All Questions
Tagged with reference-request elliptic-curves
119 questions
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138
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State of the art on attempts to solve the elliptic curve discrete logarithm problem through transfering the problem to a weaker curve
Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully ...
- 87
2
votes
1
answer
267
views
Can the number of elements of order 4 in the Tate–Shafarevich group grow arbitrarily large?
Let $E/K$ be a number field. For quadratic field extensions $L/K$, it is known that $\operatorname{Ш}(E/L)[2]$ can be arbitrarily large (cf. P. L. Clark and S. Sharif, "Period, index and ...
- 1,531
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votes
0
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128
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mod $p$ local Galois representation attached to elliptic curves
In the paper, lemma 4.4. The author gives the form of the representation of $G_p$ on $E[p]$ of the form
$$\begin{pmatrix} \varepsilon\chi & *\\0 & \chi^{-1} \end{pmatrix}.$$
Do they assumed ...
- 275
6
votes
1
answer
407
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Good reduction for the universal elliptic curve
Let $X$ be a modular curve, i.e. a quotient of the upper half plane $\mathbb{H}$ by a congruence subgroup $\Gamma$. When $\Gamma=\Gamma_0(N)$, it is known that $X$ has a smooth model denoted $\mathcal{...
- 2,907
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1
answer
118
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Reference request for the isomorphism $H^1(G_{K_v},E)[n]\cong (E(K_v)/nE(K_v))^*$ in the context of Tate-duality
Let $E/K$ be an elliptic curve over a number field $K$. Let $M_K$ be the set of all places of $K$. Let $K_v$ be a completion of $K$ at $v$.
I'm searching for a reference for the statement of the ...
- 1,531
3
votes
0
answers
87
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cubic twists of Mordell curve and their rank
Let $a$ be a non-zero integer. Consider the elliptic curve $E_a/\mathbb{Q}$ given by the equation
$$
E_a: y^2 = x^3 + a.
$$
For a cube-free integer $D$, define the elliptic curves $E_{aD^2}/\mathbb{Q}$...
- 1,283
5
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1
answer
340
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Equations for dual cubic curves
Suppose I have a cubic curve $C$ (over $\mathbb C$) in Weierstrass form $$y^2=x^3+ax+b.$$
I would like to find the degree $6$ equation for protectively dual curve $C^*$. Do you know any place where ...
- 14.3k
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0
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108
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looking for reference for two elliptic curves with equal formal group
I am looking for a reference.
In this post, @Chris Wurthrich made the following comment:
If the formal group laws (probably upon particular choice of coordinates) of two elliptic curves over any ring ...
- 195
1
vote
1
answer
142
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Algorithm for computing isogeny class of elliptic curve
Is there an algorithm for computing the entire isogeny class of a given elliptic curve $E/\mathbb{Q}$?
References/ideas are welcome. Thanks!
- 2,907
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81
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Computing elliptic periods from modular form
How are the periods of a modular elliptic curve computed as path integrals of its associated normalized weight 2 cusp form on the modular curve? Please provide specific paths for both periods and cite ...
- 2,907
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208
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Elliptic curves of rank 1 over number fields
I am interested what is known about the following statement:
For every number field $K$, there exists an elliptic curve $E$ defined over $K$ with algebraic rank equal to $1$.
Is this statement known ...
- 502
1
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1
answer
139
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Grössencharakter or Galois representation associated to a CM elliptic curve in characteristic $p$
When $E$ is an elliptic curve over a number field $L$, with complex multiplication by a maximal order in an imaginary quadratic field $K$, Silverman’s Advanced Topics in the Arithmetic of Elliptic ...
- 531
4
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306
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Gluing together the moduli stacks of elliptic curves over Z[1/2] and Z[1/3]?
I have a long-running desire to understand what is the "global" moduli stack of elliptic curves, as a stack over $\mathrm{Spec}(\mathbb{Z})$.
