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

learn more… | top users | synonyms

7
votes
3answers
177 views
+50

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 ...
-3
votes
1answer
38 views

Let $\alpha \in \mathbb{C}$. Show that $\alpha \in End(E)$?

Given some ellipctive curve $E$ and a complex number $\alpha$, how can I show $\alpha \in$ End($E$)? I understand that every complex elliptic curve had a uniformization $\mathbb{C}/\Lambda$ but I'm ...
0
votes
0answers
13 views

Rational points of the kernel of an isogeny of elliptic curves.

Let $E, E'$ be elliptic curves defined over $\mathbb{F}_{q}$ and $$ \phi:E\rightarrow E $$ an isogeny. What can we say about $E(\mathbb{F}_{q^{r}})\cap \ker\phi$? Are there any condition under which ...
3
votes
2answers
478 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 ...
0
votes
0answers
23 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
40 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 $\mathbb{Q}$) ...
2
votes
0answers
33 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
0answers
45 views

Principal homogeneous spaces of elliptic curve over Q [on hold]

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
18 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
18 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
39 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
80 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
30 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
32 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
38 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 ...
3
votes
0answers
25 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 ...
4
votes
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 ...
4
votes
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
50 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
31 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
29 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
35 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
47 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, ...
3
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 ...
5
votes
2answers
24 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 ...
4
votes
0answers
34 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
49 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 ...
4
votes
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 ...
3
votes
1answer
32 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 ...
3
votes
1answer
19 views

Convert affine coordinates to projective coordinates?

For any rational map represented by $(\frac{x^4+3y}{x^2+1}, \frac{x+1}{y})$ in affine coordinates, write down the corresponding representation $[F_1(X, Y, Z) : F_2(X, Y, Z) : F_3(X, Y, Z)]$ in ...
3
votes
0answers
42 views

Complex multiplication and ray class fields

This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve ...
1
vote
1answer
25 views

Confusion about holomorphic differential on elliptic curve

Let $(a:b)\in\mathbb{C}P^1$ and look at the elliptic curve $C$ given by $y^2=x^3+a^4x+b^6$. It is well known that on this elliptic curve we have the holomorphic differential $dx/y$. I have two ...
4
votes
0answers
41 views

How to compute explicitly the covering map in the modularity theorem?

The modularity theorem (original Shimura-Taniyama-Weil conjecture) asserts the existence of a covering (uniformization) map $\pi:X_0(N) \to E$ for every $E$, an elliptic curve defined over ...
2
votes
0answers
39 views

Show $(x,y) \rightarrow (x,-y)$ is a group homomorphism?

Show that $(x,y) \rightarrow (x,-y)$ is a group homomorphism from $E$ to itself where $E$ is an elliptic curve in Weierstrass form. So $E$ is of the form $y^2=x^3+ax+b$. Would I just show that ...
3
votes
1answer
65 views

Reference request: Fibre functor for elliptic curves is pro-representable

I am writing a project on étale fundamental groups of elliptic curves and I want to include a proof of a key theorem: the fibre functor on the category of finite étale covers of an elliptic curve is ...
0
votes
0answers
23 views

Solve for $y$ in terms of $x$ at $z=0$

Feeling like a complete idiot here (seems this should be easy)...the equation of a surface I fit is: $$ sf(x,y) = p_{00} + p_{10}x + p_{01}y + p_{20}x^2 + p_{11}xy + p_{02}y^2 $$ What is the ...
2
votes
1answer
26 views

Computing the multiplicative inverse for point addition on an elliptic curve

I'm trying to perform point addition on an elliptic for two points taken from an example in the book "Understanding Cryptography by Christof Paar & Jan Pelzl". The points I'm trying to add are: ...
2
votes
0answers
68 views

Quartic Diophantine equation $ 2 x^4 - 2 x^2 = 3 (y^2 - 1)$

About the quartic Diophantine equation: $$ 2 x^4 - 2 x^2 = 3 (y^2 - 1)$$ On oeis.org/A180445 it says that all positive solutions $(x,y)$ are: $$(1,1)\ \ (2,3)\ \ (3,7) \ \ (6,29)\ \ (91,6761)$$ ...
0
votes
2answers
35 views

Isogenous elliptic curves over finite fields have the same number of points

I'm stuck in this question, it is the first part of exercise 5.4 from Silverman - The arithmetic of elliptic curves. Let $C,D$ be two isogenous elliptic curves over a finite field $\mathbb{F}_q$. ...
0
votes
0answers
17 views

Translating from (short) Weierstrass form to a Lattice. [duplicate]

I know that given an elliptic curve in the form $\mathbb{C}/L$ for L some lattice, $\mathbb{Z}+\mathbb{Z}\tau$, then we can use the Weierstrass $\wp$ functions to turn it into short Weierstrass form, ...
1
vote
1answer
49 views

What does extra zero of an $L$-function mean?

This is a very vague question. What does an extra zero of an $L$-function mean? There are lots of papers written on this topic, investigating the extra/exceptional zeros of various $p$-adic ...
2
votes
0answers
36 views

Degree $5$ embeddings of genus 1 curves, plucker embedding of a Grassmannian…

Here is something Ravi Vakil says after 19.9F in his notes. He is talking about embeddings of a genus 1 curve $C$ by complete linear series associated to the divisors $O(nP)$. "The beautiful ...
3
votes
1answer
67 views

Is a projective curve minus finite number of points affine?

The answer to the question in the title is "yes". This is proved for example here, or here (page 5). However, I seem to be able to construct a counterexample. Can you help me find the flaw in my ...
0
votes
0answers
48 views

question about Galois theory and dimension

I try to understand the proof of the lemma 13.7 of the following article: http://www.cs.nyu.edu/courses/spring05/G22.3220-001/ec-intro1.pdf The lemma says that if $r$ is a rational function which ...
0
votes
0answers
29 views

Points of finite order : Size of the group E/E0

My curve is given by E : $y^2 = x^3-3267x+45630$. Bad primes are 2,3,17. I want to find the size of group $E/E_0$. I know that $E_0(Q_2)$ are points on $E(Q_2)$ that do not reduce to a singular point. ...
2
votes
1answer
47 views

Order of the pole of projective curve using uniformizer

I need to find divisor of functon $f=y$ i.e. $\frac{y}{z}$ for projective curve $y^2z=x^3-xz^2$ and I have some questions: For example, I have pole: $(0:1:0)$ and zeros $(0:0:1),(1:0:-i),(1:0:1)$. It ...
1
vote
0answers
49 views

Plotting Elliptic Curve over Finite Field in Maple

Not sure if I might be in the wrong section, but I am looking for guidance on how to plot an elliptic curve over a finite field in Maple. I have tried looking it up but only getting good results for ...