**0**

votes

**1**answer

231 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 ...

**10**

votes

**0**answers

227 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

125 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

242 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

172 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

193 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

143 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

225 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 $A$...

**6**

votes

**2**answers

389 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 $B$...

**2**

votes

**0**answers

102 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

223 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

207 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 $R=\textrm{End}...

**3**

votes

**0**answers

159 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 ...

**8**

votes

**0**answers

370 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 ...

**4**

votes

**1**answer

434 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

88 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 $B_K$ respectively....

**3**

votes

**3**answers

383 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 ...

**6**

votes

**0**answers

135 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

194 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 ...

**5**

votes

**0**answers

177 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

345 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

352 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 ...

**23**

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

155 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 ...

**3**

votes

**2**answers

348 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
$$
\begin{bmatrix}...

**1**

vote

**1**answer

239 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

87 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

314 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...)
...

**2**

votes

**0**answers

251 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 ...

**5**

votes

**2**answers

527 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

87 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

88 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

211 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

648 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

124 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 $N$...

**6**

votes

**1**answer

425 views

### Prym varieties as Jacobian varieties

A generic abelian variety of dimension 2 or 3 is a jacobian of a curve.
Is there a canonical way to determine a curve whose jacobian is a prym variety of a unramified double cover of a curve of genus ...

**0**

votes

**1**answer

98 views

### Is the stabilizer of an irreducible subvariety of an abelian variety irreducible ?

Let $A$ be a (semi-)abelian variety over an algebraically closed field $K$, and $X$ be a closed irreducible subvariety. Can $X$ have a non-trivial finite stabilizer ? By stabilizer, I mean the closed ...

**13**

votes

**0**answers

257 views

### Infinitely many curves with isogenous Jacobians

Let $g\geq 4$. Are there infinitely many compact genus $g$ Riemann surfaces with (mutually) isogenous Jacobians?
Does the situation change in positive characteristic?

**2**

votes

**2**answers

319 views

### Is the moduli space of ppAVs smooth?

Let $A_g$ be the moduli space of principally polarised abelian varieties of dimension $g$ over the complex numbers. (EDIT: I mean the coarse moduli space.) Is this smooth?
Since $A_g$ is the ...

**6**

votes

**1**answer

803 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

287 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 ...

**2**

votes

**0**answers

175 views

### About Alexeev and Nakamura's paper “on Mumford's construction of degenerating abelian varieites”

Is there anyone familiar with this paper? It seems to me it contains "some" typos and even some small elementary mistakes, which makes my reading very slow. Of course the key reason for my slow ...

**4**

votes

**1**answer

449 views

### Weil restriction of abelian schemes along finite étale (resp. finite locally free) morphisms

Q: Is there a simple proof of the fact that the Weil restriction of an abelian scheme along a finite étale morphism is an abelian scheme ?
Details: Let $S$ be a scheme and $f:S'\rightarrow S$ a ...

**4**

votes

**2**answers

508 views

### Moduli Spaces of Higher Dimensional Complex Tori

I know that the space of all complex 1-tori (elliptic curves) is modeled by $SL(2, \mathbb{R})$ acting on the upper half plane. There are many explicit formulas for this action.
Similarly, I have ...

**1**

vote

**0**answers

177 views

### Existence of a point on the Shimura variety of PEL-type correponding to a specific abelian variety

I have been puzzle by the following question for a while. Suppose that we have an a Shimura variety $Sh(G,h_0)$ given by some datatum $(L, V, \psi, h_0)$ such as in Section 4.9 of "Travaux de Shimura"...

**3**

votes

**0**answers

353 views

### Decomposition theorem for polarized abelian varieties in positive characteristic

In characteristic zero we have the following decomposition theorem for polarized abelian varieties: it gives an isomorphism between a PPAV and a product of PPAV's of lower dimension and is valid (as ...

**2**

votes

**1**answer

464 views

### Tate conjecture for abelian varieties over a finitely generated extension of an algebraically closed field

Let $K$ be a finitely generated extension of an algebraically closed field of characteristic zero, and $A,B$ abelian varieties over $K$.
Then is $Hom_K(A,B)\otimes \mathbb{Z_l} \cong Hom_{Gal(\bar{K}...

**0**

votes

**0**answers

170 views

### Nef and effective classes on abelian varieties

Is there any characterization of rational nef classes that don't come from effective $\mathbb{Q}$-divisors on abelian varieties? Is there any result along the lines of "Any nef $\mathbb{Q}$-divisor is ...

**0**

votes

**0**answers

205 views

### The formal Group of the dual Abelian Variety

For an abelian variety $A$ with formal group $F$, how is the formal group $F^\ast$ of the dual abelian variety $A^{\vee}$ related to $F$? In general, for a formal group $F$, is there a concept of dual ...

**13**

votes

**1**answer

854 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 $S$, $...