For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.

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Prove that a holonomic (p-recursive) difference equation returns only integral values

Consider the recurrence given by $(n+1)^2 a_{n+1} = (9n^2+9n+3)a_n-27n^2 a_{n-1}$ $a_0 = 1, a_1 = 3$. Clearly, $a_n$ is rational, but unexpectedly, the recurrence seems to output only integral ...
8
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1answer
113 views

Are all elliptic curves from $w^3 = \text{cubic}(z)$ isomorphic?

I've been playing around with Riemann surfaces of cubics, and it seems to me that all surfaces obtained as coverings of the Riemann sphere from equations of the form $w^3 = q(z)$, where $q(z)$ is a ...
8
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2answers
172 views

Completion along zero section of an elliptic curve.

I am trying to understand the intuition that I should have about the formal group of an elliptic curve. Say that I have an elliptic curve $E\to \text{Spec} R$ for some ring $R$, with section $0\colon ...
8
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1answer
112 views

Size of the group $E(\mathbb{Q}_{p})/pE(\mathbb{Q}_{p})$

Let $E$ be an elliptic curve over $\mathbb{Q}_{p}$. Do we know anything about the order of the group $E(\mathbb{Q}_{p})/pE(\mathbb{Q}_{p})$? I know that it's finite, but do we know anything else?
8
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452 views

Splitting of quaternion algebras

A rational (definite) quaternion algebra is an algebra of the form $$ \mathcal{K} = \mathbb{Q} + \mathbb{Q}\alpha + \mathbb{Q}\beta + \mathbb{Q}\alpha \beta $$ with $\alpha^2,\beta^2 \in ...
8
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123 views

Motivation for Weil pairing

The Weil pairing $$e_\phi:E[\phi]\times E'[\hat{\phi}]\to \mu_n$$ for an elliptic curve is defined as follows. Let $\phi:E\to E'$ be an isogeny of degree $n$ and $\hat\phi:E'\to E$ be the dual ...
8
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190 views

Global sections of vector bundles on a complex elliptic curve and analytic functions

Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I ...
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116 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
7
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3answers
188 views

The genus of a curve with a group structure

I'm reading Milne's Elliptic Curves and came across this statement: If a nonsingular projective curve has a group structure defined by polynomial maps, then it has genus 1. In this question a similar ...
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318 views

Diophantine equation $x^2 + 32x = y^3$

I am trying to find all solutions to the Diophantine equation $x^2 + 32x = y^3$. I think that the first step is to factorise: $x(x+32)=y^3$. If $x$ is odd, then $x+32$ is also odd. The common ...
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643 views

Geometric reason why elliptic curve group law is associative

The question title says it all. I am looking for a geometric proof for the fact that the group law defined on elliptic curves is associative. I've heard somewhere about something on the ...
7
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1answer
188 views

Integral relations in Fricke and Klein

Can someone please explain how Fricke and Klein obtain the integral relationa stated at the top of p. 34 in this book? The entire book can be previewed on Google Books. It is an old book and I do not ...
7
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2answers
149 views

$a^3+3a^2+a$ is never a perfect square.

Prove that no number of the form $ a^3+3a^2+a $, for a positive integer $a$, is a perfect square. This problem was published in the Italian national competition (Cesenatico 1991). I've been ...
7
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1answer
124 views

Do schemes help us understand elliptic curves?

I'm reading Silverman and Tate's "Rational Points on Elliptic Curves" and I'm very much enjoying learning about these objects, and in particular doing a bit of number theory. It's different to what ...
7
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1answer
294 views

Is the pushforward of the sheaf of differentials on an elliptic curve over a scheme necessarily trivial?

If $f:E\rightarrow S$ is an elliptic curve over a scheme $S$ (so $f$ is proper and smooth of relative dimension one with geometrically connected fibers of genus one, equipped with a section ...
7
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2answers
548 views

A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain ...
7
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1answer
409 views

Does the $p$-torsion of an elliptic curve with good reduction over a local field always determine whether the reduction is ordinary or supersingular?

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E/K$ an elliptic curve with good reduction. Does the $\mathbb{F}_p[\mathrm{Gal}(\overline{K})]$-module $E[p](\overline{K})$ determine whether the ...
7
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1answer
70 views

Torsion on $y^2=x^3+d$

A question that I am stuck on is: prove that the $\mathbb{Q}$-torsion subgroup of the elliptic curve $y^2=x^3+d$ has order dividing 6. Any hints on how to start would be nice. I tried saying ...
7
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2answers
181 views

Integer solutions of $x^3 = 7y^3 + 6 y^2+2 y$?

