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

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$\tau_l(P,P)$ 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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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 ...
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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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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 :...
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38 views

The bilinearity of the Cassels-Tate pairing

Let $K$ be a number field and let $A$ be an abelian variety over $K$ (I'm mostly interested in the case that $A$ is an elliptic curve). We use $v$ to denote places of $K$ and we write $H^i(k, A)$ for ...
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Representation of Frey's curve.

I read that Frey's curve is a semi-stable elliptic curve. What doe this mean ? I can find 2 dimensional representations of y^2 = x^3 + ax + b in Wikipedia. What does y^2 = x(x-a)(x+b) look like if a ...
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36 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 ...
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2answers
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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 (?): $...
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83 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 ...
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1answer
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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 ...
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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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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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38 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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33 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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1answer
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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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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?
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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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34 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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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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1answer
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Unramified cocycles and the Selmer group of an ellptic curve

In Silverman's book on elliptic curves, he gives a procedure to compute the Selmer group of elliptic curve $E$ relative to an isogeny $\phi:E\to E'$. I am confused about one step in the discussion. ...
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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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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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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 ...
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36 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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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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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
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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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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 ...
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The term “elliptic”

There are many things which are called “elliptic” in various branches of mathematics: Elliptic curves Elliptic functions Elliptic geometry Elliptic hyperboloid Elliptic integral Elliptic modulus ...
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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 ...
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Elliptic curve - perfect square discriminant

Given an elliptic curve: $$y^2 = x^3+ax+b$$ where a,b are rational, what can be said about the curve if its elliptic discriminant is a perfect square? $$discriminant= -16(4a^3+27b^2)$$ Any special ...
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358 views

Nice formulas for the lambda invariant of an elliptic curve

Where can I find some nice formulas for the lambda invariant of an elliptic curve? I vaguely recall there's a nice product formula in terms of $q$, but a Google search didn't give me much. Also, are ...
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Fundamental period of the Weierstrass $\wp$ elliptic function?

Consider the Weierstrass $\wp$ elliptic function $\wp(z, g_2, g_3)$ with the invariants $g_2\in\mathbb{R}$ and $g_3\in\mathbb{R}$: $$\wp'(z)^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3$$ According to Wikipedia ...
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224 views

Rational solutions of $y^2 = x^3 - x$

I believe that the only rational solutions of $$y^2 = x^3 - x$$ are the obvious ones $(-1,0)$, $(0,0)$, $(1,0)$, and that this was proved by Fermat using the method of descent. Can anyone outline a ...
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Finding an Elliptic Curve with 103 points

I am trying to solve the following problem: Find an elliptic curve over F101 with 103 points. I know all of the equations when needing to find alpha, and beta and all that when I am given two points ...
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1answer
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A diophantine equation of degree 3

Find the integer solutions of $y^2+6=x^3$. I guess it does not have integer solutions but I cannot prove it. By $\pmod 8$, I can know that $y$ is odd and $x\equiv7 \pmod 8$. Then what else can I do?
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If the cubic equation with rational coefficients $x^3+ax^2+bx+c=0$ has a double root, the root is rational.

This question comes from a problem in Rational Points on Elliptic Curves by Silverman. The problem asks to show that For a cubic curve $C: y^2=x^3+ax^2+bx+c$ with $a,b,c\in \mathbb{Q}$, if $S=(...
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Show that points on an elliptic curve have order 4

I am studying elliptic curves using this book and have a problem with task 4.11 which goes as follows: Let $F_q$ be a finite field of odd characteristic and let $ a,b \in F_q $ with $a \ne2b$ and $b \...
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What are the prime factors of $4^{256}+253\ $?

I search a composite number near $4^{4^4}$ with a very large smallest prime factor. A candidate is $$4^{4^4}+253=4^{256}+253$$ The number is composite and has $155$ digits, so it is in the range , ...
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1answer
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Meaning of this expression

I found a relation while studying elliptic curves, I could not understand its' meaning. $E[n]$ is a $n$-torsion subgroup then $E[n]\cong Z/nZ \oplus Z/nZ$, What does this $\oplus$ symbol mean? Thanks ...
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182 views

Rank of an elliptic curve

How could we compute the rank of an elliptic curve? I looked for a methodology in my book, but I didn't find anything. Could you give me a hint? I want to find the rank of the curve $Y^2=X^3+p^2X$ ...
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Automorphisms of an elliptic curve fixing the invariant differential?

If we consider an elliptic curve $E/k$ given in Weierstrass form $y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$, then I know that the translation maps $\tau_{P}$ with $P\in{E}$ fix the invariant ...
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How to find the solutions for the quadratic equation for conic sections $\epsilon \in (0,1)$

Going from this definition of the conic section: $\epsilon |Pl| =|PB|$, you get the following equation for the intersection with the $x$-axis: $y^2 = (\epsilon ^2-1)x^2+(B-\epsilon ^2L)2x+\epsilon ^2L^...
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do formal group laws induce group structures on schemes (as opposed to formal schemes)

Let $R$ be a ring and $f \in R[[x]]$ a commutative formal group law over $R$, meaning $f(f(x, y), z)=f(x, f(y, z))$, $\ f(x, y)=f(y, x)$ and $f(x, y)=x+y + \text{higher order terms}$. Let $G=\...
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256 views

Find all integral solutions for the Diophantine Equations $x^4 - x^2y^2 + y^4 = z^2$ and $x^4 + x^2y^2 + y^4 = z^2$.

Find all integral solutions for the Diophantine Equations $$x^4 - x^2y^2 + y^4 = z^2$$ and $$x^4 + x^2y^2 + y^4 = z^2$$ I basically think that to solve these equations we need to use the fact that ...
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26 views

Can Hecke Operators be defined on more general spaces of elliptic curves?

Classically, the Hecke Operators act as endomorphisms of $\omega^k_{\mathcal{M}_{ell}(\mathbb{C})}$, defined by noting that there is a distinguished class of covering maps $\widetilde{E}\to E$ given ...
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2answers
505 views

Real Period of an Elliptic Curve

Trying to work out what the real period of an elliptic curve is as seen in the Birch Swinnerton-Dyer conjecture. From what I've read, given an elliptic curve E over the rationals, one can associate ...
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How can I “groupify” an elliptic curve over a non-field?

The book Primes of the form $x^2+ny^2$ by David A. Cox contains the following definitions regarding an elliptic curve (by which he means an equation $y^2=4x^3-g_2x-g_3$ such that $\Delta=g_2^3-27g_3^2$...
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1answer
52 views

Homogeneous spaces of elliptic curve

Let $d>1$ be a cube free natural number and $a,b,c$ are natural numbers greater than 1 with $abc=d$. How to explain that the curve $D: ax^3+by^3+cz^3=0$ is homogeneous space (over $\mathbb{Q}$) ...