# Tagged Questions

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votes

**1**answer

167 views

### Canonical lifts from $\mathbb F_q$ and CM-theory

One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...

**5**

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**0**answers

121 views

### Are there perverse sheaves on abelian varieties with small Euler characteristic?

Let $A$ be a simple abelian variety of dimension $g$. Let $K$ be an irreducible perverse sheaf on $A$. We know that $\chi(A,K)\geq 0$. (Corollary 1.4 of Franecki and Kapranov.) How small can ...

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votes

**2**answers

100 views

### Poincaré bundle and Weil pairing for Abelian schemes

In which situations is there a PoincarĂ© bundle for Abelian schemes? In [Mumford, Abelian varieties] only the case of Abelian varieties is treated.
The same question for the Weil pairing ...

**3**

votes

**2**answers

169 views

### abelian varieties with the same CM type are isogenous

Does anybody have a reference for the following fact?
All abelian varieties with complex multiplication and same CM type are isogenous over $\overline{\mathbb{Q}}$?
Here abelian variety with ...

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vote

**1**answer

122 views

### Algebraic Hodge decomposition of CM abelian varieties

On p. 205 of Katz's paper entitled "p-adic L-functions for CM fields" Katz says that
"Shimura's algebraicity theorem, in our context, is an easy consequence of the fact that Hodge decomposition of ...

**2**

votes

**1**answer

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 ...

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vote

**0**answers

70 views

### A question on Kähler differentials and cotangent spaces on schemes

I have the following question (should be easy for those who know something about the field):
On page 92 (97 of the old edition) of Mumford's book "Abelian varieties", the author talks about an ...

**4**

votes

**1**answer

134 views

### Conductor CM abelian variety

This is probably well known but I am not an expert in the subject.
Given an abelian variety $A$ of dimension $g$ with CM by $O_K$ where $K$
is a CM field of degree $2g$, let $N_A$ be the norm of the ...

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**0**answers

128 views

### where do CM abelian varieties get good reduction?

Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ and assume that $A$ has complex multiplication by the ring of integers of a CM field $K$. Then $A$ has potentially good reduction, that is: ...

**11**

votes

**3**answers

592 views

### Why study CM abelian varieties?

I know that abelian varieties of CM type have central importance in algebraic geomtry and number theory. There are many conjectures and concepts related to them like Andre-Oort, Coleman conjecture, ...

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**0**answers

155 views

### Hodge structure of abelian surfaces

In my case, I have an abelian surface $A$ of (2,8)-polarization, and I have some finite group (may not be abelian group) $G$ acting on $A$ without fixed point. I want to understand when there is a ...

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**1**answer

150 views

### Covering of Abelian variety by product of elliptic curves

Let $A$ be an complex abelian variety of dimension $n$. Is it possible to find $n$ elliptic curves $E_1,\dots,E_n$ such that the product $E_1\times \dots \times E_n$ of the elliptic curves etale ...

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**0**answers

85 views

### Effect of Hecke transform on the Mumford-Tate group

Let $Sh_{K}(G,X)$ be a Shimura variety and $Z\subset Sh_{K}(G,X)$ be a special subvariety. $Z$ is given by a Shimura sub-datum $(H,Y)$ with $H\subset G$ an algebraic subgroup which I call the ...

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**0**answers

119 views

### Are automorphisms of abelian varieties detected by the formal group?

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$.
Assume $k$ has characteristic $p$ and denote by $A(p)$ the $p$-divisible group of dimension $g$ associated with ...

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**0**answers

291 views

### Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac ...

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votes

**1**answer

320 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 ...

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**0**answers

92 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 ...

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votes

**1**answer

155 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

252 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 ...

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**2**answers

154 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 ...

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votes

**2**answers

249 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

205 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 ...

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vote

**0**answers

93 views

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

(transcendence of canonical heights)
Is the NĂ©ron-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 ...

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votes

**1**answer

204 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 ...

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votes

**2**answers

225 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 ...

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votes

**1**answer

129 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), ...

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**0**answers

103 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 ...

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votes

**1**answer

99 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?

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125 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. ...

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votes

**1**answer

175 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 ...

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**0**answers

150 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 ...

**0**

votes

**1**answer

79 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 ...

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vote

**1**answer

209 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 ...

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votes

**0**answers

128 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 ...

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votes

**2**answers

148 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?

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**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

169 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

297 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

**1**answer

171 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

**0**answers

140 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 ...

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**0**answers

262 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

339 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**

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**0**answers

81 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

293 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 ...

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votes

**0**answers

119 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 ...

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**0**answers

159 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 ...

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**0**answers

124 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 ...

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votes

**2**answers

305 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

252 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 ...

**2**

votes

**2**answers

217 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
$$
...