All Questions
221 questions
175
votes
2
answers
66k
views
Estimating the size of solutions of a diophantine equation
A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ?
(I do not know any such numbers).
B. Suppose that $\frac{a}{b+c} + \...
- 1,881
79
votes
12
answers
13k
views
Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
- 118k
23
votes
4
answers
9k
views
Integer points of an elliptic curve
Where can I found some resources to learn how to determine the integer points of given elliptic curve? I would like to learn a method based on computing the rank and the torsion group of given curve. ...
- 261
21
votes
2
answers
2k
views
State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$
As far as I understand, both of the Diophantine equations
$$a^5 + b^5 = c^5 + d^5$$
and
$$a^6 + b^6 = c^6 + d^6$$
have no known nontrivial solutions, but
$$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$
and
$$3^...
- 151k
7
votes
1
answer
454
views
One more generator needed for a Z/6 elliptic curve
I am trying to find the next rank 8 curve with the torsion subgroup Z/6 using Kihara's family as described in https://arxiv.org/pdf/1503.03667.pdf. Meanwhile, I came across a curve generated by $t=629/...
- 913
40
votes
2
answers
3k
views
$x^4+y^4$ powerful for relatively prime $x,y$
I asked this question on the NMBRTHRY mailing list on
17 February 2014, but it remains unsolved as far as I know.
Recall that a "powerful
number" is a positive integer whose prime ...
- 79.9k
34
votes
2
answers
3k
views
The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.
I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...
- 41.4k
29
votes
4
answers
2k
views
Are most cubic plane curves over the rationals elliptic?
%This is a new version of the original question modified in the light of the answers and comments.
The word 'most' in the title is ambiguous. The following is one way of making it precise.
Question1:...
- 1,650
26
votes
7
answers
6k
views
When is a product of elliptic curves isogenous to the Jacobian of a hyperelliptic curve?
David's question Families of genus 2 curves with positive rank jacobians reminded me of a question that once very much interested me: when is a product of elliptic curves isogenous to the jacobian of ...
- 4,593
18
votes
1
answer
2k
views
Conjecture: The number of points modulo $p$ of certain elliptic curve is $p$ or $p+2$ for $p$ of form $p=27a^2+27a+7$
Numerical evidence suggests a conjecture that the number of
points of certain elliptic curve over $\mathbb{F}_p$ is
either $p$ or $p+2$ for $p$ of certain form.
Let $p$ be prime of the form $p=27a^2+...
- 25.4k
16
votes
3
answers
2k
views
Are there nonisotrivial elliptic curves over $\mathbb{G}_m$?
Is there an elliptic curve over $\mathbb{C}[t, t^{-1}]$ that has a nonconstant $j$-invariant? What is an equation for such a curve, if it exists?
- 2,663
16
votes
2
answers
1k
views
Simple proofs for the existence of elliptic curves having a given number of points
Yesterday, after he gave a nice talk, Dick Gross and I were chatting and he brought up the following annoying problem: suppose that for $p$ a prime that $H_p$ is the "Hasse interval" $[p+1- 2 \sqrt{...
- 4,646
15
votes
3
answers
3k
views
Existence of fine moduli space for curves and elliptic curves
For the moduli problem of a curve of genus $g$ with $n$ marked points, how large an $n$ is needed to ensure the existence of a fine moduli space? For this question, terminology is that of Mumford's ...
- 7,442
14
votes
3
answers
2k
views
Convergence of L-series
I remember to have read that the L-function of an elliptic curve, which a priori only converges for $\Re s > \frac{3}{2}$ also converges at $s=1$ provided that the $L$-function
satisfies the ...
- 2,810
11
votes
4
answers
12k
views
How to find all integer points on an elliptic curve?
How can I determine the integer points of a given elliptic curve if I know its rank and its torsion group?
I read same basic books on elliptic curves but as a non-professional I didn't understand ...
7
votes
1
answer
464
views
A parametric elliptic curve for $x^4+y^4+z^4 = 1$?
Noam Elkies found that $x^4+y^4+z^4 = 1$ has infinitely many rational points $xyz \neq 0$ using an elliptic curve. We use a different approach that will produce pairs of solutions and a parametric ...
- 12.6k
32
votes
4
answers
5k
views
Over which fields does the Mordell-Weil theorem hold?
According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...
- 20.8k
22
votes
2
answers
2k
views
unboundedness of number of integral points on elliptic curves?
If $E/\mathbf{Q}$ is an elliptic curve and we put it into minimal Weierstrass form, we can count how many integral points it has. A theorem of Siegel tells us that this number $n(E)$ is finite, and ...
- 41.4k
19
votes
3
answers
2k
views
Elkies' supersingularity theorem in higher dimension
The following is a theorem of Elkies:
Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero.
...
