All Questions
1,203 questions
3
votes
2
answers
754
views
Elliptic curve E and Galois representation
Assume that an elliptic curve $E$ over $\Bbb Q$ has a reducible mod $p$ representation. Then
Q: Why is the semi-simplification of $E[p]$ the direct sum of ${\Bbb Z}/p{\Bbb Z}$ and $\mu_p$?
Next
Q:...
- 1,680
2
votes
2
answers
480
views
Lifting to char 0, references and questions
Suppose that I have a surface $S$, smooth proper over an algebraically closed (perfect?)field $k$ that lifts algebraically to some $S_W$ defined over a field of char 0. I am interested in properties ...
- 4,111
0
votes
1
answer
138
views
rank of Abelian schemes under ample hypersurface section
Let $k$ be an algebraic closure of a finite field, $\ell \neq \mathrm{Char}(k)$ be prime, $S/k$ a smooth projective geometrically connected surface and $C/k$ a smooth ample connected hypersurface ...
- 4,111
0
votes
2
answers
487
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Absolute Hodge implies Galois invariant?
Let $X$ be an Abelian variety defined over a number field $K$, suppose that it has a good reduction over a fine place $\mathfrak{p}$ of $K$. Let $G_{\mathfrak{p}}$ be the local Galois group for $K_{...
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1
answer
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Number of curves over a finite field
Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points ...
- 40.3k
1
vote
1
answer
695
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Rigidity lemma over non-algebraically closed field
I would like to extend the rigidity lemma (as in Mumford's "Abelian varieties") to the case in which the base field $k$ is not algebraically closed.
I found a suitable proof in the draft of "Abelian ...
CommunityBot
- 1
12
votes
2
answers
2k
views
Modularity theorem for abelian varieties
There's alredy two posts on MO about the extension of modularity to elliptic curves over fields other than $\mathbb{Q}$ ([1], [2]), and another one about general algebraic varieties [3].
What is ...
CommunityBot
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1
vote
2
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199
views
kernel of isogeny becomes constant after base change
Let $S = Spec(O_K)$ be the spectrum of the rings of integers of a number field $K$. Let $A/S \setminus T$ be an Abelian scheme over an open subscheme $S \setminus T \subseteq S$. Does the kernel of $n$...
- 113
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votes
1
answer
367
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Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication
Let $(A,a)$ be a principally polarised (with indecomposable polarisation) Abelian variety over $\mathbb C$. Assume that End(A) contains an order $R$ of a totally real number field of degree $>1$ ...
- 9,580
4
votes
1
answer
498
views
Are Abelian varieties (sometimes) globally $F$-split?
As defined by Karen Smith here, beginning of section 3? If $E$ is an elliptic curve, then it is when $E$ is ordinary. I wonder about higher dimension cases. Any references would be greatly ...
- 351
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votes
1
answer
212
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group structure on (subsets of) tropicalizations of Abelian varieties
In this paper Vigeland shows how one can define a group law on subset of a tropical elliptic curve, so that this group is homeomorphic to $S^1$. It is not clear to me what is the relationship between ...
- 3,380
1
vote
1
answer
262
views
Linear system on an abelian surface
On a K3 surface $S$, a linear system $|C|$ is said to be hyperelliptic if the corresponding map is of degree 2 and the image is of degree $g_a(C)-1$ in $\mathbb P^{g_a}$.
For $g_a(C) > 2$, if $|C|...
- 38k
6
votes
3
answers
811
views
Torsion group of the following elliptic curve
Let $p_1=2, p_2 = 3,\ldots,$ be the prime numbers, and define $n_i = \prod_{j=1}^i p_j$. Moreover, let $E_i $ be the elliptic curve defined by $y^2 = x^3 + n_i$.
Can one compute the torsion group $...
- 22.6k
9
votes
1
answer
633
views
Motives over finite field not generated by hyperelliptic curves
So the question is that, over a finite field, does there exist an abelian variety $A$ for which there does not exist a generically one-to-one morphism from a hyperelliptic curve $C$ to $A$.
p.s. A ...
- 19.2k
3
votes
0
answers
242
views
Ampleness of Hodge bundles over complex curves
Let $C$ be a smooth, proper and connected curve over
the complex numbers $\bf C$. Let ${\cal G}\to C$ be a smooth group scheme over $C$ and let $\epsilon_{\cal G}:C\to{\cal G}$ be its
zero-section. ...
