All Questions
1,978 questions
7
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3
answers
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Atkin-Lehner involution and class number
I was told of a relation between the number of fixed points of the Atkin-Lehner involution and the class number of certain number fields.
Can someone point me to a reference where I could learn about ...
- 156k
1
vote
0
answers
332
views
Singular conics on certain algebraic surfaces
Let S be an algebraic surface in 3-dimensional complex projective space. Suppose that:
The degree of S is either 5 or 6;
The generic plane section of S is a curve of genus 1.
(Equivalently, the ...
CommunityBot
- 1
3
votes
2
answers
3k
views
Weil pairing and Miller's algorithm
I'm studying Weil pairing and its applications in cryptography. I already know that it can be defined like this:
$$w(P, Q) = (-1)^n\frac{f_P(Q)}{f_Q(P)}\frac{f_Q}{f_P}(\mathcal{O})$$
where
$\textrm{...
- 131
4
votes
1
answer
221
views
Do permutation modules of solvable groups have self-dual socle in characteristic 2?
I was searching through the small groups database in GAP to find counterexamples to a certain conjecture (which is not important here). I was checking non-nilpotent solvable groups and noticed that ...
- 44.4k
5
votes
0
answers
234
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Modular reduction of exceptional complex reflection groups
I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that $...
- 10.7k
7
votes
1
answer
2k
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Can proper-smooth base change be used to show that varieties cannot be lifted to characteristic zero?
Recall the following corollary to the proper and smooth base change theorems:
Let $\pi: X \to S$ be a proper, smooth morphism. Then the direct images $R^i \pi_* \mathcal{F}$ are locally constant ...
- 25.6k
2
votes
1
answer
555
views
How do you calculate the Euler factors of the imprimitive symmetric square at primes with bad reduction?
The reference for this question is Coates and Schmidt, Iwasawa theory for the symmetric square.
Let $G = \textrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}))$ and let $D_r \supseteq I_r$ be a ...
- 37k
5
votes
0
answers
875
views
Elliptic curves over finite fields and 2x2 matrices
Let $k$ be a finite field of order $p^a$ and characteristic $p$, and $\mathcal{C}$ a $k$ isogeny class of elliptic curves over $k$. Let $w$ be the (Galois conjugacy class of the) Weil $k$-number ...
- 2,406
11
votes
1
answer
763
views
Historical question about modularity of CM curves
I'm looking for the answer of who first proved modularity of CM curves? That is if $E$ is an elliptic curve over $\mathbb{Q}$ which has complex multiplication then there's a non-constant morphism ...
- 4,646
13
votes
2
answers
1k
views
Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?
Motivation
A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector ...
CommunityBot
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2
votes
1
answer
412
views
Torsion Points under SL-2(Z/nZ) [duplicate]
Possible Duplicate:
Torsion Points under SL_2(Z/nZ)
I would like to rephrase a question I asked three days ago and was closed for the unclear presentation.
Taking $T \in E[p]$, a point in the p-...
CommunityBot
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13
votes
1
answer
651
views
Help wanted with Chebotarev condition in characteristic 2
Having promised a longtime collaborator that I would clear my plate to finish up some joint work of ours, I am swallowing my pride and tossing up the following technical point of function field ...
- 50.7k
4
votes
2
answers
604
views
Adem-Wu relations from Bullett-Macdonald identities
Question. Let $p$ be a prime. Let $q$ be a power of $p$. Let $P^0$, $P^1$, $P^2$, ... be elements of some associative $\mathbb F_q$-algebra $A$. (Here, $P^i$ does not mean $P$ to the $i$-th power; ...
- 20.7k
5
votes
3
answers
2k
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The rank of a class elliptic curves
For elliptic curve $y^{2}=x(x+a^{2})(x+(1-a)^{2})$,($a$ is a rational number and does not equal 0,1,1/2),is its rank always 0?
- 3,268
8
votes
0
answers
833
views
Semistable Elliptic Curves and irreducible Galois representations
I am interested in the set of number fields $K$ having the property that for \emph{any semistable} elliptic curve $E$ defined over $K$, there exists a constant $c(E,K)>0$ such that
$$p>c(E,K)\...
- 37k
12
votes
5
answers
2k
views
Introducing Cryptology to Undergraduates
This summer I am going to give some lectures to some REU students. I am still tossing around ideas for what I am going to talk about, but one thing I would at least like to give one or two lectures on,...
