2
votes
1answer
100 views

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 ...
3
votes
2answers
256 views

Faltings height of a CM abelian variety

Let A be a CM abelian variety, say simple of dimension g, with $End(A) = O_K$, where $K$ is a CM field of degree $2g$. Is there an upper bound for the Faltings height $h(A)$ in terms of the ...
1
vote
0answers
93 views

algebraicity of Néron-Tate canonical height for Abelian varieties over global function fields

(transcendence of canonical heights) Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always ...
5
votes
0answers
126 views

Kernels and cokernels for morphisms of abelian schemes up to isogenies

For $S$ a noetherian scheme, let $\mathcal{A}(S)$ be the additive category of abelian schemes over $S$ and $\mathcal{A}_{\mathbb{Q}}(S)$ be the category of abelian schemes up to isogenies, i.e. ...
9
votes
0answers
152 views

Purity for abelian schemes up to $p$-isogenies

Let $S$ be a noetherian excellent regular scheme and $U\subset S$ an everywhere dense open of codimension $\geq 2$. For some fibered categories of geometric objects, it makes sense to ask whether the ...
2
votes
1answer
172 views

Purely additive reduction of Jacobian of Hyperelliptic curve

For general, let X be an abelian variety of dimension g. We say that X has 'purely additive reduction' at prime p if the dimension of the unipotent radical of the special fiber of the Neron Model of ...
4
votes
0answers
126 views

Semiabelian actions appearing in the toroidal campactification of a degenearting abelian varieties

Given a totally degenerated abelian variety $A_K$ (to make it easier) over a complete discrete valuation field $K$ with $R$, $\pi$ and $k$ the corresponding discrete valuation ring, uniformiser and ...
0
votes
0answers
97 views

How does one see level structure in the ell-adic Galois representation

This is probably a very easy question for those familiar with the arithmetic theory of abelian varieties because it is either possible or impossible for "trivial reasons". Let $A$ be an abelian ...
1
vote
1answer
243 views

Can the Albanese map be anything?

Sorry for the vague title. This question is about the Albanese map from the variety $M$ of canonically polarized varieties to the set of abelian varieties. (The variety $M$ is not of finite type...) ...
6
votes
1answer
223 views

Local Norm Mapping for Abelian Varieties

Let $A/K$ be an abelian variety defined over a nonarchimedean local field $K$ of characteristic $0$ and let $L$ be a finite extension of $K$. Consider the norm map $$A(L)\xrightarrow{N_{L/K}}A(K)$$ I ...
12
votes
1answer
701 views

Status of Grothendieck's conjecture on homomorphisms of abelian schemes

In [1] Grothendieck posits the following: Conjecture. Let $S$ be a reduced connected scheme, locally of finite type over Spec($\mathbf{Z}$) or a field $k$, $A$ and $B$ two abelian schemes over $S$, ...
2
votes
0answers
301 views

what are the possible CM-fields of PEL type shimura varieties ?

In the paper "Travaux de Shimura" section 6, Deligne had defined a PEL- type shimura variety, for the following datum $(F,E,D,\psi)$, with $F$ a totally real cubic field, and $E$ a imaginary ...
8
votes
0answers
170 views

Corresponding notion of unramified for motives (or de Rham cohomology)

The etale cohomolgoy of a variety $X$ over a number field $K$ is a Galois representation of $\mathrm{Gal}(\overline K/K)$ with some properties coming from $X$, e.g., it is unramified outside $S$ if ...
3
votes
0answers
152 views

Does the Albanese map satisfy Torelli's theorem

Let $M_h$ be the moduli space of canonically polarized varieties with Hilbert polynomial $h$. Let $M_h \to A_g$ be the Albanese map, with $g$ an integer which depends on $h$ and $A_g$ the moduli space ...
6
votes
1answer
345 views

For which fields does the isogeny theorem hold

Let $k$ be a field. We say that the isogeny theorem holds over $k$ if, for any abelian variety $A$ over $k$, there are only finitely many $k$-isomorphism classes of abelian varieties $B$ over $k$ ...
3
votes
1answer
299 views

