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

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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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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 ...
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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 ...
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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 ...
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2answers
37 views

Prerequisites for Silverman's Arithmetic of Elliptic Curves

I would like to take a course on elliptic curves using Silverman's Arithmetic of Elliptic Curves next year. I would be taking complex analysis concurrently, but it was listed as a formal prerequisite, ...
2
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1answer
171 views

Flex point on an elliptic curve

I have just started working through Pete Clark's elliptic curve notes, which are available here: http://math.uga.edu/~pete/EllipticCurves.pdf Early on, in section 2.1 on page 6, it is shown that the ...
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139 views

Why do modular curves parametrise elliptic curves?

Let $Y_1(N)=\Gamma_1(N)/H$, where $H$ is the upper half plane. In these lecture notes http://math.uga.edu/~pete/modularandshimura.pdf , the author makes the following statement: "$Y_1(N)$ ...
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1answer
31 views

Conditions of $f=a+bx+cz+dx^2+exz+fz^2+…$ such that its tangent line is $z=0$ and inflection point is at the origin.

Let $x,z$ be coordinates on $k^2$ and $f\in k[x,z]$; write $f$ as $$f=a+bx+cz+dx^2+exz+fz^2+...$$ Write down the conditions in terms of $a,b,c,...$ such that (a) $P=(0,0)\in C: (f=0)$; (b) the ...
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Show that $(Y^2-X^3)|f$ if $f$ vanishes on the curve $C: (t^2,t^3)$, and determine what property of a field $k$ will ensure that the result holds.

Let $\phi: \mathbb{R^1}\rightarrow \mathbb{R^2}$ be the map given by $t \mapsto (t^2,t^3)$; prove directly that any polynomial $f\in \mathbb{R[X,Y]}$ vanishing on the image $C=\phi(\mathbb{R^1})$ is ...
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Elliptic Curve Group and Multiplicative Inverse of an element.

Suppose $E$ be an Elliptic Curve over a field $F_q$ and $q=p^n$ where $p=$ prime. We know that the Elliptic Curve group $E(F_q)$ under addition is an Abelian/Commutative Group of order, ...
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1answer
54 views

Formal Group for the Elliptic Curve $Y^2=X^3+AX$

I'm trying to solve the following problem without resorting to a direct calculation: Let $E : Y^2 = X^3 + AX$, where $A \in \mathbb{Z}$ and $A \ne 0$. Let $F(X, Y )$ be the formal group associated to ...
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1answer
28 views

Quick question: Tensoring a 2-torsion line bundle with a rank 2 nonsplit extension over a curve of genus 1

Everything is complex algebraic. Over a curve $C$ of genus $1$, let $V$ be a rank 2 vector bundle with $\deg \det(V)=1$, which is a nonsplit extension of $\mathcal{O}_C$ and $\mathcal{O}_C(p)$, where ...
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36 views

Topics in elliptic curves over finite fields

I have to write a paper on elliptic curves over finite fields and I was wondering if anyone had any ideas of some interesting directions to take this? Like what are some subtopics within this general ...
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2k views

Group Law for an Elliptic curve

I was reading this book "Rational points on Elliptic curves" by J.Silverman, and J.Tate, 2 prominent figures in Number theory and was very intrigued after reading the first couple of pages. The ...
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23 views

The Discriminant Condition for Elliptic Curves [duplicate]

Question: Why do we need the discriminant of an elliptic curve $\Delta=-16(4a^3+27b^2)$ to be nonzero? Motivation: I am aware that when $\Delta=0$, then we obtain either a cusp (e.g. for $y^2=x^3$) ...
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128 views

Modular parametrization of elliptic curve

Let $f$ be a cusp form of weight $2$ on $\Gamma_0(N)$ and assume that $f$ is a Hecke form and a newform. Then, we easily see that $$C(\gamma)=2i\pi \int_{\tau}^{\gamma \tau}{f(\tau')d\tau'} \quad ...
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59 views

Why is $E[l]\cong\mathbb Z/l\mathbb Z\times\mathbb Z/l\mathbb Z$ for an elliptic curve $E$?