Recently I was pointed to Katz and Mazur's book, ...
- 35.5k
6
votes
1
answer
981
views
What is this huge generalization of the Modularity Theorem?
A friend of mine wrote:
The point is of course that the Modularity Theorem (as I stated it) is/should be really just a special case of some much bigger theorem which sets up a bijection between ...
- 22.3k
5
votes
1
answer
233
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Why can I take the quotient of a relative elliptic curve by a finite locally free subgroup?
I am currently reading Katz and Mazur’s Arithmetic moduli of elliptic curves and I am puzzled by a statement in the discussion of the $[\Gamma_0(N)]$ moduli problem in Chapter 3.
The authors define a $...
- 385
3
votes
1
answer
271
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Why do we get a connected 2-regular graph?
In reading "PUBLIC-KEY CRYPTOSYSTEM BASED ON ISOGENIES" by Rostovtsev and Stolbunov, they claim on page 8 that the set $U=\{E_i(\mathbb{F}_p)\}$ of elliptic curves with a specific prime $l$ ...
- 33
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220
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Elkies' family of elliptic curves of rank 19
There is a widely cited fact that Elkies had found that infinitely many curves of rank 19 in 2006, in "Z^28 in E(Q), etc. Email to the number theory mailing list
at [email protected]&...
- 161
3
votes
1
answer
320
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Counting points on elliptic curves
Consider the Legendre family of elliptic curves
$$E_a: y^2=x(x-1)(x-a).$$
Let $p$ be an odd prime.
QUESTION. Is the following true? If $p\equiv 3\pmod4$ then number of solutions to $E_2$
over the ...
- 43.2k
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81
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Potential typo in "Complete Systems of Two Addition Laws for Elliptic Curves" by Bosma and Lenstra
Here is a link to the article: https://www.sciencedirect.com/science/article/pii/S0022314X85710888?ref=cra_js_challenge&fr=RR-1.
Pages 237-238 give polynomial expressions $X_3^{(2)}, Y_3^{(2)}, ...
- 658
7
votes
1
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295
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Explicit equations for the universal vector extension of an elliptic curve
The universal vector extension $E$ of an abelian variety $A$ is an algebraic group, an extension of $A$ by a vector group $0 \to V \to E \to A \to 0$, such that any other extension of $A$ by a vector ...
- 658
1
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89
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Finiteness of elliptic curves with trivial conductor over function fields
Let $K=\mathbb{F}_q(C)$ be the function field of a smooth projective curve $C$ over a finite field $\mathbb{F}_q$ with $\text{cha}(K)>3$ and let $E$ be an elliptic curve over $E$. To $E$ we may ...
- 121
5
votes
2
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541
views
When are two elliptic curves with zero j invariant isogenous?
Consider elliptic curves of the form $E_B\colon y^2=x^3+B$ for $B\in\mathbb Q$. These are exactly the elliptic curves with zero $j$-invariant. I would like to know when are two elliptic curves $E_B$ ...
- 89
3
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122
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Description of $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for a CM elliptic curve $E$
I am looking for a specific description of the Galois groups $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for an elliptic curve $E$ with complex ...
- 309
22
votes
1
answer
770
views
Writing $3p$ when $p \equiv 1 \pmod{3}$ as a sum of two rational cubes. Is this result new? And what about its converse?
I recently discovered a proof of the following. Let $p$ be a prime that's $1 \bmod {3}$. Suppose that $p$ is not represented by the principal quadratic form $(1,9,81)$ of discriminant $-243$ (The ...
- 5,422
6
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1
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391
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Tate–Shafarevich group of Jacobian of Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$
$C/ \Bbb{Q}: 3X^3 + 4Y^3 + 5Z^3 = 0$ is known to be a nontrivial element of the Tate–Shafarevich group of the elliptic curve $E/\Bbb{Q}:X^3 + Y^3 + 60Z^3 = 0$. It is also an example of an abelian ...
