All Questions
1,203 questions
3
votes
1
answer
659
views
Exactness on rational points of algebraic groups
Let $k$ be a finite extension of the p-adic number field $Q_p$ and G be a connected algebraic (not affine) group over $k$. It is well-known (see e.g. [1] Proposition 3.1) that G decomposes as
$1\...
4
votes
1
answer
762
views
An explicit formula for Weil pairing on a complex torus
I begin by defining the Weil pairing in general (as in Oda's 1969 paper). My question is about an explicit formula for this pairing in the case of an elliptic curve over complex numbers.
Let $\lambda\...
2
votes
0
answers
191
views
degenerate abelian surfaces
I am wondering if the family of degenerate abelian surfaces constructed by K. Hulek, C. Kahn and S.H. Weintraub in "Moduli spaces of Abelian Surfaces: Compactification, Degenarations, and Theta ...
4
votes
1
answer
674
views
Inseparable Galois Cohomology
First let me give a general form of my question, and then I'll give some motivation and a more specific version of it. Let $K/k$ be a Galois extension of fields with Galois group $G$, and let $X$ be ...
5
votes
1
answer
499
views
Connected cycles of Shimura curves in $A_{g}$ not contained in larger Shimura subvarieties
Is there always a finite family of Shimura curves $(C_{i})$ in $A_{g}$ the moduli space of principally polarized abelian varieties of dimension $g(\geq 2)$, such that the union $\cup C_{i}$ is ...
1
vote
1
answer
493
views
Hasse-Weil L-Functions of CM Abelian Varieties
In Shimura's paper "On the Zeta Function of an Abelian Variety With Complex Multiplication", in his terminology, the `one-dimensional part' of the zeta function is identified with a Hecke $L$-function ...
1
vote
1
answer
522
views
Minimal polynomial of symmetric endomorphism on abelian variety
Let $(A,\Theta)$ be a principally polarized abelian variety over an algebraically closed field $k$, and let $f$ be a symmetric endomorphism of $A$ (that is, $f^\dagger=f$ where $\dagger$ denotes the ...
5
votes
1
answer
1k
views
Excellent schemes
I noticed that many results in positive characteristic assumes that the object of the theorem is excellent. I have looked up the definition of excellent and have tried to get a feeling for it, but all ...
34
votes
2
answers
3k
views
The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.
I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...
20
votes
4
answers
2k
views
Are there Néron models over higher dimensional base schemes?
Are there Néron models for Abelian varieties over higher dimensional ($> 1$) base schemes $S$, let's say $S$ smooth, separated and of finite type over a field?
If not, under what additional ...
1
vote
0
answers
117
views
Need information about particular kind of quotients of semisimple algebraic groups by free abelian discrete subgroups
Let me start with the simplest version of the question since already there I don't know anything.
For a complex number $q$, consider the quotient space $X_q:=\mathrm{SL}_2(\mathbb C)\left/\left\{\...
24
votes
1
answer
2k
views
When is "independence of l" known?
My question is for which varieties over local fields is "independence of l" known for
etale cohomology. Say $X/{\mathbb Q}_p$ is a complete non-singular variety and $W_l$ is the (complex) Weil-...
8
votes
1
answer
758
views
Do all 0-dimensional Shimura Varieties show up (as CM points) in $\mathcal{A}_g$?
Question: Let $S$ be a 0-dimensional Shimura variety. Does $S$ necessarily admit a morphism (in the category of Shimura varieties) to $\mathcal{A}_g$ for some $g\geq 1$? Here $\mathcal{A}_g$ is the ...
3
votes
0
answers
282
views
The uniform boundedness of rational torsion for traceless abelian surfaces over a function field
The function field analog of the theorem of Mazur-Kamienny-Merel (giving a universal bound in $[K:\mathbb{Q}]$ for the size of the $K$-rational torsion of an elliptic curve) is immediate just from the ...
3
votes
1
answer
193
views
Pathological behavior of Lie algebra under a map of abelian schemes
I am trying to understand Example 7.5/9 from the book "Neron models". There one has a discrete valuation ring $R'$ that is the localization of $\mathbb{Z}[\zeta_p]$ at $p$, so that the absolute ...
7
votes
1
answer
419
views
Weil height of an Abelian Variety with everywhere (potentially) good reduction
Background: Suppose that $E$ is an elliptic curve over $\mathbb{Q}$ with everywhere (potentially) good reduction. there are many ways to define the height of $E$, and I will be concerned with the ...
