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Questions tagged [elliptic-curves]

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

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Cremona 2.14.1 Why is $c_4$ and $c_6$ complex when they should be rational?

In Cremona's online book Chapter 2, in order to calculate the lattice invariant we have: $\tau=\omega_1/\omega_2$ Set $q=e^{2\pi i\tau}$ (2.14.1) $c_4 =(2\pi/\omega_2)^4(1+240 \sum_{n=1}^{\infty} n^...
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1answer
25 views

Stage 1 of Elliptic Curve Method (ECM)

Reading several texts of ECM (e.g. 20 years of ECM) the Stage 1 is described as: $Q \leftarrow P_0$ for each prime $\pi <= B_1$ $\quad$compute $k$ such that $\pi^k <= B_1 < \pi^{k+1}$ $\quad$...
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Can the coefficients of an elliptic curve be recovered from the curve itself?

Let $K$ denote an algebraically closed field. Suppose $E$ and $E'$ are elliptic curves given by Weierstrass equations with parameters $(a_1,a_3,a_2,a_4,a_6)$ and $(a_1',a_3',a_2',a_4',a_6').$ From $E =...
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1answer
31 views

Endomorphism of the elliptic curve $y^2 = x^3 - ax$

For the elliptic curve $$E:y^2 = x^3 - ax$$I know that $E\left(\mathbb{C}\right)\cong\mathbb{C}/\left(\mathbb{Z}i + \mathbb{Z}\right)$ and hence $\text{End}_{\mathbb{C}}\left(E\right)\cong\mathbb{Z}\...
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2answers
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Why is the genus of $y^2 = x^4 + 1$ not $3$ but $1$?

I saw this, but I don't understand why we can't use the genus-degree formula for this curve. I think this curve is $V(X^4 + Z^4 - Y^2Z^2)$ in $\mathbb{P}^2_k$, so by the genus degree formula, the ...
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54 views

When does an elliptic curve have accumulation points?

If the rank of an elliptic curve is greater than 1,then it has infinitely many rational points,I wonder how are these rational points distributed, especially, can we find an elliptic curve such that ...
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15 views

Identify elliptic curve from Frobenius traces

I have a quartic curve, suspect it is an elliptic curve but haven't seen how to put in normal form. Have computed numbers of (projective) points mod small primes, satisfy Hasse bound. Must be a way ...
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18 views

Field of discrete logarithms of a generator of a finite group

Suppose that $\mathbb{G}$ is finite group of order $p$ ($p$ can be a prime number if necessary) and that $g$ is a generator of $\mathbb{G}$. What is the field of discrete logarithms of $g$? I can see ...
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3answers
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Why is $y^2 = 1+x^4$ an elliptic curve?

I saw in a document that $y^2 = 1+x^4$ is (the affine equation of) an elliptic curve. Why is it the case? Typically, SAGE tells me it is isomorphic to $y^2 = x^3 - 4x$, which is an elliptic curve ...
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2answers
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What is the fastest way to estimate the Arc Length of an Ellipse?

To estimate the circumference of an ellipse there are some good approximations. $a$ is the semi-major radius and $b$ is the semi-minor radius. $$L \approx \pi(a+b) \frac{(64-3d^4)}{(64-16d^2 )},\quad ...
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Correspondence between a complex torus and an elliptic curve

In The Arithmetic of elliptic Curves, on page 156, Silverman considers the map $$\phi\colon\mathbb C/\Lambda\longrightarrow E\subset \mathbb P^2(\mathbb C)$$ $$z\longmapsto [\wp(z),\wp'(z),1]$$ where $...
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1answer
53 views

Modular Forms in Pari/GP

I'm trying out Pari's new modular forms package, and I've run into a small issue that I couldn't resolve. I want to use the modular parameterization of an elliptic curve $E$ given by the elltaniyama(...
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28 views

Elliptic curve has points over finite fields

Let $E$ be an elliptic curve with affine equation $$y^2 = x^3+ax+b$$ where $a,b \in \Bbb Z$. Let $P$ be the set of prime numbers not dividing $ab$. Can we prove directly that if $p \in P$ is at least ...
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24 views

Infinite subgroups of elliptic curves and quotients

Let $E_1$ be an elliptic curve over $\Bbb Q$ and $S \subset E_1(\Bbb Q)$ be a subgroup. Is there an elliptic curve $E_2$ with an algebraic group morphism $E_1 \to E_2$, and such that $E_2(\Bbb Q) \...
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1answer
58 views

BSD conjecture for rank 1 elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer (BSD) conjecture predicts that $$\text{ord}_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$ Thanks to the work of Gross-Zagier and ...
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1answer
73 views

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

Prob: Show that for any positive integer $a$, Diophantine equation $$x^3+x+a^2=y^2$$ has at least one solution $(x, y)$, where $x, y$ are positive integers. Source: My teacher. Attempt: First I ...
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Tate module of an Abelian Scheme

It is well known that if $A/k$ is an Abelian Variety over (the spectrum of) a field, a very important object to consider is its Tate module $T(A):=\underset{\underset{n}{\longleftarrow}}{\lim}A[p^n](\...
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How could you find the preimage of an isogeny function?

