All Questions
58 questions
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140
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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
6
votes
1
answer
407
views
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{...
- 10.7k
0
votes
0
answers
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 ...
- 18.4k
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!
- 8,945
0
votes
0
answers
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
8
votes
0
answers
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
7
votes
1
answer
1k
views
Beilinson's height pairing vs. Néron–Tate
In the literature there are several different definitions of what is often referred to as Beilinson's height pairing (see for example section 4.3.8 of Gillet and Soulé's paper Arithmetic intersection ...
- 1,446
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 ...
- 12.3k
16
votes
1
answer
2k
views
Reference to a Don Zagier result and the congruent number problem
I was looking for a reference/explanation as to how Don Zagier managed to find the side lengths of a rational right triangle with area 157. There have been many literature references to the fact that ...
- 4,713
1
vote
0
answers
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
answers
541
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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 $E_B$ ...
- 1,353
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 ...
- 4,746
3
votes
0
answers
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
7
votes
1
answer
775
views
Translations of Deuring
As the title suggests, do there exist translations of papers by Deuring?
I'm particularly interested in:
"Die Typen der Multiplikatorenringe elliptischer Funktionenkörper," Ach. Math. Sem. Hab. (1941)...
- 422
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 ...
- 50.6k
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
6
votes
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 ...
- 20.8k
4
votes
0
answers
252
views
Height pairings of Heegner points of nontrivial conductor
I am studying Gross's and Zagier's proof of the BSD conjecture for elliptic curves of rank $\leq 1.$ Their calculation essentially boils down to the following ingredients:
(1.) Finding a suitable ...
- 37k
8
votes
2
answers
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)$ ...
- 63.9k
1
vote
0
answers
234
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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\...
CommunityBot
- 1
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 ...
- 63.9k
19
votes
3
answers
1k
views
Points of elliptic curves over cyclotomic extensions
Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It ...
- 1,694
15
votes
1
answer
943
views
BSD conjecture for rank 1 elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that
$$\text{ord}_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$
Thanks to the work of Gross-Zagier and ...
- 37k
6
votes
2
answers
781
views
Books building up to the Gross-Zagier formula
I am an undergrad extremely interested in some applications of the Gross-Zagier formula for elliptic curves. I have a strong foundation in group theory and abstract algebra, and an understanding of ...
- 519
2
votes
2
answers
509
views
Question about Zeta Function of Singular Plane Curve
I am working on a project which involves learning about the zeta function (weil zeta function) for plane curves, but I do not know much algebraic geometry. (I do not know anything about schemes).
I ...
- 148k
5
votes
0
answers
317
views
Elliptic curve sequences needed for universal forgery
Elliptic Curve Digital Signature Algorithm (ECDSA) admits universal forgery (UF) if the Attacker can solve the equation
$$z=\frac{f_{k-1}(x,y)f_{k+1}(x,y)}{f_{k}(x,y)^2},$$
where $k$ is unknown, $f_{k}...
- 12.3k
0
votes
2
answers
245
views
When the $o$-th division polynomial of an elliptic curve over finite vanishes only at $x$ coordinates?
Need this for probabilistic factoring algorithm.
Let $p$ be sufficiently large prime and $E$ the elliptic curve
$E /\mathbb{F}_p: y^2=x^3+ax+b$. Let $o=\#E(\mathbb{F}_p)$.
$\psi_n$ denote the $n$-...
- 148k
8
votes
0
answers
161
views
Looking for an elliptic curve E st ${\large Ш}(\mathbb Q,E)$ cont. an element of order $p^2$ and certain other properties
I am looking for an elliptic curve $E$ with Weierstraß coefficients in $\mathbb{Q} $ so that for some prime $p$ the following conditions are satisfied:
(1) ${\large Ш}_{p^{\infty}}(\mathbb{Q},E)$ ...
- 4,746
7
votes
1
answer
1k
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Szpiro's conjecture for function fields and Mochizuki's approach to the number field case
Where can I find more details on the proof of Szpiro's conjecture for function fields, as mentioned in Minhyong Kim's answer to this MO question?
I am looking at this in the context of Mochizuki's ...
- 17.6k
7
votes
1
answer
281
views
Sato-Tate conjecture when Fourier coefficients are complex numbers
Let $k\geq 1$ and let $f=\sum_{n\geq 1}a(n)q^{n}$, $a(n)\in\mathbb{R}$, be a normalised cuspidal Hecke eigenform of weight $2k$ for $\Gamma_{0}(N)$ without complex multiplication. So the result of ...
- 20.4k
2
votes
0
answers
245
views
Help for reference of moduli stack of fake elliptic curves
I see everywhere the following:
Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$.
...
- 8,529
5
votes
0
answers
143
views
Reference request: "effective'' semistable reduction
I am looking for the origin of the following idea: suppose $m$ and $n$ are relatively prime integers $\geq 3$. Let $E$ be an elliptic curve over a number field $K$. Let $L/K$ be a finite extension ...