3
votes
1
answer
671
views
Coherent cohomology of an abelian scheme and base change
Let $f\colon A \rightarrow S$ be an abelian scheme of dimension $d$. I would like a reference or an argument for the fact that $R^1f_* \mathcal{O}_A$ is locally free of dimension $d$ and that its ...
8
votes
2
answers
657
views
Adjoining torsion points from abelian varieties
Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically ...
14
votes
0
answers
562
views
Am I missing something about this notion of Mirror Symmetry for abelian varieties?
This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s.
In the comments of the question, I was directed to the paper http://arxiv.org/abs/...
8
votes
0
answers
405
views
Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s
It is well known (cf. Dolgachev) that there is a beautiful notion of mirror symmetry for lattice-polarized K3 surfaces. That is, if we are given a rank $r$ lattice $M$ of signature $(1, r - 1)$ and a ...
7
votes
1
answer
339
views
Do complex tori contain quasi-projective open subsets?
Complex tori are not associated to projective varieties in general.
But can one find an open $U$ inside a complex torus $\mathbb C^g/L$ such that $U$ is the analytification of a quasi-projective ...
11
votes
2
answers
918
views
On a proposition in Hartshorne's paper "Ample vector bundles on curves"
In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following:
Let $A$ be an abelian variety [over an alg. closed field $...
3
votes
0
answers
337
views
Abel-Prym map for Prym-Tyurin varieties
Let $(J,\Theta)$ be the Jacobian of a smooth projective curve $C$, and let $i:P\hookrightarrow J$ be an abelian subvariety of $J$ such that $i^*\Theta\equiv e\Xi$ for some principal polarization $\Xi$ ...
24
votes
2
answers
2k
views
Have we ever proved any non-solvable case of reciprocity without the Langlands program ?
The reciprocity of the title is the following not completely well-posed problem:
Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe"
(in some sense) ...
13
votes
2
answers
768
views
Is there a proof of Warning's Second Theorem using p-adic cohomology?
Let $\mathbb{F}_q$ be a finite field, $n \in \mathbb{Z}^+$, and $f_1,\ldots,f_r \in \mathbb{F}_q[t_1,\ldots,t_n]$ with $\operatorname{deg}(f_i) = d_i$. Put $d = \sum_{i=1}^n d_i$ and suppose $d< n$...
2
votes
0
answers
442
views
Marten's proof of torelli theorem
I am trying to read the proof of torelli theorem by Henrik H.Martens "A new proof of torelli's theorem" Annals of mathematics vol78 no. 1 .The proof seems to me like using mysterious combination of 3 ...
7
votes
0
answers
294
views
Picard scheme of varieties over imperfect fields
Let $k$ be a field and $X$ a proper $k$-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a $k$-group scheme $\operatorname{Pic}_{X/k}$ which is locally of finite ...
9
votes
1
answer
863
views
Are there noncommutative extensions of $\alpha_p$ by $\mathbb{G}_m$?
Let $k$ be a field of characteristic $p > 0$ (algebraically closed, if you want; that doesn't make a difference). Consider a finite type $k$-group scheme $E$ that is a (central) extension of $\...
8
votes
1
answer
747
views
Deligne's exterior power
In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism
$$p : A^{\otimes n} \to A^{\otimes n}, ...
9
votes
1
answer
2k
views
Automorphisms of Generic Abelian Varieties
Automorphism groups of elliptic curves are very well understood. Of course, every elliptic curve has the automorphism $[-1]$ of order $2$. If we are over a (algebraically closed) field, this is the ...
6
votes
1
answer
211
views
Recursions for some binary theta series in characteristic 3
Define $A(0), A(1), A(2) \dots$ in ${\bf Z}/3[[x]]$ as follows. For $n$ in $\bf N$ let $s=3^{2n+1}$. Then $A(n) = \sum a_kx^k$ where $a_k$ is the mod 3 reduction of the number of representations of $k$...
6
votes
1
answer
2k
views
Endomorphism Ring of Simple Abelian Varieties
I know that if $A$ is a simple abelian variety over a number field $k$ with all endomorphisms defined over $K$ then $\mbox{End}(A_K)\otimes \mathbb{Q}$ is a division algebra with a positive involution....
4
votes
0
answers
209
views
Partial differential Equation over characteristic p
I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...
