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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Quadratic non-residues in elliptic divisibility sequences

Let $E: y^2 = x^3 + ax + b$ be an elliptic curve over $\mathbb{Q}$ with $a,b \in \mathbb{Z}$. Recall that any rational point $P = (x,y)$ can be written uniquely as $P = (u/d^2, v/d^3)$ with $u,v,d \in ...
Daniel Loughran's user avatar
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?
flor.ian sprung's user avatar
15 votes
5 answers
8k 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 ...
James Weigandt's user avatar
15 votes
2 answers
2k views

Does the p-adic Tate module of an elliptic curve with ordinary reduction decompose?

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E$ an elliptic curve over $K$ with good ordinary reduction. The p-adic Tate module $T_p(E)$ is (after tensoring with $\mathbb{Q}_p$) a 2-...
Martin Orr's user avatar
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15 votes
3 answers
687 views

Average rank of elliptic curves, excluding those of low rank

It's conjectured that, asymptotically, half of elliptic curves have rank 0, half have rank 1, and elliptic curves of rank $\geq 2$ have density 0. But what if we disregard elliptic curves of rank 0 or ...
Daniel Hast's user avatar
  • 1,806
15 votes
2 answers
981 views

Determining the Mordell-Weil group of a universal elliptic curve

Let $K$ be a number field and let $K(a,b)$ be the field of rational functions with two indeterminates over $K$. Consider the elliptic curve $E$ over $K(a,b)$ defined by the Weierstrass equation \begin{...
François Brunault's user avatar
15 votes
2 answers
1k views

Modular forms from counting points on algebraic varieties over a finite field

Suppose we are given some polynomial with integer coefficients, which we regard as carving out an affine variety $E$, for example: $$ 3x^2y - 12 x^3y^5 + 27y^9 - 2 = 0 \tag{$*$} $$ (We might ...
Bruce Bartlett's user avatar
15 votes
5 answers
2k views

Very strong multiplicity one for Hecke eigenforms

In Invent. math. 116, 645-649 (1994) Dinakar Ramakrishnan proves a theorem which I understand to imply that the following statement (in light of the fact that elliptic curves over $\mathbb{Q}$ are ...
Jonah Sinick's user avatar
  • 6,912
15 votes
4 answers
1k views

The ring of algebraic integers of the number field generated by torsion points on an elliptic curve

(Warning: a student asking) Let $E$ be an elliptic curve over $\mathbf Q$. Let $P(a,b)$ be a (nontrivial) torsion point on $E$. Is there an easy description of the ring of algebraic integers of $\...
Anonymous's user avatar
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3 answers
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How do you explicitly compute the p-torsion points on a general elliptic curve in Weierstrass form?

Consider the Weierstrass cubic $$y^2z = x^3 + A\, xz^2+B\,z^3.$$ This defines a curve $E$ in $\mathbb{P}^2$, which if smooth is an elliptic curve with basepoint at $[0,1,0]$. I'm interested in having ...
Charles Rezk's user avatar
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15 votes
1 answer
933 views

Congruence for the number of points in the elliptic curve $y^2 = x^3+b \pmod{p}$

Let $E$ be the elliptic curve $y^2=x^3+1$ and $p \equiv 1 \pmod{3}$ a prime. Computing the number of points mod $p$ of $E$ using the naive method gives: $$ \#E(\mathbb F_p) = 1+ \sum_{x=0}^{p-1} \left(...
Esteban Crespi's user avatar
15 votes
2 answers
829 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 ...
Mikhail Borovoi's user avatar
15 votes
2 answers
304 views

S integral points of an elliptic curve, with S of positive density

Let E be an elliptic curve over Q of non-zero rank. Let S be the union of the primes of bad reduction of E with a Chebotarev set [1]. Suppose additionally that S has density strictly less than one. ...
David Zureick-Brown's user avatar
15 votes
1 answer
1k views

What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...
Bobby Grizzard's user avatar
15 votes
1 answer
821 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 ...
Kevin Buzzard's user avatar
15 votes
1 answer
2k views

Reference for the `standard' Tate curve argument.

