**6**

votes

**1**answer

343 views

### 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 ...

**4**

votes

**0**answers

104 views

### Are torsion points in a semi-abelian variety over $\mathbb C_p$ bounded?

Let $A$ be a semi-abelian variety defined over (a subfield of) $\mathbb C_p$. Consider its $p$-adic topology with some (non-canonical) metric. Can we bound the distance of torsion points to $0$ with ...

**5**

votes

**1**answer

179 views

### Picard number of principally polarized abelian varieties

Let $A$ be an abelian variety of dimension $n$. Over $\mathbb{C}$, at least, it is known that the Picard number (that is, the rank of the Neron-Severi group of $A$) is less than or equal to $n^2$, ...

**4**

votes

**1**answer

258 views

### Fourier-Mukai transform for abelian varieties

Let $A$ be an abelian variety over $\mathbb{C}$, $L$ be a very ample line bundle on $A$, then the dual abelian variety is $\hat{A} \cong A/K(L)$ with $K(L)$ the kernel of surjective morphism $A \to ...

**5**

votes

**2**answers

159 views

### can all CM types be realized by Jacobians?

The question is kind of self contained, but let me develop a bit further.
Assume K is a CM field of degree $2g$, that is, a quadratic imaginary extension of a totally real field. A CM type of $K$ is ...

**3**

votes

**2**answers

317 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 ...

**2**

votes

**1**answer

301 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

**0**answers

102 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 ...

**0**

votes

**1**answer

321 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 ...

**0**

votes

**2**answers

232 views

### Picard number of abelian variety [closed]

I would like references or a result about the computation of the picard number of the jacobian of an algebraic curve.
What about the special case when the picard number of the Jacobian is one (is ...

**8**

votes

**1**answer

275 views

### What is the structure of the group of rational points of an abelian variety over a Laurent series field?

Let $K = \mathbb{F}_q((t))$, and let $A_{/K}$ be a nontrivial abelian variety. Then $A(K)$ is a compact $K$-adic Lie group. What can be said about its structure?
By way of comparison, if ...

**0**

votes

**1**answer

134 views

### Isogeny of abelian varieties

Suppose we have a curve $X$ (of genus $\geq 3$), and we know that $\{\phi_i : X \to E_i\ \textrm{ for } i = 1, ..., r\}$ are covers of degrees $d_i$ (with the $d_i$'s not necessarily all equal), ...

**1**

vote

**0**answers

108 views

### stably birational abelian varieties are isomorphic

Can anybody help me to prove the following result:
Proposition. Let $A$ and $B$ be abelian varieties over a field $k$ of characteristic zero. Assume that $A \times \mathbb{P}_k^n$ and $B \times ...

**2**

votes

**1**answer

109 views

### Dual of a Complex 2-Torus

Is a complex torus $A$ of dimension 2 always isomorphic to its dual torus (i.e. the torus obtained by taking the dual lattice), or are there counterexamples to this?

**5**

votes

**0**answers

139 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. ...

**0**

votes

**1**answer

180 views

### Could we construct the Jacobian variety of a smooth curve $C$ with genus $>2$ from its derived category $D(C)$?

Let's consider a smooth curve $C$ over $\mathbb{C}$. We know that the Jacobian variety $Jac(C)$ of $C$ is the moduli space of the degree $0$ line bundles on $C$. $Jac(C)$ is an abelian variety of ...

**9**

votes

**0**answers

166 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 ...

**1**

vote

**1**answer

89 views

### Degree of isogenies between (semi-)abelian schemes

Let $S$ be a connected noetherian normal scheme of dimension 0 or 1 (i.e. $S$ is a connected Dedekind scheme).
Let $f:G\to G'$ be a morphism of semi-abelian schemes over $S$. In their book on Neron ...

**1**

vote

**1**answer

213 views

### Example of non-modular elliptic surface?

In "On elliptic modular surfaces", Shioda proves some interesting theorems on smooth elliptic surfaces (admitting a section); he then focuses on "modular elliptic surfaces" and proves some more ...

