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

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35
votes
3answers
720 views

The resemblance between Mordell's theorem and Dirichlet's unit theorem

The first one states that if $E/\mathbf Q$ is an elliptic curve, then $E(\mathbf Q)$ is a finitely generated abelian group. If $K/\mathbf Q$ is a number field, Dirichlet's theorem says (among other ...
2
votes
0answers
103 views

Examples of non-minimal Weierstrass equations that make $[E(K):E_0(K)]$ arbitrarily large.

I'm taking a course on elliptic curves, in which the lecturer briefly mentioned that the Tamagawa number $c_K(E)=[E(K):E_0(K)]$ satisfies $c_K(E)=\mbox{ord}(\Delta)$ or $c_K(E) \leq 4$. He said that ...
5
votes
2answers
480 views

Turning an elliptic curve over C into a complex torus

I have been reading a lot about the Weierstrass $\wp$ function and I understand the parameterization of an elliptic curve with the elliptic function( i.e. $x=\wp(z)$ and $y=\wp^\prime(z)$). I would ...
7
votes
1answer
116 views

Short exact sequence of modules and elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve with a 3-torsion point $P$. Let $E_{d}$ denote the quadratic twist of $E$ by $d$. Then the action of $\sigma \in ...
1
vote
1answer
216 views

The Group of points on the Elliptic curve $y^2=x^3+1$ over $\mathbb{F}_5$

So I'm trying to understand the group of points of $y^2=x^3+1$ over $\mathbb{F}_5$ and for some reason I seem to be getting nonsense answers and I'm not sure what I'm doing wrong. So basically my ...
2
votes
1answer
43 views

Using the power series expansion for $w(t)$ to construct a subgroup of the rational points on an elliptic curve.

I'm doing a course on elliptic curves, and I'm stuck on a line in a proof which is supposedly using the uniqueness in Hensel's lemma. Starting with an elliptic curve ...
4
votes
1answer
219 views

Proving that the differential on an elliptic curve $E$ given by $\omega=\frac{dx}{y}$ is translation invariant

I'm taking a course on elliptic curves and I'm stuck on a line in a proof. We're assuming we're in an algebraically closed field $K$ and char($K)\not=2$. We have our elliptic curve ...
5
votes
1answer
132 views

Direct proof of the non-zeroness of an Eisenstein series

Question: Can you show directly from its formula that $G_4(i)\neq0$? Recall that the holomorphic Eisenstein series of weight $2k$ is defined by: $$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus ...
1
vote
1answer
95 views

Like Diophantine equation

The equation $x^n - ny^x-nxy$ = $0$ has solution set $(n, x, y) = (1, 1, \frac12), (2, 1, \frac14), (3, 1, \frac16), \ldots$ I would like to know/learn the following (Kindly discuss) 1) If we ...
0
votes
1answer
58 views

Equation of elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve with a 3-torsion point $T$. One can write a Weierstrass equation for $E$. If I define $C := E/\langle T \rangle$, what is the Weierstrass equation for $C$? Is ...
3
votes
1answer
55 views

Showing that the map on $\mbox{Div}^0(E)$ induced by an isogeny takes principal divisors to principal divisors.

I'm doing a course on elliptic curves. An isogeny $\phi:E_1 \rightarrow E_2$ induces a map $$\begin{array}{llll}\phi_*: & \mbox{Div}^0(E_1) & \rightarrow & \mbox{Div}^0(E_2) \\ \\ & ...
4
votes
0answers
118 views

Ribet's proof of open image for elliptic curves

In http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183555477, Ribet gives a proof of Serre's open image theorem for elliptic curves using ...
1
vote
1answer
125 views

Isogeny and minimal models of elliptic curves

Suppose I have two isogenous elliptic curves over $\mathbb{Q}$, $E$ and $E'$. Will the minimal models of $E$ and $E'$ still be isogenous?
2
votes
1answer
61 views

Change of variables for elliptic curve

Say I have an elliptic curve $y^{2} + 5xy + y = x^{3}$ with the $(0, 0)$ being a rational 3-torsion point at $(x, y) = (0, 0)$. What change of variables would I need to get it into the form $y^{2} = ...
4
votes
1answer
162 views

Discriminant of isogenous elliptic curves

Let $E$ be an elliptic curve with a rational $p$-torsion point $P$. Then $E$ is isogenous to the elliptic curve $E' := E/\langle P \rangle$ via the mod $P$ map. I know that the conductor of $E$ and ...
9
votes
2answers
226 views

Elliptic curves over Spec Z

I want to show that there are only finitely many elliptic curves over Spec $\mathbf Z$ without appealing to Siegel's theorem or Shafarevich' theorem. Firstly, I think (but I am not sure) that such an ...
2
votes
1answer
171 views

Proving the condition for two elliptic curves given in Weierstrass form to be isomorphic

