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

learn more… | top users | synonyms

1
vote
1answer
26 views

Confusion about holomorphic differential on elliptic curve

Let $(a:b)\in\mathbb{C}P^1$ and look at the elliptic curve $C$ given by $y^2=x^3+a^4x+b^6$. It is well known that on this elliptic curve we have the holomorphic differential $dx/y$. I have two ...
1
vote
1answer
59 views

What does extra zero of an $L$-function mean?

This is a very vague question. What does an extra zero of an $L$-function mean? There are lots of papers written on this topic, investigating the extra/exceptional zeros of various $p$-adic $L$-...
1
vote
2answers
41 views

Is $H^{1}(k,E[n])$ a subgroup of $H^{1}(k,E)$?

Let $E/k$ be an elliptic curve. Consider $E[n]$ which is a subgroup of $E$. Is it true that $H^{1}(k,E[n])$ is again a subgroup of $H^{1}(k,E)$ in Galois cohomology? I thought that this was true but I'...
1
vote
1answer
27 views

How to prove that $f(x)$ has a multiple root in $Q$ if and only if $disc(f) =0$

Let $f(x) = x^3 +ax+b$ contained in $Q[x]$ prove that $f(x)$ has a multiple root in $Q$ if and only if $disc(f) =0$ This is what I've so far since $f(x) = (x-A_1)(x-A_2)(x-A_3), A_1,A_2, A_3$ are ...
1
vote
1answer
36 views

Finding formula for rational point ($x$,$y$) on the y = $x^2$

How to find formula for rational point ($x$,$y$) on the $y$ = $x^2$ in term of rational parameter $t$. And also how would I write in form of ($x$,$y$) =($f$($t$),$g$($t$))
1
vote
2answers
37 views

Problem with Elliptic Curve in Montgomery form

I am trying to understand how points are added in Elliptic Curves in Montgomery form. I am working with the curve $$3y^2 = x^3 + 5x^2 + x \mod 65537$$ Adding the point $(3,5)$ with itself gives (or ...
1
vote
2answers
74 views

Finding some rational points on elliptic curves

If I am considering an elliptic curve, for example $$y^2=x^3-2$$ $$\text{Edit: and } y^2=x^3+2$$ over $\mathbb Q$, how to find rational points? What possibilities do we have to calculate some ...
1
vote
1answer
65 views

hyperelliptic curve

Please help me to solve this question: Let $H$ be a hyperelliptic curve over $\mathbb{F}_{103}$ given by the equation $ y^2 = x^5+1$. let $J$ be the jacobian of $H$ defined over $\mathbb{F}_{103}$. ...
1
vote
1answer
51 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 ...
1
vote
1answer
95 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 (\wp(z):...
1
vote
1answer
40 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 $k$ ...
1
vote
2answers
73 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 ...
1
vote
1answer
34 views

Why ins't $\mathfrak{h}$ enough to parametrize complex elliptic curves?

this a pretty idiot question and of course there is a mistake in my way of thinking. Let $E$ be a elliptic curve, $E (\mathbb{C}) \cong \mathbb{C} / \Lambda$, where $\Lambda = \langle \omega_1, \...
1
vote
1answer
46 views

(hyper) elliptic curve in characteristic two and the Jacobian criterion

Let $k$ be a field of characteristic two and let $E$ be a curve given by $$ y^2=x*(x+1)*(x^2+x+1)*(x^3+x+1)\quad\text{or}\quad y^2=f(x) $$ Now we have $dy^2/dy=2y=0$ and consider the Jacobian ...
1
vote
1answer
105 views

Mazur's theorem-abelian torsion group of rational points of an elliptic curve

I am looking at Mazur's theorem... $$E(\mathbb{Q})_{\text{torsion}} \cong \mathbb{Z}/n\mathbb{Z}, \text{ for } n=1,2, \dots ,10,12$$ means that the torsion group $E(\mathbb{Q})_{\text{torsion}}$ ...
1
vote
2answers
41 views

Equation of a non-singular cubic curve

The equation of a non-singular cubic curve in affine coordinates is $$y^2+a_1 xy+a_3 y=x^3+a_2x^2+a_4x+a_6 .$$ If $\text{ch } K \neq 2, 3$ then it is written $$y^2=x^3+ax+b .$$ Why do we write it ...
1
vote
1answer
85 views

Families of Elliptic Curves

I am looking to test some properties of elliptic curves and I would like to have a variety of different families to test. I was wondering if there was, say, a catalogue of the different interesting ...
1
vote
1answer
99 views

Parametrization of line bundles over an elliptic curve by points of that curve

Let $E$ be an elliptic curve over an algebraically closed field of characteristic zero, and let $\mathcal{L}$ be a line bundle on $E$ of degree $3$. Suppose, I can present this line bundle as $$ \...
1
vote
1answer
34 views

Elliptic curves: Can I replace a coordinate with any modularly equivalent number?

