Skip to main content

Questions tagged [elliptic-curves]

For questions about elliptic curves.

Filter by
Sorted by
Tagged with
0 votes
0 answers
24 views

What is an absolutely irreducible smooth curve and how to determine if a curve is as such over a finite field F_q?

I'm new to this cite but I will try my best to convey my question in the best way possible. I was learning about Hasse-Weil Bound from Terence Tao's website and I had many questions from the beginning....
Mustafa Dortluoğlu's user avatar
1 vote
0 answers
19 views

Analog of chord-and-tangent for elliptic curves in 3 or higher dimensions

For an elliptic curve defined by a Weierstrass equation (or any degree-3 polynomial equation) in $\mathbb{P}^2$, the group law has a geometric interpretation by the chord-and-tangent method. Suppose ...
Myath's user avatar
  • 1,093
1 vote
0 answers
30 views

reference for Lemma 1.5 of Fermat's dream book

I am looking for another reference to the following Lemma in the elliptic curve chapter 1 of the book "Number Theory 1: Fermat’s Dream" by Kato, Kurokawa, Saito. The proof was left out as ...
John Jiang's user avatar
0 votes
0 answers
22 views

Torsion subgroup of abelian variety

While studying elliptic curves I came over the question, whether for an elliptic curve $E$ and an integer $n$, we have \begin{equation} (E[n])^k = E^k[n], \end{equation} where $E^k$ is the $k$-...
HyperPro's user avatar
  • 901
2 votes
1 answer
40 views

Silverman's book, the arithmetic of elliptic curves, p.399, $E(K)/\hat{\phi}(E'(K))\cong \Bbb{Z}/2\Bbb{Z}$

Let $E: y^2=x^3+Ax^2+Bx, E': y^2=x^3-2Ax^2+(A^2-4B)x$ be elliptic curves over $\Bbb{Q}$. Let $\phi : E\to E', (x,y)\mapsto (\frac{y^2}{x^2},\frac{ y(B-x^2)}{x^2})$, $O_E,(0,0)\mapsto O_{E'}$. Let $\...
Poitou-Tate's user avatar
  • 6,377
2 votes
1 answer
38 views

Meaning of this Eisenstein series notation from Gross–Zagier paper

In equation (2.14) of their paper [1], Gross–Zagier define the following Eisenstein series \begin{align} E_N(z, s) = \sum_{\gamma \in \bigg(\begin{matrix} \ast & \ast \\ 0 & \ast \end{matrix}\...
Joseph Harrison's user avatar
1 vote
1 answer
85 views

Weierstrass Form of degree 4 equation

Take the equation $$y^2 = x^4 - 2x^3 - 2x - 1$$ I found that this is a genus 1 curve, because it is well known that for $y^2 = f(x)$ where $f$ is of even degree, the genus is $\frac{\deg{f} - 2}{2}$, ...
Ravikanth Athipatla's user avatar
2 votes
0 answers
43 views

An action of $\Gamma_0(N)$ having finitely many orbits

For positive integers $m$ and $N$ let \begin{align} M_{2, m, N}(\mathbf{Z}) = \bigg\{\gamma \in M_2(\mathbf{Z}) \; \bigg\vert \; \det(\gamma) = m, \gamma \equiv \bigg(\begin{matrix} \ast & \ast \\ ...
Joseph Harrison's user avatar
0 votes
0 answers
32 views

Modular parametrization of elliptic curves expressed as rational function of $j(\tau)$

I have learned that one can parametrize an elliptic curve using meromorphic modular functions on group $\Gamma_0(N)$ for certain level $N$. Using the example on wikipedia, the parametrization of $y^2-...
Wang Weiyi's user avatar
1 vote
1 answer
95 views

Why does the continuous homomorphism group equal the homomorphism group?

I have the following question on Silverman's book The Arithmetic of Elliptic Curves, 2nd edition. First, let $K$ be a perfect field, $G_{\overline{K}/K}$ be the absolute Galois group of $K$, and $M$ ...
user875280's user avatar
1 vote
0 answers
30 views

Order of Picard groups of non-hyperelliptic algebraic curves

Let $q$ be prime. When $E/\overline{\mathbb{F}_q}$ is an elliptic curve, it is well-known that the group of $\mathbb{F}_q$-points of $E$ is isomorphic to the Picard group of degree $0$ divisors on $E$ ...
Tejas Rao's user avatar
  • 1,950
5 votes
1 answer
80 views

Uniformization Theorem and Non Existence of Family of Elliptic Curves over Riemann Sphere

A question concerning a statement from these notes introducing/motivating period map. On first page, left column, last sentence states: Since $\Bbb{H}$ (=complex upper half plane) is biholomorphic to ...
user267839's user avatar
  • 7,663
0 votes
0 answers
27 views

Is the product of exponentiated elliptic curve basis elements invariant under FFT of scalars?

