**1**

vote

**1**answer

196 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

317 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

100 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

197 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

197 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

153 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 ...

**7**

votes

**0**answers

324 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

374 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

83 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

341 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

127 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

180 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

151 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

329 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

303 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

142 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

274 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

223 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

84 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

277 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

223 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

408 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

72 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

79 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

178 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

482 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

107 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 ...

**6**

votes

**1**answer

362 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

84 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 ...

**10**

votes

**0**answers

208 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

285 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

640 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

252 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 ...

**1**

vote

**0**answers

152 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

336 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 ...

**1**

vote

**2**answers

330 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

155 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 ...

**3**

votes

**0**answers

324 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

435 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 ...

**0**

votes

**0**answers

144 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

175 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

779 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

**1**answer

225 views

### Is the canonical height of a totally p-adic point on an abelian variety bounded away from zero?

Inspired by the result of Schinzel and Smyth that a totally real number other than $0$ and $\pm 1$ has height at least $\frac{1}{2}\log \Big( \frac{1+\sqrt{5}}{2} \Big) = 0.240659\ldots$, Bombieri and ...

**3**

votes

**0**answers

128 views

### field of definition of isogenies of abelian varieties

Let $A$ be an abelian variety over a field $k$, and let $N$ be a finite subgroup of $A$. Suppose that $N$ is also defined over $k$, or at least that all Galois automorphisms fixing $k$ leave $N$ ...

**0**

votes

**1**answer

158 views

### Reference for Complex Abelian Varieties

I am looking for a reference which explains how theta functions, algebraically independent meromorphic functions, and line bundles all fit together in the context of complex tori. More explicitly, ...

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votes

**0**answers

386 views

### Points of minimum Arakelov height and harmonic arithmetical varieties

Added. (28/2) To put it less pompously (and more vaguely, less concretely), I wanted to relate the impression that it is the general rule that an Arakelov (i.e., geometric) height on an arithmetical ...

**12**

votes

**1**answer

433 views

### The torsion point count in higher dimension

It is an easy consequence of the Serre open image theorem that for the torsion point count on elliptic curves, the following possibilities arise.
If $E/\bar{\mathbb{Q}}$ is an elliptic curve without ...

**4**

votes

**2**answers

199 views

### Theta group representation

Let $(X,L)$ be a polarized abelian variety over $k=\overline{k}$, and let $K(L)$ be the kernel of the isogeny $X\to X^\vee$ that sends $x$ to $t_x^*L\otimes L^{-1}$. The theta group $\mathscr{G}(L)$ ...

**10**

votes

**0**answers

299 views

### Average ranks of abelian surfaces

Most people nowadays believe that over a fixed global field, $50$% of the elliptic curves have $0$ rank, $50$% have rank $1$, and $0$% have higher rank. A significant advance in this direction has ...