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.
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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} + \...
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The exponent of Ш of $y^2 = x^3 + px$, where $p$ is a Fermat prime
For $d$ a non-zero integer, let $E_d$ be the elliptic curve
$$
E_d : y^2 = x^3+dx.
$$
When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD,
$$
\# Ш(E_p)...
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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 ...
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Constructing non-torsion rational points (over Q) on elliptic curves of rank > 1
Consider an elliptic curve $E$ defined over $\mathbb Q$. Assume that the rank of $E(\mathbb Q)$ is $\geq2$. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank $=$ algebraic ...
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Is there a "classical" proof of this $j$-value congruence?
Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve $\mathbf{...
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Is there a "Basic Number Theory" for elliptic curves?
Tate's thesis showed how to profitably analyze $\zeta$ functions of number fields in terms of adelic points on the multiplicative group. In particular, combining Fourier analysis and topology, Tate ...
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Definition and meaning of the conductor of an elliptic curve
I never really understood the definition of the conductor of an elliptic curve.
What I understand is that for an elliptic curve E over ℚ, End(E) is going to be (isomorphic to) ℤ or an ...
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Diophantine equation for 2016: triangular $|{\rm GL}_2({\bf F}_q)|$
For a prime power $q$ the group ${\rm GL}_2({\bf F}_q)$ has
$(q^2-1)(q^2-q)$ elements. This happens to be a triangular number for
$q=2$ (being $6 = 1+2+3$), and $-$ more notably, especially this year ...
- 77.5k
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$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 ...
- 77.5k
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Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...
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Quaternionic and octonionic analogues of the Basel problem
I asked this question in MSE around 3 months ago but I have received no answer yet, so following the suggestion in the comments I decided to post it here.
It is a well-known fact that
$$\sum_{0\neq n\...
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Why is this "the first elliptic curve in nature"?
The LMFDB describes the elliptic curve 11a3 (or 11.a3) as "The first elliptic curve in nature". It has minimal Weierstraß equation
$$
y^2 + y = x^3 - x^2.
$$
My guess is that there is some ...
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Modular curves of genus zero and normal forms for elliptic curves
This is maybe the first question I actually need to know the answer to!
Let $N$ be a positive integer such that $\mathbb{H}/\Gamma(N)$ has genus zero. Then the function field of $\mathbb{H}/\Gamma(N)...
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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 ...
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What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$?
Some experts have a hunch that for any nonnegative integer $r$ there are infinitely many elliptic curves over $\Bbb Q$ with Mordell-Weil rank at least $r$.
The best empirical evidence for this hunch ...
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Is there any theory why (for Bitcoin) the discrete logarithm problem is so hard to solve?
Note I am an active member and contributor at the sister site https://bitcoin.stackexchange.com while studying Bitcoin and as a person who studied mathematics 10 years ago there is one thing I kept ...
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Do all curves have Néron models
Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.
Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$?
By a Néron model, I mean ...
- 1,203
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Modern proof of Serre's open image theorem?
Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic Curves'...
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Elliptic curve over a scheme is a group scheme?
In Katz's article p-adic properties of modular schemes and modular forms in the Antwerp proceedings, the following definition of an elliptic curve over a base scheme $S$ is given:
By an elliptic ...
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4
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Etale cohomology and l-adic Tate modules
$\newcommand{\bb}{\mathbb}\DeclareMathOperator{\gal}{Gal}$
Before stating my question I should remark that I know almost nothing about etale cohomology - all that I know, I've gleaned from hearing off ...
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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:...
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Is the Modularity Theorem (currently) effective?
The Modularity Theorem says every elliptic curve over $\mathbb{Q}$ can be gotten from the classic modular curve $X_0(N)$ by a rational map. Here $N$ is the conductor, easily calculable from a ...
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What are the possible singular fibers of an elliptic fibration over a higher dimensional base?
An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one.
It is often required for the ...
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2
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When did people start thinking of elliptic curves as groups?
I have been reading some old papers of Cassels and Selmer from around 1950, and they talk about generators of rational solutions to elliptic curves, in the sense of Mordell–Weil, but do not appear to ...
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answers
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Does the moduli space of smooth curves of genus g contain an elliptic curve
Let $M_g$ be the moduli space of smooth projective geometrically connected curves over a field $k$ with $g\geq 2$. Note that $M_g$ is not complete.
Does $M_g$ contain an elliptic curve?
The answer ...
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Fundamental group of the moduli stack of elliptic curves
I've heard that the étale fundamental group of the moduli stack of elliptic curves (over $\mathbb{Z}$) is trivial. Is there an easy proof of that? (Note that there are plenty of étale covers once one ...
- 25.3k
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How to explicitly compute lifting of points from an elliptic curve to a modular curve?
Say $E$ is an elliptic curve over the rationals, of conductor $N$. There's a covering of $E$ by the modular curve $X_0(N)$, and if you rig it right then you can define this map over $\mathbf{Q}$: ...
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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 ...