I have a point (x, y) in an elliptic curve group. Suppose y is negative. Can I rewrite it as a positive number if that positive number is equivalent to y (modulo the characteristic of the group)? ...
1
vote
1answer
40 views

How I can calculate the algebraic rank of $C(ℚ)$

Let us consider the elliptic curve $C$ over $ℚ$ in Weierstrass form $$C:y²=x³+1$$ How I can calculate the algebraic rank of $C(ℚ)$.
1
vote
1answer
32 views

Approximating the Rank

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the $L$-...
1
vote
1answer
40 views

Surjective $p$-adic representation implies trivial $p$-primary part.

Let $E/\mathbb{Q}$ be an elliptic curve. We know that by Serre in the non-CM case, for $p\geq5$, $$\rho_p:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow Aut(T_p(E))$$ is surjective iff $$ \bar{\rho}_p:...
1
vote
2answers
112 views

Sage usage to calculate a cardinality

I would like to compute the cardinality of an elliptic curve group over the finite field $\mathbb{F}_{991}$. I'm trying to use sage but I still have an error in the syntax (I never used it before and ...
1
vote
1answer
63 views

Show that there exists a constant $c$ such that for all $n \in \mathbb{Z}_+$ one has $\#\{\omega \in L\,\vert\,n \leq |\omega| \leq n + 1\} \leq cn$.

Let $L \subset \mathbb{C}$ be a lattice (i.e. $L = \{n\omega_1 + m\omega_2 \,\vert\, \omega_1, \omega_2 \in \mathbb{C},\, \omega_1 / \omega_2 \not\in \mathbb{R}, \, n,m \in \mathbb{Z}\}$). Show that ...
1
vote
1answer
52 views

Why does an isogeny not ramify?

The following argument is, I believe, based on the premise that an isogeny (or a morphism of curves that is a group homomorphism) doesn't ramify: Considering the multiplication by $n$ map $[n]$ on a ...
1
vote
1answer
130 views

Computing the analytic $p$-adic $L$-function via modular symbols in MAGMA

I need to compute the analytic $p$-adic $L$-function of an elliptic curve at a prime $p$ via modular symbols using MAGMA. In SAGE...
1
vote
1answer
124 views

Odd torsion of elliptic curves are isomorphic

$C: Y^2=X(X^2+aX+b)$ $D: Y^2=X(X^2+a_1X+b_1)$ where $a,b,\in\mathbb Z a_1=-2a,b_1=a^2-4b,b(a^2-4b)\neq0$ Let $C_{oddtors}(\mathbb Q)$ denote the set of torsion elements of $C(\mathbb Q)$ which ...
1
vote
1answer
54 views

Why are we not allowed to rescale the variables of the equation of an elliptic curve independently one from the other?

It seems that in whatever proof of the theorem that an elliptic curve can be put in Weierstrass form that you look at, the next step after getting an equation: $$\alpha Y^2Z + a_1XY Z + a_3Y Z^2= \...
1
vote
1answer
94 views

Semistable Elliptic Curves

For a general representation $\rho: G_{\mathbb{Q}} \rightarrow \operatorname{GL}(V)$, where $V$ is a two dimensional $\overline{\mathbb{F}}_p$ vector space, the level $N(\rho)$ in Serre's conjecture ...
1
vote
1answer
90 views

Structure of $C(F_5)$ from Rational Points on Elliptic Curves

In the book Rational Points on Elliptic Curves by Silverman/Tate one examines the elliptic curve $y^2 = x^3 + x + 1$ over $F_5$. One can then easily determine the group $$ C(F_5) = \lbrace \mathcal{O}...
1
vote
1answer
119 views

Counting the number of elliptic curves with certain discriminant/conductor

I'm looking for some references regarding the above topic. To be more specific, references that address questions such as Given $D > 0$, how many elliptic curves over $\mathbb{Q}$ are there with ...
1
vote
1answer
74 views

Point multiplication in elliptic curve

Suppose $a$ is an integer and $Q$ is a point on an elliptic curve and $(x,y)$ are $x$ and $y$ coordinates of this point. My question is: Whether $a\cdot Q$ is equal to $(ax, ay)$?
1
vote
1answer
257 views

State of the art in arithmetic moduli of elliptic curves?

In trying to get into the topic of moduli spaces of elliptic curves, the following question arises: What is the state of the art in the topic right now? Deligne and Rapoport describes how the ...
1
vote
2answers
476 views

Subtraction two points on elliptic curve.

Suppose Q, T and S are three points on an elliptic curve, such that Q+T = S. With knowing Q and S, can we compute T? In other word whether exists subtraction operation on elliptic curve, or not?
1
vote
1answer
88 views

About the quotient group of degree zero divisors on $C$ by the principal divisors on $C$

Let $C$ be an elliptic curve with distinguished point $O$. My question is about a mathematical desription of this set denoted by $Pic(C)$ which is the quotient group of degree zero divisors on $C$ by ...
1
vote
1answer
76 views

Ireland-Rosen Hecke Character for $y^2=x^3-Dx$

I would like to refer you to page $310$ of Ireland-Rosen: A Classical Intro to Modern Number Theory. Firstly, to construct the Hecke character, it is enough to specify $\chi(P)$ for prime ideals $P$ ...
1
vote
1answer
185 views

What is the amount of abstract algebra needed to study elliptic curves?