10
votes
1
answer
625
views
Can a division algebra have degree divisible by its characteristic?
I apologize in advance if this is easy, but I've tried Googling, and had no luck.
I'm currently working on a proof, and I realized in the course of writing that this proof will break if out there ...
3
votes
1
answer
391
views
Embedded resolution of curves on smooth varieties
As far as I understand, embedded resolution of singularities means the following: given a variety $X$ over an algebraically closed field, and a closed subvariety $Y$, there exists a birational map $f:...
15
votes
6
answers
3k
views
Generalizations of Belyi's theorem
Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic ...
0
votes
1
answer
118
views
Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector space
Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation.
I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of ...
1
vote
0
answers
140
views
symmetric theta structures and arithmetic subgroups
A symmetric theta structure is a theta structure that commutes with (a lift of) the natural involution $\imath: A \to A$ an an abelian variety. For simplicity I will assume that $A$ is a surface.
Now,...
2
votes
1
answer
196
views
Degree of a smooth curve in an abelian variety
Let $A$ be an abelian variety, $g$ be a positive integer and $\mathcal{L}$ be an ample line bundle on $A$.
Question : Is there a real $r>0$ such that, for all smooth curve $C$ of genus $g$ in $...
8
votes
1
answer
952
views
Rank 2 vector bundle on a product of elliptic curves
Let $E$, $F$ be two complex elliptic curves, and $A=E \times F$. Let us denote by
$\pi_E \colon A \to E, \quad \pi_F \colon A \to F$
the natural projections. For all $p \in F$ let us write $E_p$ ...
-1
votes
2
answers
199
views
Interpretation of $H_1(A_\mathbb{C}^{top},\mathbb{Q})$
In the paper "Sato-Tate Distributions and Galois Endomorphism Modules in Genus 2" (arxiv: http://arxiv.org/abs/1110.6638), the authors use the singular homology $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ ($...
6
votes
0
answers
328
views
Is a semiabelian algebraic space a scheme?
Let $S$ be a scheme and let $A$ be a commutative separated smooth $S$-group algebraic space of finite presentation each of whose geometric fibers is an extension of an abelian variety by a torus. Is $...
1
vote
1
answer
229
views
Divisors on an abelian surface
Let $A$ be an abelian surface given by the quotient of a product of two generic elliptic curves $E_1 \times E_2$ by the product $T_1 \times T_2$ of two translations by $2$-torsion points. Then $A$ ...
5
votes
0
answers
785
views
Is there an excplicit number field of definition for an Abelian Variety $A/\mathbb{C}$ with CM?
Consider a simple abelian variety $A/\mathbb{C}$ with sufficiently many CMs by $\mathcal{O}$, where $\mathcal{O}$ is an order in a CM field $K$. Specifically, $K$ is a CM field of degree $2g$, where $...
4
votes
1
answer
182
views
Reference or proof for the fact that $J(X_0(N))$ splits into abelian varieties with real multiplication
It´s known that $J_0(N) = J(X_0(N))= \bigoplus_f E(f)$ splits as a sum of abelian varieties parametrized by the Hecke eingenfunctions and that it´s an elliptic curve iff the Hecke eingenvalue is an ...
3
votes
2
answers
830
views
Why is the norm map dual to restriction under Tate local duality?
Let $L/K$ be a finite Galois extension of nonarchimedean local fields, and let $A$ and $A^t$ be dual abelian varieties over $K$. Tate local duality tells us that $A^t(K)$ and $H^1(K, A)$ are ...
2
votes
1
answer
771
views
Why is the Tate local duality pairing compatible with the Cartier duality pairing?
This question is a follow up to Why is the norm map dual to restriction under Tate local duality?
Let $A$ and $B$ be dual abelian schemes over a base scheme $S$. For an integer $n \ge 1$, consider ...
36
votes
2
answers
7k
views
Why polarization of abelian varieties?
Maybe this question is not suitable for here, but I don't think I would receive a satisfactory answer in Math StackExchange.
I could never understand the intuition behind polarization of abelian ...
3
votes
1
answer
276
views
Mumford-Ramanujam examples in characteristic p [and in Arakelov geometry]
For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric ...
0
votes
0
answers
188
views
Do principally polarized abelian varieties enjoy a genus expansion?
This is a vague question from an interested outsider:
It is well known that abelian varieties which arise as Jacobian of a curve (or a bit more general as Prym variety) are distinguished by the fact ...