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

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Pole Order of Weierstrass Coordinates

I'm trying to understand a proof in Silverman's The Arithmetic of Elliptic Curves. Background: For an elliptic curve $E$, $x, y \in K(E)$ such that $\phi = [x, y, 1]: E \rightarrow C$ is a basepoint ...
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2answers
62 views

Elliptic curve $y^2= x^3 + x$ over the finite field $\mathbb{F}_p$ with $p \geq 3$.

Consider the elliptic curve $$E: y^2= x^3 + x$$ over the finite field $\mathbb{F}_p$ with $p \geq 3$. I want to show that $|E(\mathbb{F}_p)| \equiv 0 \mod 4$. I know that, if $p \equiv 3\mod 4$, ...
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1answer
70 views

The elliptic curve $y^2 = x^3 + 2015x - 2015$ over $\mathbb{Q}$

Consider the elliptic curve \begin{equation*} E: y^2 = x^3 + 2015x - 2015~\text{over}~\mathbb{Q}. \end{equation*} I want to prove that $|E(\mathbb{F}_7)| = 12$, that $|E(\mathbb{F}_{19})| = 19$ and ...
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1answer
25 views

How to calculate an elliptic curve

I need to find an elliptic curve in $F_{19}$ that has $|E(F_{19})|=18$. I am really stuck here. Can anyone help?
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2answers
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Real points of order 3 on an elliptic curve.

This comes from Silverman's Rational Points on Elliptic Curves: Consider the elliptic curve (non singular) $y^2=x^3+ax^2+bx+c=f(x)$ after some computations we can see that points of order 3 in this ...
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2answers
44 views

Elliptic curve with prescribed lattice

It's well known that there is a connection between elliptic curves and lattices. To establish such a connection one needs to use Eisenstein series. How one can one write down the explicit equation of ...
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1answer
32 views

Finding a rational point on $\mathscr{E} : y^2=x(x^2-25)$ to show $ \text{rank}(\mathscr{E})=1$

I'm trying to show that the rank of the following elliptic curve $$ \mathscr{E}: y^2=x(x^2-25)$$ is 1. Since it has a rational 2-torsion point at $(0,0)$, by considering the dual curve I've been ...
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0answers
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how to find solution(all points ) of $y^2+xy=x^3+x^2+1$? [closed]

actually ,this is elliptic curve(koblitz curve) and I want to know all the points on this curve. so would you please tell me how to find points on this curve defined over finite field(F 2^4)??
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2answers
67 views

Integral solutions to $56u^2 + 12 u + 1 = w^3.$

I would like to find all integer solutions to $$56u^2 + 12 u + 1 = w^3.$$ My computer thinks the only integral point is $(0,1).$ This problem arises from Integer solutions of $x^3 = 7y^3 + 6 y^2+2 ...
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3answers
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Semigroup law on points on the curve $f(x) = \frac{1}{x}$

Consider the positive half of the curve $f: \Bbb{R} \to \Bbb{R}, f(x) = \frac{1}{x}$. Let $A = (a,1/a), B = (b, 1/b)$ be any two points on the curve. Draw a line through them Find where this point ...
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1answer
14 views

Converting to homogenuous coordinates

Let's assume elliptic curve $E$ over $\mathbb{R}$: $y^2 = x^3 + x + 1$ How to convert this equation to homogeneous coordinates? My notes say it's $zy^2=x^3+xz^2+z^3$. Unfortunately, I have no idea ...
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2answers
160 views

Integer solutions of $x^3 = 7y^3 + 6 y^2+2 y$?

Does the equation $$x^3 = 7y^3 + 6 y^2+2 y\tag{1}$$ have any positive integer solutions? This is equivalent to a conjecture about OEIS sequence A245624. Maple tells me this is a curve of genus $1$, ...
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1answer
17 views

Proof of Theorem III, 6.2 in Silvermans Arithmetic of Elliptic curves

In the proof of part c), I cannot make sense of the sentence "Then another way of saying that $\phi:E_1\to E_2$ is an isogeny is to note that $\phi(x_1,y_1)\in E_2(K(x_1,y_1))$." (In this situation, ...
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2answers
47 views

Types of elliptic curves

I'm trying to research elliptic curves, and I always get the generic equation $$y^2 = a_0 x^3 + a_1 x^2 + a_2 x + a_3.$$ However, I'm looking for information on an equation like $$y^3 = a_0 x^3 + a_1 ...
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1answer
35 views

Interpretation of a short exact sequence from elliptic curves in terms of torsors

Consider some elliptic curve $E$ over a number field $k$. Then for any prime $p$ there is a short exact sequence $$ 0 \to E(k)/pE(k) \to H^1(k,E[p]) \to H^1(k,E)[p] \to 0. $$ Now, $H^1$ has an ...
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0answers
53 views

Some questions about reduction of elliptic curves

Let $E \rightarrow S$ be an elliptic curve (i.e, a smooth proper curve of genus 1). If $S = \text{Spec (K)}$ where $K$ is a local field, the usual way of doing a reduction at a prime $\mathfrak{p} = ...
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1answer
43 views

elliptic curve isogeny class 14.a $L$-function Dirichlet coefficients

Are the Dirichlet coefficients $a(n)$ of the $L$-function associated with isogeny class 14.a the irrationals that the inverse symbolic calculator suggests they are? The Lcalcfile suggests that they ...
2
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1answer
77 views

Riemann surfaces with Riemann Roch theorem, linear fiber over an elliptic curve

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha z)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
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1answer
36 views

Question about characteristic polynomial of the Frobenius endomorphism on elliptic curves.