**2**

votes

**0**answers

135 views

### Descent theory of line bundles on abelian varieties under isogenies (in char p>0)

I have a couple of questions regarding the descent theory of line bundles on abelian varieties under isogenies in positive characteristic.
Let $X$ be an abelian variety and $L\in Pic(X)$ a line ...

**3**

votes

**2**answers

166 views

### Quotient of an abelian surface by an antisymplectic involution

What can we say about the quotient of an abelian surface by an antisymplectic involution?

**1**

vote

**0**answers

127 views

### $k$-isogenies and $k$-subgroups of abelian varieties

Let $k$ be a field of char0, with algebraic closure $\bar{k}$. Let $A$ be an abelian variety over $k$ of positive dimension and let $d\geq 1$ be an integer.
Let $S(A,k,d)$ be the set of abelian ...

**1**

vote

**1**answer

179 views

### Nef divisors on abelian varieties

The following question stems from a question I already asked on MO:
Nakai-Moishezon theorem for abelian varieties
I would like to prove that if $L_0$ is an ample line bundle on an abelian variety ...

**5**

votes

**2**answers

305 views

### Image of abelian varieties

Let $k$ be an arbitrary field, and let $\varphi:A\to B$ be a morphism of abelian varieties over $k$.
If $k$ has characteristic zero, then $\varphi(A)$ has the structure of an abelian subvariety of ...

**2**

votes

**0**answers

97 views

### Subgroups-ideals correspondence for abelian varieties over $\mathbf{F}_p$

The question I have arose while reading Waterhouse's Thesis (Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521–560.), and motivates another question I recently asked.
...

**2**

votes

**1**answer

182 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 ...

**3**

votes

**1**answer

195 views

### Duality for rank one modules over a number ring

Let $K$ be a number field, and $R$ an order of $K$. Consider the category $\mathcal{M}$ of all finitely generated $R$-submodules of $K$. If $X$ is an object of $\mathcal{M}$ such that ...

**3**

votes

**0**answers

145 views

### alternate interpretations of Galois action on Tate module

Let $E$ be an elliptic curve over a field $K$, and let $\ell$ be a prime different from the characteristic of $K$. Consider the well-known short exact sequence of etale fundamental groups (geometric ...

**5**

votes

**0**answers

279 views

### A frustrating cohomology class on the moduli of abelian surfaces

Here's a very frustrating question that I have been stuck on for some time. I believe that my question could fit in a general framework of what happens when you restrict $L^2$-cohomology classes on a ...

**3**

votes

**1**answer

352 views

### Nakai-Moishezon theorem for abelian varieties

In Birkenhake and Lange's book, they prove a version of the Nakai-Moishezon theorem for complex abelian varieties that says that if $L_0$ is an ample line bundle on a complex abelian variety $X$ of ...

**0**

votes

**0**answers

82 views

### Morphisms of Neron models

Let $S$ be a connected Dedekind scheme with field of fractions $K$. Let $A_K$ and $B_K$ be abelian varieties over $K$, and let $A$ and $B$ be the Neron models over $S$ of $A_K$ and $B_K$ ...

**3**

votes

**3**answers

310 views

### General curves of genus 3 as plane sections of Kummer surfaces

Is it true that a general curve of genus 3 is a plane section of an appropriate Kummer surface in $\mathbb P^3$? By Kummer surface I mean image of a principally polarized Abelian surface w.r.t. the ...

**5**

votes

**0**answers

121 views

### Compactifying the space of indecomposable abelian varieties

Let $A_g$ be the moduli space of principally polarized abelian varieties and $A_g^0$ the open substack of indecomposable ones. Abstractly we know $A_g^0$ has a compactification with complement a ...

**3**

votes

**0**answers

164 views

### K-theory of categories of group schemes and abelian varieties

Let $k$ be a field (perfect, or characteristic zero if you want - I'm especially interested in when $k$ is a number field). Consider the categories $\mathsf{G}_k=\{\text{commutative affine group ...

**4**

votes

**0**answers

146 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 ...

**3**

votes

**2**answers

313 views

### Intersection multiplicity in abelian varieties

Suppose $A$ is an abelian variety, $X, Y$ are subvarieties of $A$ of complementary dimension,
Does every component of $X \cap Y$ contribute non-negatively to the intersection number?

