Abelian varieties are projective algebraic varieties endowed with an Abelian group structure. Over the complex numbers, they can be described as quotients of a vector space by a lattice of full rank. They are analogs in higher dimensions of elliptic curves, and play an important role in algebraic ...

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384 views

On morphisms to projective space arising from a linear system

Context: This question arose as I was reading the proof of Application 6.1 in Mumford's Abelian Varieties. However, I have extracted all of the relevant information below so this question should ...
6
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1answer
429 views

Abelian varieties with given endomorphism algebra

I am confused by a statement in the very classical paper of A. A. Albert "On the construction of Riemman matrices II", Ann. Math. 1935, Thm 16. If I understand what he saying, the theorem says that ...
2
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0answers
322 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 ...
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1answer
208 views

showing that abelian varieties are de Rham *without* showing that they are crystalline

If $X$ is a smooth projective variety over a $p$-adic field $K$, then Faltings' Theorem says that the etale cohomology of $X_{\overline{K}}$ is crystalline. There have been various steps towards this ...
3
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3answers
282 views

the dual abelian scheme

There are plenty of sources discussing dual abelian varieties, but I'm looking for a reference that discusses the construction and properties of the dual abelian scheme. I'm willing to accept general ...
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180 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 ...
2
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0answers
206 views

Which curves cut the Hyperelliptic locus?

Consider the moduli space $\mathcal{A}_{g,n}$ of abelian varieties with some level $n \geq 3$ structure. For simplicity, we just denote it by $\mathcal{A}_{g}$ and drop $n$. Denote the locus of ...
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1answer
899 views

Poincare line bundle

I am being stuck by the proof of the existence of Poincare line bundle of complex torus in Griffiths-Harris. Here is the question: Let $M$ be a complex torus and $M'$ be the complex torus dual to ...
3
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0answers
176 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 ...
7
votes
1answer
441 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 $k$ ...
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0answers
198 views

How does the line bundles look like on a proper model (or Néron model) of an abelian variety?

How does the line bundles look like on a proper model (or Néron model) of an abelian variety? Who knows references about this? In particular, let us work over a trait $S=\mathrm{Spec} R$, where $R$ ...
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101 views

descent of a complex of sheaves

Let X a projective variety over an algebraically closed field on which an abelian variety $A$ acts freely. Then $X/A$ is a projective variety. Let $f:X\rightarrow X/A$ Let $K\in D_{c}^{\leq ...
1
vote
1answer
270 views

complex multiplication

For an abelian variety $A$, it is said to be have $complex \ multiplication$ if $\mathrm{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$ contains a number filed $F$ of degree $2 \cdot \mathrm{dim} (A)$. ...
4
votes
1answer
335 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 ...
9
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2answers
565 views

What are some consequences of the Mumford-Tate conjecture?

Let $A/K$ be an abelian variety over a number field. On the one hand we have the singular cohomology group $$V := H^1(A(\mathbb{C}),\mathbb{Q})$$ with respect to some fixed embedding $K \subset ...
3
votes
0answers
165 views

Ordinary vs Non-ordinary for GL(2)-type Abelian Surfaces over Q

Let $A_f$ be an abelian surface over $\mathbf{Q}$ of $\mathbf{GL}_2$-type arising from a weight $2$ cuspidal eigenform $f\in S_2(\Gamma_0(N))$. What is known (or expected to be true) for the size of ...
3
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1answer
321 views

(3,3) abelian surface and k3 surfaces

SOrry for the very specific question, but curiosity bites.... So here's the story: an idecomposable principally polarized abelian surface is embedded in $P^8=|3\Theta |^* $ as a deg 18 surface A. ...
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0answers
141 views

p-divisible group over an algebraically closed field of characteristic p arises from abelian variety

It may be trivially true or trivially false, just a quick ask, if $k=\overline{k}$ and char k = p>0, X is a p-divisible group over $k$, suppose the Newton polygon of $X$ is symmetric, then there ...
10
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0answers
380 views