René Schoof's 1995 paper contains the following statement about an elliptic curve $E$ (at the bottom of page 233): [...], we use the subgroup $E[l]$ of $l$-torsion points of $E(\overline{\mathbb ...
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34 views

Elliptic curve reduction modulo $p$

While reading an introduction on elliptic curves, I stumbled upon something called reduction modulo $p$. The definition states that we want to create a group homomorphism that maps an elliptic curve ...
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36 views

Implementing odd and even functions into PARI/GP

My elliptic curve is $y^2=x^3-3267x+45630$ and I have the following code to generate $DD$ mod 1789 where DD is the sqrt(denominator) of the x-coordinate. ...
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50 views

Large initial solutions to $x^3+y^3 = Nz^3$?

Let $x,y,z$ be non-zero integers. Is it true that the initial or smallest solution (in terms of absolute value) to, $$x^3+y^3 = Nz^3\tag1$$ for $N=94$ is, $$15642626656646177^3 + ...
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637 views

Can you recommend some books on elliptic function?

I plan to study elliptic function. Can you recommend some books? What is the relationship between elliptic function and elliptic curve?Many thanks in advance!
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What is known about the numbers $M_p = \left\vert C(\mathbb{F}_p )\right\vert$?

There is a question (2.4.c) marked ** (to denote "extremely difficult/currently open problem") in Silverman and Tate's Rational Points on Elliptic Curves which I found really interesting and wondered ...
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1answer
68 views

Galois representations and isogenies of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$. For each prime $\ell$, the action of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on $E[\ell]$ (the group of $\ell$-division points of $E$) defines a ...
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1answer
69 views

How many duplication formulas exist for the Mordell curve family $Y^2-X^3=c$?

For the Mordell equation $$ Y^2-X^3 = c, $$ Bachet gave a famous duplication formula which translates one rational solution $(x_1,y_1)$ into a second rational solution $(x_2,y_2)$. Réalis gave a ...
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110 views

What is stopping every Mordell equation from having a [truly] elementary proof?

The Mordell equation is the Diophantine equation $$Y^2 = X^3-k \tag{1}$$ where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, ...
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2answers
85 views

Points on elliptic curve over finite field

Find the points on the elliptic curve $y^2 = x^3 + 2x + 2$ in $\mathbb F_{17}$. Do I have to guess a first point and then use an algorithm to spit out all other points?
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The canonical height of a point on an elliptic curve

I am struggling with exercise 3.3 in Silverman-Tate Rational Points on Elliptic Curves. Here is the paraphrased problem with necessary background: Let $C:y^2 = x^3 + a x + b$ be a nonsingular cubic ...
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2answers
188 views

Calculating the divisors of the coordinate functions on an elliptic curve

I am currently reading Silverman's arithmetic of elliptic curves. In chapter II, reviewing divisor, there is an explicit calculation: Given $y^2 = (x-e_1)(x-e_2)(x-e_3)$ let $P_i = (e_i,0),$ and $ ...
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756 views

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
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37 views

Elliptic curve cryptography order

How do I compute an order a a point P on an elliptic curve? My question is specifically in reference to the attached photo. I understand how to do part a but I am totally lost in part b. I don't know ...
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1answer
43 views

Relation between elliptic curves and Dirichlet L-series

I have read that an elliptic curve $E$ is modular if $a(n) = c(n)$ for all $n$, where $a(n)$ is the $n$-th coefficient in the Dirichlet series of $E$, $L(E,s)$, and $c(n)$ is the $n$-th coefficient in ...
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1answer
21 views

How to show rational points of finite order on an elliptic cure are closed under addition

I would like to show that rational points of finite order on an elliptic curve are closed under addition. If $P_1$ and $P_2$ are rational (actually integral) points of finite order, say $nP_1= O$ and ...
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3answers
2k views

Integral points on an elliptic curve

Let's start with an elliptic curve in the form $$E : y^2 = x^3 + Ax + B, \qquad A, B \in \mathbb{Z}.$$ I am wondering about integral points. I know that Siegel proved that $E$ has only finitely many ...
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1answer
59 views

Distribution of points in ellliptic curves over finite fileds

Let $E$ be an elliptic curve defined over a finite field ${\bf F}_p,$ where $p$ is prime. From Hasse theorem we get $p+1-2\sqrt{p} \leq |E({\bf F}_p)|\leq p+1+2\sqrt{p}.$ Now say that we choose in ...
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2answers
52 views