**4**

votes

**1**answer

267 views

### Conductor of abelian varieties

Let $A$ be a non-zero abelian variety defined over a number field $F$. Let $v$ be a finite place of $F$, and let $f_v(A)$ be the usual conductor exponent of $A$ at $v$ (defined e.g. on p.500 of the ...

**22**

votes

**3**answers

2k views

### Products of primitive roots of the unity

Let $m>2$ be an integer and $k=\varphi(m)$ be the number of $m$-th primitive roots of the unity. Let $\Phi = \{ \xi_1, \ldots, \xi_{k/2}\} $ be a set of $k/2$ pairwise distinct primitive $m$-th ...

**2**

votes

**2**answers

129 views

### Ramification in Division field of Abelian Varieties II

This is a follow-up question after this
The set-up is almost the same as before,
Let $k$ be a number field, $p$ be a rational prime. Let $A$ be an abelian variety over $k$ which has a good ...

**2**

votes

**2**answers

229 views

### Intuitive meaning of $k$-polarized Abelian surface?

Are there any good way to understand $k$-polarized Abelian surfaces? I am aware that if $A \cong \mathbb{C}^2/\Gamma$ is $k$-polarized, the lattice $\Gamma$ can be taken of the form
$$
...

**1**

vote

**1**answer

213 views

### Nef classes on abelian varieties in positive characteristic

Thomas Bauer shows in http://arxiv.org/pdf/alg-geom/9712019v1.pdf that for a complex abelian variety a nef line bundle is numerically equivalent to an effective divisor (this is shown in Lemma 1.1). ...

**0**

votes

**0**answers

78 views

### field of definition of abelian varieties with extra endormorphism

Let $A$ be a complex abelian variety such that $\mathrm{End}(A)$ is strictly bigger than $\mathbb{Z}$.
Question: Is is true that $A$ is defined over $\overline{\mathbb{Q}}$?
This is of course what ...

**1**

vote

**1**answer

256 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...)
...

**1**

vote

**0**answers

189 views

### a reference for Kummer theory, with proofs ?

What is a standard reference for Kummer theory of semi-Abelian varieties ?
I need a complete exposition with detailed proofs. Also in prime characteristic,
although I am not sure what the statement ...

**4**

votes

**2**answers

347 views

### etale cohomology of an abelian variety and its dual

Let $A$ an abelian variety over a field $k$ and $A^{*}$ the dual abelian variety.
How can we relate the étale cohomology of $A$ with etale cohomology of $A^{*}$?

**1**

vote

**0**answers

66 views

### Bound for field of definition (vs field of moduli) of an abelian variety

Let $A$ be a principally polarised abelian variety over $\mathbb{C}$ of dimension $g$.
Let $K$ be the field of moduli of $A$.
Proposition. $A$ has a model defined over an extension $L$ of $K$ such ...

**2**

votes

**0**answers

75 views

### do commutative groups torsors have a point in an Abelian extension of the base field?

Let $A$ be a principal homogeneous space for a commutative algebraic group defined over a field $k$ that contains all roots of unity. Is it true that $A$ has a $K$-point for an extension $K \supset k$ ...

**3**

votes

**1**answer

166 views

### Simple abelian varieties over non algebraically closed fields.

I was wondering what people would normally mean by a simple abelian variety $A$ where $A$ is defined over a field $k$ that is not algebraically closed.
The definition I found in for example on the ...

**0**

votes

**1**answer

382 views

### When is an ample line bundle on an abelian variety base point free?

So, any line bundle $L$ on an abelian variety $X$ determines a type $(d_1,\ldots,d_g)$ where $d_i|d_{i+1}$. It's well known that if $d_1\geq 3$ then $L$ defines an embedding, that if $L$ has no fixed ...

**0**

votes

**1**answer

99 views

### Ramification in Division field of Abelian Varieties

This might be a very simple question, and that might be the reason that I could not find any reference on this.
My question is
Let $A$ be an abelian variety defined over a number field $k$, and ...