All Questions
1,203 questions
10
votes
3
answers
3k
views
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
1
vote
1
answer
569
views
references for abelian schemes
Hi,
I have a very basic question.
I am looking for references explaining how to construct explicitily a Jacobian starting from a curve or examples of projective equations for an abelian scheme. I ...
- 231
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
5
votes
1
answer
571
views
Selmer of an abelian variety versus that of its dual.
What is the precise relationship between the Selmer group of an abelian variety and that of its dual? For instance, does the vanishing of one not imply the same for the other?
To fix ideas, let $A$ ...
- 1,141
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
- 1
5
votes
1
answer
901
views
Tamagawa numbers of abelian varieties and torsion.
Let $A$ be an abelian variety defined over a number field $K$. Fix a prime $v \subset \mathcal{O}_K$, with underlying rational prime $p$. What relationship, known or conjectural (if any), should there ...
- 1,141
5
votes
0
answers
352
views
Why is Pic^0(C) of a curve C a variety?
Let $C$ be an abstract non-singular curve.
I'm having a hard time finding a reference for why $\text{Pic}^0(C)$ is a variety.
Any pointers towards a reference would be appreciated.
- 4,586
5
votes
1
answer
796
views
moduli space of abelian varieties of CM-type
Fix a CM-field $K$ of degree $2g$, and a natural number $n$ which is a multiple of $g$. Write
$\tau_1, \tau_2, \ldots, \tau_g, \rho \tau_1, \rho \tau_2, \ldots, \rho \tau_g$
for the different ...
- 1,832
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) ...
- 423
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
views
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
0
votes
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 91
8
votes
3
answers
762
views
possible CM-types of abelian varieties
Fix a CM-field $K$ of degree $2g$. Let $A$ be a polarized abelian variety of dimension $n$ over $\mathbb{C}$, with an isomorphism $\theta : K \to End_{\mathbb{C}}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$. (...
- 65.4k
12
votes
1
answer
721
views
Galois action on one-dimensional quotients of l-adic cohomology
Let $A$ be an abelian variety of dimension $g$ over a number field $K$, and $\ell$ be a rational prime. Suppose that the Galois action on the $\ell$-adic cohomology $H^k(A, \mathbb{Z}_\ell) \otimes_{\...
- 1,832
5
votes
0
answers
129
views
global units on moduli spaces of abelian varieties
This is a question from a colleague.
Let $A_{g,d,n}$ be the coarse moduli space over $\mathbb Z$ (of the moduli stack, or assume $n\ge3$ if you like) of abelian schemes of relative dimension $g,$ ...
- 4,265
1
vote
2
answers
393
views
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
288
views
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
0
votes
1
answer
175
views
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
9
votes
3
answers
1k
views
Is there an intrinsic way to define the group law on Abelian varieties?
On an elliptic curve given by a degree three equation y^2 = x(x - 1)(x - λ), we can define the group law in the following way (cf. Hartshorne):
We note that the map to its Jacobian given by $\...
- 6,825
6
votes
1
answer
305
views
What is the closure of product loci in A_g?
Let $A_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$. For any partition of $g$, one can consider the corresponding locus inside $A_g$ of products of lower-...
- 42.9k
6
votes
1
answer
737
views
Tate models for semistable algebraic varieties with mixed reduction over a local field
It's known that if $A$ is an abelian variety of totally multiplicative reduction over a p-adic field K, then, after taking a finite field extension, it becomes isomorphic, as a rigid analytic group, ...
- 7,972
2
votes
0
answers
1k
views
deformation of abelian varieties
$k$ is a field of characteristic p, $C_k$ is the category of all artinian local rings with residue field an extension of $k$. $A$ is a dim-$g$ abelian variety over $k$, $L$ is a CM field with $[L:\...
- 1,091
7
votes
1
answer
807
views
$2$-torsion line bundles on abelian varieties
Let $\mathcal{A}_{g,D}$ be the moduli space of abelian varieties of dimension $g$ and polarization $D$ of type $(d_1, \ldots, d_g)$.
Let $\mathcal{M}$ be the moduli space parametrizing pairs $(A, \...
- 66.3k
31
votes
4
answers
5k
views
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
6
votes
1
answer
2k
views
Abelian subvarieties of abelian varieties --- reference request
This question may be too naive, in which case I apologise in
advance. Anyway, it is a well-known fact (see e.g. Milne's notes)
that any abelian variety A has only finitely many direct factors
up to ...
CommunityBot
- 1
10
votes
2
answers
1k
views
Failure of Theorem of the Cube?
I am trying to understand the theory of cubical structures and am interested in knowing if a disconnected commutative group variety whose identity component is a semi-abelian variety satisfies the ...