I am working with an elliptic curve defined over a finite field $\mathbb{F}_p$ and have a basis set of points ${g_0, g_1, \ldots, g_n}$. When I perform the FFT on these points, I obtain a new basis ${...
Nerses Asaturyan's user avatar
0 votes
0 answers
42 views

Example II.3.5 in Arithmetic of Elliptic curves

The example is Let $C$ be a smooth curve, let $f \in \overline{K}(C)$ be a nonconstant function, and let $f:C\rightarrow \mathbb{P}^1$ be the corresponding map (II.2.2). Then directly from the ...
Choon Lee Yan's user avatar
-1 votes
0 answers
55 views

Twist of elliptic curve by degree $n$ extension

Let $E/K : y^2=x^3+ax+b$ be an elliptic curve over number field $K$. For quadratic extension $L=K(\sqrt{D})/K$, $E_D/K : Dy^2=x^3+aX+b$ is called a twist of $E/K$ by $D$. This curve $E_D$ has ...
Poitou-Tate's user avatar
  • 6,377
1 vote
0 answers
92 views

Determining rational solutions to $y^2 = 2(x^4 - 2x^3 - 2x - 1)$

I am trying to find all rational solutions to $y^2 = 2(x^4 - 2x^3 - 2x - 1)$ So far, I have tried to show that rational solutions to the above form an isomorphism with rational solutions to an ...
Ravikanth Athipatla's user avatar
1 vote
0 answers
13 views

Relating discriminants of hyperelliptic curves to discriminants of their defining polynomials

Let $C$ be a hyperelliptic curve defined by an equation of the form $$ C: y^2=f(x) $$ where $f$ is a polynomial of prime degree $p\geq3$, over a complete field $K$ of residue characteristic $p$. ...
did's user avatar
  • 323
-1 votes
1 answer
78 views

Detailed Proof of Proposition 2.12 a) from Diamond, Darmon, Taylor, "Fermat's Last Theorem."

I seek a highly detailed proof of this statement in the split multiplicative reduction case. The result can be found on page 57 here. That is, if $E/\mathbb Q$ has split multiplicative reduction at $p$...
Johnny Apple's user avatar
  • 4,429
1 vote
0 answers
48 views

Elliptic curves with same $a_p$ for every $p$ are isogenous

If two curves E,F satisfy $\#E(\mathbb{F}_p) = \#F(\mathbb{F}_p)$ for each large prime then E and F are isogenous (conversely, two isogenous curves must have the same values of $\#E(F_p)$ for every $p$...
Rodrigo's user avatar
  • 1,043
1 vote
0 answers
48 views

Computing degree of $x$ map for elliptic curve given by Weierstrass equation

Suppose $E$ is an elliptic curve given by the Weierstrass equation $$ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6 $$ I want to calculate the degree of the map $$ \varphi\colon E\to\mathbb{P}^1\qquad\quad[x,y,1]...
Navid's user avatar
  • 147
1 vote
0 answers
29 views

Proof of Grothendieck's Semistability Criterion for Elliptic Curves

We have the following result: Let $E/\mathbb Q$ be an elliptic curve and let $p$ be a prime. Then $E$ is semistable at $p$ if and only if $\rho_{E,\ell}|_{I_p}$ is unipotent for some (all) prime(s) $\...
Johnny Apple's user avatar
  • 4,429
3 votes
1 answer
62 views

Proof of theorem $V.4.1(c)$ in Silverman Elliptic Curves, How to Deduce $H_p(t)$ has no multiple roots?

I am trying to understand the proof of theorem $V.4.1(c)$ in Silverman Arithmetic of Elliptic Curves. Let $p$ be a prime, $q=p^n$, and $m=(p-1)/2$. Define the polynomial $$H_p(t)=\sum_{i=0}^m \binom{m}...
Snacc's user avatar
  • 2,417
3 votes
3 answers
276 views

Changing equation into elliptic curve

We can change $s^2 = \frac{t^2 - 1}{2t}$ to $$2ts^2 = t^2 - 1$$ $$16t^2s^2 = 8t^3 - 8t$$ and making the change of variables $(x, y) = (2t, 4ts)$, we get the elliptic curve $y^2 = x^3 - 4x$ Can we do ...
Ravikanth Athipatla's user avatar
0 votes
0 answers
32 views

Intersection of two quadrics is elliptic curve

I am currently studying Jacobi intersection models of elliptic curves, and so far I have found that whenever you have a lattice in $\mathbb{C}$ we have an embedding of $\mathbb{C}/\Lambda$ into $\...
MarvinsSister's user avatar
1 vote
1 answer
28 views

If the analytic rank is one then the sign in the functional equation is -1?

