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

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6
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1answer
90 views

Sum and product of rational numbers is unity

Consider the system of equations: $$\sum_{i=1}^n X_i = 1$$ $$\prod_{i=1}^n X_i = 1$$ It is reasonably simple to show that for $n\ge 4$, this system admits a rational solution $(x_1, \dots, x_n) \...
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Show that the set $\{ P \in E(\mathbb{Q})\ |\ h(P) \le M \}$ is finite, for any constant $M$.

Show that the set $\{ P \in E(\mathbb{Q})\ |\ h(P) \le M \}$ is finite, for any constant $M$. Here $h(P)$ is logarithmic height of $P$, that is, $h(P):=\log H(P)$ and $H(P)=H(x)$, for $P=(x,y) \in E(...
2
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1answer
36 views

Geometric Intuition of Group Structure on Elliptic Curve

I am reading Number Theory 1: Fermat's Dream by Kato. In Chapter 1 he defines the group structure on a general elliptic curve $$y^{2} = ax^{3} + bx^{2} + cx + d$$ (where $a \neq 0$, and the cubic ...
1
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0answers
46 views

Can you explain Tate's algorithm by the example?

Consider the elliptic surface $\mathcal{E}$ over $\mathbb{P}_k^1$ with homogenous coordinates $t$, $s$ and a field $k$ of even characteristic: $$ \mathcal{E}\!: s^{10}(y^2z + yz^2) = t^4s^6x^3 + (t^...
0
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1answer
17 views

Schoof's algorithm: when we find a factor of $\psi_\ell$, why do its roots correspond to the kernel of some endomorphism?

I'm busy self-studying Schoof's algorithm from Andrew Sutherland's notes. In section 9.6, he states that when we happen to find some factor $g$ of the division polynomial $\psi_\ell$, then the roots ...
8
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73 views

How many elliptic curves have complex multiplication?

Let $K$ be a number field. Suppose we order elliptic curves over $K$ by naive height. What is the natural density of elliptic curves without complex multiplication? More generally, suppose we order $...
3
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2answers
95 views

Finding integer solutions to $y^2=x^3-2$

I have the equation: $$y^2=x^3-2$$ It seems to be deceivingly simple, yet I simply cannot crack it. It is obviously equivalent to finding a perfect cube that is two more than a perfect square, and a ...
2
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1answer
24 views

Modular forms under isogenies

Let $f$ be a modular form of level 1, say an Eisenstein series to simplify. If $n$ is a natural number, what can we say about $f(n\tau)$ in terms of $f(\tau)$?
3
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1answer
92 views

Is there an elliptic curve with exactly one rational point?

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Is there an example of such an $E$ such that the only rational point in $E(\mathbb{Q})$ is the point at infinity? In other words, consider the ...
6
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1answer
81 views

Quotients of Elliptic Curves

I am fairly inexperienced with elliptic curves so there might be aspects of my question that may need better wording but let me know if there are any issues: Question: Say I have an elliptic curve ...
3
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0answers
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Add $P$ to itself $N$ times on elliptic curve $y^2 = f(x)$, end up with expression in denominator of $x$ vanishing iff $NP$ is point at infinity?

See the second to last paragraph from page 39 of Koblitz's Introduction to Elliptic Curves and Modular Forms. Why is it that when we add a point $P$ to itself $N$ times on an elliptic curve $y^2 = ...
3
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1answer
69 views

Mordell curves with many integral points

For $k\in{\mathbb Z},k\neq 0$, denote by $f(k)$ the number of integral points on the Mordell curve $y^2-x^3=k$. According to the data at http://tnt.math.se.tmu.ac.jp/simath/MORDELL , the largest value ...
5
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1answer
97 views

1-1 correspondence of class group of an order '$\mathcal{O}$' and elliptic curves having complex multiplication by $\mathcal{O}$

I came across these two results Let $\mathcal{O}$ be an order in an imaginary quadratic field.There is a 1−1 correspondence between the ideal class group $C(\mathcal{O})$ and the homo-thety classes ...
4
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1answer
47 views

On Schoof's proof of deterministically finding $\sqrt{x} \bmod p$ when $p \neq 1 \bmod 16$

I am reading Schoof's paper in which he gave a polynomial time algorithm for counting points on an Elliptic curve over Finite field there he gave as an application an algorithm for deterministically ...
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0answers
68 views

How to determine redundant elliptic curves?

