Questions tagged [elliptic-curves]
For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.
1,886
questions
4
votes
0answers
20 views
Values of Grössencharacter attached to CM elliptic curve
Let $E$ be an elliptic curve defined over a number field $L$, having CM by by the ring of integers $\mathcal{O}_K$ for $K$ quadratic imaginary. If $K \subseteq L$, then (as constructed in Silverman's ...
2
votes
0answers
42 views
Confusion in Proposition 10.4 of Silverman's Advanced Topics in the Arithmetic of Elliptic Curves
My question concerns the proof of Proposition 10.4. in chapter II of Silverman's book Advanced Topics in the Arithmetic of Elliptic Curves. The situation is the following.
Let $E/L$ be an elliptic ...
0
votes
0answers
17 views
Models for the modular curve $X_{0}(N)$
I think my question can be summarised as follows:
I have seen the construction of the canonical model for $X_0(N)$ "starting from the bottom". By that I mean I started with the definition of the ...
2
votes
1answer
14 views
How to graph path of parametric equations
I am having a lot of trouble answering the following question as it is required to consider the value of t also. If t did not have to be considered, it would have been an easy ellipse.
Let an ...
2
votes
1answer
40 views
Intuition behind ramification index of a map between smooth curves.
I'm studying from The Arithmetic of Elliptic Curves (Silverman) and I'm having a hard time understanding the intuition behind the $\textit{ramification index}$ concept.
In the book, we let $\phi:C_1 ...
2
votes
1answer
46 views
Can I find an endomorphism of an elliptic curve with a specific kernel size?
Say I want to find an endomorphism of an ordinary elliptic curve $E$ with kernel size of a prime $l$ that divides the cardinality of $E$. Is this possible in its endomorphism ring and what is the ...
0
votes
0answers
28 views
Non-vanishing Taylor coefficients and Poincaré series
I'm studying these algebraic numbers in the table below and have succesfully shown what i wanted with the given values I had.
The table is found in the book "The 1-2-3 of Modular Forms" by Jan ...
2
votes
0answers
74 views
Map between curves and integral points
Suppose we have an equation of two variables $x,y$, for example $E: x^3+y^3=d$, and we have a bijective map
\begin{align*}
\varphi: E &\to F\\
(x,y) &\mapsto \left(\frac{3d}{x+y},-\frac{9dy}{x+...
2
votes
1answer
85 views
Isogeny between j-invariants
I am studying $l$-isogeny graphs (volcanoes). As I understand these graphs have $j$-invariants as vertices but I am having a hard time understanding the edges. The following is not clear to me:
...
0
votes
1answer
21 views
Prove j(E) is an integer for an elliptic curve with CM by a quadratic field of class number 1
If $E$ has CM by an imaginary quadratic ring $\mathcal{O}_K$ such that $h(\mathcal{O}_K)=1$, how would we show that $j(E)$ is an integer? (or, equivalently, that $j(\frac{1+\sqrt -t}{2})\in\mathbb{Z}$ ...
1
vote
1answer
34 views
Endomorphisms of an elliptic curve
If I am correct, an endomorphism is a homomorphism from one group to itself. In the case of an elliptic curve, we'd need a map $\phi: E \rightarrow E$. Ive seen in some places that you can get such a ...
0
votes
0answers
30 views
Generators of a Graded Algebra
I have a question about a statement from P. Wagreich's paper "Elliptic Singularities of Surfaces" (page 425):
We consider an elliptic curve $X$ and a line bundle (=invertible sheaf) $L$ on $X$.
This ...
4
votes
1answer
57 views
Mod $p$-representation ($p$-torsion points ) of elliptic curve (over number fields) with CM can be irreducible?
Let $E/K$ be an elliptic curve with complex multiplication and let $E[p]$ be the group of $p$-torsion points of $E$ where $K$ is a number field and $p$ is a prime number.
My question is: as a $(\...
0
votes
1answer
64 views
Find the identity element of elliptic curve
I try to understand the definition of Identity element of elliptic curve.