I have another possibly trivial question about elliptic curves. A lot of papers I've seen state that the characteristic polynomial of the Frobenius endomorphism of an elliptic curve over a finite ...
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1answer
43 views

A question about endomorphism rings of elliptic curves

This is probably a very trivial question, but I haven't been able to find a rigorous explanation anywhere so far or at least haven't understood it. Assume we have an elliptic curve $E$ over ...
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1answer
63 views

Elliptic curves and Weierstrass $\wp$ function - an example

Let $E: y^2 = 4x^3 - b$ an affine equation of an elliptic curve in $\mathbb{P}^2_{\mathbb{C}}$. Let $b$ be chosen such that the map $f: \mathbb{C} \rightarrow \mathbb{P}^2$ given by $z \mapsto ...
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1answer
33 views

Benefit from local coordinates

I am reading Elliptic Curves by Anthony Knapp. Its the first time that I am dealing with local coordinates. In page 21 he introduces them as follows: Let $[x_0,y_0,w_0]\in \mathbb P_2(k)$ where $k$ ...
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1answer
28 views

Question on why short Weierstrass can't be used for curves with char=2

An elliptic curve given by $E: y^2=x^3+ax+b$ with $a,b \in K$ and $Δ(E)=-16(4a^3+27b^2) \neq 0$ is adequate for elliptic curves with $char\neq2,3$ Because of the factor -16 in the definition of ...
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1answer
38 views

Topics in elliptic curves over finite fields

First of all, sorry if I didn't put this question in the correct category. This a paper aimed for undergraduate math majors. So I am writing a general paper explaining about elliptic curves over ...
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0answers
76 views

Mordell Equation $y^2 = x^3 - 20$. [closed]

Prove that the only integral solutions to $y^2 = x^3 − 20$ are $(x, y) = (6, \pm14)$.
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2answers
23 views

Weierstrass equation long vs. normal form

So I am studying elliptic curves over finite fields and I am a little confused about something. In some texts I see a "long" Weierstrass equation and in some I see a "short" Weierstrass equation, what ...
2
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1answer
24 views

A clarification of addition on elliptic curves over the complex numbers

I am trying to prove that the order of the two points $P_{\pm}=(0,\pm\sqrt{-g_3})$ is three on the elliptic curve $y^2=4x^3-g_3$, for $g_3 \not= 0$, defined over $\mathbb{P}^2_{\mathbb{C}}$. Here's an ...
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1answer
187 views

Mordell-Weil rank in elliptic surfaces

Suppose that an elliptic smooth K3 surface $X$ defined over a number field $k$ has arithmetic Picard rank $r$ and assume that it is equipped with a $k$ fibration over $\mathbb{P}^1$ that has a section ...
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0answers
38 views

$x^3+y^3+z^3 = 0$ is isomorphic to $\mathbb{C}/\Lambda$, where $\Lambda = \{n+m\omega \mid n,m \in \mathbb{Z}, \omega^3=1, \omega \not= 1\}$

I am rather stuck trying to prove that $x^3+y^3+z^3 = 0$ in $\mathbb{P}^2_{\mathbb{C}}$ is isomorphic to $\mathbb{C}/\Lambda$, where $\Lambda = \{n+m\omega \mid n,m \in \mathbb{Z}, \omega^3=1, \omega ...
2
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1answer
23 views

Elliptic curve notation

This might be kind of a silly question about notation. I know: $E$: an elliptic curve $\mathbb{F_q}$: finite field But I recently ran across the notation $E/\mathbb{F_q}$ for the first time, so ...
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1answer
34 views

Hasse's Theorem for Elliptic Curves over Finite Fields + proof clarification

I need a little help understanding Hasse's theorem for elliptic curves over finite fields, as well as the proof of this theorem. (Sorry about my editing) Hasse’s Theorem: Let $E$ be an elliptic ...
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0answers
48 views

Genus of the product of two elliptic curves

In trying to understand the trichotomy of the genus of algebraic curves, I first consider the following two elliptic curves (over $\mathbb{Q}$), well-known to be of rank $2$, $ y^2 = x^3+17$ and $ ...
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2answers
151 views

Diophantine equation $x^2 + xy + y^2 = \left({{x+y}\over{3}} + 1\right)^3$.