Properties of subvarieties of a simple abelian variety

Let $A$ be a simple abelian variety over a field $k$. (For simplicity, we assume char $k =0$.) Let $X$ be a smooth projective geometrically connected variety over $k$ of positive dimension. Suppose ...
4
votes
0answers
245 views

Dieudonné modules over rings of charateristic zero

Dear Colleagues, would appreciate if you could recommend references, if such a theory exits, for the following question. Let $A$ be an Abelian scheme over $\text{Spec}(R)$, where $R$ is a subring of ...
3
votes
2answers
246 views

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 ...
6
votes
2answers
397 views

Jacobians defined over smaller fields

Let $L/K$ be an extension of number fields. Let $X$ be a curve over $L$ which can not be defined over $K$. Let $J(X)$ be the Jacobian of $X$ over $L$. In general, the Jacobian $J(X)$ probably ...
8
votes
1answer
357 views

Modularity of higher dimensional abelian varieties

In another question I asked about strategies for giving an effective version of the Shafarevich conjecture for abelian varieties over $\mathbb{Q}$. For elliptic curves, one can give a proof using ...
17
votes
3answers
1k views

Over which fields does the Mordell-Weil theorem hold?

According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...
3
votes
0answers
197 views

Does semi-stable reduction behave well with Weil restriction of scalars

Let $A$ be an abelian variety over a number field $K$ with semi-stable reduction over $O_K$. Does the Weil restriction $\textrm{Res}_{K/\mathbf{Q}}A$ of $A$ to $\mathbf{Q}$ have semi-stable reduction ...
8
votes
1answer
380 views

Mordell-Weil group of the universal abelian scheme

Let $n>2$ and let $k$ be either $\bf Q$ or a finite field whose characteristic is prime to $n$. Let $A_{g,n}$ be the moduli scheme, which represents the functor, which with every $k$-scheme $S$ ...
0
votes
2answers
246 views

Does the self-product of a $g$-dimensional abelian variety contain an abelian variety of dimension smaller than $g$ at some point

Let me be more precise than the title. (This will be my last attempt to do something with abelian varieties. Sorry for all the basic questions. The answers have been great!) Let $A$ be a simple ...
0
votes
1answer
200 views

Is any simple abelian variety covered by a non-simple abelian variety

Let $A/k$ be a simple abelian variety. Does there exist a non-simple abelian variety $B/k$ and a finite homomorphism $f:B\to A$ over $k$? I don't need $f:B\to A$ to be etale.
2
votes
1answer
215 views

Are abelian varieties degree two covers of some projective space

Let $A$ be an abelian variety over a field $k$ of dimension $g\geq 2$. There exists a finite morphism $A\to \mathbf{P}^g_k$. Here's the question. Does there exist a finite morphism $A\to ...
4
votes
1answer
305 views

Albert classification of rational endomorphism rings of simple Abelian varieties over finite fields

Recall the Albert classification of rational endomorphism rings with involution of simple Abelian varieties over arbitrary fields: Type I: totally real, trivial involution Type II and III: ...
2
votes
0answers
223 views

deRham cohomoloy of CM liftings of Jacobians

Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
4
votes
0answers
222 views

false elliptic curves and principal polarizations

Hi, Let $\Delta$ be a quaternion algebra over $\mathbf Q$ and let $\mathcal O_\Delta$ be a maximal order in $\Delta$. Recall that a false elliptic curve over a field $K$ is a pair $(A/K,i)$ ...
3
votes
1answer
307 views

Construction of Kummer map for abelian variety

Let $A$ be an abelian variety over the rational numbers $\mathbf{Q}$. Let $V=T_p A \otimes \mathbf{Q}_p$ be the $\mathbf{Q}_p$-Tate module of $A$. Let $G$ be the absolute Galois group of $\mathbf{Q}$. ...
2
votes
0answers
162 views

CM abelian variety from an algebraic Hecke character?

