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

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7
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2answers
161 views

Clarifying a comment of Serre

Let $\rho_{\ell}$ be the "mod $\ell$" Galois representation associated to an elliptic curve $E/K$ (i.e., corresponding to the action of Galois on the $\ell$-torsion points). Serre proved that in the ...
7
votes
1answer
104 views

Proof in Kummer Theory - why is this subgroup finite?

I'm doing a course on elliptic curves. We're working with a field $K$ with $\mu_n \subset K$ ($K$ contains all $n$th roots of unity) and $\mbox{char}(K)\not|\;n$. I'm trying to understand the proof ...
7
votes
1answer
175 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
votes
1answer
346 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
votes
1answer
126 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
votes
1answer
133 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 ...
7
votes
0answers
136 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 ...
7
votes
1answer
146 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. ...
7
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0answers
73 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 ...
7
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0answers
109 views

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 ...
6
votes
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
votes
2answers
330 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 $E(K)$ ...
6
votes
3answers
282 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 ...
6
votes
2answers
442 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 internet about ...
6
votes
2answers
78 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
votes
2answers
233 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 ...
6
votes
2answers
426 views

Weierstrass $\wp$ function doubly periodic

I'm working my way through Silverman and Tate's Undergraduate Introduction to Elliptic Curves. I haven't yet been able to study complex analysis, so it comes as no surprise that I'm having a tough ...
6
votes
1answer
208 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
votes
1answer
91 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" ...
6
votes
1answer
239 views

Euler Product of Modular L-series

This question is probably really easy to someone that knows the theory of modular forms well, so I apologize if this is obvious. Suppose $E_1$ and $E_2$ are elliptic curves over $\mathbb{Q}$ and the ...
6
votes
1answer
144 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
votes
1answer
92 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
votes
1answer
134 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
votes
1answer
77 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 ...
6
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0answers
109 views

Some questions related to Iwasawa invariants of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$. Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...
6
votes
0answers
45 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 ...
6
votes
0answers
103 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 ...
6
votes
1answer
109 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 ...
5
votes
3answers
171 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 ...
5
votes
1answer
381 views

Universality of Tate-conjectures

We all know that Prof.John Tate proposed a set of conjectures(along with Prof.Emil Artin) formally spread under the name of "Tate conjectures", they have a wide range of influence on various fields of ...
5
votes
4answers
285 views

upper bound on rank of elliptic curve $y^{2} =x^{3} + Ax^{2} +Bx$

I was told the following "Theorem": Let $y^{2} =x^{3} + Ax^{2} +Bx$ be a nonsingular cubic curve with $A,B \in \mathbb{Z}$. Then the rank $r$ of this curve satisfies $r \leq \nu (A^{2} -4B) +\nu(B) ...
5
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1answer
206 views

A question about modular curves and base change

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Suppose that the curve $X\times_{K,\sigma} \mathbf{C}$ is a modular curve for some $\sigma:K\to \mathbf{C}$. Can ...
5
votes
1answer
460 views

Number of 3-torsion points on an elliptic curve

If we take our elliptic curve over $K$ to be the zero set of $$ F(X_1, X_2, X_3) = X_2^2 X_3 - (X_1^3 + AX_1X_3^2 + BX_3^2), $$ which is in projective form with $X = X_1, Y = X_2, Z=X_3$, then I ...
5
votes
2answers
208 views

Find the rational points on $1 + 18 x + 81 x^2 + 44 x^3 = y^2$ with Sage

I'm trying to use Sage on-line,but I meet some trouble with the code of it. I want to find the rational points on an ellipse curve,such as $$1 + 18 x + 81 x^2 + 44 x^3 = y^2,\tag1$$ I know that ...
5
votes
1answer
59 views

Exact rank of Elkies curve

A naïve question. We definitely know an elliptic curve of rank $28$ or more exists by Elkies but no one knows exactly what the rank is for this curve (and for similar examples given previously). ...
5
votes
2answers
196 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 ...
5
votes
2answers
61 views

Property of the $p^n$-Selmer group

Consider the $p^n$-Selmer group of an elliptic curve $E/\mathbb{Q}$, $\operatorname{Sel}_{p^n}(E)$. Must we always have $\operatorname{Sel}_{p^n}(E) \cong (\mathbb{Z}/p^n\mathbb{Z})^{s}$ for some $s$? ...
5
votes
2answers
122 views

Computing rank using $3$-Descent

For an elliptic curve $E$ over $\Bbb{Q}$, we know from the proof of the Mordell-Weil theorem that the weak Mordell-Weil group of $E$ is $E(\Bbb{Q})/2E(\Bbb{Q})$. It is well known that $$ 0 \rightarrow ...
5
votes
1answer
183 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 ...
5
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1answer
50 views

deg functions and maps

For any map $f$ between curves $C_1$ and $C_2$, one defines $\mathrm{deg}(f) = [K(C_1) : f^*K(C_2)]$ as given in "The Arithmetic of Elliptic Curves" by Silverman. For algebraic functions on elliptic ...
5
votes
1answer
88 views

Discriminants and Weierstrass form of elliptic curves

I'm confused by what appears to be contradictory information. In this post, the claim is made that "Every elliptic curve over $\mathbb{Q}$ can be written in the form $y^{2}= x^{3}+ax+b$ where ...
5
votes
1answer
128 views

Finding the completion of a coordinate ring

Consider $A=\mathbb C[x,y]/(y^2-x(x+1))$, and consider the $\mathfrak m$-adic completion, where $\mathfrak m =(x,y)$. I want to show that this completion is isomorphic to $\mathbb C[[u,v]]/(uv)$, ...
5
votes
1answer
147 views

Family of elliptic curves with trivial torsion

I'm wondering, if it is true that the torsion subgroup of $y^2=x^3+p$ (for $p$ some prime, greater than 2), is always trivial?. I was trying to prove this using Lutz-Nagell, but I can't quite get it. ...
5
votes
2answers
342 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 ...
5
votes
1answer
120 views

Direct proof of the non-zeroness of an Eisenstein series

Question: Can you show directly from its formula that $G_4(i)\neq0$? Recall that the holomorphic Eisenstein series of weight $2k$ is defined by: $$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus ...
5
votes
2answers
221 views

Relation involving the conductor of an elliptic curve

Consider an elliptic curve $E: y^{2} = x^{3} + ax + b$. Then the quadratic twist by a squarefree $d$ is given by $E^{d} : dy^{2} = x^{3} + ax + b$. What is the relationship between the conductor of ...
5
votes
1answer
107 views

Complex elliptic curve for the “conjugate” lattice

Let $\Lambda$ be a lattice in $\mathbb{C}$, and $E=\mathbb{C}/\Lambda$ the corresponding complex elliptic curve. Let $\bar{\Lambda}$ be the "conjugate" lattice, i.e. the one obtained by conjugating ...
5
votes
1answer
68 views

Minimal degree of a field extension to obtain an elliptic curve

Let $K$ be a number field and let $X$ be a smooth projective geometrically connected curve over $K$ of genus $1$. There exists a number field $L/K$ such that $X$ has a $L$-rational point. Let $L$ be ...
5
votes
1answer
58 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 ...
5
votes
1answer
616 views

How do I show that this curve has a nonsingular model of genus 1?

Let $C$ be the projective closure of $Z(f) \subset \mathbf{A}^2$ where $f$ is an irreducible polynomial of degree 4 in $x$ and degree 2 in $y$, so $C = Z(f^*) \subset \mathbf{P}^2$ where $f^*$ is the ...