- 1,531
26
votes
0
answers
567
views
Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
- 13.3k
7
votes
2
answers
615
views
Reference request for recurrence relation of division polynomials
The recurrence relations for division polynomials of elliptic curves are well known:
$$\Psi_{2n} = \Psi_n \left( \Psi_{n+2} \Psi_{n-1}^2 - \Psi_{n-2} \Psi_{n+1}^2 \right) / \ 2y$$
$$\Psi_{2n+1} = \...
- 265
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220
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Proof of $L(E,1)/Ω(E)=1/8$ for elliptic curve $E:y^2=x^3-x/ \Bbb{Q}$?
Let
$E:y^2=x^3-x$ be an elliptic curve over $ \Bbb{Q}$ and
$ω_E=dx/2y=dx/2\sqrt{x^3-x}$.
Then
$$
\begin{split}
\Omega(E)&=\int_{E(\Bbb{R})} ω_E\\
\\
&=2\int\limits_1^{+\infty} dx/\sqrt{x^3-x}...
- 1,531
1
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0
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100
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Iterated integrals on higher dimensional Calabi-Yau manifolds?
I recently read about the construction of closed quasi-periodic differential forms on elliptic curves (1-dim Calabi-Yaus) via the Kronecker-Eisenstein series. I now wonder if similar constructions are ...
- 221
6
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2
answers
1k
views
Reference for universal elliptic curves
I've seen the following sentence come up in a few papers:
Consider the modular curve $Y_1(N)$ and let $E$ be the universal elliptic curve over $Y_1(N)$.
This comes up in Deligne's construction of ...
- 1,949
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votes
0
answers
90
views
Counting points on Elliptic Curves with CM by $\mathbb{Q}[\sqrt{-d}]$, $d=1,3$ (CM ring with non-trivial units)
Consider an elliptic curve $E/H$ with CM by the entire ring of integers $\mathcal{O}_K$ of $K=\mathbb{Q}[\sqrt{-d}]$ (and such that $j(E)=j(\mathcal{O}_K)$) such that $H$ is the Hilbert class field of ...
- 101
1
vote
1
answer
385
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Cohomology of the dual Abelian variety
I am interested in the (degree $1$) Betti cohomology of the dual $A^\vee$ of an Abelian variety $A$ (say, over $\overline{\mathbb{Q}}$). We can even assume $A$ to be an elliptic curve, if this makes ...
- 533
2
votes
0
answers
1k
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What's the best reference for Abelian varieties?
I am curious about learning about Abelian varieties, specifically how they are in some ways generalizations of elliptic curves.
I know of the two sources: https://www.jmilne.org/math/CourseNotes/AV....
- 101
4
votes
1
answer
234
views
integral points on elliptic curves in terms of discriminant
I am curios where in the literature was the first time written the following conjecture.
Say we have we have an elliptic curve $E$ given by the Weierstrass equation $y^2=x^3+AX+B$ with $A,B\in \...
- 847
1
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0
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104
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A subgroup of the $n$-Selmer group
Let $p$ be an odd prime and for the purpose of this question let $n$ be an integer which is a power of $p$.
Let $E$ be an elliptic curve over a number field $F$.
The $n$-Selmer group, denoted by $S_n(...
- 1,283
3
votes
1
answer
440
views
Moduli space of genus 1 curves with a degree n divisors
I am sure this is well known, but I don't know what to search for:
Consider $M_{1,n}$, the moduli space of genus 1 curves with $n$ marked points. The symmetric group on $n$ letters acts on this space ...
- 7,746
2
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0
answers
155
views
Reference for the $3$-series of an elliptic formal group law
The $3$-series of the formal group law of the Weierstrass curve $y^2 = x^3 + a_2 x^2 + a_4 x$ begins
$$
[3](z) = 3 z - 8 a_2 z^3 + (24 a_2^2 - 96 a_4) z^5 - (72 a_2^3 - 288 a_2 a_4) z^7 + (216 a_2^4 - ...