- 17.6k
1
vote
0
answers
74
views
Equivalent of Lauricella $F_D$ on an elliptic curve?
Lauricella's hypergeometric function $F_D$ is related to (weighted) configurations of points on $\mathbb{P}^1$. I am looking for generalizations to weighted point configurations on an elliptic curve. ...
- 19.6k
7
votes
1
answer
646
views
Serre's surjective theorem importance
I'm studying Serre's paper in wich he shows the following theorem:
Let K be a number field, $E$ an elliptic curve over K without CM. Then the representation $$\rho_{\ell}:\mathrm{Gal}(\bar K/K)\...
- 313
3
votes
1
answer
233
views
Addition law for elliptic curves of the form $x^2y^2+a(x+y)+b=0$
Did anybody consider addition law for elliptic curves of the form $$x^2y^2+a(x+y)+b=0\,?$$ Does this form have any specific name?
- 12.3k
2
votes
1
answer
514
views
Heegner points on elliptic curves
I want to know about Heegner point computations for a CM elliptic curve. What is the best reference book/paper for reading?
- 519
10
votes
1
answer
1k
views
Rank of Elliptic Curves
Recently, I have heard of some heuristics that would suggest that the rank of elliptic curves are bounded (specifically in the congruent number family). I always though that the best way to prove ...
- 8,945
0
votes
1
answer
246
views
Does the modified Szpiro conjecture require minimal model?
The modified Szpiro conjecture is described in
Wikipedia
and here and here.
The modified Szpiro conjecture states that: given $\varepsilon > 0$, there exists a constant $C(\varepsilon)$ such that ...
CommunityBot
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4
votes
0
answers
242
views
Evaluation of $E_{\ell,2}$ on supersingular curves over $\mathbb{F}_{p^2}$
As mentioned in an answer to Modularity of $E_2$ on congruence subgroups, there exist modular forms $E_{\ell,2}$ of level $\Gamma_{0}(\ell)$ and weight 2, with $q$-expansion $E_{\ell,2}(q)=E_{2}(q)-\...
CommunityBot
- 1
3
votes
0
answers
680
views
Birch/Swinnerton-Dyer "Notes on Elliptic Curves II"
I would like to know if any of you know if there is a more general treatment to what Birch and Swinnerton-Dyer did in "Notes on Elliptic Curves II" (http://www.ams.org/mathscinet-getitem?mr=179168).
...
- 521
8
votes
0
answers
595
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A property of supersingular $j$-invariants (reference request)
Edit 2: For those who understandably don't want to read such a long post, I think Voloch's suggestion reduces the problem to asking whether $j$-invariants of supersingular curves are 3rd powers in $\...
- 1,537
11
votes
1
answer
1k
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Best results regarding the Lang-Trotter conjecture
Let $E$ be an elliptic curve over $\mathbb Q$, which does not have complex multiplication over the algebraic closure of $\mathbb Q$. For $x>0$, let $P(x)$ be the number of primes $p < x$ such ...
- 79.9k
15
votes
0
answers
591
views
For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?
An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...
CommunityBot
- 1
4
votes
2
answers
655
views
Intersection of Hilbert class fields of imaginary quadratic fields
In this question Hilbert class field of Quadratic fields it is mentioned that if $d\equiv 1 \mod 4$ then the Hilbert class field of $\mathbb{Q}(\sqrt{-d})$ contains $\mathbb{Q}(i,\sqrt{d})$.
Could ...
CommunityBot
- 1
6
votes
1
answer
1k
views
Must the $j$-invariant of an elliptic curve with an isogeny be integral?
Let $K$ be a quadratic field, and $E/K$ a non-CM elliptic curve with a $K$-rational $p$-isogeny, for $p$ a prime. I would like to say the following:
For large enough $p$, the $j$-invariant $j(E)$ ...
- 19.2k
5
votes
3
answers
3k
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Effective way of finding generators on the curve and the rank conjecture
Hello everyone,
I have never heard of a polynomial time running algorithm that finds the generators of elliptic curves efficiently. I do know that Nagell-Lutz theorem is useful in computing the ...
CommunityBot
- 1
4
votes
1
answer
338
views
Reference for Rank Distribution Conjecture.
I am currently writing my master's thesis and I was wondering if the rank distribution conjecture was ever formally written down. Recall that it says that:
Half of all elliptic curves have rank $0$, ...
22
votes
3
answers
7k
views
A recommended roadmap to Fermat's Last Theorem
I was inspired to undertake math as a career after watching a documentary on the proof of Fermat's Last Theorem. As such it's been a small goal of mine to understand Wiles et al's proof.
In a ...
CommunityBot
- 1
11
votes
1
answer
1k
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Finiteness of Tate-Shafarevich
Does anyone happen to know who conjectured the finiteness of the Tate-Shafarevich group?
We recall the conjecture. Let $E/K$ be an elliptic curve where $K$ is a number field. Then $Ш(E/K)$ is finite.
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