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $L(E, s)$ denote its $L$-function over $\mathbf{Q}$. Also $f$ denotes the weight two cusp form associated to $E$, but this shouldn't be ...
Joseph Harrison's user avatar
0 votes
0 answers
65 views

Reference Request: Studying elliptic curves in algebraic geometry

I am an undergraduate student who has studied some algebraic geometry. I want to read about elliptic curves from an algebraic-geometric perspective. I have read Hartshorne Chapter 1, and I know a ...
Sardines's user avatar
  • 759
1 vote
1 answer
53 views

Why is $E$ an elliptic curve, and how to calculate its $j$-invariant?

I am reading an example of Springer fiber written by Bernstein and Kazhdan. Let $V$ and $V'$ be two $\mathbb C$-vector spaces, both of dimension $3$. Let $\lambda_1,\lambda_2,\lambda_3 \in \mathbb C$ ...
sunkist's user avatar
  • 1,155
0 votes
1 answer
37 views

Structure of $p$-primary Abelian groups without Divisorial Elements [closed]

Let $A$ be a $p$-primary (in particular, torsion) Abelian group ($p$ prime). Assume that $A[p]=\{a \in A \vert pa=0\}$ is finite and that there are no nontrivial infinitely $p$-divisible elements, ie $...
user267839's user avatar
  • 7,663
5 votes
1 answer
82 views

The $j$-invariant on the 'critical line' and its zeros

It is well-known that for the $j$-invariant, we have $j\left(\frac{1+\sqrt{-n}}{2}\right)\in\mathbb R$ whenever $n>0$. Moreover, $j\left(\frac{1+\sqrt{-n}}{2}\right)=0$ for $n=3$ at the cusp. In ...
Wolfgang's user avatar
  • 1,042
1 vote
1 answer
30 views

Isomorphism as $\Bbb{F}_p$ vector space and Proof of local Tate--Duality $H^1(G_K,E)[p]\cong (E(K)/pE(K))^*$

This is a question regarding Theorem $1.4$ of https://kskedlaya.org/kolyvagin-seminar/duality.pdf. Let $E/K$ be an elliptic curve over number field $K$. Let $p$ be a prime number. The goal of Theorem $...
Poitou-Tate's user avatar
  • 6,377
0 votes
0 answers
49 views

Why restricted product $\prod'$ is $\varinjlim_{S\subset I \text{ runs finite subset of} I} (\prod_{i\in S} X_{i}\times \prod_{v\in I-S}Y_i)$

This is a question related to this page. https://ncatlab.org/nlab/show/restricted+product . Let $I$ be a directed set. Let $X_i(i\in I)$ be a group. Let $\prod'_{i\in I}(X_i,Y_i)$ be a restricted ...
Poitou-Tate's user avatar
  • 6,377
0 votes
1 answer
68 views

Can associativity of elliptic curve group law be verified by hand?

On p.15 of Silverman's Rational Points on Elliptic Curves, after sketching the proof of the associativity law of point addition on an elliptic curve $y^2=x^3+ax^2+bx+c$ via Bezout's theorem, the ...
183orbco3's user avatar
  • 1,461
2 votes
0 answers
84 views

How should the conjecture associated with Taniyama, Shimura, etc. be referred to?

I'm writing an article in which I need to refer to the Modularity theorem before it was a theorem. I always knew it as the first on the list below, but I'm aware there are alternatives, so I need to ...
tkp's user avatar
  • 133
0 votes
0 answers
42 views

Proof that the order of elements stays the same under Frobenius endomorphism

Let $E: Y^2=X^3+Ax+B$ be an elliptic curve, defined over $\mathbb{F}_p$ where $p$ is a prime. Define: $$\phi: E(\bar{\mathbb{F}}) \rightarrow E(\bar{\mathbb{F}})$$ by $$\phi(P) = \left\{ \begin{array}...
Yvonne's user avatar
  • 11
1 vote
2 answers
95 views

How do I tell if two elliptic curves are isogenous?

How do I tell if two elliptic curves are isogenous? For instance, a paper I'm reading claims $y^2=x^3+1$ and $y^2=x^3-27$ are isogenous over $\mathbb{Q}$ , but how do I figure this out? And how do I ...
usr0192's user avatar
  • 3,223
1 vote
1 answer
175 views

Relation between direct sum $\bigoplus$ and restricted product $\prod'$ of Galois cohomology

This is a question about the relation between directed sum $\bigoplus_{v \in M_K} H^1(\text{Gal}(\overline{K}/K), M)$ and restricted direct product $\prod'_{v \in M_K} H^1(\text{Gal}(\overline{K}/K), ...
Poitou-Tate's user avatar
  • 6,377
1 vote
0 answers
25 views

The galois group of $p^{k}$-torsions of an elliptic curve defined over a global function field of characteristic $p \ge 5$

