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

learn more… | top users | synonyms

1
vote
0answers
31 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
0answers
41 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
9 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
19 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 ...
0
votes
0answers
19 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 ...
14
votes
2answers
189 views

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 ...
4
votes
1answer
68 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
32 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 ...
3
votes
0answers
85 views

Birationally transforming a quartic elliptic curve [migrated]

Consider the elliptic curve $$y^2=ax^4+cx^2+dx+f$$ I am aware that there are algorithmic methods for birationally transforming a nondegenerate cubic curve into the Weierstrass canonical form ...
0
votes
1answer
356 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 ...
5
votes
1answer
112 views

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 ...
8
votes
2answers
215 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
67 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
69 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 ...
0
votes
1answer
29 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
63 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
30 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 ...
3
votes
2answers
180 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 $Y^2=X^3+p^2X$ ...
4
votes
1answer
105 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 ...
0
votes
2answers
12 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 ...
1
vote
1answer
40 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 ...
9
votes
3answers
243 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 ...
2
votes
0answers
22 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 ...
3
votes
2answers
489 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 ...
1
vote
0answers
27 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 ...
2
votes
1answer
49 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 $\mathbb{Q}$) ...
0
votes
0answers
20 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 ...
2
votes
1answer
41 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
81 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 ...
1
vote
0answers
32 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 ...
1
vote
1answer
34 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}: ...
1
vote
1answer
40 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
22 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
35 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 $n>0$. ...
1
vote
2answers
53 views

Elliptic curves with trivial Mordell–Weil group over certain fields.

I am looking for elliptic curves $E,E'$ defined over $\mathbb{F}_{3}$ and $\mathbb{F}_{4}$ respectively and given by a Weierstrass equation such that their Mordell-Weil group is trivial, i.e. such ...
1
vote
1answer
24 views

The cardinal of the Mordell-Weil group is prime for certain elliptic curves over $\mathbb{F}_{q}$ for certain $q$.

Let $p\in\{2,3\}$ and $r\in\mathbb{Z}_{\geq 2}$. I would like to find if there exists an elliptic curve defined over $\mathbb{F}_{p}$ such that $|E(\mathbb{F}_{p^{r}})|$ is a prime number. If $p>3$ ...
0
votes
0answers
32 views

How are non-homogenous elliptic curves projective varieties?

So if I am given an elliptic curve such as $Y^2Z=X^3$ then I see how it can be realized as the projective variety $Proj(k[X,Y,Z]/(Y^2Z-X^3))$. But, given an elliptic curve like $Y^2 = X^3 + X$, then ...
0
votes
1answer
26 views

Help needed in determining the singularity

Can someone teach me how to determine the singularity of algebraic curve $y^2 =x^3+x^2$. I'll be really grateful and thanks in advance
2
votes
0answers
31 views

Relation between pullbacks of a degree zero line bundle on an elliptic curve

Let $E$ be an elliptic curve over a field $k$. Let $$\mu:E \times_k E \to E$$ be the addition map on $E$. Furthermore let $p_1,p_2:E \times_k E \to E$ be the two canonical projections and let ...
0
votes
1answer
39 views

help to parametrize $y^2 = x^3 -x^2$

I appreciate if someone could help me to parametrize this equation $y^2 = x^3 -x^2$. Thanks in advance. I used maple to find the solution as $(x,y) = ((t^2-1),(t(t^2-1))$
1
vote
0answers
48 views

$p$ adic modular forms and wide open neighbourhood (e.g. Coleman primitive): is it possible to obtain a holomorphic function?

It is well known that a modular form (of weight $k$ and level $N$) is in particular also a classical modular form; this can be seen both using Serre's definition with $q$-expansion and the Katz's one, ...
0
votes
1answer
17 views

Show 2-torsion subgroups are equivalent?

Let $E/k$ be an elliptic curve defined by the Weierstrass form $y^2=x^3+ax+b$. Let $c$ be a nonzero squarefree element in $k$. Let $E_c/k$ be a curve defined by $cy^2=x^3+ax+b$. Show that the ...
2
votes
2answers
26 views

Simple automorphism proof with lattice elliptic curves?

For a lattice $\Lambda_1=\mathbb{Z}+\mathbb{Z}i$, find Aut($E_1$) where $E_1(\mathbb{C})=\mathbb{C}/\Lambda_1$. So I know that End($E$) $ =\{\beta \in \mathbb{C} $| $\beta\Lambda \subseteq ...
1
vote
0answers
35 views

Elliptic curve n-torsion point?

For an elliptic curve $E/k$, let $\alpha$ be any endomorphism over $\bar{k}$ in End($E$) and let $[n]$ be the multiplication-by-n endomorphism. Show that for any n-torsion point $P \in E[n]$, ...
0
votes
0answers
52 views

What is more amazing?

$S$ denotes the set of rational points of any curve in the plane. What is more amazing between a) and b)? a) $S$ is dense in the curve $y^2=x^3-2^4\cdot3^3\cdot7^2$ b) $S=\emptyset$ in the curve ...
1
vote
1answer
21 views

How to show elliptic curve endomorphism is commutative?

For an elliptic curve $E/k$, let $\alpha$ be any endomorphism over $\bar{k}$ in End($E$) and let $[n]$ be the multiplication-by-$n$ endomorphism. Show that $[n] \bullet \alpha=\alpha \bullet ...
2
votes
0answers
13 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 ...
0
votes
1answer
34 views

Show an elliptic curve is a twist of another curve?

Let $E/k$ be an elliptic curve defined by the Weierstrass form $y^2=x^3+ax+b$. Let $c$ be a nonzero square free element in $k$. Let ${E_c}/k$ be a curve defined by $cy^2=x^3+ax+b$. Using a ...