I'm taking a course on elliptic curves and trying to understand the proof of Proposition 3.2. Let $E$, $E'$ be elliptic curves over $K$ in Weierstrass form: ...
8
votes
1answer
111 views

Size of the group $E(\mathbb{Q}_{p})/pE(\mathbb{Q}_{p})$

Let $E$ be an elliptic curve over $\mathbb{Q}_{p}$. Do we know anything about the order of the group $E(\mathbb{Q}_{p})/pE(\mathbb{Q}_{p})$? I know that it's finite, but do we know anything else?
3
votes
0answers
57 views

Is there a construction known for associating a K3 surface to a curve or cover of curves

Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian. Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
8
votes
1answer
658 views

Automorphism group of the elliptic curve $y^2 + y = x^3$

Consider the elliptic curve $E : y^2+y = x^3$ over $\overline{\mathbb{F}_2}$. It has the biggest automorphism group $G$ among all elliptic curves, namely with order $24$. What is the structure of $G$? ...
1
vote
2answers
147 views

$N^2=2M^4-2p^2e^4$ has no integer solution

If $\gcd(M,e)=\gcd(N,e)=1$ and $p$ is prime and $p‎\equiv 5 \mod(16)$ then how I can show that $N^2=2M^4-2p^2e^4$ has no integer solution.
4
votes
1answer
83 views

What is Weil paring computing really?

I have trouble in understanding Weil paring on $N$-torsion points on an elliptic curve. Please see Wikipedia for the definition of Weil paring. I would like to know what Weil paring is computing ...
4
votes
1answer
166 views

Cohomology group and elliptic curve

Let $E$ be an elliptic curve with a 3-torsion point $P$ and $G = \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Let $X = \{O, P, -P\}$ where $O$ is the point at infinity and $X$ is a ...
3
votes
1answer
301 views

The Process of Choosing Projective Axes to Put an Elliptic Curve into Weierstrass Normal Form

I'm reading the book "Rational Points on Elliptic Curves" and on page 23 the author takes an arbitrary (non-singular) elliptic curve in the projective plane and finds a rational point $O$, referring ...
9
votes
1answer
297 views

Where does this elliptic curve come from?

In Zeta functions of an infinite family of K3 surfaces, Scott Alhgren, Ken Ono and David Penniston compute the zeta functions (given a good reduction restriction mentioned below) of the K3 surfaces ...
2
votes
0answers
127 views

Distribution the points in Elliptic curves over a finite field F_p where p is prime.

I want to know why the distribution the points in Elliptic curves over a finite field $\mathbb{F}_p$ where $p$ is prime is uniform. That means the number of points in elliptic curve $E$ with ...
3
votes
1answer
71 views

Unramified at certain places and Selmer groups

Consider the $p$-Selmer group of an elliptic curve $E/\mathbb{Q}$ denoted by $\operatorname{Sel}_{p}(E/\mathbb{Q})$. Why does showing that $E(\mathbb{Q}_{\ell})/pE(\mathbb{Q}_{\ell}) = 0$ (for $\ell ...
1
vote
1answer
433 views

Elliptic curves mod p

I am currently revising for an exam and need some help with a question. Below is an example from my notes which I am trying to understand. I can fill out the table just fine, but I can't figure out ...
1
vote
2answers
66 views

What does the $[(0 : 3 : 1)]$ means in Sage.

I tried to solve the integer points of $y(y+2)=x^3+(x+3)(x+5)$ by using Sage's command E.integral_points(). Its output was $[(0 : 3 : 1)]$. I tried that ...
1
vote
0answers
68 views

Elliptic curve terminology confusion

I've been reading a paper that says "Let $E(K)$ be an elliptic curve..." where $E(K)$ means the $K$-rational points of $E$ (where $K$ is a number field). I've seen phrases like "Let $E/K$ be an ...
4
votes
1answer
124 views

A question on level structures on elliptic curves

I have a question on $\mathbb{H}/\Gamma(N)$, which parametrizes level $N$ structures on elliptic curves. Let $Y(N)$ be the set of isomorphism classes of such objects, then, according to Fact 2 on page ...
1
vote
0answers
139 views

Computing the degree of an isogeny

Let $E$ be an elliptic curve with a $p$-torsion point. Denote this point by $P$. Why does is isogeny $\phi: E \rightarrow E/\langle P \rangle$ of degree $p$? I do know that if $\phi$ is separable, ...
3
votes
2answers
157 views

Property of elliptic curves with a torsion point

Let $E/\mathbb{Q}$ be an elliptic curve with a $p$-torsion point. Does this imply that $E/pE$ is isomorphic to $E$? If not, are there any conditions I can assume such that this is true?
0
votes
0answers
75 views

Moduli space of elliptic curves with $C_n$ action

I would like to construct moduli space of elliptic curves with cyclic group $C_n$-action. In other words, I want to classify a pair $(C,\phi)$, where $C$ is an elliptic curve and $\phi:C_n\rightarrow ...
2
votes
1answer
203 views

Help in finding curve equation.

