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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6
votes
1answer
397 views

realizing uniform boundedness of Galois representations associated to elliptic curves

This is less of a direct question and more of an argument that I've been worried about for a while and want to check (apologies for the length and if my writing is unclear). Suppose I have an ...
24
votes
4answers
834 views

Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$

David Speyer commented the following here. I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials ...
6
votes
1answer
314 views

Analytic continuation for $L$-functions of elliptic curves

Let $E$ be an elliptic curve over a number field. When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ ...
10
votes
2answers
560 views

BSD and congruent numbers

Let $n$ be a positive integer, and let $E_n$ denote the elliptic curve $y^2=x^3-n^2x$. By work of Tunnell, it's known that if $E_n$ satisfies the BSD conjecture, then there is an algorithm to decide ...
-1
votes
1answer
86 views

Is there an algorithm to find a linear dependence between points on elliptic curves?

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ of characteristic $p$. Let $P,Q\in E(\mathbb{F}_q)$, such that $Q=mP+n\tau(P)$, where $\tau$ is the p-th power of frobenious map and $m$ ...
3
votes
2answers
268 views

Congruent numbers and elliptic curves

Who first explicitly stated the link between $N$ being a congruent number and the existence of rational points of infinite order on $y^2=x(x^2-N^2)$?
1
vote
0answers
119 views

Elliptic curves with potential good reduction over a prescribed extension

Notation: Let $K/\mathbb{Q}$ be a quadratic number field; let $p\geq 3$ be a rational prime and let $\mathfrak{p}$ denote a prime lying above $p$; let $K_{\mathfrak{p}}$ denote the completion of $K$ ...
4
votes
1answer
206 views

Checking whether two rational points of infinite order are generating the torsion free part of an elliptic curve

Let an elliptic curve be given. As the title says I want to know if we can show that two independent points $P$ and $Q$ are generators of the torsion free part of $E$. For instance let ...
1
vote
1answer
146 views

Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvable extension?

Are there counterexamples to the following: Given two varieties $A$, $\tilde{A}$, both defined over $\mathbb{Q}$, one of which, say $A$, is a Shimura variety. Then, every isomorphism defined over ...
8
votes
1answer
239 views

Intuitive reasons for the existence of modular parametrizations

Whenever I encounter anything about modular parametrizations, I have a feeling it is something very unnatural: you have some kind of moduli space and all of a sudden it parametrizes an object ...
13
votes
1answer
629 views

Reference to a Don Zagier Result and the Congruent Number Problem

I was looking for a reference/explanation as to how Don Zagier managed to find the side lengths of a rational right triangle with area 157. There have been many literature references to the fact that ...
7
votes
2answers
452 views

Galois cohomologies of an elliptic curve

I asked this question at math stackexchange but did not get any answer and I was suggested to post the question here. I am studying basic theory of elliptic curves. I encountered Galois cohomology. ...
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'} ...
6
votes
1answer
594 views

Why are integer points on elliptic curves interesting and useful?

I read some papers which dealed with integer points on elliptic curves. One of these papers are http://projecteuclid.org/euclid.rmjm/1214947612. My question is: Why are integer points on elliptic ...
1
vote
2answers
340 views

For what integer $n$ are there infinitely many $-a+nb+c = -d+ne+f$ where $a^6+b^6+c^6 = d^6+e^6+f^6$?

(Much revised for clarity.) I was considering the system of equations, $$-a+nb+c = -d+ne+f\tag1$$ $$a+b+c = d+e+f\tag2$$ $$a^2+b^2+c^2 = d^2+e^2+f^2\tag3$$ $$a^6+b^6+c^6 = d^6+e^6+f^6\tag4$$ ...
4
votes
0answers
93 views

Minimal discriminant of an elliptic curve in terms of its Galois representation

From the Galois representation of an elliptic curve $E$ we can read the conductor of $E$, and further some information about the minimal discriminant. So is there any more information about the ...
4
votes
2answers
241 views