When we enumerate elliptic curves $y^2 = x^3 + ax + b$ over a finite field, how do we determine redundant ones, i.e. ones that are equivalent to others?
2
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29 views

when do two elliptic curves over finite fields have a common point

Given two elliptic curves over $\overline{\mathbb{F}}_p$ in the form of trivariate degree $3$ polynomial we want to find whether they have a common intersection point. This is a decision problem(I don'...
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0answers
36 views

elliptic k3 surface and Shioda Inose structure

We know that suppose given two elliptic curves $E$ and $E'$, there is a Kummer surface $km(E,E')$. And I'm curious suppose we know a $K3$ surface is kummer, how do we recover the pair $(E,E')$? For ...
2
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1answer
117 views

Points on the elliptic curve for Ramanujan-type cubic identities

Given the rational Diophantine equation, $$t^3 - t^2 - \tfrac{1}{3}(n^2 + n)t - \tfrac{1}{27}n^3=w^3\tag1$$ Two points are, $$t_0 = 0\tag2$$ $$t_2 = \frac{-(1 + 2 n) (1 + 11 n + 42 n^2 + 14 n^3 + 13 ...
1
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1answer
44 views

Silverman AEC 11b

Some search on the internet and this site didn't result in any topic about this question of Silverman's The Arithmetic of Elliptic Curves: Let $W \subset \mathbb{P^n}$ be a smooth algebraic set, each ...
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29 views

Use of Galois cohomology in elliptic curves

I'm studying elliptic curves on the book of Silverman The Arithmetic of Elliptic Curves. In the appendix the author describes the cohomology groups for finite and profinite groups . In the first case ...
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119 views

Generalizing Ramanujan's cube roots of cubic roots identities

(This extends this post.) Define the function, $$\sqrt[3]{G(t)} = \sqrt[3]{t+x_1}+\sqrt[3]{t+x_2}+\sqrt[3]{t+x_3}\tag1$$ where the $x_i$ are roots of the cubic, $$x^3+ax^2+bx+c=0\tag2$$ While $G(t)...
7
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2answers
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For which $n$ can $(a, nb, c)$ and $(b, c, d)$ be Pythagorean triples?

Fermat proved that if $(a, b, c)$ is a Pythagorean triple, then $(b, c, d)$ cannot be a Pythagorean triple. Suppose $(a, nb, c)$ form a Pythagorean triple. Can $(b, c, d)$ be a Pythagorean triple? ...
1
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1answer
19 views

$\tau_l(P,P)$ is a primitive $l$th root of unity

I'm trying to understand the Frey-Rück attack on elliptic curves, in particular the following lemma ($\tau_l$ being the Tate-Lichtenbaum pairing, $E(\mathbf{F_q})[l]$ the set of elements of $E(\mathbf{...
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0answers
36 views

What does the '#' sign mean in elliptic curves?

My question is regarding specifically elliptic curves. I have seen the notation $\#E(\mathbb{F}_{q})$ used over and over again (especially in the description of Hasse's theorem). I know that sometimes ...
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0answers
41 views

Representation of Frey's curve.