What I have is length of the bottom line $L$ and area under parabolic curve $S$. How can I find this parabolic curve equation, depending on area under it? The following picture illustrates the ...
4
votes
0answers
133 views

Complex multiplication - Ray class fields

I'm pretty new to complex multiplication and am struggling with Corollary 5.20 in Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer by Rubin. According to ...
2
votes
0answers
51 views

Different elliptic curves over given $\mathbb{F}_q$ can have different orders?

As the title says: For a given $q \in \mathbb{N}$, in $\mathbb{F}_q$, is it true that different curves on it can have different group orders? I assume yes. Let $q:=5$. Wikipedia gives $9$ points for ...
2
votes
1answer
147 views

Nef divisors on the compactified modular curve level $N$

Consider the compactified modular curve with full level structure $X=\overline{\Gamma(N)\setminus \mathcal{H}}$. We know the Hodge bundle (the extension of the hodge bundle to the compactification) ...
5
votes
2answers
560 views

Prove that the equation $y^2=x^3-73$ has no integer solutions

Prove that there are no integers $x,y$ such that $y^2=x^3-73$. Thank you.
3
votes
0answers
123 views

Why does Lenstra ECM work?

I came across Lenstra ECM algorithm and I wonder why it works. Please refer for simplicity to Wikipedia section Why does the algorithm work I NOT a math expert but I understood first part well enough ...
-1
votes
1answer
52 views

Does 0 lie on elliptic curve?

Does $0$ lie on an elliptic curve, where $0$ is the identity (e.g. $p + 0 = p$)?
3
votes
1answer
49 views

Conditions such that $\# E_{\mathrm{ns}}(\mathbb{F}_{\ell})\not\equiv 0 \bmod{p}$

Let $E/\mathbb{Q}$ be a semistable elliptic curve. Let $\ell$ be a prime of multiplicative reduction and consider $\# E_{\mathrm{ns}}(\mathbb{F}_{\ell})$. Given a prime $p \neq \ell$, are there any ...
5
votes
0answers
120 views

Why is this a characterization of isogenies of elliptic curves? (From Silverman)

In the proof of Theorem III.6.2 (c) in Silverman's The Arithmetic Of Elliptic Curves it says: Let $x_1, y_1 \in K(E_1)$ and $x_2, y_2 \in K(E_2)$ be Weierstrass coordinates. We start by looking at ...
1
vote
0answers
80 views

Addition law on moduli space of curves

Dislaimer: I know very little about this, so if parts of my question don't make sense, please feel free to edit in a way that does, or ask me to clarify. Let $\mathcal{M}_{1,2}$ be the moduli space ...
0
votes
1answer
139 views

Prove that there are $p+1$ points on the elliptic curve $y^2 = x^3 + 1$ over $\mathbb{F}_p$, where $p > 3$ is a prime such that $p \equiv 2 \pmod 3$.

Let $p > 3$ be a prime such that $p \equiv 2 \pmod 3$. Define the elliptic curve $E$ over $\mathbb{F}_p$ by $y^2 = x^3 + 1$. Prove that $E(\mathbb{F}_p)$ consists of $p+1$ points. Using Fermat's ...
1
vote
0answers
81 views

Elliptic curves with form$ y^2$=$x^3$+$p^2$$x$

We Know that from a conjecture by Goldfeld says that half of all elliptic curves have rank zero. Are there any known infinite families of elliptic curves in form $y^2=x^3+p^2x$ where p is prime with ...
2
votes
0answers
113 views

Do K3-surfaces have Weierstrass equations

I've been wondering a bit about K3-surfaces and their analogy to elliptic curves. I've just started so this might be a very silly question. Do all K3-surfaces have a Weierstrass equation (up to ...
1
vote
1answer
135 views

Prove that there are no natural numbers, $i, j$ such that $ 3i^2+3i+7=j^3$

I'm not sure if this is true but, I've tried with many different values of $i, j$ and didn't get any contradictions. The question again, here Prove that there are no natural numbers, $i, j$ such ...
2
votes
1answer
148 views

A question regarding tensor product and isogenies of elliptic curves

Let $E_1$ and $E_2$ be elliptic curves and $T_l(E_i)\cong \mathbb{Z}_l \oplus \mathbb{Z}_l$ the $l$-adic Tate module. Given $ \varphi \in Hom(E_1,E_2)$ this induces $\varphi_l \in ...
5
votes
2answers
319 views

Relation involving the conductor of an elliptic curve

Consider an elliptic curve $E: y^{2} = x^{3} + ax + b$. Then the quadratic twist by a squarefree $d$ is given by $E^{d} : dy^{2} = x^{3} + ax + b$. What is the relationship between the conductor of ...