Questions tagged [elliptic-curves]
For questions about elliptic curves.
3,181
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Are the Springer MyCopy Softcover books worth it? [closed]
I am a graduate student who is trying to start his bookshelf of Math textbooks I have used or will use. Of course having the hardcover version of the textbook would be really nice, but for half the ...
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Quadratic twist of elliptic surface as automorphism?
I am struggling to understand the notion of quadratic twists for elliptic surfaces. For elliptic surfaces, the singular fibres are classified by Kodaira's classification. A quadratic twist of an ...
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Finding slope of a given point on the elliptic curve given the points x and y coordinates
How to find the slope of a given point on the elliptic curve provided i have it's x and y coordinates without knowing how the x and y coordinate were formed.
The equation for the curve am looking for ...
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$n$-torsion parts of rank 2 Drinfeld modules and elliptic curves over function fields
I'm studying Drinfeld modules to study elliptic curves over function fields.
(We assume the characteristic $p$ is large enough if we need.)
Simply, we consider rank $2$ Drinfeld $A$-module $\rho$ over ...
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About calculating isogeny between two elliptic curves
I'm trying to understand Vélu formulas for calculating isogenies. I took an elliptic curve $E: y^2 = x^3 + 3x + 5$ over $GF(7)$. So I've got the following points on this curve:
\begin{equation}
\{\...
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How do I translate the intersection of two affine curves in a plane into a statement about ideals in $k[X, Y]$?
Let $E : y^2 = x^3 + Ax + B$ be an elliptic curve over a field $k$ and let $x = x_0$ be a line that intersects $E$ in $(x_0, \pm y_0)$.
According to Lemma 10 of this paper, this is expressed ...
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Twist of Elliptic curve $E_D: Dy^2=x^3+ax+b$ is not isomorphic to $E:y^2=x^3+ax+b$?
Let $E/K$ be an elliptic curve over a field $K$.
Let $E/K: y^2=x^3+ax+b$ be an elliptic curve.
Let $E_D: Dy^2=x^3+ax+b$ is called a quadratic twist of $E/K$ by a square free integer $D$.
This curve is ...
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Does the relationship between the L function of the elliptic curve and a quadruple series hold?
Let $E_{X_0(11)}$ be the elliptic curve (over $\bf Q$) of conductor $11$ defined by
$$y^2+y=x^3-x^2-10x-20.$$
First, some theorems and formulas are introduced as follows.
The modularity theorem (Slow ...
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Fixpoints of the frobenius homomorphism in the algebraic closure of a finite field
I have a questiion about a statement about Galoistheory in Silvermans book on elliptic curves. In particular i want to know why
this statement holds. of course, one direction is clear, but i cant ...
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Blow up and hyperelliptic curves.
Let $C'$ be a non singular affine curve $y^2=x^5+3$ over $\mathbb{C}$. $C'^\#$ be its projective closure : $Y^2Z^3=X^5+3Z^5$. It has singular point at $\mathcal{O}'=(0:1:0)$.
On the other hand, let $C$...
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Two copied of $\mathbb{P}^2_k$ glued by a non-torsion point on an elliptic curve - how to prove all invertible sheaves are trivial?
This is 19.11.11 in Vakil's Foundations of Algebraic Geometry, July 31 2023 version.
Let $E$ be an elliptic curve, $p$ be a non-torsion point, and $\infty$ be the point at infinity. Consider the usual ...
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How can I find $P$ s.t. $2P=\infty$?
Let me consider the elliptic curve $C: y^2=x^3+2x$. Now I want to finde $P\in C(\Bbb{Q})$ s.t. $2P=\infty$.
I thought about writing $P=\left(\frac{a}{b}, \frac{c}{d}\right)$, then I can use the ...
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Isomorphism between $2y^2=x^4-17$ and $y^2=x^3+17x$ over quadratic field
Isomorphism between $2y^2=x^4-17$ and $y^2=x^3+17x$ over quadratic field.
