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

learn more… | top users | synonyms

0
votes
0answers
19 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 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 ...
2
votes
1answer
44 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 ...
1
vote
0answers
27 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? Where did ...
2
votes
0answers
30 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
votes
2answers
139 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 (?): $...
5
votes
0answers
54 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
votes
0answers
33 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 $y^2=x^3+x$ ...
1
vote
2answers
47 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 ...
1
vote
0answers
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 ...
0
votes
0answers
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 ...
-1
votes
0answers
45 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
votes
1answer
46 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?...
4
votes
0answers
34 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 ...
1
vote
0answers
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 ...
1
vote
0answers
30 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 $E(\mathbb{C})$ ...
3
votes
0answers
24 views

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=-\...
1
vote
0answers
29 views

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}\...
1
vote
0answers
20 views

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
votes
0answers
34 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 ...
0
votes
1answer
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 ...
1
vote
0answers
38 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 ...
1
vote
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 $...
1
vote
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 ...
1
vote
0answers
26 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
44 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
73 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 ...
0
votes
0answers
36 views

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 ...
8
votes
2answers
220 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 ...
1
vote
0answers
76 views

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 ...
0
votes
1answer
34 views

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?
3
votes
2answers
79 views

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=(...
0
votes
0answers
66 views

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 , ...
0
votes
1answer
31 views

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 ...
0
votes
2answers
14 views

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^...
1
vote
1answer
43 views

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=\...
2
votes
0answers
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 ...
1
vote
0answers
30 views

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 $\Delta=g_2^3-27g_3^2$...
4
votes
1answer
128 views

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 ...
2
votes
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 $\mathbb{Q}$) ...
1
vote
1answer
32 views

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 \...
0
votes
0answers
23 views

clarification on the inertia group in the proposition 1.5 chap VIII of Silverman's Arithemic of elliptic curves

Let K be a number field. Let $ Q\in \mathbb {P}^2 (\overline {K})$ and define $K (Q)=$ fixed field of $\{\sigma\in G_{ \overline {K}/K } :Q^\sigma=Q \}$ where $G_{ \overline {K}/K }$ is the ...
1
vote
0answers
34 views

Derivative of the Klein j-invariant

To prove that the field of meromorphic functions on $X(1)$ is generated by the Klein j-invariant, we need to show that the derivative of $j(\tau)$ does not vanish. (It only has simple zeroes). But I ...
2
votes
1answer
48 views

Can a separable isogeny of elliptic curves have an inseparable dual?

Let $\phi: E_1\to E_2$ be an isogeny of elliptic curves over a field $K$ of characteristic $p>0$. Suppose that $\phi$ is separable and let $\hat{\phi}: E_2\to E_1$ denote the dual isogeny. Then $\...
1
vote
0answers
84 views

Intersection of algebraic curves at a point with given multiplicity

I don't know if this question is too basic for MO, so I put it here, but if you think I should migrate the question to MathOverflow please suggest me. Let $C/k$ be a smooth curve over a perfect ...
9
votes
3answers
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 ...
1
vote
1answer
35 views

Homomorphism between elliptic curves $C: y^2=x^3+ax^2+bx$ and $\bar{C}: y^2=x^3-2ax^2+(a^2-4b)x$.

I am reading Rational Points on Elliptic Curves by Silverman and Tate. In Section 3.4, Page 76, the authors defined two elliptic curves elliptic curves $C: y^2=x^3+ax^2+bx$ and $\bar{C}: y^2=x^3-2ax^...
1
vote
1answer
47 views

Find all points of finite order on the elliptic curve $y^2+7xy=x^3+16x$.

I am studying Rational Points on Elliptic Curves by Silverman and Tate. This is Problem 2.12 (h). Determine all of the points of finite order on the elliptic curve $y^2+7xy=x^3+16x$. Also ...
0
votes
0answers
26 views

$2 \times 2$ matrix representing elliptic curve?

Suppose we have $E/\mathbb{C}$ and we let $E/\mathbb{C}=\mathbb{C}/\Lambda$ for a lattice $\Lambda=\mathbb{Z}+\mathbb{Z}\sqrt{5}i$. Suppose also that $\alpha=10+3\sqrt{5}i$. For a basis $\{\...
1
vote
0answers
23 views

Trace and degree of elliptic curve endomorphism?

Let $E/\mathbb{C}=\mathbb{C}/\Lambda$ for a lattice $\Lambda = \mathbb{Z} + \mathbb{Z}\sqrt{5}i$. Let $\alpha=10+3\sqrt{5}i$. Show that $\alpha \in$ End$(E)$ and compute the trace and degree of $\...
1
vote
1answer
36 views

Elliptic curve linear recurrence proof

Let $E/\mathbb{F}_q$ be an elliptic curve with $q=p^m$ for some prime $p$. Let $a_n=q^n+1-\#E(F_{q^n})$ and by convention we let $a_0=2$. Prove that $a_{n+2}=a_1a_{n+1}-qa_n$ for all $n>0$. ...