- 19.6k
2
votes
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
6
votes
1
answer
276
views
Can the simplicity of abelian varieities be implied by the reduction
A is an abelian variety over number field K, with simple good reduction at a finite field $\kappa$, can we deduce that $A$ itself is simple?
- 11.1k
2
votes
0
answers
163
views
The automorphism group of a particular ppav
Let $E$ be the elliptic curve $y^2=x^3-x$ defined over $\mathbb F_5.$ It is ordinary with $j=-2$ and $End_{\mathbb F_5}(E)=\mathbb Z[i],$ where $i:(x,y)\mapsto(-x,2y).$ So $Aut_{\mathbb F_5}(E\times E)...
- 4,265
11
votes
0
answers
1k
views
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
6
votes
2
answers
1k
views
About isogenies of abelian varieties
Why it is true that, over an algebraically closed field, any abelian variety is isogenous to a principally polarized abelian variety?
- 13.1k
5
votes
1
answer
842
views
an exercise about elliptic surface in Beauville's book
In Beauville's "Complex Algebraic Surfaces", given an elliptic surface $f : X \to C$ with a generic fiber $E$. Then either $\text{Alb}(X) \cong \text{Jac}(C)$ or there is an exact sequence of abelian ...
- 839
4
votes
1
answer
660
views
isogenies between abelian varieties that induce isomorphisms?
Let $\varphi : A \to B$ be an isogeny between 2 abelian varieties of dimension $g$. Are there known conditions for the $\ker\varphi$ so that this induces an isomorphism between $A$ and $B$? For ...
- 65.4k
0
votes
0
answers
555
views
étale cohomology with values in the $\ell$-torsion of an Abelian scheme
Let $S/\mathbf{F}_q$ be a $d$-dimensional smooth projective variety and $A/S$ be an Abelian scheme. Is there an easy description of $H^0(S, A(\ell)(d-1))$?` ($A(\ell)$ = union of $A_{\ell^n}$)
- 227
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
1
vote
1
answer
787
views
Morphism between polarized abelian varieties
Is it true that if an isogeny between two principally polarized abelian varieties respects the polarization, then it is in fact an isomorphism?
- 6,290
23
votes
1
answer
2k
views
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 ...
- 50.7k
2
votes
1
answer
428
views
equation for abelian varieties with a given polarization
Let $A$ be an abelian variety of dimension g and a polarization $L$ of type $(d_1,.....,d_g)$ (let alone the case $d_i=d_j,$ $\forall i, j$). What is the degree of the generators of the homogeneous ...
- 16k
2
votes
0
answers
290
views
quasi-trigonal curves
I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory ...
- 3,779
16
votes
1
answer
684
views
Characterizations of complex Abelian varieties (especially 3-folds) among projective nonsingular varieties?
If $X$ is a complex Abelian variety of dimension $g$, then
The canonical sheaf is trivial
$\dim {\rm H}^i(X; \mathcal{O}_X) = \binom{g}{i}$.
When $g =1,2$, then any connected, projective ...
- 5,137
5
votes
3
answers
2k
views
Modular forms reference
If f is a weight 2 newform on $\Gamma_1(N)$ then there exists an abelian variety Af whose endomorphism algebra is isomorphic to the field generated by the coefficients of f.
I've seen this proven in ...
- 65.4k
36
votes
1
answer
9k
views
Fontaine-Mazur for GL_1
For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified ...
- 21.3k
8
votes
0
answers
649
views
Automorphic representations attached to abelian varieties
Let $A$ be an abelian variety defined over $\mathbb{Q}$, of dimension $d$. It is widely expected that there is an automorphic representation $\pi_A$ of $GL(2d)/\mathbb{Q}$ whose L-function agrees ...
- 13.1k
7
votes
3
answers
3k
views
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
answers
2k
views
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 ...
- 52.9k
7
votes
1
answer
701
views
$p$-torsion in the Mordell-Weil group of Abelian varieties injecting in reduction
Let $K$ be a number field and $\mathfrak{p}$ be a place of good reduction. It is easy to see that the reduction map on prime-to-$p$ torsion $A(K)[p'] \hookrightarrow A_{\mathfrak{p}}(\kappa(\mathfrak{...
- 57.6k
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 ...
- 12.4k
4
votes
2
answers
923
views
What is the correct formulation of the CDE triangle?
The CDE triangle with C Cartan matrix and D decomposition matrix is well-known for finite groups. I have not seen a full account of this for finite dimensional algebras which I find surprising as ...
- 44.7k
2
votes
1
answer
837
views
Betti Cohomology of singular Kummer Surface
Let $A$ be a complex torus of (complex) dimension 2 and $X$ the associated Kummer variety $A/\sigma$, where $\sigma(x)=-x$. I would like to compute the cohomology of $X$ with $\mathbb{Z}$ coefficients....
- 22.6k