I heard $2y^2=x^4-17$ and $y^2=x^3+17x$ are isomorphic curves over a certain quadratic number field (what we call twist).
Over ...
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Example of a quotient of an elliptic curve by a finite group being rational
I am interested in an example of the following situation, over an algebraically closed field o zero characteristic.
Let $E$ be an elliptic curve, and $G$ a finite group of automorphisms of $E$ (as an ...
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Why is this construction of an affine curve not uniformization?
I'm learning the Shimura curve. When I reading the note of Pete L. Clark (SC2-Fuchsian.pdf (uga.edu)), I was stuck on a thinking question.
First, there is a theorem (Uniformization Theorem) about ...
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Why are all real inflection points on a cubic projective algebraic curve on 1 line?
Say we have $C\subset \mathbb{CP}^2$, a smooth curve of degree 3. I am aware of the group structure on cubics, what I don't get, is why are all inflection points with only real coordinates lie on a ...
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Twist of 2-Selmer group can be arbitrary large
Let $Sel^2(E/\Bbb{Q})$ be a 2-Selmer group of an elliptic curve $E/\Bbb{Q}$.
For $D\in \Bbb{Z}$, let $E_D/\Bbb{Q}$ denote its quadratic twist by $D$.
It is known that
$Sel^2(E_D/\Bbb{Q})$ can be ...
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How to calculate effectively computable constant for number of solutions of elliptic curve
I was studying M. Bhargava's seminal paper titled "Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves"
And came across a very fascinating observation ...
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Mordell's Theorem and Fermat's Last Theorem for $n=4$: A general method?
In his book Elliptic Curves, A. Knapp illustrates the close relationship between the proof of Mordell's Theorem and Fermat's proof (both via infinite descent) that the equation $u^4 + v^4 = w^2$ has ...
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Isomorphism of lattices/complex tori
This is essentially a reference request (apologies if it is a duplicate): it is known that every lattice $\Lambda$ in $\mathbb{C}$ is isomorphic to one of the form $\mathbb{Z} \oplus \mathbb{Z}[\tau]$ ...
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Let $E/\Bbb{Q}$ be an elliptic curve. Let $K=\Bbb{Q}(\sqrt{D})$ be an quadratic number field.
Let $E/\Bbb{Q}$ be an elliptic curve.
Let $K=\Bbb{Q}(\sqrt{D})$ be an quadratic number field.
There is a trace map $E(\Bbb{Q}(\sqrt{D}))\to E(\Bbb{Q})$ given by $P\to P+P^{\sigma}$, $\sigma$ is a ...
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Does there exist infinitely many quadratic field such that rank of elliptic curves gains at most 2?
Let $L/K$ be a quadratic extension of number field.
Most elliptic curves $E/K$ will have the property that $rank(E/K) \le rank(E/L) +2$.
Is it known for an arbitrary $E/K$, does there exists ...
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How to transform points in subgroup to coset points (quotient group points)?
Elliptic curve parameters: y^2 = x^3 + 3 and prime p = 5119
Let G be the full-group of order 5053.
Let H be the prime-order subgroup (of G) of order = 163.
Let G/H be the quotient group/left coset (...
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What is the natural map $E_D(\Bbb{Q})\times E(\Bbb{Q})\to E(\Bbb{Q}(\sqrt{D}))$?
Let $E/\Bbb{Q}$ be an elliptic curve. Let $K=\Bbb{Q}(\sqrt{d})$ be a quadratic field.If $E:y^2=x^3+ax+b$, let $E_D:Dy^2=X^3+ax+b$.
In this page, professor Silverman reads there is a natural map $E_D(\...
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How to find 3P of Elliptic curve when slope is a decimal?
Hi I was soliving Elliptic Curves and I found out $2P$ but when it came to $3P$ I became a little bit stumped. My equation is
$$y^2 =x^3 + 2x + 9 \qquad(\text{mod } 23)\ .$$
The values I got for $P$ ...
