All Questions
1,978 questions
18
votes
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About isogeny theorem for elliptic curves
$K$ a number field, $G_K$ its Galois group, $E_1, E_2$ two elliptic curves defined over $K$. The isogeny theorem says that if for some prime number $\ell$, The Tate modules (tensored with $\mathbb{Q}$)...
- 1,503
8
votes
0
answers
485
views
Binary quadratic forms attached to supersingular elliptic curves over F_p?
The question that I have is a more precise version of an earlier one (1), posted by myself on MO a little bit ago. Sorry for repeating myself.
Let $p$ be prime number which for simplicity shall be ...
- 2,406
7
votes
0
answers
504
views
Constructing large rank elliptic curves by multiplying quadratic imaginaries by cubes so that all have same imaginary part
I've been thinking of the relation between elliptic curves of large rank and quadratic imaginary fields with large 3-rank class groups. There are quite a few papers constructing infinitely many ...
- 4,593
2
votes
2
answers
334
views
For which composite $N$ does $X_0(N)$ possess a non-cuspidal rational point?
According the the introduction to Mazur's Rational Isogenies of Prime Degree the following question was open in 1978:
Let $N$ be one of the integers 39, 65, 91, 125, or 169. Does the modular ...
- 3,268
2
votes
0
answers
321
views
Dimension of fibres of moment maps in characteristic $p$
Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that ...
- 3,545
22
votes
1
answer
2k
views
Which elliptic curves over totally real fields are modular these days?
As the title says. In particular, every elliptic curve over $\mathbb{Q}$ is modular; but what is the current state of the art for general totally real number fields? I assume the answer is ...
- 13.1k
19
votes
1
answer
1k
views
Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup?
I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf ...
- 3,625
1
vote
1
answer
693
views
compute the Kähler moduli of an elliptic curve
Say given elliptic curve $ \{ (x,y) | y^2 = (x^2-1)(x^2-k^2) \}$, what is the right form of the K$\ddot{a}$hler form and how to compute the K$\ddot{a}$hler moduli of this elliptic curve? Thank you.
- 103
2
votes
1
answer
660
views
Point at infinity on the moduli space of elliptic curves over finite field
Let $A_{1,1}$ be the j-line for elliptic curves over $\mathbb{C}$, $A_{1,1}\otimes \mathbb{F}$ is the mod p reduction(here $\mathbb{F}$ is the algebraic closure of a finite field), then can I say the ...
- 1,091
34
votes
7
answers
3k
views
What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$?
Some experts have a hunch that for any nonnegative integer $r$ there are infinitely many elliptic curves over $\Bbb Q$ with Mordell-Weil rank at least $r$.
The best empirical evidence for this hunch ...
- 3,268
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
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
4
votes
1
answer
679
views
A bound for the Manin constant
I recall that the Manin constant for a strong elliptic curve is a rational integer $c_E$ such that, for a modular parametrization $\phi: X_1(N) \to E$, one has $\phi^*(\omega_E)= 2\pi i c_E f(z)\...
- 310
15
votes
1
answer
2k
views
Fermat's Bachet-Mordell Equation
Fermat once claimed that the only integral solutions to $y^2 = x^3 - 2$ are $(3, \pm 5)$.
Fermat knew Bachet's duplication formulas (more precisely, Bachet had a formula for computing what we call $-...
- 32.6k
9
votes
1
answer
1k
views
Isogenies between Tate curves
Let $q$ and $q'$ be complex numbers with $0<|q|,|q'|<1$, and let $m$ and $n$ be positive integers.
Suppose that $q^m={q'}^n$. Then the map
$$
f:\mathbb{C}^\times/q^{\mathbb{Z}} \to \mathbb{C}^\...
- 27.2k
17
votes
1
answer
3k
views
How do you calculate the group scheme of E[p] for a an elliptic curve E in characteristic p?
I know that the answer is $\mu_p \times \mathbb{Z}/p\mathbb{Z}$ if $E$ is ordinary, and $\alpha_p$ if $E$ is supersingular, where $\mu_p$ and $\alpha_p$ are the kernels of Frobenius on $\mathbb{G}_m$ ...
- 821
16
votes
4
answers
1k
views
Geometric meaning of fiber of modular parameterization over a point of an elliptic curve?
Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, parameterization $\psi : X_0(N) \rightarrow E$, and a point $P \in E$, take the fiber $\psi^{-1}(P)$. Its points, being on $X_0(N)$, correspond ...
- 4,593
5
votes
3
answers
942
views
Square of an elliptic curve and projective plane
Let's assume one takes $E = \mathbb{C}^* / \langle p \rangle$ an elliptic (Tate) curve over the complex field ($p = e^{2 \pi i \tau}$ where $1, \tau$ are the 2 periods in additive notation; $\Im \tau &...
- 73
8
votes
3
answers
731
views
Fourier coefficients for elliptic curves on average
Fix a prime p, and look at elliptic curves in some family (e.g. all elliptic curves ordered by height). How often do the Fourier coefficients a_p occur? Are there any conjectures?
- 1,022
26
votes
7
answers
6k
views
When is a product of elliptic curves isogenous to the Jacobian of a hyperelliptic curve?
David's question Families of genus 2 curves with positive rank jacobians reminded me of a question that once very much interested me: when is a product of elliptic curves isogenous to the jacobian of ...
- 4,593
9
votes
1
answer
2k
views
CM rational points on modular curves
Dear MO Community,
I am trying to understand Mazur's 1976 notes "Rational points on Modular Curves" (which can be found in Springer Lecture Notes in Mathematics 601).
Let N be a prime number, and ...
- 2,827
17
votes
1
answer
1k
views
If p is a prime congruent to 9 mod 16, can 4 divide the class number of Q(p^(1/4))?
When $p$ is a prime $\equiv9\bmod16$, the class number, $h$, of $\mathbb Q(p^{1/4})$ is known to be even. In
[Charles J. Parry,
A genus theory for quartic fields.
Crelle's Journal 314 (1980), 40--71]...
- 5,422
6
votes
2
answers
754
views
Elliptic curves — general structure of the group
Let $K$ be a field and $E$ be an elliptic curve defined over $K$. It well understood the $K$-points on $E$ forms an abelian group. What is the structure of this group?(Depending on char($K$)?) Is it a ...
- 513
6
votes
0
answers
971
views
Curious propositon in "Les schemas de modules de courbes elliptiques"
Currently I am reading "Les schemas de modules de courbes elliptiques" by Deligne and Rapoport and I got myself seriously confused about the following proposition (in English translation):
(II ...
- 839
14
votes
2
answers
1k
views
Can a reductive group act non-linearly on a vector group?
Let $k$ be a field; I'm going to discuss linear algebraic groups over $k$. The question I'll pose is only interesting when the characteristic is $p>0$.
1. Some motivation
A vector group is an ...
- 3,246
17
votes
2
answers
2k
views
Quaternary quadratic forms and Elliptic curves via Langlands?
The content of this note was the topic of a lecture by Günter Harder at the School on Automorphic Forms, Trieste 2000. The actual problem comes from the article
A little bit of number theory by ...
- 32.6k
29
votes
0
answers
3k
views
What are the possible singular fibers of an elliptic fibration over a higher dimensional base?
An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one.
It is often required for the ...
- 3,022
0
votes
0
answers
520
views
Motivation of proof of Riemann-Roch for elliptic curve and generalizations
Given a lattice $L \subseteq \mathbb{C}$, Alain Robert defines a theta function as a meromorphic function such that $\theta(z+\omega)=a(\omega) e^{\pi h(\omega)(z+\frac{\omega}{2})} \theta(z)$ for all ...
- 15.4k
8
votes
3
answers
570
views
Variations on a theme of O'Bryant, Cooper and Eichhorn concerning power series over $\mathbb Z/2\mathbb Z$
Define 2 power series over the field $\mathbb Z/2\mathbb Z$ by $f=1+x+x^3+x^6+\dots$, the exponents being the triangular numbers, and $g=1+x+x^4+x^9+\dots$, the exponents being the squares. Write $f/g$...
- 5,422
2
votes
0
answers
381
views
elliptic curves with CM and hecke L-series
I know that if you have an elliptic curve E with complex multiplication, then the Hasse Weil L-series attached to it can be expressed in terms of Hecke L-series. Is there anything that can be said ...
- 21
2
votes
2
answers
570
views
Elliptic curves over proper variety over $\mathbf{F}_q$ isotrivial
Why is every elliptic curve over a proper (edit: smooth and geometrically connected) base over $\mathbf{F}_q$ isotrivial, i.e. is constant after base changing with $\bar{\mathbf{F}}_q$? If the moduli ...
