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

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2
votes
1answer
197 views

Inverse Scalar Multiplication of a point over elliptic curve

I was implementing point arithmetic operation, and was exploring the properties of point arithmetic, and I am unable to conclude whether $$ k^{-1}(kP) = P $$ where P is a point over elliptic curve $ ...
7
votes
3answers
311 views

Diophantine equation $x^2 + 32x = y^3$

I am trying to find all solutions to the Diophantine equation $x^2 + 32x = y^3$. I think that the first step is to factorise: $x(x+32)=y^3$. If $x$ is odd, then $x+32$ is also odd. The common ...
10
votes
3answers
413 views

Integer solutions for $x^3+2=y^2$?

I've heard a famous result that $26$ is the only integer, such that $26-1=25$ is a square number and $26+1=27$ is a cubic number.In other words, $(x,y)=(5,3)$ is the only solution for $x^2+2=y^3$. ...
2
votes
0answers
134 views

Relations between elliptic curves and topological quantum field theory

I heard that there are relations between elliptic curves and topological quantum field theory (TQFT). I googled and found that something called "elliptic genus" might be the key word to relate these ...
2
votes
0answers
79 views

Two quartic polynomials to be made a square?

Given two generally non-square quartic polynomials that are to be simultaneously made squares for particular values of $x$, $$c_1x^4+c_2x^3+c_3x^2+c_4x+c_5 = y_1^2$$ ...
1
vote
1answer
54 views

What is the upper bound of order of $ E(F_{599})$

Can you please help me to slove this problem: Let E be the elliptic curve $y^{2}=x^{3}+1$ over the finite field $F_{599}$. Using Hasse's theorem find What is the upper bound of order of $ E(F_{599})$ ...
4
votes
2answers
197 views

Rank of the elliptic curve $y^2=x^3+px$

I need to prove that the rank of the curve $y^2=x^3+px$ is $0$, if $p\equiv 7 \pmod {16}$ is a prime. Using the standard technique, we need to show that none of the following two equations admits an ...
7
votes
1answer
99 views

Why is $H^1(\text{Gal}(\overline{K}/K),A)=\lim_{\rightarrow}H^1\left(L/K, A^{\text{Gal}(\overline{K}/L)}\right)$

I'm doing a course on elliptic curves, and I'm stuck on this. Here are the definitions: Let $G$ be a group and let $A$ be a $G$-module (that is, a $\mathbb{Z}[G]$-module). We have the "cochains" ...
4
votes
1answer
143 views

Elliptic curve question

Let $P$ be a point on an elliptic curve over $\mathbb{R}$. Give a geometric condition that is equivalent to P being a point of order (a) $2$ , (b) $3 $ , (c) $ 4$ . Could someone explain this to ...
4
votes
3answers
245 views

The rational points on the curve: $y^2=ax^4+bx^2+c$.

I wonder how to find the rational points on the curve: $y^2=ax^4+bx^2+c$. Is there infinite rational points on this curve? For example:$y^2=x^4+3x^2+1.$If we set $y=x^2+k$,then $2kx^2+k^2=3x^2+1$, ...
4
votes
1answer
504 views

Computing the divisors of a meromorphic function defined by a hyperelliptic curve.

Let $X$ be a hyperelliptic curve defined by $y^2=h(x).$ Let $\pi : X\to \mathbb{P}^1$ be the double covering map sending $(x,y)$ to $x$. Let $\omega=\pi^{*}(dx/h(x)).$ Compute div$(\omega)$. I ...
1
vote
1answer
2k views

Is it possible to compute order of a point over Elliptic curve?

In the elliptic Curve cryptography, it is said that the order of base point should be a prime number, and order of a point $P$ is defined as $k$, where $kP = \mathcal{O}$. And to compute the order we ...
12
votes
2answers
221 views

Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube

Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube. I found $x=0$, any others ?
2
votes
2answers
307 views

order of elliptic curve $y^2 = x^3 - x$ defined over $F_p$, where $p \equiv 3 \mod{4}$

It is said that the elliptic curve $y^2 = x^3 - x$ defined over a prime field $\mathbb{F}_p$, where $p \equiv 3 \mod{4}$ has an order $p + 1$. When I tried to get the elements of $E = \{(x,y) \in ...
0
votes
1answer
68 views

Find lift($E_{p^2}$) of an elliptic curve $E_p$ defined in field $F_p$ where $p$ is a prime

How to find $E_{p^2}$ of an elliptic curve $E_p$ defined over finite field $F_p$ where $p$ is a prime number?
1
vote
3answers
107 views

Combine two given Elliptic Curves

I want to combine two Elliptic curves such $E_p$ (defined in the field $F_p$) and $E_q$ (defined in the field $F_q$) i.e to find $E_n$ where $n=pq$. Is there any method to do it?
3
votes
1answer
104 views

Torsors under elliptic curves splitting over the same fields

I have a question somewhat related to my last question. Suppose $C$ and $C'$ are two genus $1$ curves (smooth, projective, geom conn.) over a perfect field $k$ with no $k$-rational points and that $C$ ...
7
votes
1answer
122 views

Proof in Kummer Theory - why is this subgroup finite?

I'm doing a course on elliptic curves. We're working with a field $K$ with $\mu_n \subset K$ ($K$ contains all $n$th roots of unity) and $\mbox{char}(K)\not|\;n$. I'm trying to understand the proof ...
2
votes
1answer
77 views

Testing to see if $\ell$ is of split or nonsplit multiplicative reduction

Suppose an elliptic curve $E/\mathbb{Q}$ has multiplicative reduction at $\ell$. Are there any other ways of seeing if $\ell$ is of split or nonsplit reduction aside from computing ...
10
votes
1answer
112 views

If $E/\mathbf Q$ is an elliptic curve and $n$ is odd, then the $n$-torsion $E(\mathbf Q)[n]$ is cyclic; elementary proof?

I know that this follows from the existence and non-degeneracy of the Weil pairing. A consequence of the existence of the Weil pairing is that, if the whole $n$-torsion is defined over $\mathbf Q$, ...
35
votes
3answers
704 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
470 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
114 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
207 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
42 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
214 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
131 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
94 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
54 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
113 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
124 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
60 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
158 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
219 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
165 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
641 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
82 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
164 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
292 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
255 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
120 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
70 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
415 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
65 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 ...