- 156k
19
votes
1
answer
3k
views
Questions about the "universal elliptic curve" over the affine $j$-line punctured at 0 and 1728
So my question refers to families of elliptic curves over the $\mathbb{A}^1_\mathbb{C}\setminus\{0,1728\}$ whose fiber above a point $j$ has $j$-invariant equal to $j$ (I understand it's not universal)...
- 10.7k
18
votes
3
answers
2k
views
More elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
(Note: See also the $a^4+b^4+c^4 = 1$ version in this old MSE post.)
The equation discussed in a paper by Jacobi and Madden,
$$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 = z^4\tag1$$
or equivalently,
$$(p-2q + ...
- 12.6k
15
votes
6
answers
6k
views
bad reduction for elliptic curves
Why do elliptic curves have bad reduction at some point if they are defined over Q, but not necessarily over arbitrary number fields?
- 1,022
14
votes
2
answers
2k
views
Surjectivity of reduction maps of elliptic curves over Q
Let $E/\mathbf{Q}$ be an elliptic curve of rank $>0$. It is easy to see that there is a positive-density set of primes $p$ such that the reduction map $\mathrm{red}_p : E(\mathbf{Q}) \rightarrow \...
- 4,745
13
votes
2
answers
2k
views
Extensions of the modularity theorem
In 1995 (if I'm not mistaken) Taylor and Wiles proved that all semistable elliptic curves over $\mathbb{Q}$ are modular. This result was extended to all elliptic curves in 2001 by Breuil, Conrad, ...
- 1,458
13
votes
2
answers
1k
views
On Euler's elliptic curve for $A^4+B^4 = C^4+D^4$?
To solve,
$$A^4+B^4 = C^4+D^4$$
we use Euler's method. Let,
$$(p+q)^4+(r-s)^4=(p-q)^4+(r+s)^4$$
and define $p = (a^3 - b),\, q = a y,\, r = b (a^3 - b),\, s = y.\,$ The equation above transforms to ...
- 12.6k
11
votes
2
answers
679
views
Z/8Z elliptic curve with a missing generator
We are searching for the rank $6$ elliptic curves with the torsion subgroup $\mathbb{Z}/8\mathbb{Z}$ using the families similar to Allan MacLeod's as described in
A. J. MacLeod, A Simple Method for ...
- 913
9
votes
3
answers
3k
views
Tate module of CM elliptic curves
This is an exercise in Silverman's book "the arithmetic of elliptic curves". Ex 3.24, page 109.
E/K CM elliptic curve, Prove for $\ell \neq char(K)$, the action of $Gal(\bar{K}/K)$ on $T_{\ell}(E)$ ...
- 1,503
9
votes
3
answers
866
views
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z[t]$ be unbounded as n varies?
We've recently seen this question: Can the number of solutions $ab(a+b+1)=n$ for $a,b,n \in \mathbb{Z}$ be unbounded as $n$ varies? It appears initially plausible that the answer is yes, but evidently ...
- 30.1k
9
votes
4
answers
3k
views
reduction of CM elliptic curves
Can someone indicate how to prove the following equivalences for a CM elliptic curve $E$:
(i) $p$ is inert in End($E$)
(ii) $E_p$ is supersingular
(iii) The trace of the Frobenius at $p$ is $0$ [...
user19475
8
votes
1
answer
747
views
Deligne's exterior power
In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism
$$p : A^{\otimes n} \to A^{\otimes n}, ...
- 63.1k
8
votes
2
answers
3k
views
congruent number problem [closed]
I am studying the congruent number problem
and I heard that there is a paper by Kazuma Morita
which claims to solve this problem from my colleague.
I saw the paper on his homepage but it is very ...
- 115
8
votes
1
answer
904
views
Hard: One more generator needed for a Z/6 elliptic curve
We are searching for rank 8 elliptic curves with the torsion subgroup $\mathbb{Z}/6$ using newly discovered families similar to Kihara's as described in
A. Dujella, J.C. Peral, P. Tadić, Elliptic ...
- 913
7
votes
1
answer
5k
views
Chevalley's Theorem on Constructible Sets
I'm having a hard time understanding the theorem in the title, more specifically the proof of the related fact that the image of a dominant morphism contains a dense open set of it's closure. (My ...
- 75
6
votes
3
answers
4k
views
Re: Mordell's equation $y^2 = x^3 + k$ and perfect numbers
I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on.
My question is: When does Mordell's equation
$$Y^2 = X^3 + K$$...
6
votes
2
answers
483
views
Difficult elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
Similar to the case $x^4+y^4+z^4 = 1$ discussed in this MO post, define the system,
$$x^4+y^4+z^4+1 = (x+y+z+1)^4\tag1$$
$$\frac{x^2+x+1}{(x+y+1)(x+z+1)}=u\tag2$$
$$\frac{y^2+y+1}{(y+z+1)(y+x+1)}=v\...
