All Questions
58 questions
0
votes
0
answers
140
views
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{...
- 2,907
0
votes
0
answers
108
views
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
views
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
0
votes
0
answers
81
views
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
views
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
3
votes
1
answer
320
views
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
1
vote
0
answers
89
views
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
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
votes
0
answers
122
views
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
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:
$$\Psi_{2n} = \Psi_n \left( \Psi_{n+2} \Psi_{n-1}^2 - \Psi_{n-2} \Psi_{n+1}^2 \right) / \ 2y$$
$$\Psi_{2n+1} = \...
- 265
6
votes
2
answers
1k
views
Reference for universal elliptic curves
I've seen the following sentence come up in a few papers:
Consider the modular curve $Y_1(N)$ and let $E$ be the universal elliptic curve over $Y_1(N)$.
This comes up in Deligne's construction of ...
- 1,949
8
votes
2
answers
643
views
Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers
I am interested to know if a similar theorem that shows this answer of the post
Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
1
vote
0
answers
212
views
Generate algorithmically an elliptic curve with its exact class group structure?
Is it possible to generate an elliptic curve $E$ (randomly), together with knowing its class group $\mathrm{Cl}(\mathcal{O})$ structure? where $\mathcal{O}$ is its endomorphism rings $\mathsf{End}(E)$ ...
- 181
1
vote
0
answers
234
views
Why is the $\mathbb{Z}_p$-corank of $\operatorname{Sel}_{p^\infty}(E/\mathbb{Q})$ finite?
I'm interested on the Mordell-Weil rank of an elliptic curve over $\mathbb{Q}$. I read that the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group $\operatorname{Sel}_{p^\infty}(E)\doteq\...
user105226
8
votes
1
answer
408
views
Max order of an isogeny class of rational elliptic curves is 8?
I am looking for a reference for the proof of the following question following Theorem 5 in Mazur's Rational Isogenies of Prime Degree.
Theorem 5 There is a constant $C$ such that every elliptic ...
- 163
15
votes
1
answer
943
views
BSD conjecture for rank 1 elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that
$$\text{ord}_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$
Thanks to the work of Gross-Zagier and ...
- 253
19
votes
3
answers
1k
views
Points of elliptic curves over cyclotomic extensions
Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It ...
- 2,305
6
votes
2
answers
782
views
Books building up to the Gross-Zagier formula
I am an undergrad extremely interested in some applications of the Gross-Zagier formula for elliptic curves. I have a strong foundation in group theory and abstract algebra, and an understanding of ...
- 169
2
votes
2
answers
509
views
Question about Zeta Function of Singular Plane Curve
I am working on a project which involves learning about the zeta function (weil zeta function) for plane curves, but I do not know much algebraic geometry. (I do not know anything about schemes).
I ...
- 53
5
votes
0
answers
317
views
Elliptic curve sequences needed for universal forgery
Elliptic Curve Digital Signature Algorithm (ECDSA) admits universal forgery (UF) if the Attacker can solve the equation
$$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$-...
- 25.4k
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)$ ...
- 1,805
7
votes
1
answer
1k
views
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 ...
- 3,309
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)...
- 521
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 ...
- 433
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 ...
- 1,805
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 ...
- 51
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 ...
- 163
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)\...
- 205
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 $D$.
...
- 21
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
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. ...
- 111
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?
- 151
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 ...
- 103
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 ...
- 25.4k
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)-\...
- 1,537
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
views
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
views
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 ...
- 26k
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 ...
- 605
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 ...
- 1,905
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 $j(E)$ ...
- 2,827
5
votes
3
answers
3k
views
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 ...
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 $0$, ...
- 1,458
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 ...
- 1,458
7
votes
2
answers
2k
views
Tamagawa Number of Elliptic Curves over $\mathbb{Q}$
I am currently reading a paper by De Weger and one theorem in it proves a bound for the Tamagawa number of any elliptic curve defined over $\mathbb{Q}$.
I was wondering if anyone has any good ...
- 1,458
1
vote
1
answer
435
views
Elliptic subfields of a function field
Let $C$ be a curve and $K(C)$ be its function field of genus 2, where $K$ = $\mathbb{C}$.
The number of essential elliptic subfields of $K(C)$ is 0 or 2 or $\infty$.
Edit: I am looking for a proof. ...
- 471