- 3,867
4
votes
2
answers
694
views
Ample line bundle and Frobenius morphism on smooth toric variety
Let $k$ be an algebraically closed field of $\mathrm{char}(k)=p>0$, $X$ a smooth toric projective variety of $\dim X=n$, $F_X:X\rightarrow X$ the absolute Frobenius morphism of $X$. Then for any $\...
- 16.2k
3
votes
0
answers
282
views
Find the canonical subgroup of a CM curve with ordinary reduction!
Let $p$ be a prime and $K$ a quadratic imaginary field in which $p$ splits. Let $\mathcal{O}$ be an order in $K$ and $A$ an elliptic curve with CM by $\mathcal{O}$. Then $A$ can be defined over the ...
- 786
14
votes
3
answers
2k
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A question on K_1 of an elliptic curve
Consider an elliptic curve $E/ \mathbb{Q}$, with a regular model $\mathcal{E} / \mathbb{Z}$. We have (Beilinson regulator) maps
$$ K_1(\mathcal{E})^{(2)} \to K_1(E)^{(2)} \to H_D^3(E_{/ \mathbb{R}} , \...
- 20.8k
2
votes
1
answer
332
views
Ample bundle under Frobenius morphism
Let $k$ be an algebraically closed field of char($k$)=p>0, $X$ a smooth projective variety over $k$, $F:X\rightarrow X^{(1)}$ the relative Frobenius morphism. Let $E$ be an ample vector bundle on $X$. ...
- 35.2k
9
votes
3
answers
3k
views
Elliptic Curves over Global Function Fields
I am currently thinking about a problem, and I feel that by knowing more about elliptic curves over extensions of $\mathbb{F}_q(T)$, for $q$ a power of $p$ say, might lead to insight. I am also ...
- 406
2
votes
2
answers
802
views
Elliptic curves with arbitrarily large conductor
Recall that the analytic rank $r^{\rm an}(E)$ of a (modular)
elliptic curve $E$ is defined to be the order of vanishing of its
Hasse-Weil $L$-function $L(E,s)$ at $s=1$. A conjecture due to
Ralph ...
- 786
3
votes
1
answer
759
views
Mordell-Weil Group of Elliptic Surface
Suppose $ X \to \mathbb{P}^1 $ is an elliptic surface with section, with Weierstrass model defined over $ \mathbb{Q} $. If $ \sigma: \mathbb{P}^1 \to X $ is a torsion section with order $n$, then for ...
- 3,338
0
votes
0
answers
352
views
Liftability in positive characteristic
What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?
- 85
7
votes
2
answers
513
views
Tameness for the Galois closure of a map of curves
Say we are working over some $K=\overline{K}$, of characteristic $p>0$. Let $\phi: Y\rightarrow X$ be a nonconstant map of smooth projective curves. To this map we can associate a map $\psi: Z\...
- 839
15
votes
6
answers
6k
views
bad reduction for elliptic curves
Why do elliptic curves have bad reduction at some point if they are defined over Q, but not necessarily over arbitrary number fields?
CommunityBot
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3
votes
0
answers
340
views
Rank of Subgroup of Elliptic Curve
I'm currently looking at two rational points $ p, q $ on an elliptic curve $E$ over $ \mathbb{Q} $. SAGE tells me that $E$ has rank 5 and no torsion, and that $p$ and $q$ both have infinite order. ...
- 309
19
votes
3
answers
3k
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Regulators of Number fields and Elliptic Curves
There is supposed to be a strong analogy between the arithmetic of number fields and the arithmetic of elliptic curves. One facet of this analogy is given by the class number formula for the leading ...
- 13k
0
votes
0
answers
524
views
DeRham cohomology
The Poincare lemma fails in positive characteristic, since pth powers vanish under differention. My question is : is there still some kind of resolution of the local system k by considering some ...
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10
votes
1
answer
1k
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Which primes can divide orders of Tate-Shafarevich groups?
Heuristic arguments due to (I believe) Delauney predict that every prime divides the order of the Tate-Shafarevich group of infinitely many elliptic curves over $\mathbf{Q}$. However, is it even ...
- 65.4k
10
votes
3
answers
3k
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Why can projective varieties just have abelian group operations?
I just started to read Shimura - Automorphic forms and number theory (Lecture notes in mathematics, 54). On page 20 or so, he mentions that every projective variety which is an algebraic group, is ...
- 66.3k
4
votes
2
answers
339
views
Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular?
Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular?
- 20.5k
6
votes
1
answer
825
views
More on universal homeomorphisms
I would like to understand this notion better; where could I find some examples? In particular, I am interested in the following questions (and references for the answers).
Is a universal ...
