# Questions tagged [elliptic-curves]

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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 ...
1 vote
22 views

### 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 ...
27 views

### 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 ...
14 views

### $n$-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 ...
54 views

### 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} \{\...
1 vote
37 views

### 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 ...
1 vote
89 views

### 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 ...
1 vote
40 views

### 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 ...
20 views

### 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 ...
103 views

### 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 $C$...
1 vote
42 views

### 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 ...
80 views

### 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 ...
1 vote
44 views

### 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 ...
45 views

### 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 ...
23 views

### 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 ...
1 vote
101 views

### 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 ...
42 views

### 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 ...
1 vote
49 views

### 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 ...
217 views

### 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 ...
1 vote
73 views

### 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 $\mathbb{Z} \oplus \mathbb{Z}[\tau]$ ...
1 vote
34 views

### 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 ...
29 views

### 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 ...