- 9,263
8
votes
2
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643
views
Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers
I am interested to know if a similar theorem that shows this answer of the post
Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
1
vote
0
answers
212
views
Generate algorithmically an elliptic curve with its exact class group structure?
Is it possible to generate an elliptic curve $E$ (randomly), together with knowing its class group $\mathrm{Cl}(\mathcal{O})$ structure? where $\mathcal{O}$ is its endomorphism rings $\mathsf{End}(E)$ ...
- 181
4
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163
views
References for the computation of the Mordell-Weil group of an elliptic curve
I am reading about the Mordell-Weil group of an elliptic curve over a number field using primarily Silverman's AEC. While the book is excellent in discussing materials prior to Chapter 8, I think ...
- 401
1
vote
0
answers
234
views
Why is the $\mathbb{Z}_p$-corank of $\operatorname{Sel}_{p^\infty}(E/\mathbb{Q})$ finite?
I'm interested on the Mordell-Weil rank of an elliptic curve over $\mathbb{Q}$. I read that the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group $\operatorname{Sel}_{p^\infty}(E)\doteq\...
user105226
1
vote
0
answers
83
views
Can someone suggest some references for rank discussion of elliptic curves (bonus if it systematic)?
First, I apologise if such a question has been asked before. Please feel free to refer me to the previous question, if it answers my current query then I will delete this post.
I am reading the ...
- 401
4
votes
0
answers
540
views
Formula for Neron Severi group of product surfaces
Let $A=E\times E'$ be a surface which is a product of two elliptic curves. Then it is claimed that there is an isomorphism:
$$\mathbb Z \oplus {\rm Hom}(E, E')\oplus \mathbb Z \to {\rm NS}(A)$$ ...
- 3,472
4
votes
1
answer
412
views
Motivations of families of modular forms, elliptic curves and Galois representations?
I want to know some reference, why do some number theorists study the families of the elliptic curves, modular forms or Galois representations? As far as I know, I always consider the Galois ...
user141691
2
votes
1
answer
435
views
Explicit semi-stable theorem for elliptic curves over $p$-adic fields
In this paper of Maja Volkov, the authur metions a number called "défaut de semi-stabilité" on page 9. It is defined as $\text{dst}(E)=\frac{12}{\text{pgcd} (12,v_p(\Delta_E))}$ where $E$ is ...
user141691
8
votes
1
answer
408
views
Max order of an isogeny class of rational elliptic curves is 8?
I am looking for a reference for the proof of the following question following Theorem 5 in Mazur's Rational Isogenies of Prime Degree.
Theorem 5 There is a constant $C$ such that every elliptic ...
- 163
6
votes
4
answers
814
views
Texts on moduli of elliptic curves
I want to study FLT (Fermat's Last Theorem), and now I'm studying moduli of elliptic curves.
I've heard that Deligne-Rapoport, Katz-Mazur, Mazur's "Modular curves...", and Katz's "p-adic..." are very ...
- 1,364
4
votes
1
answer
531
views
Can someone help in how to approach reading Mordell-Weil Theorem for abelian varieties?
I was thinking to start reading the proof of Mordell-Weil Theorem for abelian varieties over number fields, after getting done with the proof in the case of elliptic curves over number field and I ...
- 401
7
votes
0
answers
166
views
$\lambda$-invariants in cyclotomic $\mathbb{Z}_p$ extensions
The idea that Selmer groups and class groups are related is not new. More recently, we understand that the growth patterns of fine Selmer groups are very similar to that of class groups in cyclotomic $...
- 1,283
3
votes
0
answers
191
views
anomalous primes and CM elliptic curves
Let $E$ be an elliptic curve defined over a number field $F$ and suppose $E$ has CM by $\mathcal{O}_K$ where $K$ is an imaginary quadratic field. What can we say about the non-anomalous primes of such ...
- 1,283