Explicit $2$-descent on elliptic curves

Let $k$ be a field of characteristic $0$ and let $$E: y^2 = f(x)$$ be an elliptic curve over $k$, with $\mathrm{deg}(f) = 3$. Kummer theory yields a map $$\varphi:\mathrm{H}^1(k, E[2]) \to ...
8
votes
1answer
222 views

Integral points on elliptic curves of the form $y^2=x^3+px$

As the title says. Can we determine all the integral points on elliptic curves of the form $$y^2=x^3+px$$ for a prime $p$? If yes, can someone explain me how? A good reference would also be ...
2
votes
1answer
213 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 ...
1
vote
0answers
104 views

Generalized “elliptic integrals”

I am interested in evaluating the following type of integrals. Here we a polynomial $q(x)$ of degree $d \geq 2$ with no non-negative roots. Then is there a name for integrals of the shape ...
11
votes
1answer
232 views

What is the chromatic number of the “conic hypergraph” on a non-singular plane cubic?

Can we color the points of a complex non-singular plane cubic curve with finitely many colors so that no conic intersects the curve at 6 distinct points of the same color? If so, what is the smallest ...
6
votes
2answers
485 views

confounding riddle about fine moduli schemes and twists of elliptic curves

I've encountered a strange situation while thinking about modular curves... Consider the modular curve $Y(3)$ parametrizing elliptic curves with a symplectic basis for their 3-torsion. This curve has ...
8
votes
1answer
165 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 ...
11
votes
1answer
234 views

$\pi_1$ of the moduli of G-bundles on elliptic curves and the double affine braid group

For a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, the fundamental group of $\mathfrak{h}_\text{reg}/W$ (where $\mathfrak{h}$ is the Cartan subalgebra, $\mathfrak{h}_\text{reg}$ is the subset ...
4
votes
2answers
261 views

What is the fastest algorithm for counting points in elliptic curves mod n?

I need an algorithm for getting the order of the group in random elliptic curves mod n, being n a composite module. As far as I know, usual algorithms like Schoof's algorithm only works with prime ...
2
votes
2answers
180 views

Finite orbits on an elliptic curve with two generic involutions

Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor. Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an ...
3
votes
1answer
181 views

Veronese embeddings of elliptic curves in weighted projective space

Let $E$ be an elliptic curve and $D_k=kp$ a divisor on $E$, where $p\in E$, for $k\in\mathbb{N}$. Then we can reconstruct $E$ from the graded ring $R(D_k)=\bigoplus_{n\geqslant0}\mathcal{L}({nD_k})$: ...
7
votes
2answers
320 views

Order of reduction of infinite order rational point on an Elliptic Curve

Let $E/$ℚ be an elliptic curve and $P$ ∈ $E($ℚ$)$ a rational point of infinite order. Does the reduction of $P$ mod $p$ generate a maximal cyclic subgroup of $E(\mathbb{F}$$p$$)$ for ...
6
votes
3answers
676 views

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 ...
10
votes
1answer
412 views

Ramification of the map from the stack of elliptic curves to the $j$-line

Let $\mathcal{M}_{1, 1}$ be the stack of elliptic curves. Its coarse moduli space is $\mathbb{A}^1_{\mathbb{Z}}$ with the map $\mathcal{M}_{1, 1} \rightarrow \mathbb{A}^1_{\mathbb{Z}}$ given by the ...
7
votes
0answers
191 views

On discriminants of elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$ and $\Delta_E$ denote the discriminant of $E$. We say an elliptic curve has entanglement fields if the intersection of the $m_1$ and $m_2$ ...
10
votes
1answer
443 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 ...
7
votes
1answer
332 views

Serre's surjective theorem importance

I'm studying Serre's paper in wich he shows the following theorem: Let K be a number field, $E$ an elliptic curve over K without CM. Then the representation $$\rho_{\ell}:\mathrm{Gal}(\bar ...
2
votes
1answer
114 views

The $p$-th power of the invariant derivative on an elliptic curve in characteristic $p$

