paper?
There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...
- 3,208
6
votes
1
answer
346
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Adèlic points and algebraic closure
Consider $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$.
Let $F/K$ vary over all finite Galois number ...
user335418
5
votes
1
answer
190
views
Are there orthogonal Cauchy-like matrices with rational entries for any given size?
This is inspired by a recent question about the existence of orthogonal Cauchy-like matrices. It is proved that there are indeed such matrices, i.e. there are vectors $x,y,r,s\in\mathbb R^n$ such ...
- 13.2k
5
votes
1
answer
926
views
Elliptic curves over the complex numbers: everything "well known"?
This may be a very naive question. We always hear about elliptic curves over the rational numbers, or over other arithmetically significant fields or rings.
But, are there open problems or recent ...
- 22.7k
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3
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1k
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How to generate the n-torsion group in an elliptic curve
Let $E$ be an elliptic curve over a field $K$.
I was curious about the following sentence: "then the $n$-torsion on $E(\overline{K})$ has known structure, as a Cartesian product of two cyclic ...
- 131
5
votes
3
answers
488
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Is there a way to find any non-trivial $\mathbb{F}_p(t)$-point on the given elliptic curve?
Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$) and the elliptic curve
$$
E\!:y^2 = x^3 + (t^6 + 1)^2
$$
over the univariate ...
- 1,801
5
votes
1
answer
723
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Upper bound for Hall's conjecture on separation of squares and cubes
Hall's (weak) conjecture is the statement that for all $\varepsilon > 0$ there exists a positive number $c(\varepsilon) > 0$ such that for all $x,y \in \mathbb{Z}$ with $y^2 \ne x^3$, that
$$\...
- 25.5k
5
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0
answers
165
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Touchard / van der Pol's identity for the sum of divisors and an elliptic curve for perfect numbers
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$, satisifies the following recurrence relation ($n>1$):
$$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
- 213
5
votes
2
answers
613
views
Growth of Mordell-Weil Rank of Elliptic Curves over Field Extensions
I'm a graduate student just checking to make sure that what he's researching isn't already known.
Let $\mathbb{F}$ be a number field, and let $E$ be an elliptic curve defined over $\mathbb{F}$. Is ...
- 1,266
5
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2
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249
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What integer value can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$?
Fix a positive integer $g$. What positive integer $N$ can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$ ?
For example, as there is no abelian varieties over $\mathbb Z$, $N$ ...
- 6,158
5
votes
2
answers
763
views
Is there a presentation of the cohomology of the moduli stack of torsion sheaves on an elliptic curve?
Let $E$ be your favorite elliptic curve, and let $Tor^m$ be the moduli stack of torsion sheaves of degree $m$ on $E$. This sounds horrible, but it's not so bad; it's a global quotient of a smooth ...
- 44k
5
votes
1
answer
289
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The name of the equianharmonic curve
I have found several references where the elliptic curve $y^2=x^3-1$ is called the equianharmonic curve, and, more often, where the half-period of this curve
$$ \omega_1 = \frac{\Gamma(1/3)^3}{4\pi} $$...
- 836
5
votes
1
answer
456
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Solutions of a general diophantine equation
So it turns out that there exist positive integers a, b, c and n, such that $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=n.$ See Estimating the size of solutions of a diophantine equation
Now I am ...
5
votes
1
answer
460
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On the elliptic curve $x(x+a^2)(x+b^2) = y^2$
Ajai Choudhry showed that special cases of the elliptic curve,
$$x(x+a^2)(x+b^2)=y^2\tag1$$
can be used to prove that,
$$u_1^7+u_2^7+\dots + u_9^7 = 0\tag2$$
has an infinite number of primitive ...
- 12.1k
5
votes
1
answer
519
views
Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$
According to a conjecture there are no three
consecutive powerful numbers.
Necessary condition for this is integer solution of
$$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$
What are integer solutions of ...
- 24.2k
4
votes
1
answer
293
views
Szpiro ratios of elliptic curves over $\mathbb{Q}$
For an elliptic curve $E/\mathbb{Q}$, let us denote by $\Delta_{\min}(E)$ the minimal discriminant of $E$ and $N(E)$ the conductor of $E$. Then it is well-known that $N(E) | \Delta_\min(E)$.
