An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry in addition; note also the tag arithmetic-geometry as well as some related tags such as rational-points, ...

learn more… | top users | synonyms

1
vote
0answers
130 views

Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$

According to a conjecture there are no three consecutive powerful numbers. Necessary condition for this is integer solution of $$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$ What are integer solutions ...
4
votes
2answers
313 views

Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$

This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better. I believe it is a theorem of Grauert that any holomorphic vector ...
4
votes
3answers
149 views

Quadratic twist of an elliptic curve given by non-Weierstrass model

Suppose $f(x)$ is a polynomial of degree 4 with integer coefficients and nonzero discriminant. Let $C$ be the hyperelliptic curve of genus 1 defined by $y^2=f(x)$. If we assume that $C$ has a rational ...
1
vote
1answer
98 views

Elliptic curves with square conductor

Is there a characterization of elliptic curves over $\mathbb Q$ whose conductor is a square? Does this property have a geometric meaning?
-2
votes
0answers
57 views

What does this compute to $s=(3(16)^2+9)\cdot(2\cdot 5)^{-1}\bmod{23}$? [closed]

Sorry for asking such a n00b question but what does the following compute to? $s=(3(16)^2+9)\cdot(2\cdot 5)^{-1}\bmod{23} = 11$ In an online resource, I saw this computes to $11$ but whenever I do ...
-1
votes
0answers
43 views

Trace of Frobenius of elliptic curve is integer [migrated]

I recently started to read the book "Arithmetic of Elliptic Curves" by Silverman. And I can't solve an exercise 5.10. Let $E/\mathbb F_q$ be an elliptic curve and $\phi$ is Frobenius endomorphism, ...
0
votes
1answer
156 views

Kernel of a 3-isogeny between two elliptic curves

Suppose $E_1$ and $E_2$ are two elliptic curves defined over $\mathbb{Q}$ and there exists a 3-isogeny $\varphi$: $E_1 \longrightarrow E_2$. If $E_1$ has no $\mathbb{Q}$-rational point of order 3, ...
1
vote
0answers
113 views

Ratio of periods for elliptic curves in an isogeny class

Let $E \to E^\prime $ be an isogeny of elliptic curves defined over $\mathbb{Q}$. Then what is the definition of ratio of periods for elliptic curves in the isogeny class and how to calculate the ...
1
vote
2answers
121 views

Anomalous elliptic curves over finite rings

I was wondering if it is possible to solve the discrete logarithm on an Elliptic Curve E(Z/nZ) (defined over the ring of integers modulo a composite n) with #E(Z/nZ)=n by applying a method analogous ...
1
vote
1answer
201 views

On $x^3-y^2=1728 \text{ unit}$ in number fields

Consider solution of $$x^3-y^2=1728 \text{ unit} \qquad (1)$$ in a number field. This is related to the discriminant of elliptic curve in terms of $c_4,c_6$. Via elliptic curves it might have ...
5
votes
1answer
482 views

Main conjecture for elliptic curves

Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$, and that $p$ is a prime where $E$ has good ordinary reduction. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ ...
2
votes
1answer
148 views

Counting curves of degree 4 in $\mathbb{P}^{3}$

Let $p_1,...,p_8\in\mathbb{P}^{3}$ be points in linear general position. Then there exists a unique elliptic curve $C$ of degree $4$ passing through $p_1,...,p_8$. I am interested in what happens for ...
20
votes
1answer
769 views

Is the Modularity Theorem (currently) effective?

The Modularity Theorem says every elliptic curve over $\mathbb{Q}$ can be gotten from the classic modular curve $X_0(N)$ by a rational map. Here $N$ is the conductor, easily calculable from a ...
10
votes
1answer
413 views

Examples of elliptic curves over $\mathbb{Q}$

I need examples of two non-isogenous elliptic curves $E_{1}, E_{2}$ over $\mathbb{Q}$ having the following 2 properties - 1) $E_{1}, E_{2}$ have no rational torsion points. 2) $E_1[9] \cong E_2[9]$ ...
1
vote
1answer
108 views

A special curve with points of order 3(or 6)

Could you please tell me what is the points of order 3 (or 6) Hon the elliptic curve $y^2=x^3+sx^2-x$ where $$s = -\frac{1}{432}\frac{(81k^8-2592k^4-6912)}{k^6}$$ and $k$ is rational?
20
votes
3answers
627 views