Hi, Given an algebraic Hecke character $\chi$ of a number field $k$ there should be a "rank 1 CM-motive" $M$ with $\overline Q$-coefficients such that $L(s,M) = L(s,\chi)$. This follows from the ...
1
vote
0answers
216 views

modern reference for Néron's “Quasi-fonctions et Hauteurs sur les Varietes Abeliennes”

Is there a modern reference for Néron's "Quasi-fonctions et Hauteurs sur les Varietes Abeliennes" http://www.jstor.org/pss/1970644 i.e. using Grothendieck's language of schemes and in English?
8
votes
1answer
247 views

finiteness of torsion points of an abelian variety over a totally real field?

Ribet has shown the following result: if $A$ is an abelian variety over a number field $E$, then the torsion subgroup of $A(E^{cyc})$ is finite. Here $E^{cyc}$ is the union $\bigcup_nE(\mu_n)$, ...
3
votes
2answers
272 views

Almost Northcott properties for heights of abelian varieties

Let $h$ be a function on the moduli space of abelian varieties of dimension $g$ over $\overline{\mathbf{Q}}$. Let $K$ be a number field and let $g\geq 2$ be an integer. Fix a real number $C$. Does ...
9
votes
1answer
472 views

Can we always find a curve which doesn't have semi-stable reduction

Let $K$ be a number field and let $g$ be a positive integer. Does there exist a smooth projective geometrically connected curve $X/K$ of genus $g$ such that $X$ does not have semi-stable reduction ...
5
votes
1answer
289 views

O-linear Weil-pairing on abelian varieties with real multiplication

Let $A/k$ be an abelian variety with real multiplication by some ring of integers $\mathcal O \subset F$. Let $n$ be an integer prime to the characteristic of $k$. We have the standart $e_n$ pairing ...
7
votes
2answers
413 views

Global Sections of the Identity Component of Neron model

Let $A$ be an abelian variety over a number field $K$ and consider the Neron model $\mathcal{A}$ of $A$ over $X=Spec{\mathcal{O}_K}$. If $\mathcal{A}^0$ is the identity component of $\mathcal{A}$, ...
3
votes
1answer
306 views

Maps on the identity components of Neron models

Any map $A \to B$ of abelian varieties of the same dimension over a global field $K$ induces a map $\mathcal{A} \to \mathcal{B}$ on the corresponding Neron models over $X$ (where ...
8
votes
1answer
787 views

About the Serre-Tate theorem

It is somehow a general principle that the (infinitesimal) local behavior of a representable moduli functor $X$ at some point $x$ is closely related to the deformation problem of the structure ...
4
votes
1answer
352 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
vote
1answer
422 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 ...
1
vote
2answers
463 views

Shimura datum of family of fake elliptic curves

Suppose we have a PEL type $(H,\phi ,*;T,O,V)$ where H is a rational nonsplit quaternion algebra, $\phi$is an embedding of Q-algebra $\phi : H-->M(2,R)$, and * is a positive anti involution of H; ...
0
votes
0answers
535 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 $A_{\ell^n}$)
22
votes
1answer
3k 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 ...
6
votes
1answer
447 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 ...
4
votes
0answers
530 views

p-divisible groups of superspecial abelian varieties

Let $p$ be a prime and $F$ be an algebraic closure of the field with p elements. I will consider abelian varieties over F up to prime-to-$p$ isogeny. Principal polarizations will be $Q$-homogeneous ...
10
votes
1answer
1k views

Crystalline cohomology of abelian varieties

I am trying to learn a little bit about crystalline cohomology (I am interested in applications to ordinariness). Whenever I try to read anything about it, I quickly encounter divided power ...
2
votes
4answers
962 views

reduction of CM elliptic curves

Can someone indicate how to prove the following equivalences for a CM elliptic curve $E$: (i) $p$ is inert in End($E$) (ii) $E_p$ is supersingular (iii) The trace of the Frobenius at $p$ is $0$ ...
15
votes
2answers
800 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 ...
3
votes
2answers
997 views

Brauer-Manin obstruction and Tate-Shafarevich group of an Abelian variety

I read that the Brauer-Manin obstruction $A(\mathbb{A}_K)^{\mathbf{Br}}$ of an Abelian variety $A$ over a number field $K$ equals (naturally?) its Tate-Shafarevich group $\mathrm{III}(A)$. Is this ...