Does the equation $$x^3 = 7y^3 + 6 y^2+2 y\tag{1}$$ have any positive integer solutions? This is equivalent to a conjecture about OEIS sequence A245624. Maple tells me this is a curve of genus $1$, ...
7
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1answer
80 views

How to test if a given elliptic curve has complex multiplication

Is there a general, reasonably easy to understand, algorithm for testing whether an elliptic curve has CM? For example, consider the curve $y^2=x^3+\frac{27}{1727}x+\frac{54}{1727}$ This has ...
7
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1answer
101 views

Why is $H^1(\text{Gal}(\overline{K}/K),A)=\lim_{\rightarrow}H^1\left(L/K, A^{\text{Gal}(\overline{K}/L)}\right)$

I'm doing a course on elliptic curves, and I'm stuck on this. Here are the definitions: Let $G$ be a group and let $A$ be a $G$-module (that is, a $\mathbb{Z}[G]$-module). We have the "cochains" ...
7
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1answer
194 views

How can I determine in practice whether two elliptic curves over $\mathbb{Q}$ have isomorphic $p$-torsion?

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ with good, ordinary reduction at an odd prime $p$. I'm wondering how to determine whether $E_1[p]$ and $E_2[p]$ are isomorphic ...
7
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1answer
367 views

Special privilege enjoyed by Elliptic Curves with Complex Multiplication

I think after reading the title one may understand the intention of me, this question is concerned about the Elliptic curves having a Complex Multiplication. I have been reading many theorems, ( ...
7
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1answer
136 views

Challenge from Fermat

Fermat challenged Frenicle with finding a pythagorean triple (a,b,c) where $(a-b)^2-2b^2$ is itself a square. By making the substitution $a=m^2-n^2$, $b=2mn$, and $c=m^2+n^2$ into $(a-b)^2-2b^2=d^2$ ...
7
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1answer
125 views

Prove that $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p

I am trying to prove that the congruence $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p. I proved it using primitive root, but my professor in number theory told me that it can be ...
7
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1answer
168 views

Dimension of family of hyperelliptic curves

Suppose we have an elliptic curve E with a point $P$ of order $5$ over a field of characteristic $0$. Denote $E'$ the curve $E/\langle P\rangle$. Now let $x$ (resp. $x'$) be a function on $E$ (resp. ...
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89 views

Ex. $2.30$ in Silverman Adv. Topics

I would like to refer you to Exercise $2.30(c)$ in Silverman's Advanced Topics in Elliptic Curves. Question: Let $E/L$ be an EC with CM by $K$. Assume that $K\nsubseteq L$, and let $L'=LK$ and let ...
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Surfaces ruled over elliptic curves

Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve. Suppose $E$ is an elliptic ...
7
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1answer
116 views

Short exact sequence of modules and elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve with a 3-torsion point $P$. Let $E_{d}$ denote the quadratic twist of $E$ by $d$. Then the action of $\sigma \in ...
6
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3answers
2k views

Group Law for an Elliptic curve

I was reading this book "Rational points on Elliptic curves" by J.Silverman, and J.Tate, 2 prominent figures in Number theory and was very intrigued after reading the first couple of pages. The ...
6
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3answers
140 views

Fact check: global geometry / topology of moduli space of curves

Question: Is the moduli space of smooth complex curves of genus $g\geq2$ isomorphic to the affine space $\mathbb A_{\mathbb C}^{3g-3}$? (Note: I am not asking about the compactification of this ...
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722 views

References for elliptic curves

I just finished reading Silverman and Tate's Rational Points on Elliptic Curves and thought it was very interesting. Could any of you point me to some more references (ex. books, articles) on ...
6
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2answers
1k views

Local-Global Principle and the Cassels statement.

In a recent article I have read, i.e. " Lecture notes on elliptic curves ", Prof.Cassels remarks in page-110 that There is not merely a local-global principle for curves of genus-$0$, but ...
6
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287 views

Purpose of cusps

In the theory of of modular forms, there is the set of of cusps defined by $\mathbb{P}^1 (\mathbb{Q})= \mathbb{Q} \cup \{\infty\}$. For an subgroup $\Gamma < \text{SL}_2(\mathbb{Z})$ of finite ...
6
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2answers
377 views

The Néron-Tate canonical height on elliptic curves

I have been trying to understand the Néron-Tate global canonical height of algebraic points on elliptic curves. Let $K$ be a number field, $E$ an elliptic curve (over $\mathbb{Q}$, say), and $E(K)$ ...
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255 views

Injection of $E(\mathbb{Q})_{\text{tors}}$ into $\tilde{E}(\mathbb{F}_p)$?

