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 in addition; note also the tag arithmetic-geometry as well as some related tags such as rational-points, ...

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20
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2answers
1k 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 ...
29
votes
4answers
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. ...
21
votes
2answers
1k 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 ...
10
votes
3answers
2k 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 ...
3
votes
0answers
531 views

Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve

Edited after Noam Elkies' comment: From what I understand (very few actually), there exist elliptic curves defined over some number fields $\mathbb{K}$ Galois over $\mathbb{Q}$ which are isogenous to ...
13
votes
2answers
878 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 ...
12
votes
1answer
801 views

What's wrong with my understanding of the scheme $\text{Isom}(E_\lambda, E_{\lambda'})$?

Let $\mathcal{M}_{1,1}$ be the moduli stack of elliptic curves (over the complex numbers). Define $$\begin{eqnarray*} X &:=& \Bbb{A}^1_{\lambda} - \{0,1\}\\ X' &=& \Bbb{A}^1_{\lambda'} ...
9
votes
2answers
868 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, ...
49
votes
3answers
5k views

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 ...
26
votes
3answers
2k views

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 ...
18
votes
6answers
2k 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 ...
10
votes
6answers
2k views

Generalizations of Belyi's theorem

Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent: 1) $X$ is defined over $\overline{\mathbb{Q}};$ 2) There exists a meromorphic ...
9
votes
5answers
1k views

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 ...
24
votes
3answers
2k views

Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?

Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ 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 (assuming n ...
12
votes
4answers
1k views

Torsion subgroups in families of twists of elliptic curves

Here is the short version: Fix an elliptic curve $E/\mathbb{Q}$. How does the torsion structure $E_d(\mathbb{Q})_{tors}$ vary, as $E_d$ runs through the quadratic twists of $E$? Here is the ...
13
votes
1answer
525 views

components of E[p], E universal in char p.

I have just realised that a group scheme I've known and loved for years is probably a bit wackier than I'd realised. In this question, in Charles Rezk's answer, I erroneously claim that his ...
21
votes
3answers
2k 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 ...
13
votes
3answers
2k views

Rational Points on $y^2=x^3-86069^5$

The analytic rank of the Mordell elliptic curve $y^2=x^3-86069^5$ indicates that it has rank 2. However, deriving a set of generators, and hence the regulator, is proving to be a little bit of an ...
11
votes
4answers
5k views

Is the ABC conjecture known to imply the Riemann hypothesis?

I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...
10
votes
1answer
886 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 ...
3
votes
2answers
527 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 ...
3
votes
4answers
1k 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 $0$ ...
15
votes
2answers
852 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 ...
12
votes
4answers
831 views

Geometric meaning of fiber of modular parameterization over a point of an elliptic curve?

Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, parameterization $\psi : X_0(N) \rightarrow E$, and a point $P \in E$, take the fiber $\psi^{-1}(P)$. Its points, being on $X_0(N)$, correspond ...
10
votes
1answer
448 views

Converse to Modularity I: weight 2 newforms

Since 2008 we have the following remarkable correspondence: Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1 newforms note: all Galois representations in this question are ment ...
10
votes
3answers
1k views

A question on K_1 of an elliptic curve

Consider an elliptic curve $E/ \mathbb{Q}$, with a regular model $\mathcal{E} / \mathbb{Z}$. We have (Beilinson regulator) maps $$ K_1(\mathcal{E})^{(2)} \to K_1(E)^{(2)} \to H_D^3(E_{/ \mathbb{R}} , ...
6
votes
0answers
688 views

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 ...
21
votes
1answer
1k views

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 ...
11
votes
2answers
674 views

Best bounds toward Serre's uniformity conjecture

If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the Galois representations ...
9
votes
1answer
521 views

Parametric Families for Large Torsion Subgroups

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 ...
8
votes
3answers
789 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 ...
4
votes
2answers
340 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 ...
3
votes
3answers
2k 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 ...
7
votes
1answer
290 views

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 ...
20
votes
1answer
1k views

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 ...
13
votes
1answer
1k views

Fermat's Bachet-Mordell Equation

Fermat once claimed that the only integral solutions to $y^2 = x^3 - 2$ are $(3, \pm 5)$. Fermat knew Bachet's duplication formulas (more precisely, Bachet had a formula for computing what we call ...
11
votes
1answer
409 views

Elements of arbitrary large order in the first Galois cohomology of an elliptic curve

Let $E$ be an elliptic curve over $k=\mathbb{Q}$. Consider $H^1(k,E)$. In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ...
9
votes
1answer
347 views

A reference for $\mathbb{A}^1_R$ being a coarse moduli space of the stack of elliptic curves

Let $R$ be a ring and let $\mathcal{M}_{1, R}$ be the algebraic $R$-stack of elliptic curves (over $R$-schemes as bases). One knows that the coarse moduli space of $\mathcal{M}_{1, R}$ is supposed to ...
8
votes
1answer
168 views

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 ...
8
votes
1answer
579 views

how many consecutive integers $x$ can make $ax^2+bx+c$ square ?

The following problem was raised in a Mathlinks thread: If $a,b,c\in\mathbb Z$ such that $a\ne0$ and $b^2-4ac\ne 0$, for how many consecutive integers $x$ can $ax^2+bx+c$ ba a perfect square ? The ...
7
votes
0answers
237 views

Is the compositum of all quadratic extensions of the rationals an ample field?

Let $K$ be the compositum of all quadratic extensions of $\mathbb{Q}$, that is $$K = \mathbb{Q}(\sqrt{d} \ : \ d \in \mathbb{Q}).$$ Is there a (geometrically irreducible) smooth variety ...
3
votes
1answer
727 views

Abelian varieties with CM

In this site, I looked at a paper of Kazuma Morita claiming the BSD conjecture for the CM case posted on his homepage (he made a mistake three years ago for full BSD). But, I am interested in this ...
2
votes
1answer
242 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? ...
1
vote
1answer
149 views

Real points $a∈ℝ$ such that the equation $f^{(k)}(s)=a$ have a finite number of real solutions $s$ for some $k$

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the ...
1
vote
0answers
258 views

Does the property (P) holds true for the derivatives of $L$?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros ...
10
votes
2answers
852 views

Legitimacy of reducing mod p a complex multiplication action of an elliptic curve?

I scoured Silverman's two books on arithmetic of elliptic curves to find an answer to the following question, and did not find an answer: Given an elliptic curve E defined over H, a number field, ...
5
votes
2answers
555 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 ...
4
votes
1answer
139 views

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 ...
4
votes
2answers
262 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 ...
3
votes
1answer
165 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 ...