- 3,076
6
votes
1
answer
519
views
Are there perverse sheaves on abelian varieties with small Euler characteristic?
Let $A$ be a simple abelian variety of dimension $g$. Let $K$ be an irreducible perverse sheaf on $A$. We know that $\chi(A,K)\geq 0$. (Corollary 1.4 of Franecki and Kapranov.) How small can $\chi(A,K)...
- 149k
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answer
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Why are torsion points dense in an abelian variety?
Hi everyone,
let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$ of characteristic $p\geqslant 0$. I'm trying to prove that the subgroup $A'$ which is the union of ...
- 4,745
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votes
0
answers
82
views
Abelian varieties/$p$-divisible groups are an integral category
A preabelian category is called integral if epimorphisms are stable
under pullbacks and monomorphisms are stable under pushouts.
A major property of integral category is that by inverting bimorphisms ...
- 31
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votes
1
answer
2k
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The Tate conjecture for abelian varieties
Let $k$ be a number field. Recall that Faltings proved the famous Tate conjecture, which states that for any abelian variety $X$ over $k$ and any prime $\ell$, the natural map
$$\mathrm{End}(X) \...
- 35.2k
14
votes
0
answers
1k
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Lifting Abelian Varieties to p-adic fields
Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...
- 1,453
7
votes
1
answer
425
views
showing that abelian varieties are de Rham *without* showing that they are crystalline
If $X$ is a smooth projective variety over a $p$-adic field $K$, then Faltings' Theorem says that the etale cohomology of $X_{\overline{K}}$ is crystalline.
There have been various steps towards this ...
- 11
2
votes
1
answer
801
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Canonical lifts from $\mathbb F_q$ and CM-theory
One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...
- 1,254
4
votes
2
answers
402
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Defining isogenies over smaller fields
I'm having some issues with abelian varieties and fields of definition. This already became clear in my previous question on Jacobians. Here's another question. If somebody can explain some nice facts ...
- 1,196
1
vote
0
answers
147
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Ampleness of the Canonical Bundle for Siegel Modular Varieties
Background
Throughout I only work with varieties over $\mathbb{C}$.
For $p$ a prime number, Let $Y(p)$ denote the modular curve parametrizing elliptic curves together with full $p$-torsion structure,...
- 2,834
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 $W_k$ ...
- 8,998
4
votes
1
answer
280
views
isomorphism of extensions by abelian varieties
Let $A$ and $G$ be abelian varieties over $\mathbb{C}$. An element $P$ of $\text{Ext}(A, G)$ is an exact sequence
$0 \to G \to P \to A \to 0$,
here one can give $P$ the structure of an abelian ...
- 2,834
14
votes
7
answers
4k
views
Is there any rational curve on an Abelian variety?
Is that true that there is no rational curves contained in an Abelian variety? If it's true, is that because abelian varieties are not uniruled? How do I know whether an abelian variety is not ...
- 8,998
2
votes
0
answers
160
views
Must the coordinates of a polynomial iteration have about the same size?
Original post. The following statements seem plausible (not to say intuitively obvious), but I do not see how to prove them.
Let us say that a polynomial mapping of $\mathbb{C}^2$ is reducible if ...
- 13.8k
0
votes
2
answers
610
views
Poincaré bundle and Weil pairing for Abelian schemes
In which situations is there a Poincaré bundle for Abelian schemes? In [Mumford, Abelian varieties] only the case of Abelian varieties is treated.
The same question for the Weil pairing $\mathscr{A}[...
CommunityBot
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11
votes
1
answer
675
views
Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)
Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in Goldman-...
- 40.3k
7
votes
0
answers
236
views
Invariant theory of $SL_2$ over a field of positive characteristic
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W$ be a finite dimensional $SL_2$-module over $k$. Let $V$ be the natural representation of $SL_2$.
What can be said - in ...
- 605
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votes
2
answers
484
views
abelian varieties with the same CM type are isogenous
Does anybody have a reference for the following fact?
All abelian varieties with complex multiplication and same CM type are isogenous over $\overline{\mathbb{Q}}$?
Here abelian variety with ...
