All Questions
224 questions
7
votes
1
answer
1k
views
Modular polynomials for elliptic curves point counting
The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" ...
- 647
7
votes
1
answer
569
views
Discriminant of elliptic curve (Frey-Hellegouarch), j-invariant and positive definite kernels, similarities?
Consider the Frey-Hellegouarch curve given $a,b$ natural numbers:
$$y^3= x(x-\frac{a}{\gcd(a,b)})(x+\frac{b}{\gcd(a,b)})$$
Then the discriminant is given by $\Delta = \Delta(a,b) = 16 \left(\frac{ab(a+...
user6671
7
votes
1
answer
543
views
A constructive proof of the theorem of the cube
Do you know a constructive proof of the theorem of the cube ? More precisely, let $X$, $Y$, $Z$ be projective varieties (e.g., over an algebraically closed field $k$) with points $x$, $y$, $z$ ...
- 1,833
7
votes
1
answer
630
views
Cubic twist of elliptic curves and its rank
Let $E/\mathbb{Q}$ be an elliptic curve defined by $E: y^2 = x^3 + b$ (where $b \in \mathbb{Q}$).
Let $E_D$ be an elliptic curve defined by $E_D: y^2 = x^3 + bD^2$.
$E$ and $E_D$ are isomorphic over $\...
- 1,541
7
votes
1
answer
502
views
Is an elliptic curve that is isomorphic to its Frobenius conjugate defined over $\mathbb{F}_p$?
Let $p$ be prime and $q = p^n$. Let $E$ be an elliptic curve over $\mathbb{F}_q$, and let $E^{(p)}$ be the pullback of $E$ by the $p$-power Frobenius of $\mathbb{F}_q$. If $E$ is isomorphic (over $\...
- 2,663
7
votes
3
answers
908
views
Do there exist elliptic curves over schemes which have all primes as residue characteristics?
It's well known that there are no elliptic curves over Spec $\mathbb{Z}$, but it's unclear (to me at least) if the proof generalizes.
My question is: If $S$ is a connected scheme such that has every ...
- 10.7k
6
votes
0
answers
936
views
Can you get Siegel's theorem "for free" from modularity and Mazur's Eisenstein Ideal paper?
There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...
- 3,268
6
votes
1
answer
351
views
Adèlic points and algebraic closure
Consider $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$.
Let $F/K$ vary over all finite Galois number ...
user335418
6
votes
1
answer
1k
views
Generic Smoothness Type of Results in Positive Characteristic
Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f_*\mathcal{O}_X=\mathcal{O}_Y$, and $X$ is smooth.
We know that ...
- 313
6
votes
1
answer
367
views
Lifting of Frobenius on torsors over abelian varieties
This is related to my previous question Assume that $A$ is an abelian variety over a field $k$ of characteristic $p$, $\mathcal{L}$ is a line bundle on $A$. Assume that $A$ is ordinary and $\mathcal{L}...
- 7,377
6
votes
3
answers
2k
views
Direct proof of special case of Hasse's theorem for elliptic curves
Consider the elliptic curve $y^2 = x^3 + x$ over $\mathbb{F}_p$, where $p \equiv 1 \pmod 4$.
If memory serves correctly, the number of points (excluding the point at infinity) is $p - a$ where $a$ is ...
- 1,810
5
votes
2
answers
250
views
What integer value can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$?
Fix a positive integer $g$. What positive integer $N$ can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$ ?
For example, as there is no abelian varieties over $\mathbb Z$, $N$ ...
- 6,622
5
votes
1
answer
526
views
Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$
According to a conjecture there are no three
consecutive powerful numbers.
Necessary condition for this is integer solution of
$$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$
What are integer solutions of ...
- 25.4k
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 53
5
votes
3
answers
497
views
Is there a way to find any non-trivial $\mathbb{F}_p(t)$-point on the given elliptic curve?
Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$) and the elliptic curve
$$
E\!:y^2 = x^3 + (t^6 + 1)^2
$$
over the univariate ...
- 1,833
5
votes
1
answer
318
views
The name of the equianharmonic curve
I have found several references where the elliptic curve $y^2=x^3-1$ is called the equianharmonic curve, and, more often, where the half-period of this curve
$$ \omega_1 = \frac{\Gamma(1/3)^3}{4\pi} $$...
- 854
5
votes
2
answers
794
views
Is there a presentation of the cohomology of the moduli stack of torsion sheaves on an elliptic curve?