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Special Elliptic Curves Families
In Washington’s “Elliptic curves number theory and cryptography” section 4.4, he mentions a special family of curves. This family is of the form $y^2=x^3-kx$ over a prime field $F_p$. The interesting ...
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Rank of elliptic curves does not decrease under field extension?
Let $E/\Bbb{Q}$ be an elliptic curve.
Rank of elliptic curve over quadratic extension $L=K(\sqrt{D})/K$ is calculated by a formula $rank(E/L)=rank(E/K)+rank(E_D/K)$(this is easy to prove).
In ...
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Check if Two Elliptic Curves have the Same Number of Points
Given two elliptic curves $E_1$, $E_2$ over $F_q$.
Assume that the two curves’ order is unknown. Is there a way to tell if they have equal order without counting it?
Meaning, if $N_i(A_i,B_i,q)$ is ...
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Finding Elliptic Curves with Provably Different Number of Points
Is there a way to find $k$ elliptic curves over $F_q$, all with provably different number of points?
And doing so without counting the number of points on the curves.
For really small $k$ (2,3) I know ...
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Rank of elliptic curves and number prime factors of discriminant
Let $E/\mathbb{Q}$ be an elliptic curve. Its rank, denoted by $\text{rank}(E/\mathbb{Q})$, is a mysterious object. However, there exists a well-known very rough embedding
$$
E(\mathbb{Q})/2\mathbb{Q} \...
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A question on an elliptic curve having endomorphism ring Z[i]
While reading Silverman's beautiful book The Arithmetic of Elliptic Curves, I was stuck trying to understand some elements of Example III.4.4:
First, I don't see why the case of characteristic 2 is ...
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Can we make the rank of twist of a given elliptic curve over number field $0$ by infinitely many ways?
Let $E/K$ be an elliptic curve over a number field $K$.
Does there exists an infinitely many $D\in \Bbb{Z}$ such that rank of $E_D/K$ is $0$ ?
Maybe this is an open problem, so
weak problem can be :In ...
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Group cohomology $H^1(G,M)$ , where $G$ is finite and $M$ is finitely generated
Let $G$ be a finite group of order $n$.
Let $M$ be a finitely generated abelian group generated by $m_1,m_2,・・・,m_k$.
Let $H^1(G,M)$ be a group cohomology.
Because $nH^1(G,M)=0$ from restriction and ...
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Residual theorem for ellipse perimeter integral
Trying to use residual theorem for integrating ellipse perimeter.
Can I use the residual theorem for the ellipse perimeter?
The calculation process I've followed so far
If it cannot use the residual ...
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Does Galois group acts trivially on kernel of reduction?
Let $L/K$ be a unramified quadratic extension of local fields.
Let $E/K$ be an elliptic curve over $K$.
We can consider reduction map $E(L)\to E(l)$ where $l$ is a residue field of $L$. Suppose ...
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Genus $1$ curve over $\Bbb{F}_q$ always has rational points, exercise $10.6$ in Silverman
I want to understand the proof of the fact
Genus $1$ curve over $\Bbb{F}_q$ always have rational points
The result was originally proven by F.K. Schmidt and is currently presented as Exercise $10.6$ ...
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Calculating unit vectors B , T , N a (probably ellipse like) curve
$$r(t) = a\cos(t)i+b\sin(t)j+ctk$$
So this is the equation of a curve for which I've had trouble calculating it's unit vectors, especially
$B$ and $N$.
I know that $T = v/|v|$ and that also $B = v × a ...
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Finding all rational points on elliptic curve $y^2=x^3-\frac{13}{3} x+\frac{70}{27}$
Find all rational points on the elliptic curve
$$ y^2=x^3-\frac{13}{3} x+\frac{70}{27}$$
The problem is essentially equivalent to the following problem;
Find al pairs $(x,y)$ of rational numbers ...