I'm looking at Example VII.3.3.3 (p.193, 2nd ed.) of Silverman's The Arithmetic of Elliptic Curves. We have the elliptic curve $E:y^2=x^3+x$, with discriminant $\Delta=-64$, so there is good reduction ...
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2answers
73 views

The mod $p$ Galois representation of the Frey curve is unramified away from $2, p$

Given a hypothetical solution to Fermat's last theorem for $p \ge 5$ $$a^p + b^p + c^p = 0$$with $a \equiv -1 \pmod 4$, $b$ even, we can write down the Frey Curve$$E: y^2 = x(x-a^p)(x+b^p)$$which has ...
6
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154 views

References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
6
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2answers
264 views

Other ways to compute the torsion subgroup of elliptic curves

Suppose I have a family of elliptic curves $E_{n}/\mathbb{Q}$. I would like to determine the torsion subgroup of $E_{n}(\mathbb{Q})$ denoted by $E_{n}(\mathbb{Q})_{\textrm{tors}}$. Two ways to do this ...
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1answer
149 views

Why do modular curves parametrise elliptic curves?

Let $Y_1(N)=\Gamma_1(N)/H$, where $H$ is the upper half plane. In these lecture notes http://math.uga.edu/~pete/modularandshimura.pdf , the author makes the following statement: "$Y_1(N)$ ...
6
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1answer
279 views

What do the involutions of an elliptic curve look like?

Every automorphism $\varphi \in \mathrm{Aut}(E)$ of an elliptic curve $E$ (with base point $O$ over a field $k$) can be written $\varphi = \tau_Q\phi$ where $\phi \in \mathrm{Aut}(E,O)$ is an isogeny ...
6
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232 views

UPDATE: How to find the order of elliptic curve over finite field extension

I want to find the order of elliptic curve over the finite field extension $\mathbb{F}_{p^2}$, where $E(\mathbb{F}_{p^2}):y^2=x^3+ax+b $ I am using the method illustrated by John J. McGee in his ...
6
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1answer
170 views

Computing the trace of the following automorphism of the elliptic curve $y^2 = x^3+x$

Consider the elliptic curve $E$ defined by $y^2z= x^3 +xz^2$ over an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$. Consider the endomorphism $f:E\to E$ given by $(x:y:z)\mapsto ...
6
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3answers
70 views

Semigroup law on points on the curve $f(x) = \frac{1}{x}$

Consider the positive half of the curve $f: \Bbb{R} \to \Bbb{R}, f(x) = \frac{1}{x}$. Let $A = (a,1/a), B = (b, 1/b)$ be any two points on the curve. Draw a line through them Find where this point ...
6
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2answers
491 views

Turning an elliptic curve over C into a complex torus

I have been reading a lot about the Weierstrass $\wp$ function and I understand the parameterization of an elliptic curve with the elliptic function( i.e. $x=\wp(z)$ and $y=\wp^\prime(z)$). I would ...
6
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1answer
129 views

transform into weierstrass-form

How can I transform the elliptic curve $E/\mathbb{C}$ of the form $$y^2=4(x-e_1)(x-e_2)(x-e_3)$$ with $e_1>e_2>e_3\in\mathbb{R}$ roots of $E$ into a Weierstrass-Form $y^2=x^3+ax+b$?
6
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1answer
196 views

If six points of an elliptic curve are contained in a conic, then their sum is $O$.

Let $C$ be a projective cubic without singular points and $O\in C$ an inflexion point. We consider the addition in $C$ with $O$ as neutral element. If $R_{1},...,R_{6}\in C$ are different points such ...
6
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1answer
91 views

Intersection of two quadrics

How to understand (maybe, informally) why the intersection of two quadrics in general position in $\mathbb{CP}^3$ is an elliptic curve? It is obvious that it is a compact 2-manifold, i.e. a sphere ...
6
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1answer
152 views

explict form of the equation of elliptic curve

Let $E(\mathbb{F}_{q^2})$ is elliptic curve with #$E(\mathbb{F}_{q^2}) =q^2 + q + 1$. Can we write equation of this curve in the explicit form?
6
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1answer
84 views

Reduction of kernel of isogenies in the CM case

Let $F$ be a number field and $E/F$ an elliptic curve with CM by an order $\mathcal{O}$ in a quadratic imaginary field $K$. Let us suppose that $K\subseteq F$. Let $p$ be a prime that splits in ...