I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following: Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$...
David Zureick-Brown's user avatar
15 votes
2 answers
1k views

Structure of $E(Q_p)$ for elliptic curves with anomalous reduction modulo $p$

For simplicity, take $p\ge7$ a prime and $E/\mathbb{Q}$ an elliptic curve with good anomalous reduction at $p$, i.e., $|E(\mathbb{F}_p)|=p$. There is a standard exact sequence for the group of points ...
Joe Silverman's user avatar
15 votes
2 answers
1k views

Computing endomorphism rings of supersingular elliptic curves

I would like to know what algorithms there are to compute the linearly independent generators $(1,i,j,k)$ for quaternion algebra containing the endomorphism ring of a supersingular curve. The curve in ...
BlackAdder's user avatar
15 votes
1 answer
2k 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 $-...
Franz Lemmermeyer's user avatar
15 votes
0 answers
586 views

For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?

An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...
Lloyd Yu-West's user avatar
14 votes
3 answers
1k 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?
Lisa S.'s user avatar
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14 votes
6 answers
3k 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 ...
14 votes
3 answers
4k views

Transforming a Diophantine equation to an elliptic curve

I heard that the following problem lead to determine the rational points of an elliptic curve: For which integers $n$ there are integers $x,y,z$ such that $x/y+y/z+z/x=n$. Could anyone show me why ...
Student's user avatar
  • 183
14 votes
3 answers
4k views

Recent progress toward Birch and Swinnerton-Dyer conjecture

Has there been any progress toward the Birch and Swinnerton-Dyer conjecture after The current status of the Birch & Swinnerton-Dyer Conjecture
guest's user avatar
  • 141
14 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 ...
Anweshi's user avatar
  • 7,272
14 votes
3 answers
714 views

The boundedness of the rank of twists of a fixed curve

It is conjectured that there are do not exist elliptic curves over $\mathbb Q$ of arbitrarily high rank. I was wondering wether someone made a similar conjecture if one restricts to a fixed $j$-...
Maarten Derickx's user avatar
14 votes
4 answers
4k views

Supersingular elliptic curves

I've read that an elliptic curve is supersingular if and only if its endomorphism ring is an order in a quaternion algebra. Does anyone have a simple explanation of this (or a good reference)?
Jonathan Wise's user avatar
14 votes
3 answers
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 ...
Kevin Acres's user avatar
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 ...
wood's user avatar
  • 2,714
14 votes
3 answers
778 views

Order of torsion group

What can one say about the order of the torsion group of an elliptic curve defined over the compositum of all quadratic extensions of $\mathbb{Q}$ ?
Suman's user avatar
  • 1,211
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 \...
R.P.'s user avatar
  • 4,745
14 votes
2 answers
511 views

Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank specialization?

Let $\mathcal{E}/\mathbb{Q}(t)$ be given by $$y^2=x^3+A(T)x+B(T)$$ for some $A(T),B(T)\in\mathbb{Q}[T]$ and assume $\mathcal{E}$ is non-isotrivial (the $j$-invariant $\frac{6912 A(T)^3}{4A(T)^3 + 27B(...
Jonathan Love's user avatar
14 votes
1 answer
862 views

BSD conjecture for rank 1 elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that $$\text{ord}_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$ Thanks to the work of Gross-Zagier and ...
baobab's user avatar
  • 243
14 votes
2 answers
2k views

Elliptic Curves with CM and Class Field Theory

Let $K$ be an imaginary quadratic field with Hilbert class field $H$, and let $E$ be an elliptic curve defined over $H$ with complex multiplication by the ring of integers $O_K$ of $K$. It is known ...
abourdon's user avatar
  • 235
14 votes
3 answers
2k 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}} , \...
Andreas Holmstrom's user avatar
14 votes
1 answer
2k views

Reduction mod $p$ map is injective?

