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

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$a^3+3a^2+a$ 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
267 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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3answers
505 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 ...
7
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2answers
539 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
340 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
61 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
176 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 ...
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1answer
93 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" ...
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1answer
105 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
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1answer
183 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 ...
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1answer
255 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 ...
7
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1answer
354 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
129 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
144 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 ...
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0answers
78 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 ...
7
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1answer
154 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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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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152 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 ...
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1answer
111 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 ...
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3answers
165 views

What does the Tate module of an elliptic curve tell us?

I started studying elliptic curves, and I see that it is rather common to take the Tate module of an elliptic curve (or, of the Jacobian of a higher genus curve). I'm having a hard time isolating the ...
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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 ...
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2answers
352 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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3answers
294 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
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2answers
243 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
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2answers
115 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 ...
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1answer
238 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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1answer
195 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
154 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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1answer
102 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
75 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
137 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
80 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 ...
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0answers
119 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 ...
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0answers
110 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 ...
5
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3answers
276 views

The elliptic curve $y^2 = 23328x^3-890273x^2+14755570x-7^7$

The elliptic curve, $$y^2 = 23328x^3-890273x^2+14755570x-7^7 \tag{1}$$ has the small solution $x = 58$. I know how to find other rational points, but the number of digits in the denominator gets ...
5
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3answers
206 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 ...
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1answer
394 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 ...
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4answers
307 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) ...
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1answer
214 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 ...
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1answer
511 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 ...
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2answers
1k views

What is a primitive point on an elliptic curve?

While working with elliptic curves for cryptography reasons, I found the notion of a primitive point, but no definition. For example, $P(0,6)$ is a primitive point on the elliptic curve $y^2\equiv ...
5
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2answers
237 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 ...
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1answer
67 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
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2answers
214 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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2answers
64 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 $s$? ...
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2answers
130 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 ...
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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 ...
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1answer
117 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 ...
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1answer
138 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 $\mathbb C[[u,v]]/(uv)$, ...
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1answer
110 views

Cokernel of morphism of Tate module of elliptic curves

Let $K$ be a field, and $\phi: E_1\to E_2$ be an isogeny of elliptic curves over $K$. Given a prime $\ell$ different from the characteristic of $K$, $\phi$ induces an injection $T_\ell \phi: T_\ell ...