For the following elliptic curve over $k = \mathbb{F}_5$:
$$y^{2} = x^{3} + 1$$
The points that in $E$ are shown in the ...
1
vote
1answer
48 views
elliptic curve $~y^2 = x^3 + 3x +8~ (mod~ 13)~$ - torsion groups
Corollary $6.4.$ Let $E$ be an elliptic curve and let $m \in \mathbb{Z}$ with $m \neq 0$.
$(b)$ If $m \neq 0$ in $K$, i.e., if either $\operatorname{char}(K) = 0$ or $p := \operatorname{char}(K) &...
0
votes
0answers
36 views
Points on an Elliptic Curve, how to interpret $(x(2P):z(2P))$?
This is from J.S. Milne's book 'Elliptic Curves'-
With $E(\mathbb{Q}) : Y^2Z = X^3 +aX Z^2+bZ^3$ and given a point on $P =(x,y)$ on it's Weierstrass equation dehomogenized $E: y^2= x^3+ax+b$ where $a,...
2
votes
0answers
88 views
A difficult step in the proof of projective elliptic curve group law
I'm formalizing this paper in a proof assistant. On page 14, you can find a proof of a property called dichotomy. It is useful to stablish the group law for this curves. However, I'm not understanding ...
1
vote
1answer
36 views
Determinant representation of Tate module and cyclotomic character
Let $E$ be an elliptic curve over a field $K$, then $G_K:=G_{K^s/K}$ acts on the Tate module of $E$. This is a 2-dim representation of $G_K$. I have heared that we can prove the determinant ...
0
votes
0answers
60 views
Continuity of the “logarithm” $u \mapsto \frac{du}{udx}$ on function fields
Let $k$ a field and $F$ a finite extension of $k(x)$. Let the rational 1-forms $$Fdx = \{ f dx, f \in F\}=\{ f dg, f,g \in F\} $$
(obeying to the rules of $F$-modules, of $k$-linearity and $d1=0$, $d(...
1
vote
1answer
48 views
A question on Serre' open image theorem and p-adic Lie group
Recently I read Serre's wonderful book Abelian l-Adic Representations and Elliptic Curves
. There is a theorem in section2.2 chapterIV.
Theorem: If $E$ has no complex multiplication, then $g_l=End(...
1
vote
0answers
40 views
Dual curve of elliptic curve
I've asked about this already here (thanks to user10354138 for the quick and good answer!) but when I showed the solution to my instructor, he said I should follow first his procedure for getting the ...
2
votes
0answers
39 views
Group law for projective Edwards curves
I'm reading this paper and in section 4 the author proposes projective elliptic curves.
I have a doubt on why it is assumed implicitly that $c = 1$. This is done explicitely in the Mathematica ...
0
votes
1answer
21 views
Finding preimage of point for isogenies between elliptic curves
Let's say one has an isogeny $\alpha:E_1\to E_2$ between two elliptic curves, and that $\ker\alpha$ is known. If there is a point $S_2\in E_2$, is there an efficient way to find its preimage $S_1\in ...
1
vote
0answers
34 views
Line intersecting at 3 points on Elliptic curve? [closed]
I'm reading about elliptic curves and how drawing a line between two points on the curve will always intersect with a third point. It seems pretty easy, though, to draw a line such that it only ...
0
votes
0answers
22 views
Why do we ask $w\nmid F$ in order for $i\left (\left (0:1:0\right ),w,F\right )$ to be defined?
Knapp's book on elliptic curves, on the fourth section of its second chapter, has a confusing argument which I cannot understand.
For a cubic $F\in k\left [x,y,w\right ]$ (homogeneous third-degree ...
0
votes
1answer
49 views
elliptic curve ${X^3+Y^3=AZ^3}$
consider the elliptic curve $X^3+Y^3=AZ^3$ and $A$ in $K*$ with $O=(1,-1,0)$. show the $j$ invariant of this elliptic curves is $0$. (part d of Silverman exercise page 104 Q3.3 )
I can compute the $j$...
