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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5
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1answer
182 views

Is there $t\in\operatorname{Gal}(\overline{K}/K)$ s.t. $\operatorname{rank}_{\mathbf{Z}_p}((t-1)E_{p^\infty}(\overline{K}))=1$?

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\varinjlim_{k\in\mathbb{Q}[\mu_{p^\infty}]} \mathbb{Q}\left[\mu_{p^\infty},k^{1/p^\infty}\right]$$ and ...
5
votes
1answer
281 views

Division by $n$ in elliptic curves

Let $E/\mathbb F_{p^m}$ be an arbitrary elliptic curve over the Galois field $\mathbb F_{p^m}$, and let $$[n]^{-1}(P)\cap E(\mathbb F_{p^m})=\{Q\in E(\mathbb F_{p^m})\mid nQ=P\}.$$ Also let ...
5
votes
1answer
535 views

Elliptic curves and prime numbers

Let $p_n$ be the $n^{th}$ prime number. Suppose $E(F_{p_n})$ denotes an elliptic curve over the Galois field $GF(p_n)$ which is defined by $y^2=x^3+ax+b$. Is the below claim true? For each integer ...
17
votes
2answers
875 views

elliptic curves and group cohomology

Recently, I've been trying to understand Jacob Lurie's 2-equivariant elliptic cohomology a bit better than I had in the past. From what I can tell, the fragment of the story that only deals with ...
8
votes
2answers
403 views

What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?

What is known about the structure of $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}}]$? More generally, what do we know about $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}},k^{1/p^{n}}]$, where ...
19
votes
0answers
212 views

“High-concept” explanation for proof of a theorem of Ochanine?

See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here. Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative ...
10
votes
1answer
421 views

Galois representations for the curve $y^{2} = x^{3} - x$

Let $E / \mathbb{Q}$ be the elliptic curve given by $y^{2} = x^{3} - x$. I would like to know explicitly what the field of all $2$-power torsion looks like, as well as the image in ...
6
votes
1answer
247 views

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 ...
7
votes
1answer
308 views

What are some open problems regarding elliptic curves in finite fields?

I accept that my question seems so vague and broad, and I already looked into some similar questions in MO. But I would like to learn specifically about some open problems and conjectures regarding ...
4
votes
0answers
121 views

Ranks of elliptic curves over Q(t)

I have an elliptic curve $E/\mathbb{Q}(t)$, and I want to compute its rank. Does knowing the rank over $\mathbb{F}_p(t)$ for some prime of good reduction give a bound on the rank over ...
5
votes
1answer
369 views

The Weil numbers and modulus of an elliptic curve

I have an ignorant question about elliptic curves which I'll be slightly imprecise about. If I have an elliptic curve $X$ defined over $\mathbb Z$, I can base change to $\mathbb C$, and then ...
0
votes
1answer
161 views

Rational maps between elliptic curves [closed]

I am studying Silverman's "The Arithmetic of Elliptic Curves" and I got the following question: In the first chapters he defines rational between projective varieties (see the first definition in ...
6
votes
1answer
400 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
841 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
322 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
568 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
273 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
120 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
209 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
147 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
240 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
644 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
454 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
596 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
345 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
251 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 ...
9
votes
1answer
300 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
215 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
234 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
489 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
167 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
236 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
266 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
184 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
323 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
677 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
464 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
207 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
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 ...
7
votes
1answer
336 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
116 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
337 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
289 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
117 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 ...