How do you know if an isogeny is surjective or not, and how do you tell how many points on E maps to E'? Does the answer lie in the degree of the isogeny function?
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How can the Neron-Tate height pairing be considered an intersection number?

Given the Neron-Tate height $\hat h$ on an elliptic curve we defined the associated bilinear form: $$\langle P,Q\rangle = \frac{1}{2} \bigl( \hat h(P+Q) - \hat h(P) - \hat h(Q) \bigr) $$ My professor ...
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1answer
49 views

$\wp(z)$ and points of order $3$

This problem comes from Koblitz's book Introduction to Elliptic Curves and Modular Forms page 41 problem 2 and it says Let $$f_N(z) = N \Pi(\wp(z)-\wp(u))$$ Where the product is taken over nonzero $...
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4answers
107 views

What is the group-like structure on $x^2+y^2+z^2-2xyz=1$?

(Background: this is inspired by Chebyshev polynomials and expanding a function as a Chebyshev series.) Solving for $ z $ gives $$ z=xy \pm \sqrt{(1-x^2)(1-y^2)}, $$ where $-1\leq x,y \leq 1$. Now ...
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1answer
42 views

What is the rank of an elliptic curve?

Is it the amount of rational points on the curve(That aren't integers)? So if an elliptic curve has a rank of 1, does that mean it has only 1 rational point on the curve?(In both X and Y) I've been ...
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1answer
107 views

Rational points on the elliptic curve $y^2=x^3 -x$ [duplicate]

I've been trying to find rational points on the elliptic curve $y^2=x^3 -x$ but I can't find anything else apart from $(-1,0), (0,0), (1,0)$. I have like an 'edgy' proof that it may not be possible to ...
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1answer
185 views

A nicer form for $\sum_{r=1}^{m}\sum_{d|r}(-1)^{r+d} d^3$?

Specific Question Let $R$ be a whole number. Is it possible for any whole number $c>0$ that there is a nicer form for $\sum_{r=1}^{R^c}\sum_{d|r}(-1)^{r+d} d^3$? I can hope that we can just ...
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0answers
44 views

polynomial representation of binary in computer memory and its efficiency

I have noticed that in some cases 'polynomial representation' of a number is the preferred method to represent numbers in computer programs, especially those involving modular arithmetic and ...
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0answers
46 views

How can I find the Weierstrass form of an elliptic curve given as a singular quartic?

Suppose you have an elliptic curve given as a singular quartic and you want to find the Weierstrass form of this elliptic curve. In particular you know that the singular points of this quartic are ...
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Effective Cartier divisors defined by a section

Let $S$ be a scheme, and $C$ be a smooth curve over $S$. It is known that any section $s\in C(S)$ of the curve defines a (relative) effective Cartier divisor of degree $1$, often denoted $[s]$ (see ...
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2answers
34 views

(Reference request) How to show elliptic curve has positive Mordell-Weil rank

I know there must be a lot of ways to show an elliptic curve has positive Mordell-Weil rank if it really does. And I guess that I am supposed to collect them by myself. But since I am not working in ...
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1answer
57 views

Elliptic Curve Isomorphism

I read that an elliptic curve $E({\mathbb R})$ is isomorphic to ${\mathbb R}/{\mathbb Z}$ if $x^3 + ax + b$ has only one real root, but what is the exact map? Does it come from the Weierstrass ...
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56 views

Is the “naive” full level $N$ moduli problem on elliptic curves $\Gamma(N)$ étale over $\mathbf{(Ell)}$?

Following the notations and notions developped in Katz and Mazur's book "The arithmetic moduli of elliptic curves", we denote $\mathbf{(Ell)}$ the category whose objects are elliptic curves $E/S$ over ...
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Unicity of subgroup-scheme of a supersingular elliptic curve of a given rank

Let $E/k$ be a supersingular elliptic curve over an algebraically closed field $k$ of characteristic $p>0$. Then for any positive integer $r$, there is a unique closed subgroup-scheme of $E$ of ...
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1answer
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Proof of Velu's formulas in Washington's Elliptic Curves

The proof of Velu's formulae in Washington's "Elliptic Curves" uses two exercises (Ex. 12.6 and Ex.12.8). One part in Ex.12.6 is the following: Let $E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$ be an ...
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1answer
40 views

Why is the kernel of the Verschiebung morphism either étale either connected?

Consider an elliptic curve $E$ over an algebraically closed field $k$ of characteristic $p>0$. We have the Fröbenius morphism $$F:E\rightarrow E^{(p)}$$ and the Verschiebung morphism $V:E^{(p)}\...
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26 views

What is the universal formal deformation of a supersingular elliptic curve?