To be more specific, how much is needed to understand the book 'Rational points on elliptic curves' by Silverman?
1
vote
1answer
277 views

Understanding whether a ramified covering ramifies over infinity

Let $C$ be the (smooth) curve in $\mathbb{C}^2$ defined by $y^2 = x^4 - 1$, and let $\pi : C \to \mathbb{C}$, $\pi(x,y) = x$. $\pi$ is a ramified cover, ramified over $\pm 1, \pm i$. $C$ is a non-...
1
vote
1answer
209 views

Drawing elliptic curve

Consider an elliptic complex curve in $\mathbb{C}^2$ given by equation $w^2 = (z-a)(z-b)(z-c)$ where $a,b,c$ are complex mutually distinct constants. It is a $2$-dimensional surface in $4$-dimensional ...
1
vote
2answers
161 views

Regarding a notation related to divisors & elliptic curves

Section 5.8 of the book An Introduction to Mathematical Cryptography defines the divisor of a rational function $f(X,Y)$ defined on an elliptic curve $E: Y^2 = X^3 + AX + B$ as the formal sum: $\text{...
1
vote
1answer
306 views

Divisor of a function on a curve

Let $k$ be an algebraic closed field with character not equal to 2, $a,b,c\in k$ be distinct numbers, and consider the curve $C: y^2=(x-a)(x-b)(x-c)$. Let $P=(a,0),P_{\infty}$ for the point at ...
1
vote
1answer
207 views

Tamagawa number conjecture

I heard somewhere that the above formulation of conjecture is for predicting the exact leading term of a L-function at an integer. But i didnt find any reference about how it is stated, anyone please ...
1
vote
1answer
100 views

Determining elliptic curve's parameters from addition procedure

Given a procedure that adds two points on an unknown elliptic curve, is it possible to determine curve's parameters, treating this procedure as a black box? We are given two points on this curve $P$ ...
1
vote
2answers
47 views

Elliptic curves over $\mathbf{F}_q$ with $q = p^{2m}$

I am reading Washingtons book about elliptic curves and struggling with an exercise there (4.9), which is the following: Let $E$ be an elliptic curve over $\mathbf{F}_q$ with $q = p^{2m}$. Suppose ...
1
vote
1answer
32 views

Show that points on an elliptic curve have order 4

I am studying elliptic curves using this book and have a problem with task 4.11 which goes as follows: Let $F_q$ be a finite field of odd characteristic and let $ a,b \in F_q $ with $a \ne2b$ and $b \...
1
vote
1answer
35 views

Homomorphism between elliptic curves $C: y^2=x^3+ax^2+bx$ and $\bar{C}: y^2=x^3-2ax^2+(a^2-4b)x$.

I am reading Rational Points on Elliptic Curves by Silverman and Tate. In Section 3.4, Page 76, the authors defined two elliptic curves elliptic curves $C: y^2=x^3+ax^2+bx$ and $\bar{C}: y^2=x^3-2ax^...
1
vote
1answer
47 views

Find all points of finite order on the elliptic curve $y^2+7xy=x^3+16x$.

I am studying Rational Points on Elliptic Curves by Silverman and Tate. This is Problem 2.12 (h). Determine all of the points of finite order on the elliptic curve $y^2+7xy=x^3+16x$. Also ...
1
vote
1answer
25 views

The cardinal of the Mordell-Weil group is prime for certain elliptic curves over $\mathbb{F}_{q}$ for certain $q$.

Let $p\in\{2,3\}$ and $r\in\mathbb{Z}_{\geq 2}$. I would like to find if there exists an elliptic curve defined over $\mathbb{F}_{p}$ such that $|E(\mathbb{F}_{p^{r}})|$ is a prime number. If $p>3$ ...
1
vote
1answer
22 views

How to show elliptic curve endomorphism is commutative?

For an elliptic curve $E/k$, let $\alpha$ be any endomorphism over $\bar{k}$ in End($E$) and let $[n]$ be the multiplication-by-$n$ endomorphism. Show that $[n] \bullet \alpha=\alpha \bullet [n]$, ...
1
vote
1answer
54 views

Elliptic Curves over Finite Fields as Two Cyclic Groups

Let $E$ be an elliptic curve over $\mathbb{F}_q$. I want to show $E(\mathbb{F}_q) \cong (\mathbb{Z}/m_1\mathbb{Z}) \times (\mathbb{Z}/m_2\mathbb{Z})$ where $m_1,m_2 \in \mathbb{Z}$ are such that $...