- 227
2
votes
1
answer
510
views
hyperalgebras (positive characteristic)
The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$.
How to ...
- 829
9
votes
1
answer
777
views
Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0
In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in ...
- 139
4
votes
1
answer
4k
views
How is the period of an elliptic curve defined exactly?
I sometimes read $\int_{E(\mathbf{R})} \frac{dx}{2y + a_1x + a_3}$ and sometimes $\int_{E(\mathbf{R})} |\frac{dx}{2y + a_1x + a_3}|$. Furthermore, one has to choose an orientation on $E(\mathbf{R})$.
...
user19475
2
votes
1
answer
278
views
What is the family derived from the absolute Frobenius on the Hilbert scheme?
Let $f$ be a Hilbert polynomial, and $X := Hilb_h(P^d_{F_p})$ a Hilbert scheme defined over $F_p$. Then there is an absolute Frobenius map $F: X \to X$. I'm even interested in the case $f \equiv 1$, ...
- 27.9k
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
7
votes
0
answers
491
views
Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
- 4,450
14
votes
3
answers
4k
views
Transforming a Diophantine equation to an elliptic curve
I heard that the following problem lead to determine the rational points of an elliptic curve:
For which integers $n$ there are integers $x,y,z$ such that $x/y+y/z+z/x=n$. Could anyone show me why ...
- 183
17
votes
1
answer
2k
views
Hecke operators acting as correspondences?
This question is inspired by Relation between Hecke Operator and Hecke Algebra
I remember having heard of yet another way of looking at Hecke operators acting on the spaces of modular forms for ...
- 23.5k
15
votes
2
answers
2k
views
Does the p-adic Tate module of an elliptic curve with ordinary reduction decompose?
Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E$ an elliptic curve over $K$ with good ordinary reduction.
The p-adic Tate module $T_p(E)$ is (after tensoring with $\mathbb{Q}_p$) a 2-...
- 1,500
14
votes
1
answer
986
views
P-adic L-functions of nonabelian twists of elliptic curves
Let $E$ be an elliptic curve and $\rho$ an Artin representation of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. Then there is a "twisted L-function" $L(E, \rho, s)$, corresponding to the ...
- 37k
12
votes
5
answers
2k
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Introducing Cryptology to Undergraduates
This summer I am going to give some lectures to some REU students. I am still tossing around ideas for what I am going to talk about, but one thing I would at least like to give one or two lectures on,...
- 4,842
14
votes
3
answers
2k
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Convergence of L-series
I remember to have read that the L-function of an elliptic curve, which a priori only converges for $\Re s > \frac{3}{2}$ also converges at $s=1$ provided that the $L$-function
satisfies the ...
- 2,810
18
votes
3
answers
3k
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Lifting varieties to characteristic zero.
If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ ...
- 4,015
37
votes
3
answers
5k
views
Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...
- 45.7k
4
votes
1
answer
899
views
How to compute div(dx)
Let $C$ be an elliptic curve defined by $y^2=(x-e_1)(x-e_2)(x-e_3)$.
My question is how to determine the order of the differential $dx$ at infinity, $ord_{\infty}(dx)$.
- 141
9
votes
1
answer
1k
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Images of action of Galois on the Tate module of Elliptic Curve,
Let E be an elliptic curve over the rationals, and let $TE = \lim_\leftarrow E[n]$ be the Tate module of the elliptic curve. The action of the Galois group of $\bf Q$ gives rise to a representation $\...
- 818
8
votes
4
answers
3k
views
Class Field Theory for Imaginary Quadratic Fields
Let $K$ be a quadratic imaginary field, and $E$ an elliptic curve whose endomorphism ring is isomorphic to the full ring of integers of $K$. Let $j$ be its $j$-invariant, and $c$ an integral ideal of $...
- 2,827
10
votes
1
answer
1k
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Are there "reasonable" criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic?
Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite
dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus ...
- 52.9k
23
votes
1
answer
2k
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Wanted: Quadratic Space in Characteristic 2 as a Counterexample to a Theorem of Arf
Hi. Peter Roquette sent me an email asking for an example of a quadratic space in characteristic 2 having certain features. I have no idea on this, but maybe someone reading this does.
He would ...
- 50.6k