I'm looking for an analogy of Corollary 7.5.3 in 『A first course in modular forms』(Diamond and Shurman) over a global function field. The colloary follows: For the elliptic curve $$ E: y^{2} = x^{3} -...
jawlang's user avatar
  • 705
0 votes
1 answer
44 views

Self intersection number of diagonal

Suppose I have an elliptic curve $E$. How would I calculate the self intersection in $A_*(E \times E)$ of the diagonal $\Delta$? It seems the formula I need to use is $\Delta . \Delta = c_1(N_{\Delta/...
Slim Shady's user avatar
2 votes
0 answers
59 views

Line Bundles on Elliptic Curves lift From Quotient

Suppose I have an elliptic curve $E$ and take its quotient by some finite group $G$. Is there some relation between the line bundles on $E$ and the line bundles on $E/G$? For example does every line ...
Slim Shady's user avatar
1 vote
1 answer
55 views

Base Case of Theorem III.5.2 in Silverman's AEC

I am studying Silverman's Arithmetic of Elliptic Curves, and I am a little confused on the base case of the following theorem: Theorem 5.2. Let $E$ and $E'$ be elliptic curves, let $\omega$ be an ...
Ryder Pham's user avatar
0 votes
0 answers
31 views

How can I be certain of the existence of elliptic curves of certain order when the parameter a is fixed?

My question came up while researching an attack on Elliptic Curve Cryptography (described in Computer Security - ESORICS 2015. I'm given an elliptic curve $E$ defined by $y^2=x^3+ax+b$ over the finite ...
Yvonne's user avatar
  • 11
1 vote
0 answers
44 views

Silverman AEC Exercise 7.4

The following problem in chapter 7 of Silverman's book has been bothering me: Here's my progress: I first tried to solve it in the $\text{char}(k) \ne 2,3$ case, wherein we can assume the equation is ...
Aditya Khurmi's user avatar
0 votes
0 answers
30 views

Scalars that are both Additive and Multiplicative inverses on secp256k1

I believe I have found two scalars (a) and (b) that are both additive and multiplicative inverses on the secp256k1 elliptic curve. So scalars (a) and (b) meet the following criteria: $[ a + b \equiv 0 ...
TrialAndError's user avatar
1 vote
1 answer
54 views

Find isomorphism of elliptic curves in Weierstrass form

I have the following two elliptic curves over an algebraically closed field of characteristic distinct from 2: $$E:y^2=x^3+4x^2+2x\quad \quad E':y=x^3-8x^2+8x$$ I want to find an isomorphim $\psi:E'\...
kubo's user avatar
  • 2,067
0 votes
0 answers
32 views

Alternative way to compute degree of Frobenius endomorphism

Let $p$ be a prime, $q=p^r$, $K=\mathbb{F}_{q}$, $E/K$ an elliptic curve and let $\phi$ be the $q^{th}$ Frobenius endormorphism i.e. $\phi=(x^q,y^q,1)$. I want to show that $\phi$ has degree $q$. I am ...
kubo's user avatar
  • 2,067
1 vote
1 answer
87 views

Computing divisors of elliptic curves

I recently have a tough time trying to compute divisors of functions on elliptic curves. This is a part of exercise 11.1. from Washington: Find the divisor of $g(x,y)=\frac{y^4}{(x^2+1)^3}$ over $\...
HyperPro's user avatar
  • 901
0 votes
0 answers
36 views

Degree of an elliptic function

I'm attending a course about elliptic functions (over complex) The professor defined the degree of an elliptic function as the number of poles (counted with multiplicity) in the fundamental domain (...
cespun's user avatar
  • 94
0 votes
0 answers
56 views

Degree of $[\rho]^2-[\rho]$ with $\rho^3=1$

I have the elliptic curve $E:y^2=x^3+B$ where $B\in K^\times$ and $K$ is a field of characteristic distinct than $2,3$. I have the map $\mu:=[\rho]^2-[\rho]$ and I want to compute its degree. My ...
kubo's user avatar
  • 2,067
0 votes
0 answers
27 views

Chords and Tangents on Elliptic Curves Over Arbitrary Fields

I've been reading some books (e.g. Definition 3 in this chapter on elliptic curve cryptography or Group Law Algorithm 2.3 from The Arithmetic of Elliptic Curves) which define the group operation on ...
simon's user avatar
  • 11
3 votes
1 answer
139 views

Are these elliptic curves over $\mathbb{Q}(\sqrt{2})$ isogenous?

I have the following elliptic curves: $$E:y^2=x^3-\sqrt{2}x \quad \quad E':y^2=x^3-2x$$ over $\mathbb{Q}(\sqrt{2})$. I want to determine whether they are isogenous. I have a few strategies to do this ...
kubo's user avatar
  • 2,067

1
2 3 4 5
68