0
votes
0answers
32 views
A problem on counting the number of points of an elliptic curve [duplicate]
I am studying " Arithmetic Of Elliptic Curves " of Silverman and trying to solve some of its exercises. My question is in chapter 5 :
Prove that for every $ p\geq 3$ , the elliptic curve
$$ E : y^2 ...
0
votes
1answer
42 views
Solving a problem without using the Riemann - Roch theorem
I am studying elliptic curves and I faced with the concept of the divisors and Riemann Roch Theorem. My reference is " The Arithmetic of The Elliptic Curves " of Silverman .
I tried to solve one of ...
0
votes
1answer
21 views
Montgomery form elliptic curve with given j-invariant
For a given $j\in K$, I would like to find a Montgomery form elliptic curve $M_{A,B}: By^2=x^3+Ax^2+x$ such that j-invariant of $M_{A,B}$ is j (if it is possible). In the form $y^2=x^3+Ax^2+x$ is also ...
1
vote
1answer
48 views
Prove that two groups for elliptic curves are isomorphic
I was asked to calculate all the possible groups for elliptic curves and their order in $\mathbb{F}_5$. There are $p^2-p$ groups that respect $\Delta \neq 0$, so there are $20$ groups.
Some of them ...
1
vote
1answer
38 views
Determine groups for elliptic curves over a finite field
I have the following question:
Determine all possible groups $E(F_5)$ for elliptic curves over $F_5$. What are
their orders?
I am completely in the dark here about calculating the groups and their ...
0
votes
0answers
32 views
Number of complex embeddings of a certain number field
Let $K$ be an imaginary quadratic field with class number 1 and $E/K$ be an elliptic curve with CM by $\mathcal{O}_K$. Consider the number field $F=K(E[p])$. Can $r_2(F) =1$?
2
votes
0answers
33 views
On the definition of Edwards curves group addition
I'm working with Hales The Group Law of Edwards Curves where he defines the addition $z_1 \oplus z_2 = (\frac{x_1x_2 - c y_1y_2}{1 - d x_1x_2y_1y_2}, \frac{x_1y_2+y_1x_2}{1+dx_1x_2y_1y_2})$ and he ...
1
vote
1answer
19 views
To prove that the Elliptic modular function is invariant under the modular transformation
I am not being able to understand that the Elliptic modular function $J(\tau)=\frac{g_2(w_1,w_2)^3}{g_2(w_1,w_2)^3-27g_3(w_1,w_2)^2}$ is invariant under the modular transformation $\tau\mapsto \frac{a\...
0
votes
1answer
44 views
Rank of Elliptic Curve, $Y^2=x^3+px$ where $p$ is prime is either $0,1,2$
I am following book "Rational Points on Elliptic Curves" by Silverman-Tate(basic version not the "The Arithmetic of Elliptic Curves" by Silverman-Tate) And I am trying to solve for cubic curve, $y^2=x^...
-1
votes
1answer
21 views
Compact surface with closed 1-form [closed]
Why a compact surface with closed a nowhere vanishes complex 1-form, giving $T^{2}$?
3
votes
0answers
19 views
Tamagawa measure of elliptic curve at a place dividing 2
There's quite a bit of set-up for this, but please bear with it!
Let $K$ be a number field, and let $E/K$ be an elliptic curve. Let $v$ be a finite place of $K$, and suppose that at $v$, $E$ has ...
0
votes
1answer
46 views
Moduli Space of Elliptic curves
I am trying to see that:
The moduli space of Riemann surfaces of genus 1 with one marked point is $\mathbb{H}/PSL(2,\mathbb{Z})$.
I know the facts that $PSL(2,\mathbb{Z})$ acts on the upper half ...
2
votes
1answer
31 views
Deriving Bachet's duplication formula
Let $y^2-x^3 = c$ be Bachet's equation and pretend $(x,y)$ is a solution.
The tangent at $(x,y)$ of Bachet's curve is going to intersect it in a unique new point whose coordinates are supposed to ...