Let $E$ be your favorite elliptic curve, and let $Tor^m$ be the moduli stack of torsion sheaves of degree $m$ on $E$. This sounds horrible, but it's not so bad; it's a global quotient of a smooth ...
- 44.7k
5
votes
0
answers
387
views
Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems?
In this question Joel Bellaiche constructed an algebra, M, of modular forms for
gamma_0 (N) in finite characteristic (which he called p, but I'll call ell) and asked to know its structure. Matt ...
- 5,422
5
votes
1
answer
462
views
Solutions of a general diophantine equation
So it turns out that there exist positive integers a, b, c and n, such that $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=n.$ See Estimating the size of solutions of a diophantine equation
Now I am ...
5
votes
0
answers
171
views
Touchard / van der Pol's identity for the sum of divisors and an elliptic curve for perfect numbers
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$, satisifies the following recurrence relation ($n>1$):
$$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
- 213
5
votes
1
answer
948
views
Elliptic curves over the complex numbers: everything "well known"?
This may be a very naive question. We always hear about elliptic curves over the rational numbers, or over other arithmetically significant fields or rings.
But, are there open problems or recent ...
- 23.4k
5
votes
3
answers
2k
views
How to generate the n-torsion group in an elliptic curve
Let $E$ be an elliptic curve over a field $K$.
I was curious about the following sentence: "then the $n$-torsion on $E(\overline{K})$ has known structure, as a Cartesian product of two cyclic ...
- 131
5
votes
0
answers
243
views
Map associated to linear system onto curve is morphism
In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...
- 61
5
votes
1
answer
750
views
Upper bound for Hall's conjecture on separation of squares and cubes
Hall's (weak) conjecture is the statement that for all $\varepsilon > 0$ there exists a positive number $c(\varepsilon) > 0$ such that for all $x,y \in \mathbb{Z}$ with $y^2 \ne x^3$, that
$$\...
- 27k
5
votes
1
answer
471
views
On the elliptic curve $x(x+a^2)(x+b^2) = y^2$
Ajai Choudhry showed that special cases of the elliptic curve,
$$x(x+a^2)(x+b^2)=y^2\tag1$$
can be used to prove that,
$$u_1^7+u_2^7+\dots + u_9^7 = 0\tag2$$
has an infinite number of primitive ...
- 12.6k
4
votes
2
answers
864
views
When is the period of elliptic curve over the rationals transcendental?
Given an elliptic curve $E/\mathbf{Q}$, when is its period transcendental/algebraic?
user19475
4
votes
1
answer
326
views
Szpiro ratios of elliptic curves over $\mathbb{Q}$
For an elliptic curve $E/\mathbb{Q}$, let us denote by $\Delta_{\min}(E)$ the minimal discriminant of $E$ and $N(E)$ the conductor of $E$. Then it is well-known that $N(E) | \Delta_\min(E)$.
The ...
- 27k
4
votes
1
answer
415
views
3-, 6-, 12-descent for Z2xZ6 elliptic curves
We are trying to write a snippet of Magma code to clarify the steps in the simplified procedure of applying $3$-, $6$-, $12$-descent and hopefully resolve the missing generator of the following $\...
- 913
4
votes
1
answer
423
views
A generator needed for a Z/6 elliptic curve
We are searching for rank $8$ elliptic curves with the torsion subgroup $\mathbb{Z}/6$ using newly discovered families similar to Kihara's as described in
A. Dujella, J.C. Peral, P. Tadić, ...
- 913
4
votes
1
answer
827
views
Frobenius at ramified primes
Let $E$ be an elliptic curve defined over $\mathbf{Q}$, fix an odd prime $p>3$, let $T_p$ denote the $p$-adic Tate module of $E$, and let $V_p = T_p \otimes \mathbf{Q}_p$.
If the action of $G_\...
- 1,422
4
votes
1
answer
569
views
Lifting of Frobenius on semi-abelian varieties
Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume ...
- 7,377
4
votes
2
answers
699
views
About equivalent statements of the Birch and Swinnerton-Dyer Conjecture [closed]
The Birch and Swinnerton-Dyer Conjecture is well known in the current literature
http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture
My question is about the possible equivalent ...
- 49
4
votes
1
answer
918
views
Heisenberg group in characteristic two
I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V \...
- 3,605
4
votes
1
answer
176
views
$\#E(\mathbb{F}_p) \in \{p,p+2\}$ iff $p$ is of the form $27a^2+27a+7$
Related to this question.
Let $E / \mathbb{F}_p : y^2=x^3+2$.