- 41
1
vote
1
answer
344
views
Algebraic Hodge decomposition of CM abelian varieties
On p. 205 of Katz's paper entitled "p-adic L-functions for CM fields" Katz says that
"Shimura's algebraicity theorem, in our context, is an easy consequence of the fact that Hodge decomposition of ...
- 7,609
3
votes
1
answer
339
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Potential good reduction of abelian varieties
In Corollary 3 on page 498 of the article "Good reduction of abelian varieties" it says that, under some specified conditions, the minimal subextension $L/K$ of $\overline{K}/K$ over which an abelian ...
- 47.4k
3
votes
0
answers
225
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A question on Kähler differentials and cotangent spaces on schemes
I have the following question (should be easy for those who know something about the field):
On page 92 (97 of the old edition) of Mumford's book "Abelian varieties", the author talks about an ...
- 39
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votes
1
answer
249
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Conductor CM abelian variety
This is probably well known but I am not an expert in the subject.
Given an abelian variety $A$ of dimension $g$ with CM by $O_K$ where $K$
is a CM field of degree $2g$, let $N_A$ be the norm of the ...
- 3,597
19
votes
3
answers
2k
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Elkies' supersingularity theorem in higher dimension
The following is a theorem of Elkies:
Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero.
...
- 3,597
7
votes
0
answers
444
views
where do CM abelian varieties get good reduction?
Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ and assume that $A$ has complex multiplication by the ring of integers of a CM field $K$. Then $A$ has potentially good reduction, that is: ...
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1
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763
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What is the motivation for defining the conductor of an abelian variety?
Let $K$ be a $p$-adic field, and let $A$ be an abelian variety over $K$. The conductor of the abelian variety is often defined as $2u+t+\delta$, where $u$, $t$ and $\delta$ are invariants related to ...
- 47.4k
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votes
3
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1k
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Why study CM abelian varieties?
I know that abelian varieties of CM type have central importance in algebraic geomtry and number theory. There are many conjectures and concepts related to them like Andre-Oort, Coleman conjecture, ...
- 10.3k
5
votes
0
answers
287
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Nori fundamental group and etale fundamental group in positive characteristic
Let $X$ be a smooth projective surface over an algebraically closed field of char $p > 0$. Suppose that $\pi_{1}^{et}(X) = \{1\}$. Can Nori fundamental group scheme of $X$ be non-trivial?
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576
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What's known about the mod 2 reduction of the level l Jacobi modular equation?
Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...
- 5,422
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votes
1
answer
1k
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On morphisms to projective space arising from a linear system
Context: This question arose as I was reading the proof of Application 6.1 in Mumford's Abelian Varieties. However, I have extracted all of the relevant information below so this question should ...
- 123
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votes
0
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247
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Hodge structure of abelian surfaces
In my case, I have an abelian surface $A$ of (2,8)-polarization, and I have some finite group (may not be abelian group) $G$ acting on $A$ without fixed point. I want to understand when there is a ...
- 3,472
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votes
1
answer
1k
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the group of all biholomorphic group automorphisms of complex tori
My background is complex geometry, but when I confront complex tori, I feel it is easier to consider it as a compact connected complex Lie group although I just know the definition of Lie group.
Let $...
- 149k
4
votes
1
answer
354
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Covering of Abelian variety by product of elliptic curves
Let $A$ be an complex abelian variety of dimension $n$. Is it possible to find $n$ elliptic curves $E_1,\dots,E_n$ such that the product $E_1\times \dots \times E_n$ of the elliptic curves etale ...
- 66.3k
8
votes
1
answer
331
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If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion?
If $C$ is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center $Z(C)$ is fusion. More generally, if $C$...
- 28.1k
5
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0
answers
154
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Effect of Hecke transform on the Mumford-Tate group
Let $Sh_{K}(G,X)$ be a Shimura variety and $Z\subset Sh_{K}(G,X)$ be a special subvariety. $Z$ is given by a Shimura sub-datum $(H,Y)$ with $H\subset G$ an algebraic subgroup which I call the $Mumford-...
- 2,221
12
votes
0
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464
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Are automorphisms of abelian varieties detected by the formal group?
Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$.
Assume $k$ has characteristic $p$ and denote by $A(p)$ the $p$-divisible group of dimension $g$ associated with ...
- 12.1k
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0
answers
1k
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Katz--Mazur for abelian varieties
Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac 1N]...
- 18.7k