3
votes
0answers
67 views
Can someone suggest a path to study Mordell-Weil theorem for someone studying on their own?
So, I want to read the proof of Mordell-Weil theorem and so, I picked up the book 'Arithmetic of Elliptic Curves' by J. Silverman and J.S. Milne's Elliptic Curves book. But after going through both ...
0
votes
0answers
42 views
elliptic curve with Teichmüller parameter $τ=i$
Let the complex form $z_{1}+dz_{1}\wedge dz_{2}$ on $D^{2}\times T^{2}$ ,where $D^2$ is a unit disk un $z_{1}$-plane. here $z_{1}$ and $z_{2}$ is a standard coordinates on $\mathbb{C^2}$. I read on ...
2
votes
1answer
46 views
Descent by 2-isogeny
Im practicing with a exercise about 2-isogenies, but struggling a bit. Im doing the following exercise:
Given two elliptic curves over $\mathbb Q$.
\begin{equation}E: y^2 = x(x^2-5) \quad E':y^2 = x(...
0
votes
0answers
32 views
Change of variables for elliptic curves
My main aim is to understand the basic facts about elliptic curves (including the notion of isomorphism) without having to use advanced results. So I will recap here some basic facts. To avoid ...
2
votes
0answers
51 views
Group structure of an elliptic curve over a finite field
I'd be interested in a short proof of and a reference (first source, not a textbook as e.g. one of Silverman's monographs) for the following result:
Let $q$ be a prime power and $C/F_q$ be an ...
2
votes
0answers
67 views
Isomorphism of elliptic curves
One of the first things discussed in books on elliptic curves is the change of variables that "puts the curve" in Weierstrass form. To give a concrete example, the curve $6y^2=2x^3+3x^2+x$ (the "...
1
vote
1answer
19 views
Isomorphism between elliptic curves over $\mathbb Q$ and $\mathbb F_5$
Given are the Elliptic curves $E_1 : y^2 = x^3+x$ and $E_2 = y^2 = x^3+3x$. Are these isomorphic over
a) $\mathbb Q$?
b) $\mathbb F_5$?
I see they are isomorphic over $\mathbb C$, as they have the ...
0
votes
0answers
18 views
isogenous elliptic curves have same rank
This is based on exercise 14.3 from Cassels, Lectures on Elliptic Curves. Let $$E:y^2=x(x^2+ax+b), E':y^2=x(x^2+a_1x+b_1)$$
be two elliptic curves over $\mathbb{Q}$, with $a_1=-2a$, $b_1=a^2-4b$. We ...
0
votes
0answers
33 views
Elliptic curve $Y^2=X^3+X$
Let $p$ be prime s.t. $p=a^2+4b^2, a\equiv 1\,mod\,4$.
Then, for $f=X^3+X$, $a_E(p)=2a$?
(Here, $a_E(p):=p+1-\#\{(x,y)\in\{0,...p-1\}^2|x^3+x\equiv y^2\,mod\,p\}$)
0
votes
1answer
32 views
“No way to put a topology on $\mathbb{Q}^*/\mathbb{Q}^{*2}$ such that the map $\alpha$ is continuous.”
Let
\begin{align*}
\mathbb{Q}^{*2}:=\{x^2\;|\;x\in\mathbb{Q}^*\}.
\end{align*}
and
\begin{align*}
\alpha:\;E(\mathbb{Q})&\to\mathbb{Q}^*/\mathbb{Q}^{*2},\\
\alpha(x,y)&=x\mod\mathbb{Q}^{*2}\...
2
votes
1answer
43 views
Solutions to $y^2 = x^3 + x$ in $\mathbb{P}^2(\mathbb{F_p})$. [duplicate]
Consider the elliptic curve $$\mathcal{C}: y^2 = x^3 + x.$$
Let us consider the reduction of $\mathcal{C} \mod{p}$.
Some explicit computation show that $\# \tilde{\mathcal{C}}(\mathbb{F}_3) = 4$, $...