- 27k
4
votes
1
answer
655
views
Period integrals of the fiber of elliptically fibered K3 manifolds
Suppose I have a smooth elliptically fibered K3 manifold
over $\mathbb{P}^1$ defined by the Weierstrass equation,
\begin{equation}
y^2=x^3+f(z)x+g(z)
\end{equation}
where $x,y,z$ are local ...
- 424
0
votes
1
answer
1k
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Elliptic curve over finite field: scalar multiplication
I'm implementing arithmetics for elliptic curves over secp256r1 as a homework assignment.
For scalar multiplication, the assignment specifically specifies that $k$ is "any hexidecimal encoded integer"...
- 3
6
votes
1
answer
393
views
finite quotients of fundamental groups in positive characteristic
For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental ...
CommunityBot
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3
votes
2
answers
1k
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supersingular elliptic curve in char. 2 or 3
Let $p=2$ or 3, and let $k$ be an algebraically closed field of char. $p.$ Let $E$ be the supersingular elliptic curve over $k$ (with $j=0$). Let $G$ be the automorphism group of $E,$ which has order ...
- 45.7k
2
votes
2
answers
852
views
Second stage of elliptic curve factorization via random walk/Pollard's rho in constant (or low) memory?
The second stage of elliptic curve factorization has the drawback of large memory usage.
Let $n=pq$, $E(\mathbb{Z}/n\mathbb{Z})$ is elliptic curve and $P$ point on $E(\mathbb{Z}/n\mathbb{Z})$.
On $...
16
votes
3
answers
2k
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Are there any rational solutions to this equation?
I am not sure if this is an appropriate question, but I was asked this by a colleague today and do not know how to answer it.
1) Are there any rational solutions to the following equation:
$$x^3-8x^2+...
- 5,422
32
votes
4
answers
4k
views
Modular curves of genus zero and normal forms for elliptic curves
This is maybe the first question I actually need to know the answer to!
Let $N$ be a positive integer such that $\mathbb{H}/\Gamma(N)$ has genus zero. Then the function field of $\mathbb{H}/\Gamma(N)...
- 818
11
votes
2
answers
863
views
Valuations and separable extensions
Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...
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6
votes
3
answers
730
views
When is an affine part of an elliptic curve isomorphic to an affine part of a norm equation?
Given a cubic number field and a basis $\{\gamma_1,\gamma_2,\gamma_3\}$ for it over the rationals, we can write down the norm equation $N(x_1\gamma_1+x_2\gamma_2+x_3\gamma_3)=1$. For almost all ...
- 4,593
1
vote
2
answers
582
views
Modular interpretation of an action of the linear group SL_2 on the cohomology of an elliptic curve
Let $E$ be an elliptic curve and $x,y \in H^1(E, \mathbb{Q})$ be a basis for the first rational cohomology group of $E$. There is an action of the linear group $SL_2(\mathbb{Q})$ on $H^*(E,\mathbb{Q})$...
- 23.5k
18
votes
5
answers
3k
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Why does the group law commute with morphisms of elliptic curves?
I know this should be pretty simple, but right now the only way I can see how to prove it is to sit down and write out explicit formulae for the group law, and see that everything works out. What's ...
- 40.3k
2
votes
1
answer
528
views
Is there an easy proof of the fact that the intermediate image functor respects weights?
It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j_{!*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially ...
- 1,576
4
votes
2
answers
402
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lower bound for torsion of abelian varieties
Let $A$ be an abelian variety defined over a field $K$ of characteristic $p>0$. Let $A[\ell]$ be the group of $\ell$-torsion points, $\ell\neq p$ a prime. Are there positive constants $C, \eta$ ...
- 5,050
12
votes
1
answer
708
views
Is there a canonical height on the Weil-Chatelet group?
Jon Hanke and I were just chatting and realized we didn't know the answer to the following question. If E is an elliptic curve over a number field, is there in any sense a "canonical height" on the ...
- 13.3k
0
votes
2
answers
2k
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non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 91
12
votes
3
answers
1k
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The order of the discriminant of a good-reduction elliptic curve
Notation. Let $p$ be a prime number, $K$ a finite extension of
$\mathbb{Q}_p$ and $E|K$ an elliptic curve which has good reduction.
The discriminant $d_{E|K}$ of $E|K$ is an element of the
...
- 18.7k
9
votes
2
answers
1k
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Counting isomorphism classes of elliptic curves with specific torsion
Generally speaking, I am interested in counting the number of $\mathbb{F}_p$- isomorphism classes of elliptic curves containing a specific torsion subgroup, and I was wondering if there were any ...
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