Solve in integers the equation$$x^2 + xy + y^2 = \left({{x+y}\over3} + 1\right)^3.$$
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1answer
12 views

Sato-tate conjecture for elliptic curves over finite fields

I am doing a research project about elliptic curves over finite fields and I am across the Sato-tate conjecture, but I am having some difficulty understanding it. What I (think) I took away from the ...
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1answer
37 views

$j$-invariants of isogenous elliptic curves

Suppose that $E,E'$ are isogenous smooth complex elliptic curves - is there some relation between their $j$-invariants?
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2answers
36 views

Chinese Remainder theorem on Elliptic Curve group

I read somewhere (Blake, Seroussi, Smart: Elliptic Curves in Cryptography, p.160) that one can use the Chinese Remainder theorem to split $E(\mathbb{Z}/N\mathbb{Z})$, where $N$ is a composite number. ...
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1answer
16 views

Point at Infinity of E.C. in Jacobian Coordinates

I am reading some notes about elliptic curves right now and the author mentions the alternative Jacobian projective coordinates, where one establishes the equivalence $(x,y,z)\sim (\lambda^2 x, ...
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2answers
66 views

A special modular function: $ j $-invariant.

It is known that j invariant $$j(\tau)= 1728 \frac{g_2^3(\tau)}{\Delta(\tau)} $$ $\tau \in \mathbb{H}$ attains every complex value , Can someone guide me its proof.?? where $L(\tau ) = \{\tau m ...
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1answer
33 views

Fourier Series of Eisenstein series [closed]

$$G_{2k}(\tau)= 2\zeta(2k)+2\frac{(2\pi i)^{2k}}{(2k-1)!}\sum_{n\geq 1}\frac{n^{2k-1}q^n}{1-q^n}$$ where $q =e^{2\pi i \tau}$ and $G_k=\sum_{\omega \in L , \omega \neq 0}\frac{1}{\omega^k} $ $L(\tau ...
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1answer
17 views

Subgroups of points of order 2 in an elliptic curve

Depending on the roots of $y^2 - x(x^2+ax+b) = 0$ being real or not, we can have 2 subgroups of points of order 2 for a given elliptic curve- Kelin-4 group or a cyclic group of order 2. How does one ...
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1answer
20 views

Is this simple proof that the Frobenius endomorphism of an elliptic curve defined over $\mathbb F_q$ is surjective valid?

I am quite sure that the following "proof" is flawed, but I don't see why: Let $E$ be an elliptic curve defined over $\mathbb F_q$. Since $E$'s ideal is generated by a polynomial in $\mathbb ...
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1answer
31 views

Jacobians and ranks of a curve

I would like to know the following: How to find Jacobian and rank of an hyper elliptic curve like $x^5-x= y^2-y$? High regards Rosy
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2answers
53 views

Arithmatic on modular curves

I had tried to read the first few pages of Glenn Stevens's$\,$ Arithmetic on Modular Curves, but it is somehow extremely unreadable to me, the text format is odd and stating too much facts without ...
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0answers
29 views

Galois action on the fibre of a morphism determined by a linear system

If $X$ is an elliptic curve, let $P,Q\in X$, then $|P+Q|$ determines a morphism $g:X\to \mathbb{P}^1$. It is easy to see $K(X)/K(\mathbb{P}^1)$ is a Galois extension of degree 2. Let $\sigma$ be the ...
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1answer
66 views

Does there have to be a point on elliptic curve over $\mathbb{C}(t)$

Let $E$ be an elliptic curve over $\mathbb{C} (t)$ (rational functions). I require $E$ to be defined by the following equation. $$ y^2 = x^3 + A x + B$$ Where $A, B \in \mathbb{C} (t)$. Question: ...
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2answers
64 views

Diophantine equation resembling FLT

I was wondering if the equation $x^p+y^p=2z^p$ has been studied. For small cases it is seen that the only solutions are trivial: $x=y=z$. There are probably methods to solve this for regular ...
4
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1answer
108 views

Sums of three cubes in arithmetic progression equal to a cube $x^3+(x+y)^3+(x+2y)^3 = z^3$

Using exhaustive search, small positive and primitive integer solutions to, $$x^3+(x+y)^3+(x+2y)^3 = 3 x^3 + 9 x^2 y + 15 x y^2 + 9 y^3= z^3\tag1$$ are, $$x,y = 3,1,\quad x+y =2^2$$ $$x,y = ...
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35 views

Calculate line integral $\frac{-y}{x^2+2y^2}dx +\frac{x}{(x^2+2y^2)}dy$

I have this question in my calculus course: Calculate the line integral $\int \frac{-y}{x^2+2y^2}dx +\frac{x}{(x^2+2y^2)}dy$ over the curve a) $x^2+y^2=1$ in the positive direction b) ...
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1answer
23 views

Pole of elliptic function

Let $f:C→P1$ be such that $f(z+1)=f(z+i)=f(z)$ for all z∈C. Let $Γ=\{m+ni:m,n∈Z\}$. Show that if $f$ is holomorphic on $C∖Γ$, and $z⋅f(z)$ is bounded in a neighbourhood of $z=0$, then $f$ is ...
2
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1answer
47 views

What are the zeros of the j-function?

Recall that, for a complex number $\tau$ with positive imaginary part, the $j$-invariant is given by $j(\tau)=1728 \frac{g_2(\tau)^3}{g_2(\tau)^3-27g_3(\tau)^2}$ where $g_2(\tau)=60 ...