Numerical evidence up to $2 \cdot 10^5$ suggests:
Conjecture: $\#E(\mathbb{F}_p) \in \{p,p+2\}$ iff $p$ is of the form
$27a^2+27a+7$ ...
- 25.4k
4
votes
1
answer
627
views
Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?
Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$.
For any $k$-vector space $V$, consider the canonical ...
- 33.8k
4
votes
1
answer
800
views
Abelian image of l-adic representation
For a number field F, let E/F be an elliptic curve with CM by a quadratic field K. Let $\rho_\ell: \text{Gal}_F \to \text{Aut}(T_{\ell}E)$ be the $\ell$-adic representation associated to E for some ...
- 235
4
votes
1
answer
600
views
Is it expected that every natural number is the rank of some elliptic curve over the rationals?
It is a well-known problem on the theory of elliptic curves that the rank of an elliptic curve (the number of generators of the free part of the Mordell group of the elliptic curve) cannot be ...
- 27k
3
votes
1
answer
343
views
Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism
MOTIVATION
Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...
- 5,422
3
votes
1
answer
2k
views
Fundamental characters of level 1,2, etc
I'm trying to understand the concept (definition) of the fundamental characters of level 1,2, etc. as Serre defined those in his Inventiones'72 paper and how they are related to Serre's conjecture. ...
- 31
3
votes
1
answer
213
views
5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians
I would like to know if there is something I can read to compute the following:
Let $H$ be a hyperelliptic curve of genus $2$ given by $y^2=x^5 + 10$ and let $J$ be its Jacobian.
How can I prove ...
3
votes
3
answers
483
views
Growth of the coefficients of the inversion of the $j$-invariant function
We have the $j$-invariant defined as
I have that
$$
j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k,
$$
where $q=e^{-2\pi t}$ ($\tau=it$) and $c_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$.
The inversion ...
- 515
3
votes
2
answers
412
views
Infinitely many elliptic curve with twist rank more than $1$ in specific case
Let $E/\Bbb{Q}$ be an elliptic curve. Let $D$ be a square free negative integer.
It is conjectured that 50％ of twist of elliptic curve $E_D$ has rank $0$ and $50％$ has rank $1$.
But is some particular ...
- 1,541
3
votes
0
answers
94
views
Dimension of a kernel of a cocycle map
Inspired by a previous question (Dimension of a kernel of a linear map) and some of the answers I was given I thought wheter I can generalize the question to the following:
Compute the kernel (or at ...
- 911
3
votes
1
answer
808
views
Conductor of a CM elliptic curve and its Grössencharacter
For a CM elliptic curve $E$ and its Grössencharakter, their conductors are both supported on bad primes of $E$. Moreover, by comparing their functional equation, there should be some obvious relations....
- 871
3
votes
1
answer
941
views
Abelian varieties with CM
In this site, I looked at a paper of Kazuma Morita claiming the BSD conjecture for the CM case
posted on his homepage (he made a mistake three years ago for full BSD).
But, I am interested in this ...
- 31
3
votes
1
answer
530
views
a question on CM elliptic curves
Let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$ defined by an equation
$y^2=4x^3-g_2x-g_3$
and let $\omega=\int_\gamma \frac{dx}{y}$ be the integral of the regular differential form $\...
- 31
3
votes
1
answer
321
views
Is there an infinite family of primes $q_{1},q_{2},...$ so that the rank of $E(\mathbb{Q}(\sqrt{-q_{i}}))$ equals that of $E(\mathbb{Q})$?
It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
Much less is known if $K$ is infinite-...
- 1,805
3
votes
1
answer
366
views
Descent for étale covers of proper regular models of elliptic curves
Let $K$ be a complete (but think Henselian suffice for purposes of this question) local field of characteristic $0$ with residue field $k$ of characteristic $p>0$ and ring of integers $R=\mathcal{O}...
- 5,986
3
votes
0
answers
126
views
FLT and integral points on elliptic curves
For integers $x,y,z,t,n$ define $S_n : xy(x+y)=t^n$.
For $ n > 2$, Fermat's Last Theorem implies there are no integral
solution on $S_n$ with $x,y$ coprime and $xy(x+y) \ne 0$ since $x,y,x+y$ are
...
- 25.4k
3
votes
1
answer
821
views
Formal group and formal completion of an elliptic curve
Let $A$ be a ring, $S$ the spectrum of $A$, $f : E \to S$ an elliptic curve.
Then assuming $f_*\Omega_{E/A}$ is free over $S$, $\hat{E}$ (the formal completion along the $0$-section.) $ \cong \...
- 1,364