The ...
- 25.5k
4
votes
1
answer
750
views
Frobenius at ramified primes
Let $E$ be an elliptic curve defined over $\mathbf{Q}$, fix an odd prime $p>3$, let $T_p$ denote the $p$-adic Tate module of $E$, and let $V_p = T_p \otimes \mathbf{Q}_p$.
If the action of $G_\...
- 1,412
4
votes
1
answer
404
views
A generator needed for a Z/6 elliptic curve
We are searching for rank $8$ elliptic curves with the torsion subgroup $\mathbb{Z}/6$ using newly discovered families similar to Kihara's as described in
A. Dujella, J.C. Peral, P. Tadić, ...
- 913
4
votes
1
answer
548
views
Lifting of Frobenius on semi-abelian varieties
Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume ...
- 7,027
4
votes
2
answers
823
views
When is the period of elliptic curve over the rationals transcendental?
Given an elliptic curve $E/\mathbf{Q}$, when is its period transcendental/algebraic?
user19475
4
votes
1
answer
242
views
Does the modular form associated to cubic twist of a elliptic curve $E$ corresponds to some twist of $f_E$?
Let $E$ be an elliptic curve defined over $\Bbb Q$ and $f_E$ be the modular form associated with the elliptic curve $E$.
Suppose the elliptic curve $E^D$ is a quadratic twist of $E$.
I understand that ...
- 723
4
votes
1
answer
736
views
Abelian image of l-adic representation
For a number field F, let E/F be an elliptic curve with CM by a quadratic field K. Let $\rho_\ell: \text{Gal}_F \to \text{Aut}(T_{\ell}E)$ be the $\ell$-adic representation associated to E for some ...
- 235
4
votes
1
answer
407
views
3-, 6-, 12-descent for Z2xZ6 elliptic curves
We are trying to write a snippet of Magma code to clarify the steps in the simplified procedure of applying $3$-, $6$-, $12$-descent and hopefully resolve the missing generator of the following $\...
- 913
4
votes
1
answer
175
views
$\#E(\mathbb{F}_p) \in \{p,p+2\}$ iff $p$ is of the form $27a^2+27a+7$
Related to this question.
Let $E / \mathbb{F}_p : y^2=x^3+2$.
Numerical evidence up to $2 \cdot 10^5$ suggests:
Conjecture: $\#E(\mathbb{F}_p) \in \{p,p+2\}$ iff $p$ is of the form
$27a^2+27a+7$ ...
- 24.2k
4
votes
2
answers
690
views
About equivalent statements of the Birch and Swinnerton-Dyer Conjecture [closed]
The Birch and Swinnerton-Dyer Conjecture is well known in the current literature
http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture
My question is about the possible equivalent ...
- 49
3
votes
2
answers
322
views
Infinitely many elliptic curve with twist rank more than $1$ in specific case
Let $E/\Bbb{Q}$ be an elliptic curve. Let $D$ be a square free negative integer.
It is conjectured that 50％ of twist of elliptic curve $E_D$ has rank $0$ and $50％$ has rank $1$.
But is some particular ...
- 1,405
3
votes
1
answer
569
views
Is it expected that every natural number is the rank of some elliptic curve over the rationals?
It is a well-known problem on the theory of elliptic curves that the rank of an elliptic curve (the number of generators of the free part of the Mordell group of the elliptic curve) cannot be ...
- 25.5k
3
votes
1
answer
510
views
The existence of elliptic curves with prescribed supersingular primes
For a given infinite set of primes, not too big, eg, satisfying Lang-Trotter conjecture, can we always find an E.C. with supersingular reduction (at least) at these primes? How about E.C. without CM?
- 851
3
votes
1
answer
745
views
Conductor of a CM elliptic curve and its Grössencharacter
For a CM elliptic curve $E$ and its Grössencharakter, their conductors are both supported on bad primes of $E$. Moreover, by comparing their functional equation, there should be some obvious relations....
- 851
3
votes
1
answer
315
views
Is there an infinite family of primes $q_{1},q_{2},...$ so that the rank of $E(\mathbb{Q}(\sqrt{-q_{i}}))$ equals that of $E(\mathbb{Q})$?
It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
Much less is known if $K$ is infinite-...
- 1,727