In Katz and Mazur's book "Arithmetic moduli of elliptic curves" (available here), the previously undefined notion of "universal formal deformation" is stated and I fail to understand how it is defined ...
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1answer
65 views

Locus of a vertex of a triangle inside an equilateral one, under an integer constraint

Given an equilateral triangle $ABC$, we choose a point $D$ inside it, determining a new triangle $ADB$. We draw the circles with centers in $A$ and in $B$ passing by $D$, determining the new points $...
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Character sum related to an elliptic curve

Let $M$ be an odd prime and $\chi$ be a primitive character $\bmod M$. What is the evaluation of the character sum, $$\sum_{b\bmod M}\chi(b)\left(\frac{b^2-1}{M}\right), $$ where $(\frac{.}{M})$ is ...
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81 views

Involution On elliptic curve

In an elliptic curve E given by $y^2=x^3 + ax^2 +bx+ c$ and origin at the point of infinity, why does the map $i$ sending $(x, y)$ to $(x, -y)$ send $i(P)=-P$, where $-P$ denotes the inverse of $P$ ...
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1answer
51 views

Any representable moduli problem of elliptic curves is rigid

EDIT: As I now tried to provide an answer to this problem, I would appreciate any proof verification of my reasoning. Thank you very much! In Katz and Mazur's book (available here, page 109, page 60 ...
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51 views

Equivalent definitions of $\Gamma _1(N)$-structures on elliptic curves

In Katz and Mazur's book "Arithmetic moduli of elliptic curves" (available here), two equivalent definitions of a $\Gamma _1(N)$-structure (point of exact order $N$) are given on page 99 (page 55 in ...
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1answer
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Weierstrass equation for a family of elliptic curves, pushforward of sheaf

Let $f:E \to S$ be a (family of) elliptic curve. Let $[0]:S \to E$ be the zero section and $I:=I([0])$ its ideal sheaf. Why is $f_*(I^{-n})$ locally free, as claimed by Hida in the following excerpt ...
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Pushforward of sheaf of relative differentials in family of elliptic curves [duplicate]

Update: never mind, This question has been asked before here Let $f:E \to S$ be an elliptic curve (precise definition is given below from Hida’s book Geometric Modular Forms and Elliptic Curves). Is $...
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Proof explanation of Katz and Mazur's “autoduality theorem” for elliptic curves

I am currently studying Katz and Mazur's book "Arithmetic Moduli of Elliptic Curves" and I face difficulties understand a part of the proof of theorem 2.5.1, page 77 in the book (available here, page ...
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Question about this reduction to the “universal elliptic curve”

My question arises from the reading of Katz & Mazur's book "Arithmetic Moduli of Elliptic Curves", which can be found here. Given an arbitrary scheme $S$, an elliptic curve over $S$ is a proper ...
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0answers
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The image of a homomorphism on the group of $F_p$-rational points on an elliptic curve

Let $p$ be prime and let the elliptic curve $E/\mathbb F_p$ be given an affine equation \begin{equation} y^2 = f(x) = (x - e_1)(x - e_2)(x - e_3), \quad e_i \in \mathbb F_p. \end{equation} Let the $...
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Is an elliptic curve geometrically connected?

According to Katz & Mazur's book "Arithmetic Moduli of Elliptic Curves", an elliptic curve $E$ over an arbitrary scheme $S$ is a proper smooth curve over $S$ with geometrically connected fibers ...
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27 views

Strategy to prove the isogeny with arbitrary kernel is unique

I encountered a theorem saying that, Given an elliptic curve $E_1$ and arbitrary subgroup $H$, there only exist an isogeny $\phi:E_1\rightarrow E_2$ with $\ker\phi=H$ up to isomorphism of its image. ...
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49 views

Katz & Mazur's construction of generalized Weierstraß equation for a family of elliptic curves

I am reading Katz and Mazur's book "Artihmetic moduli of elliptic curves" (available here) and I have some questions about the construction of the generalized Weierstraß equation for a family of ...
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1answer
51 views

Katz and Mazur - Derived pushforward and invertible sheaves

Reading Katz & Mazur's book "Arithmetic moduli of elliptic curves" (available here), I came across the following statement that I fail to understand (p.66). Let $E$ be an elliptic curve over an ...
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0answers
30 views

$2-$descent over $\mathbb{Q}[i]$

I am trying to get my head round the concept of complete $2-$descent in general number fields. Over $\mathbb{Q}$, it is not difficult to understand, and there are several examples, such as this paper ...
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65 views

Points of elliptic curves over different fields

Let $f : E \to E'$ be an isogeny between elliptic curves (or abelian varieties), defined over a field $k$. Let $X$ be the kernel of $f$. Let $L \supset k$ be an algebraically closed field. Is it ...