I am not an expert in elliptic curves at all, so my question may naive and/or obvious. Let $E$ be an (affine) elliptic curve defined over a finite (or perfect) field of characteristic $p$. Since its ...
6
votes
1answer
333 views

Derivatives of theta functions at zero

Let $L$ be a line bundle over complex elliptic curve, $\deg L = k>0$. Theta functions $$ \theta_s(z;\tau)_k=\sum_{r\in \mathbb{Z}} e^{\pi i [(\frac{s}{k} + r)^2 k \tau + 2kz(\frac{s}{k}+r)]}, ...
3
votes
2answers
288 views

Is it normal surface of general type to have infinitely many positive rank elliptic curves?

Cross-posted from MSE. I am not good at algebraic geometry and almost surely am misunderstanding something. Got an alleged argument against Bombieri-Lang conjecture and would like to know what the ...
2
votes
0answers
152 views

Help for reference of moduli stack of fake elliptic curve

I see everywhere says the following: Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$. ...
0
votes
0answers
112 views

Polynomial identities for congruent numbers and Bunyakovsky's conjecture

Bunyakovsky's conjecture states that polynomial with integer coefficients takes infinitely many prime values unless there are obvious reasons not to. It appears to imply something about polynomial ...
5
votes
0answers
204 views

On $a+b+c= abc = n$, elliptic curves, and solvable Galois groups

Solving $a+b+c = abc = 6$ in the rationals entails solving, $$-24a+36a^2-12a^3+a^4=z^2\tag1$$ which is birationally equivalent to an elliptic curve. It can be shown that if $a$ is a solution, then ...
2
votes
0answers
108 views

Number of CM lifting of an ordinary elliptic curve

Before asking my questions I will start with an example: There are two CM elliptic curves over $\mathbb{Q}$ with CM field $\mathbb{Q}(\sqrt{-7})$, whose $j$-invariants are $-3^3.5^3$ and $3^3. 5^3. ...
8
votes
3answers
670 views

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?
2
votes
1answer
219 views

Congruence Number of 197A1

It is reported in this paper by Zagier, as well as in Sage, that the elliptic curve $E=197A1$ has congruence number 10. (Since $E$ has prime conductor, a theorem of Ribet ensures that the congruence ...
2
votes
0answers
156 views

An elliptic curve trivial over any extension unramified outside 7 and infinity?

Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?
1
vote
0answers
130 views

Possible counterexample to a conjecture of Granville about automorphisms of twists of hyperelliptic curves

This might be a counterexample to a conjecture of Granville about automorphisms of twists of hyperelliptic curves. In this paper, the quadratic twist of $f(x)=y^2$ is denoted by $C_d : d y^2=f(x)$ ...
7
votes
0answers
234 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 ...
0
votes
1answer
266 views

A particular interesting elliptic curve

Given the elliptic curve $E:y^2=x^3-4x+4$. (a) How to prove that the group of rational points $E(\mathbb{Q})$ is generated by $P=(2,2)$. (b) If we consider the piece of curve on the region ...
4
votes
1answer
502 views

Weierstrass form of genus one $y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0$

Related to the n-conjecture. We are looking for Weierstrass form and map from it of the genus one curve: $$ y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0 $$ It is ...
3
votes
4answers
509 views

Integral points on a particular family of curves

This is a follow-up to this question (and comments thereon). Namely, it follows from Felipe Voloch's comment that for any $n>2$ there is a finite set of integral $(x, y),$ such that $$ ...
9
votes
1answer
594 views

Remark 4.23.4 in Hartshorne

Crosspost from math.stackexchange, since it's quite possible I might not get a response there. Remark 4.23.4 in Chapter IV of Hartshorne's Algebraic Geometry references a paper by Elkies that ...
4
votes
1answer
183 views

Equidistribution of representations by a binary cubic form

Let $f(x,y)\in\mathbb{Z}[x,y]$ be a binary cubic form with nonzero discriminant, and for a positive integer $m$ consider the integral representations $f(x,y)=m$. Assume that the number of ...