All Questions
1,978 questions
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Could the Kunneth decomposition of a motif depend on the choice of $l$?
Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ${\...
- 1,576
3
votes
1
answer
167
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Can the number of solutions $x(y^2-x-1)=n$ in $\mathbb{Z}$ (or $\mathbb{Z}[t]$) be unbounded?
Solutions of $x(y^2-x-1)=n$ are easy to enumerate assuming $n$ is factored. Appears a quadratic must have solutions $\mod \text{divisors of } \frac{n}{x}$ for each solution $(x,y)$. If this is ...
- 47.4k
3
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1
answer
288
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Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?
I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}_l$-adic cohomology, one can consider $Ch_l(M)=\sum (-1)^i\dim_{\mathbb{Q}_l} H^i(...
- 10.9k
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votes
1
answer
175
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An inseparable lift of a regular variety.
Let $X$ be a variety over an (imperfect) field $k$, that is regular as a scheme. Let $k'/k$ be an algebraic inseparable extension (I am interested in $k'$ being the perfection or the algebraic closure ...
- 19.6k
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1
answer
1k
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Supersingular elliptic curve dilemma
Let $E$ be a supersingular elliptic curve over a finite field of
characteristic $p$, and $\mathbb{F}_q\supset \mathbb{F}_{p^2}$ be a finite field large enough such
that all (absolute) endomorphisms of ...
- 57.6k
9
votes
3
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866
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Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z[t]$ be unbounded as n varies?
We've recently seen this question: Can the number of solutions $ab(a+b+1)=n$ for $a,b,n \in \mathbb{Z}$ be unbounded as $n$ varies? It appears initially plausible that the answer is yes, but evidently ...
CommunityBot
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5
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Writing down minimal Weierstrass equations
Let $E$ be an elliptic curve over $\mathbb Q_p$. It is possible that $E$ has bad reduction but then when you see $E$ as a curve over a finite extension $K$ of $\mathbb Q_p$, it obtains good reduction. ...
CommunityBot
- 1
9
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1
answer
650
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Neron models of elliptic curves with level N structure?
In the Deligne-Rapoport paper entitled "Les schemas de modules de courbes elliptiques" the following is written (I translated in english):
Let $E$ be an elliptic curve with $\Gamma(N)$-level ...
- 47.4k
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1
answer
434
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Choosing tau for elliptic curves over the rational numbers with prescribed ramification data
Let $r>2$ and let $b_1,b_2,\ldots,b_r$ be in $\mathbf{P}^1(\mathbf{Q})$. Let $B$ be the divisor $$B:= \sum [b_i].$$ We consider this data to be fixed. For $d>1$, we define $\textrm{Ell}(b_1,b_2,\...
- 9,497
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377
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Would an oracle for integral points on elliptic curves be a factoring oracle?
Oracle finding all integral points on genus 0 curves is a factoring oracle (e.g. $xy=n$ and $x^2-y^2=n$
I asked Can the number of solutions $xy(x−y−1)=n$ for x,y,n∈Z be unbounded as n varies? and ...
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0
answers
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elementary question on ECDLP
If $\mathbb{F}_q$ is a finite field and the elliptic curve E is defined over finite field $\mathbb{F}_p$ such that ECDLP is hard in E($\mathbb{F}_p$), where $q, p$ are prime and $q \ll p$. Let T $\in ...
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4
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The Frobenius morphism
I found the following list on the "Frobenius Page" by David Ben-Zvi, described by the author as "an outdated collection of intuitive ways to think about raising to the p-th power".
Generates a ...
- 10.8k
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1k
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Fake CM elliptic curves
Suppose one has an elliptic curve $E$ over $\mathbb{Q}$ with conductor $N < k^3$ for some (large) positive $k$, with the property that its Fourier coefficients satisfy
$$
a_p=0, \; \mbox{ for all }...
- 2,381
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273
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Hilbert space from the Tate pairing
Fix an elliptic curve $E$ over ${\Bbb Q}$ (or if you prefer, something more general over something more general). For each extension $F$ of ${\Bbb Q}$, the Néron-Tate height pairing gives
an inner ...