- 4,563
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2
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Why do the $2$-Selmer ranks of $y^2 = x^3 + p^3 $ and $y^2 = x^3 - p^3 $ agree?
I was playing around with sage, when I found that the Mordell-Weil ranks (over $\mathbb{Q}$) of the elliptic curves $y^2=x^3+p^3$ and $y^2=x^3-p^3 $ almost always agree, for $p$ prime. The first few ...
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Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
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votes
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Possible counterexample to a theorem assuming Lang's conjecture
Looks like I found a counterexample to a theorem assuming Lang's conjecture,
but not sure it is correct.
Boundedness of Mordell–Weil ranks of certain elliptic curves and Lang’s conjecture
p. 2
...
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What are the recommended books for an introductory study of elliptic curves?
I am currently doing a self study on algebraic geometry but my ultimate goal is to study more on elliptic curves. Which are the most recommended textbooks I can use to study? I need something not so ...
- 513
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The class number formula, the BSD conjecture, and the Kronecker limit formula
If K is a number field then the Dedekind zeta function Zeta_K(s) can be written as a sum over ideal classes A of Zeta_K(s, A) = sum over ideals I in A of 1/N(I)^s. The class number formula follows ...
- 6,942
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Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in\mathbb Z$ be unbounded as $n$ varies?
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in\mathbb Z$ be unbounded as $n$ varies?
$x,y$ are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of $n$ (...
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Conceptual understanding of the Gross-Zagier theorem.
The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more ...
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Universal homotheties for elliptic curves
Let $K$ be a number field and $E_1, \cdots, E_n$ elliptic curves over $K$. Let $\ell$ be a prime. Then there exists an element $\sigma \in \text{Gal}(\overline{K}/K)$ such that $\sigma$ acts on $T_\...
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Algorithms for finding rational points on an elliptic curve?
I am looking for algorithms on how to find rational points on an elliptic curve $$y^2 = x^3 + a x + b$$ where $a$ and $b$ are integers. Any sort of ideas on how to proceed are welcome. For example, ...
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How can I see the relation between shtukas and the Langlands conjecture?
The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about.
Drinfeld ...
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answers
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Fastest way to factor integers < 2^60
I've been running a search for Mordell curves of rank >=8 for about 12 months and have identified approximately 280,000 curves in our archivable range, amongst many millions that aren't.
For this ...
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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 ...
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Can one prove complex multiplication without assuming CFT?
The Kronecker-Weber Theorem, stating that any abelian extension of $\mathbb Q$ is contained in a cyclotomic extension, is a fairly easy consequence of Artin reciprocity in class field theory (one just ...
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Writing $3p$ when $p \equiv 1 \pmod{3}$ as a sum of two rational cubes. Is this result new? And what about its converse?
I recently discovered a proof of the following. Let $p$ be a prime that's $1 \bmod {3}$. Suppose that $p$ is not represented by the principal quadratic form $(1,9,81)$ of discriminant $-243$ (The ...
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Which elliptic curves over totally real fields are modular these days?
As the title says. In particular, every elliptic curve over $\mathbb{Q}$ is modular; but what is the current state of the art for general totally real number fields? I assume the answer is ...
- 13k
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votes
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answers
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Fake CM elliptic curves
Suppose one has an elliptic curve $E$ over $\mathbb{Q}$ with conductor $N < k^3$ for some (large) positive $k$, with the property that its Fourier coefficients satisfy
$$
a_p=0, \; \mbox{ for all }...
- 2,381
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votes
4
answers
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Integer points of an elliptic curve
Where can I found some resources to learn how to determine the integer points of given elliptic curve? I would like to learn a method based on computing the rank and the torsion group of given curve. ...
- 241
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3
answers
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A recommended roadmap to Fermat's Last Theorem
I was inspired to undertake math as a career after watching a documentary on the proof of Fermat's Last Theorem. As such it's been a small goal of mine to understand Wiles et al's proof.
In a ...
- 1,438
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votes
4
answers
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Does p-adic $L$- function determine the $L$ function
Let $E_1$ and $E_2$ be two Elliptic curve defined over $\mathbb Q$ . Let $p$ be an fixed given odd prime of $\mathbb Q$ at which both the curves have good ordinary reduction. Moreover p-adic $L$-...
- 213
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2
answers
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State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$
As far as I understand, both of the Diophantine equations
$$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^...
- 149k
21
votes
3
answers
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views
Consecutive square values of cubic polynomials
Let $P(x)$ be a cubic polynomial with integer coefficients. Does there exist a constant $c$ such that at least one of the following values $P(0),P(1),...,P(c)$ is not a square?
It is known that the ...
- 321
21
votes
2
answers
1k
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CM $j$-invariants in $p$-adic fields
I'm trying to understand the $p$-adic distribution of $j$-invariants for elliptic curves with complex multiplication.
Specifically, suppose $\sigma$ is some embedding $\sigma:\overline{\mathbb Q}\to \...
- 1,021