I read that Frey's curve is a semi-stable elliptic curve. What doe this mean ? I can find two dimensional representations of $y^2 = x^3 + ax + b$ in Wikipedia. What does $y^2 = x(x-a)(x+b)$ look like ...
2
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1answer
64 views

Silverman, arithmetic of EC, I1.9 no nonconstant morphisms $P^m \to P^n$ for m>n

This topic goes about problem 9 of the first chapter of Silverman, arithmetic of EC: If $m>n$, prove that there are no nonconstant morphisms $P^m \to P^n$. A solution can be found for example at ...
2
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52 views

Derivation of Frey equation from FLT

I understand, on a layman's level, Fray's motivation to write an elliptic equation corresponding to an assumed solution to FLT. My question is, how technically is Frey's equation derived? $1.$ FLT :...
2
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51 views

Proof of the Ribet's theorem

My question is very simple : My goal is to read a proof the proof of the epsilon conjecture proven by Ken Ribet (1986) which is an ingredient of the proof of the Fermat Last Theorem (I want the ...
7
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2answers
181 views

Is the equality $1^2+\cdots + 24^2 = 70^2$ just a coincidence?

I have read a question (written in Korean) that the equality $$1^2+2^2+\cdots + 24^2 = 70^2$$ is just a coincidence or not. It is a related to the integral points of the following elliptic curve (?): $...
6
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0answers
197 views

Prime Powers and Differences of Consecutive Cubes

I am wondering if it has been proven that there does not exist a prime $p$ and an integer $r \ge 3$ such that $p^r = (n + 1)^3 - n^3$ for some integer $n$. Note that this is a special case of Beal's ...
4
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0answers
38 views

Show that division polynomials of elliptic curve $y^2=x^3+x$ are in $\mathbb{Z}[x,y]$.

This is an exercise from Rational Points on Elliptic Curves by Silverman and Tate. Define a sequence of division polynomials $\psi_n\in \mathbb{Z}[x,y]$ for the elliptic curve $y^2=x^3+x$ ...
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2answers
50 views

Elliptic curves over $\mathbf{F}_q$ with $q = p^{2m}$

I am reading Washingtons book about elliptic curves and struggling with an exercise there (4.9), which is the following: Let $E$ be an elliptic curve over $\mathbf{F}_q$ with $q = p^{2m}$. Suppose ...
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0answers
39 views

Hasse-Weil bound for even degree polynomial

Consider the equation $y^2 = f(x)$, in the finite field $\mathbb{F}_q$($q$ is a prime power) where $f(\cdot)$ is a monic polynoimal of even degree (greater than or equal to $4$) with integer ...
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36 views

Sheeted-covering space degree 2 of Riemann Surfaces

In Milne - Elliptic curves, one finds the following on page 92: Branched-covering maps are not local isomorphisms at the ramified points; so could somebody explain to me what Milne means by 'a ...
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52 views

Covering maps of schemes.

A curve $X$ is modular if there is a finite covering $X_0(N)\rightarrow X$. What does covering mean in this context, and for more general morphisms of schemes? Just covering as topological spaces?
2
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1answer
50 views

Solve Elliptic Curve equation

Suppose you have an elliptic curve $E_{p}$: $y^{2} = x^{3} + Ax + B \mod p$ and points $P$ and $Q$ which lie on $E_{p}$. Does there always exist $n$ such that $nP=Q$? If so, how do we solve for it?...
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38 views

Prove that there are only finitely many rational numbers $p/q$ satisfying $|\frac{p}{q}-\sqrt[d]{b}|\leq \frac{C}{q^3}$.

This is a problem from Silverman & Tate's Rational Points on Elliptic Curves. The following is the Diophantine Approximation theorem by Thue which is proved in Chapter 5: Theorem. Let $b$ be a ...
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0answers
36 views

Inertia of an elliptic curve with potentially good reduction

Let $E/\mathbb{Q}_p$, $p\geq5$ be an elliptic curve with additive potentially good reduction. Then there is a unique, minimal, finite and totally ramified extension $K$ such that $E/K$ has good ...
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0answers
32 views

How do I show the tangent to an elliptic curve over the complex numbers meets the elliptic curve at another point?