Consecutive square values of cubic polynomials

Let $P(x)$ be a cubic polynomial with integer coefficients. Does there exist a constant $c$ such that at least one of the following values $P(0),P(1),...,P(c)$ is not a square? It is known that the ...
0
votes
1answer
154 views

Elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$

How to find out examples over elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$ $?$
5
votes
2answers
312 views

Salmon's proof that tangents to a cubic from a point on it have the same cross-ratio

In Higher plane curves, nr 167, Salmon proves that the cross-ration of the four tangents to a non-singular plane cubic, drawn from a point on the curve, is independent of the point. A proof can be ...
1
vote
0answers
129 views

Average rank of elliptic curves over function fields

De Jong showed in 2002 if the finite field $\mathbb{F}_q$ has characteristic not equal to 3, then the limsup of the average of 3-Selmer rank is bounded above, where the average is taken over the ...
3
votes
2answers
130 views

Root number of a quadratic twist of an elliptic curve

Could someone provide a reference for the following fact which is stated without proof in section 4.3 of Alice Silverberg's survey "Open Questions in Arithmetic Algebraic Geometry"? Let E be an ...
4
votes
1answer
373 views

Point of order 5 over an elliptic curve

For this curve $y^2=x^3+b^2x^2-a^2b^2x$ where $a \neq b$ and $a,b$ are rational. I can prove that if $b^2+4a^2$ is square then torsion group of curve is $\mathbb Z2 \times \mathbb Z2$, and when ...
3
votes
1answer
58 views

Algebraicity of isogenies as maps of lattices

Let $E_i\colon y^2=4x^3+A_ix+B_i$, for $i=1,2$ be two elliptic curves where $A_i,B_i \in \mathbb C$ are algebraic over $\mathbb Q$. For $i=1,2$ let $\Lambda_i\subseteq \mathbb C$ be the unique lattice ...
6
votes
1answer
323 views

Explicit calculation of Weil Deligne representations

According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations. However, given a galois representation, it is usually difficult ...
7
votes
2answers
308 views

Is there a largest prime p such that J_0(p) completely splits into elliptic curves

The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny ...
7
votes
1answer
216 views

Algebraic equations for modular parameterizations

I was wondering if there some place where for some small $N$ I can find explicit modular parameterizations in an algebraic way. One type of model for $X_0(N)$ is just given by a single algebraic ...
11
votes
1answer
316 views

$S$-Tate-Shafarevich groups of elliptic curves

Let $S$ be a finite set of places of a number field $k$ and let $E$ be an elliptic curve over $k$. Define the ''$S$-Tate-Shafarevich group" of $E$ to be $$Ш(E,S) = \ker\left(H^1(k,E) \to \prod_{v ...
3
votes
1answer
333 views

$\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2

Consider the elliptic curves - $ E_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $ $ E_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $ with both good ...
5
votes
0answers
133 views

When are Galois representations with open image attached to elliptic curves?

Let $K$ be a number field with absolute Galois group $G_K$. Let $\rho:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$ be a Galois representation such that the image of $\rho$ is open in ...
5
votes
1answer
382 views

What did Shimura say about $y^2 + y = x^3 - x$?

From the introduction of Ribet-Stein: Shimura showed that if we start with the elliptic curve $E$ defined by the equation $y^2 +y = x^3 −x^2$ then for “most” $n$ the image of $\rho$ is all of ...
3
votes
2answers
150 views

Hesse pencil and Schrodinger representation of Heisenberg group

Let $E$ be a smooth elliptic curve over an algebraically closed field of characteristic zero. Let $\mathcal{L}$ be a line bundle of degree $3$. Heisenberg group $H_3$ acts on global sections of ...
3
votes
1answer
110 views

What is the complexity of finding a point with a given height on elliptic curve?

It is well known that there exists a canonical height function $\hat{h}:E(\mathbb{Q})\longrightarrow\mathbb{R}$. My question is: I have a real number $h$ and elliptic curve $E$. Is there a feasible ...
3
votes
2answers
183 views

Example of elliptic curve with CM (complex multiplication) by \sqrt{-7}

Can someone give me an example of elliptic curve with CM by sqrt(-7) with the action. I've found a list of examples in the following link but not the action. ...
3
votes
2answers
184 views

$\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 1

1) What are the examples of elliptic curves over $\mathbb{Q}$ with good reduction and $\mu$-invariant $\geq 2$ at $p = 3$ and how to find them $?$ 2) Let $\Lambda = \mathbb{Z}_{p}[[T]] $ and $ ...
2
votes
0answers
122 views

Finite Heisenberg groups action on cohomology of line bundles

Let $E$ be a smooth elliptic curve over algebraically closed field $k$ of characteristic zero, $\mathcal{L}$ is a line bundle over $E$, $\operatorname{deg}(\mathcal{L})=n \geq 1$. Then I define the ...
3
votes
1answer
226 views

Are elliptic Kummer extensions big?