Tate-Shafarevich groups and Hasse principle (reference)

I'm looking for a proof of the fact that the Hasse local-global principle holds for an elliptic curve $E$ defined over $Q$ if and only if the Tate-Shafarevich group of $E$ vanishes. I just need to ...
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1answer
65 views

Points on an elliptic curve over $\mathbb F_p$

Let $E$ be an elliptic curve over $\mathbb F_p$ (the finite field of $p$ elements) defined by $y^2=x(x-n)(x-m)$ where $p\nmid nm(n-m)$. Let $N$ be the number of $\mathbb F_p$-valued points of $E$. ...
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1answer
59 views

Why does an elliptic curve have genus one?

I read that one definition of an elliptic curve goes as follows: Let $k$ be a field. We define the elliptic curve over $k$ be a smooth projective curve $E$ over $k$, isomorphic to a closed subvariety ...
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78 views

In cryptography, why do we reduce elliptic curves over finite fields?

What's wrong with real numbers? Is the continuous logarithm problem "easy" to solve for elliptic curves? Here's what I believe: elliptic curves over the real numbers have infinitely many points, many ...
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Find a point $P$ on an elliptic curve, given $2P$

Let $E$ be the Elliptic curve given by $Y^2=x^3+5x-6$ and suppose $P$ is a point on $E$ over $\mathbb F_{65537}$ with $2P=(7283,24272)$. Find $P$. I approached this question as follows. ...
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32 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, ...
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77 views

How to test if a given elliptic curve has complex multiplication

Is there a general, reasonably easy to understand, algorithm for testing whether an elliptic curve has CM? For example, consider the curve $y^2=x^3+\frac{27}{1727}x+\frac{54}{1727}$ This has ...
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1answer
18 views

Multiplication by n on E(K) is surjective

What's the easiest way to see this? I can imagine a proof for $n=2^k$ since for some $P \in E(K)$ you can just move a line intersecting P round the curve till it's tangent, then that point, say $Q ...
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45 views

Proof of a Proposition regarding the reduction of N-torsion groups on elliptic curves

In Diamond-Shurman A first course in Modular forms p.334 Prop. 8.4.4. It is stated, For E elliptic curve over $\bar{\mathbb{Q}}$ with good reduction at the prime ideal $\mathfrak{p}$ the reduction ...
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1answer
37 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 ...
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Unable to find any information regarding this fact (Frey, elliptic curves)

Frey states in 'Links between stable elliptic curves and certain Diophantine equations' the following "The most important fact about elliptic curves with reduction of muItipIicative type is due to ...
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1answer
50 views

Elliptic curves $\mathbb C/\Gamma , \mathbb C/\Gamma'$ are isomorphic iff $\Gamma=\lambda\Gamma'.$

Let, $\Gamma, \Gamma'$ be $lattices$ of $\mathbb C$, define $elliptic$ $curves$ by $\mathbb C/\Gamma , \mathbb C/\Gamma'$, then $\mathbb C/\Gamma , \mathbb C/\Gamma'$ are isomorphic ...
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66 views

The mod $p$ Galois representation of the Frey curve is unramified away from $2, p$

Given a hypothetical solution to Fermat's last theorem for $p \ge 5$ $$a^p + b^p + c^p = 0$$with $a \equiv -1 \pmod 4$, $b$ even, we can write down the Frey Curve$$E: y^2 = x(x-a^p)(x+b^p)$$which has ...
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38 views

Singularity of $V(Y^2-X^3-X)\to\mathbb{P}^1$

In "The Arithmetic of Elliptic Curves, in example I.3.7, Silverman define $\Phi:V(Y^2Z-X^3-X^2Z)\to\mathbb{P}^1$ with $\Phi(X,Y,Z)=[Y,X]$. He says that $\Phi$ is not regular at $[0,0,1]$. How to prove ...
6
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1answer
193 views

If six points of an elliptic curve are contained in a conic, then their sum is $O$.

Let $C$ be a projective cubic without singular points and $O\in C$ an inflexion point. We consider the addition in $C$ with $O$ as neutral element. If $R_{1},...,R_{6}\in C$ are different points such ...