If $E(\mathbb{C})$ is an elliptic curve given by $y^2=ax^3+bx+c$ for $a,b,c\in \mathbb{C}$, and $\ell$ is a line tangent to $E(\mathbb{C})$ at some point $p$, then why does $\ell$ meet $E(\mathbb{C})$ ...
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Show $|E_1(\mathbb{F_q})|+|E_2(\mathbb{F_q})|=2(q+1)$

...under the assumption that $E_1,E_2$ are elliptic curves over $\mathbb{F_q}$ and that there is a (surjective) isogeny $\pi:E_1\rightarrow E_2$ defined over $\mathbb{F_{q^2}}$ obeying $\pi\phi_1=-\...
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0answers
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Finding a $\gamma$ to define a Kummer extension like $E=\mathbb{Q}(\zeta_5)(X^5-\gamma)$

Previous theory: All the cyclic extensions of order $5$ are $\mathbb{Q}(\zeta_5)(\sqrt[5]{\gamma})/\mathbb{Q}(\zeta_5)$ where $\zeta_5$ is the generator of the group $\left(\mathbb{Z}/5\mathbb{Z}\...
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Is $H^{1}_{Sel}\left(K,E_{p^{n}}\right)\rightarrow\prod_{q \text{ a nonarchimedean prime of }K}\left(K_{q},E_{p^{n}}\right)$ an injection?

If $K$ is a number field, $E$ an elliptic curve and $p$ a prime, does the Selmer group $$H^{1}_{\operatorname{Sel}}\left(K,E_{p^{n}}\right)$$ always inject into $$\prod_{q \text{ a nonarchimedean ...
2
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38 views

Congruent Numbers and Integral Points on Elliptic Curves

As you probably know, congruent numbers $N$ and elliptic curves of the form $$E_N:y^2=x^3-N^2x$$ are intimately connected. While playing around with curves of this form, I found that $E_N$ will have ...
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1answer
37 views

Group of $\mathfrak a$-torsion points

Silverman defines the Group of $\mathfrak a$-torsion points of an elliptic curve $E/\mathbb C$ (with $\mathfrak a$ an ideal in $\mathrm{End}(E)$) in Advanced topics of elliptic curves as $$E[\mathfrak ...
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40 views

Normalizing an elliptic curve to find integer solutions

I have an elliptic curve $$ c_1y^2 + a_1xy + a_3 = c_2x^3 + a_2x^2 + a_4x + a_6 $$ with integers $a_1,a_2,a_3,a_4,a_6,c_1,c_2$ and I would like to find all integer solutions of this elliptic curve. I ...
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1answer
55 views

Proof of finiteness of Selmer groups in Silverman's Arithmetic of Elliptic Curves

I'm trouble in understanding the proof of Lemma X.4.3 in Silverman's Arithmetic of Elliptic Curves (2nd edition), that claims $H^1(G_{\bar{K}/K}, M; S)$ is finite. In page 334, the book state the map $...
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1answer
10 views

Lines intersections distance on the asymptotes

Like in picture we have two lines. Lenght of one of them is 2E and other's lenght 2C and also ellipse asymptotes are A and B and its center is on origin(0,0) I want to find D and F How can I ...
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27 views

Comparison of discrete logarithms.

Additive discrete logarithm: In $\Bbb Z_n^+$ we have to find $z$ in $zg=h\bmod n$ where $g$ generates $\Bbb Z_n^+$. $z$ is unique upto $z \bmod n$. Multiplicative discrete logarithm: In a cyclic ...
2
votes
1answer
82 views

Splitting of a prime and $p$-divisibility on an elliptic curve

Let $K$ be a quadratic imaginary field and let $\lambda$ a prime of norm $l^2$, for a rational prime $l$. We consider $E$ to be an elliptic curve such that $E[p](K)$ is trivial, where $p\neq l$ is a ...
4
votes
1answer
77 views

History and future of algebraic curves and the like?

Now that Fermat's last theorem has been proven, and also elliptic curves see widespread use in simple everyday applications, I would love to learn how the related theories came into beeing, how they ...