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good or multiplicative reduction at $p$. How do I see that the reduction mod $p$ map from the endomorphisms of $E$ over $\overline{\mathbb{Q}}$ to ...
user101037's user avatar
14 votes
2 answers
2k views

Galois theory and rational points on elliptic curves

I am in search of a concrete example [a concrete elliptic curve in Weierstrass form] of how Galois theory helps to find rational points on an elliptic curve. Chapter VI of Silverman and Tate discusses ...
14 votes
1 answer
704 views

$S$-Tate-Shafarevich groups of elliptic curves

Let $S$ be a finite set of places of a number field $k$ and let $E$ be an elliptic curve over $k$. Define the ''$S$-Tate-Shafarevich group" of $E$ to be $$Ш(E,S) = \ker\left(H^1(k,E) \to \prod_{v \...
Daniel Loughran's user avatar
14 votes
1 answer
962 views

P-adic L-functions of nonabelian twists of elliptic curves

Let $E$ be an elliptic curve and $\rho$ an Artin representation of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. Then there is a "twisted L-function" $L(E, \rho, s)$, corresponding to the ...
David Loeffler's user avatar
14 votes
0 answers
1k views

Explicit example of elliptic curve of the kind needed for IUTT

At the nLab, we are currently trying to illustrate the definition of initial theta-data in Mochizuki's first IUTT paper by means of an explicit example. The exposition should end up at the following ...
user124294's user avatar
14 votes
0 answers
530 views

Am I missing something about this notion of Mirror Symmetry for abelian varieties?

This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s. In the comments of the question, I was directed to the paper http://arxiv.org/abs/...
Simon Rose's user avatar
  • 6,240
13 votes
2 answers
4k views

Example of connected-etale sequence for group schemes over a Henselian field?

Can someone give a really concrete example of such a sequence? I am looking at several notes related with such things, but haven't seen any well-calculated example. And I'm really confused at this ...
natura's user avatar
  • 1,503
13 votes
2 answers
1k views

Galois cohomologies of an elliptic curve

I asked this question on Mathematics Stack Exchange 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 ...
user avatar
13 votes
3 answers
3k views

$j$-invariants of elliptic curves over finite fields

Let $K$ be a finite field, and $\overline{K}$ its algebraic closure. It is well known that two curves are isomorphic over $\overline{K}$ if and only if they have the same $j$-invariant. If two such ...
Calodeon's user avatar
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13 votes
1 answer
1k views

Is there a substitution that relates every Fermat curve to an elliptic curve?

I asked this question on MSE but didn't get any response, so I'm asking here. I apologize in advance if this question is not research level. A Fermat Curve of degree $n$ is the set of solutions to $x^...
YiFan's user avatar
  • 236
13 votes
2 answers
901 views

Sums of two cubes

In his solution of the equation $x^3 + dy^3 = 1$, Nagell comes across the equation $$ u^3 + 6u^2v + 3uv^2 - v^3 = w^3. $$ He then observes that $$ (u^3 + 6u^2v + 3uv^2 - v^3) U^3 = V^3 + W^3 $$ for $$...
Franz Lemmermeyer's user avatar
13 votes
2 answers
1k 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 $\...
Joël's user avatar
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13 votes
2 answers
1k views

Are Kato's zeta elements integral?

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $T$ the $p$-adic Tate module of $E$. Kato's Euler system, constructed in the paper "P-adic Hodge theory and values of zeta functions of modular forms"...
David Loeffler's user avatar
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 ...
Tito Piezas III's user avatar
13 votes
2 answers
781 views

GRH and the rank of elliptic curves

I have been using the Magma calculator recently, and while calculating ranks of elliptic curves with very big coefficients, there is a possibility to assume GRH is true, which signaficantly speeds up ...
FusRoDah's user avatar
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