- 12.6k
6
votes
2
answers
1k
views
Root number of a quadratic twist of an elliptic curve
Could someone provide a reference for the following fact which is stated without proof in section 4.3 of Alice Silverberg's survey "Open Questions in Arithmetic Algebraic Geometry"?
Let E be an ...
- 1,021
6
votes
0
answers
343
views
Are all stabilizer groups of the co-adjoint action smooth?
Let $k$ be a (non-archimedean) local field of positive characteristic $p$ and $\mathfrak{n}$ be any finite-dimensional nilpotent Lie algebra over $k$ with nilpotence length $l<p$. It is well-known ...
- 1,702
5
votes
1
answer
289
views
Transforming the Kondo quintic $5T2$ into the Lehmer quintic $5T1$?
I. Kondo-Brumer quintic
The deceptively simple solvable quintic,
$$x^5 + (a - 3)x^4 + (-a + b + 3)x^3 + (a^2 - a - 1 - 2b)x^2 + b x + a=0$$
is quite important for imaginary quadratic fields. For ...
- 12.6k
5
votes
3
answers
2k
views
The rank of a class elliptic curves
For elliptic curve $y^{2}=x(x+a^{2})(x+(1-a)^{2})$,($a$ is a rational number and does not equal 0,1,1/2),is its rank always 0?
- 51
4
votes
2
answers
408
views
Argument for unboundedness of integral points of elliptic curves over number fields
Probably this is well known to those who know it.
Got an argument and numerical support that over
number fields elliptic curves in minimal models
might have unbounded number of integral points,
the ...
- 25.4k
4
votes
0
answers
506
views
Euler Systems and Coleman’s Conjecture
I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what ...
- 99
4
votes
2
answers
2k
views
Example of elliptic curve with CM (complex multiplication) by \sqrt{-7}
Can someone give me an example of elliptic curve with CM by sqrt(-7) with the action.
I've found a list of examples in the following link but not the action.
http://planetmath.org/...
- 91
3
votes
0
answers
578
views
On Choudhry's $x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+y_4^k$ for $k=7$?
I. Fifth Powers
The Diophantine equation,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$
for $k=5$ is quite well-explored. It has an infinite number of primitive solutions (as points on an elliptic ...
- 12.6k
2
votes
0
answers
7k
views
Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve
$\DeclareMathOperator\Aut{Aut}\newcommand{\alg}{\mathrm{alg}}\newcommand{\an}{\mathrm{an}}$Edited after Noam Elkies' comment: From what I understand (very little actually), there exist elliptic curves ...
- 6,982
2
votes
1
answer
184
views
Centralizers of Cartan subgroups
Let $E$ be an elliptic curve with CM by an order $\mathcal O$ in an imaginary quadratic field $K$. Choose a basis for $E[N]$ to get an isomorphism $\operatorname{Aut}(E[N])\cong \operatorname{GL}_2(\...
- 2,375
1
vote
1
answer
241
views
locally closed orbits in metric Hausdorff topology
I learned the following fact from Bruhat and Tits's paper "Homomorphismes “abstraits” de groupes algebriques simples" Section 3.18 that
Let $k$ be a local field. Suppose that a $k$-group $H$ acts $k$...
- 1,702
1
vote
0
answers
95
views
Zeros of modular functions and automorphisms
Let $F_N$ be the field of modular functions of level $N$ and with Fourier coefficients in $\mathbf Q(\zeta_N)$. We have
$$F_N=\mathbf Q(j, f^{(r,s)}_N),$$
where the $f^{(r,s)}_N$ are the Fricke ...
- 2,375
1
vote
1
answer
591
views
Argument for non-existence of elliptic curve over $\mathbb{C}[t, t^{-1}]$
I have a couple of questions about this answer by Noam D. Elkies showing that there exist no elliptic curve $E$ over $\mathbb{C}[t, t^{-1}]$ having nonconstant $j_E$-invariant.
The strategy is to ...
- 5,986
0
votes
1
answer
242
views
Primes of the form $p=3a^2+3ab+b^2$ or $p=27a^2+27ab+7b^2$ and the number of points of $y^2=x^3+2$ modulo $p$
Trying to generalize this answered question based on limited numerical evidence.
Let $E / \mathbb{F}_p : y^2=x^3+2$.
Conjecture 1 Let $p=3a^2+3ab_0+b_0^2$ be prime and $a,b_0$
positive integers.
...
- 25.4k
75
votes
9
answers
17k
views
Why is an elliptic curve a group?
Consider an elliptic curve $y^2=x^3+ax+b$. It is well known that we can (in the generic case) create an addition on this curve turning it into an abelian group: The group law is characterized by the ...
- 9,366