- 17.6k
3
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0
answers
713
views
Tunnell's theorem
Is it possible in some way, to use Tunnells's theorem to determine how long it will take a computer, to determine whether a number n, is a possible area of a rational right triangle?
- 8,467
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1
answer
646
views
Quotient by p-th roots of unity in characteristic p
Let $X$ be a variety over $k$ of characteristic $p>0$ (you can assume $k$ algebraically closed and $X$ normal) with an action of the group scheme of $p$-th roots of unity $\mu_p = {\rm Spec}\ k[\...
- 12.4k
2
votes
0
answers
415
views
bounds on regulator of elliptic curve,
Let E be an elliptic curve over Q with positive rank and trivial torsion structure. Is there any sort of upper bound (conjectural or unconditional) on the regulator of E in terms of the conductor of E?...
- 818
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2k
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Mirror symmetry for elliptic curves
Lets $E_{\tau}^{\rho}$ be the elliptic curve with complex structure given by $\tau$ in upper half plane and complexified Kahler form $\rho \frac{dz\wedge d\bar{z}}{2}$.( $\rho$ is in upper half plane ...
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Do the Standard Conjectures imply parts of the "Weil II" Riemann Hypothesis?
It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...
- 1,764
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1
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811
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Is the direct limit of Weil restriction of an elliptic curve a scheme?
In a discussion today on the Shafarevich-Tate group of an elliptic curve, the following structure and question came up. I will abuse many notations and be very vague about some things, but am very ...
- 7,108
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5
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Analysis of a quadratic diophantine equation
Hi! This is my first post on Math Overflow. I have two equations: $a(3a-1) + b(3b-1) = c(3c-1)$ and $a(3a-1) - b(3b-1) = d(3d-1)$. I'm trying to find properties of $a$ and $b$ that lead to solutions, ...
- 1,545
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1
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How do you describe vector bundles on elliptic curves?
Throughout "curve" means smooth projective curve over an algebraically closed field.
Motivation and Background
I read somewhere that Atiyah has classified vector bundles on elliptic curves. My ...
- 3,968
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1
answer
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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 ...
- 57.6k
18
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1
answer
5k
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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}$)...
- 22.6k
8
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answers
485
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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 ...
CommunityBot
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0
answers
504
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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
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2
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334
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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 ...
CommunityBot
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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
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1
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693
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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.
- 3,267
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1
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660
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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 ...
- 30.5k
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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 ...
CommunityBot
- 1
4
votes
1
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679
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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)\...
- 57.6k
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1
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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 $-...
- 8,945
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4
answers
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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 ...
- 8,945
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1
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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}^\...
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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 ...
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3
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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 &...
- 4,394
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2
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Legitimacy of reducing mod p a complex multiplication action of an elliptic curve?
I scoured Silverman's two books on arithmetic of elliptic curves to find an answer to the following question, and did not find an answer:
Given an elliptic curve E defined over H, a number field, ...
- 2,368
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3
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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?
- 4,671
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4
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The class number formula, the BSD conjecture, and the Kronecker limit formula
If K is a number field then the Dedekind zeta function Zeta_K(s) can be written as a sum over ideal classes A of Zeta_K(s, A) = sum over ideals I in A of 1/N(I)^s. The class number formula follows ...
- 13.6k
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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 ...
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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 ...
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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
4
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1
answer
899
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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, $ord_{\infty}(dx)$.
- 601
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520
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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 ...
- 12.6k
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1
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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 $E(\mathbf{R})$.
...
- 8,945
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2
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570
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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 ...
- 839
2
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0
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381
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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 ...
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votes
3
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3k
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congruent to 1 mod p
This is a somewhat vague question: for a prime number p, we often see that various counts come out to be 1 modulo p. What are the possible reasons for this?
Here are some I've encountered:
For some ...
- 39.9k
16
votes
3
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2k
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On Category O in positive characteristic
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$. In the case that $k$ has characteristic 0, there has been intensive study of the BGG category O of representations of ...
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