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$\bar{S}=\left\{[z,w,t]\in \mathbb{C}P^2 : w^2t^{k-2}=(z-a_1t)\ldots (z-a_kt)\right\}$ is a Riemann surface iff $k \leq 3$
I know that if $P \in \mathbb{C}[z]$ is a monic polynomial with distinct roots,
$$P(z)=(z-a_1) \ldots (z-a_k),$$
then the set
$$S= \left\{ (z,w)\in \mathbb{C}^2 : w^2=P(z) \right\}$$
is a connected ...
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Consecutive Mordell Curves
Are there infinitely many $k$ such that the Mordell curves $x^3 = y^2 + k$ and $x^3 = y^2 + k + 1$ both have no solutions?
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Is the order of cokernel of trace map of elliptic curve a power of 2?
Let $L/K$ be a quadratic extension of a number field.
Let $E/K$ be an elliptic curve.
There is a trace map $E(L)\to E(K)$ given by $P\to P+P^{\sigma}$, where $\sigma$ is a generator of $Gal(L/K)$.
Why ...
- 5,565
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The theta function on Weierstrass normal form
We want to write down the theta function of the point on Weierstrass normal form.
Now, we can embed an elliptic curve to projective plane by$E=\mathbb{C}/(\tau\mathbb{Z}+\mathbb{Z})\to \mathbb{P}^2$
...
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Coordinates after point multiplication not in elliptic curve.
When calculating $2P$ where $P = (7, 11)$ on the elliptic curve E: $y^2 = x^3 + x + 1 \mod 23$.
I get $$ \lambda = \frac{3 * 49 + 1}{2 * 22} = \frac{74}{11} \mod \ 23 = 10.$$
Then when I calculate $ ...
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Real points of elliptic curves and Weierstrass $\wp$-function
Let $\Lambda = \mathbb{Z} + \mathbb{Z}\tau \subset \mathbb{C}$ be a lattice generated by $1$ and $\tau$. Weierstrass $\wp$-function $\wp_\Lambda: \mathbb{C} \to \mathbb{CP}^1$ gives an isomorphism
$$
\...
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How to prove that $f(x,y)=xy(x+y)$ isn't surjective as $f:\mathbb{Q}^2\to\mathbb{Q}$.
I would like to show that the rational polynomial function $f:\mathbb{Q}^2\to\mathbb{Q}$ defined as $f(x,y)=xy(x+y)$ isn't surjective. To do this, I tried proving that the algebraic plane curve
$$K:xy(...
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Map between Points of two different Elliptic Curves of same order
Let $E_1$ and $E_2$ be two elliptic curves defined over $F_{p1}$ and $F_{p2}$ respectively and with same number of points $\#E_1(F_{p1}) = \#E_2(F_{p2})$. Since number of points of two curves are same,...
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Reduction of elliptic curve and trace map, $E_1(L)/\text{trace}(E_1(K))\to E(K)/\text{trace}(E(L))\to \tilde{E}(l)/d\tilde{E}(l)\to 0$?
Let $l/K$ be a finite(degree $d$) ramified Galois extension of local fields and $E/K$ be an elliptic curve.Let $l$ be a residue field of $L$.
Let $E_1$ be a reduction of kernel.
Why does there exists ...
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Generators of the function field of an elliptic curve
I came across the following claims (from https://aghitza.github.io/publication/translation_velu/, the very beginning of section 1).
Let $E$ be an elliptic curve (in Weierstrass form) over over an ...
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Weierstrass sigma function identity - Silverman AEC 6.3
In Silverman's AEC Chapter VI we define the Weierstrass $\wp$ and $\sigma$ functions, particularly for $\Lambda = \mathbb{Z} + \tau \mathbb{Z}\subset \mathbb{C}$ a lattice, $$\wp(z) = \wp(z,\Lambda) =...
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Why an R-valued point on an arithmetic surface can be specialized to the generic fiber?
Let $R$ be a Dedekind domain with $K=Frac(K)$, $\mathcal{C}$/$R$ be an arithmetic surface,
and let $C/K$ be the generic fiber of $\mathcal{C}$.
Why any point in $\mathcal{C}(R)$ can be specialized to ...
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