Loosely speaking, are elliptic Kummer extensions big? More concretely: Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime, and let $F$ be a subfield of $\overline{\mathbb{Q}}$ ...
0
votes
0answers
121 views

Inflection points on elliptic curves over a field of characteristic 2

I'm looking at the elliptic curve $C:={\cal Z}(XY^2+ZX^2+YZ^2)$ in the field $k:=\overline{\mathbb{F}_2}$. I want to prove that this curve has 9 inflection points. Since the characteristic of $k$ is ...
0
votes
1answer
110 views

Does the modified Szpiro conjecture require minimal model?

The modified Szpiro conjecture is described in Wikipedia and here and here. The modified Szpiro conjecture states that: given $\varepsilon > 0$, there exists a constant $C(\varepsilon)$ such ...
1
vote
1answer
204 views

Elliptic units and Euler system

Maybe this question is quite obscure and ambiguous. I am really sorry for such ambiguity. My question is, what is the good thing we get from defining elliptic units and Euler system? There are lots ...
2
votes
1answer
240 views

Mordell-Weil and finiteness of rational points

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order, that is, $\mathrm{End}_K(E)\cong \mathcal{O}_K$. Suppose the class number of $K$ is equal to $1$. Let ...
3
votes
1answer
146 views

The rank of $y^2=x^3\pm i$

How Can I calculate the rank of curves $y^2=x^3\pm i$ over Q(i)? Is there any soft function to do it?
8
votes
0answers
255 views

divisibility of Tamagawa numbers

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$. Let $p\ge11$ be a prime of good ordinary reduction for $E$ and assume that $p$ does not divide the degree of a minimal modular parametrization ...
1
vote
0answers
107 views

Cube-root of j-invariant [closed]

Cube-root of j-invariant is a modular function of level 3, does it have similar property as j-invariant? How about the its minimal polynomial? In particular, are it's coefficients much smaller like ...
1
vote
1answer
82 views

The relative sizes of coordinates of a point on projective genus 1 curve (second try)

Hopefully this is better than what I asked yesterday and Milton solved. Let $ C : F(x,y,z)=0$ be a projective genus $1$ curve over $\mathbb{Q}$ with no restriction on the degree. Write a point $P = ...
2
votes
1answer
306 views

Invariants for isogenous elliptic curves

How to prove that for elliptic curves, the $\lambda$-invariant is always unchanged by an isogeny ?
5
votes
2answers
348 views

Mazur's torsion theorem on elliptic curves and its generalisations

I want to study Mazur's torsion theorem for elliptic curves over $Q$ and its generalizations for number fields, i.e., papers by Kamienny, Kenku & Momose, Filip Najman. So please suggest to me what ...
5
votes
2answers
160 views

Rational points and torsion points of CM elliptic curve

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order i.e., $\mathrm{End}_K(E)\cong \mathcal{O}=\mathcal{O}_K$. Let $\mathfrak{p}$ be a prime of $K$ such that ...
1
vote
1answer
92 views

The relative sizes of coordinates of a point on projective genus 1 curve

Let $ C : F(x,y,z)=0$ be a projective genus $1$ curve over $\mathbb{Q}$ with no restriction on the degree. Write a point $P = (X , Y , Z)$ with the smallest coprime integers $X,Y,Z$. Is it true that ...
1
vote
2answers
142 views

Is there an efficient algorithm to solve ECDLP over global field?

Let E be an elliptic curve over $\mathbb{Q}$. Is there an efficient algorithm which can solve an elliptic curve discrete logarithm in E?
27
votes
0answers
495 views

The exponent of Ш of y^2 = x^3 + px, where p is a Fermat prime

For $d$ a non-zero integer, let $E_d$ be the elliptic curve $$ E_d \colon y^2 = x^3+dx. $$ When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD, $$ \# ...
1
vote
2answers
207 views

Isogeny classes and elliptic curves over finite fields

Fix a conductor and a prime $p$. Then 1) Do the elliptic curves in the same isogeny class after reduction modulo $p$ have the same number of points over the finite field $\mathbb{F}_{p} ?$ 2) Do the ...