About the Bloch conjecture on entire curves

The Bloch conjecture states the following: Bloch's conjecture. Let $X$ be a compact complex Kähler variety such that the irregularity $q = h^0(X,\Omega^1_X)$ is larger than the dimension $n = \dim ...
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0answers
189 views

Canonical forms for elliptic fibrations with Mordell-Weil group of rank 1 and zero torsion

Consider an elliptic fibration given by the following Weierstrass model: $$ E: y^2 + a_1 x y + a_3 y =x^3 + a_2 x^2 + a_4 x + a_6,\quad a_6=a_2 a_4. $$ ( I work with characteristic zero). With the ...
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0answers
256 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 ...
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0answers
372 views

Grothendieck-Messing theory

Hello, I would like to work out some examples of deformation of isogenies via Grothendieck-Messing theory. Let's take an easy example: Let A be an abelian variety over $k=\overline{\mathbb{F}_p}$ and ...
5
votes
1answer
314 views

Genus 2 curves vs Abelian surfaces

In the Satake compactification of abelian surfaces we have the following degeneration of a family of abelian surfaces in $\mathbf{H}_2$ $lim_{t \to \infty}\begin{pmatrix} it & b \\\ b & ...
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3answers
440 views

Another question related to the isogeny theorem for elliptic curves

I was reading the following question: About isogeny theorem for elliptic curves and was interested in the following statement at the end of Torsten Ekedahl's answer: "Note also that the situation is ...
7
votes
1answer
416 views

On simple factors of modular jacobians: endomorphism ring and simplicity of mod p reduction

Let $A_f$ be the abelian variety over $\mathbf{Q}$ arising as a $\mathbf{Q}$-simple factor of the Jacobian $J_0(N)$ of the modular curve associated to a normalized newform $f$ of weight $2$ on the ...
4
votes
2answers
270 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
412 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
406 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 ...
18
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
1answer
232 views

Explicit period lattices for abelian surfaces

Given an explicit description (as an intersection) of an abelian surface $A$ is there an algorithm for computing the period lattice of the surface? For the specific examples that I am interested in, ...
10
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1answer
313 views

What is the Brauer group of the moduli space of (p.p.) abelian varieties?

What is the Brauer group of the moduli space of principally polarized abelian varieties of a given dimension? I am primarily interested in the "open" moduli space, i.e. not a compactification. The ...
20
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1answer
2k views

Modern proof of Serre's open image theorem?

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic ...
2
votes
1answer
209 views

Can we control the size of the intersection of two abelian subfactors of an abelian variety ?

Let $A$ be an abelian variety over an algebraically closed field $K$, and $B$ an abelian subvariety. We know (for instance, from Milne course on AV) that there is an abelian "cofactor" $B'\subset A$ ...
1
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1answer
308 views

Serre-Tate canonical lifts for finite fields

A result of Serre-Tate states that we can canonically lift an ordinary abelian variety over a perfect field $k$ of positive characteristic to an abelian scheme over the ring of Witt vectors of $k$ and ...
3
votes
0answers
211 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
413 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 $S$ ...
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0answers
136 views

Construction of RM abelian variety from eigenform

Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations ...
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votes
2answers
254 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
208 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
223 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 ...
0
votes
0answers
112 views

Frobenius eigenvalues of abelian variety

Let $A$ be an abelian variety over a finite field $\mathbb{F}_q$ and $x_i$ the Frobenius eigenvalues on $H^1$. Does $x_i \mapsto q/x_i$ permute the $x_i$, and why? It should follow from Poincare ...
6
votes
2answers
407 views

Ample vector bundles on complex tori

Let $X$ be a $n$-dimensional complex torus and $\omega$ a Kähler form on $X$. Then, it is well known that a real $(1,1)$-class $[\alpha]\in H^{1,1}(X,\mathbb R)$ is a Kähler class if and only if for ...
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vote
0answers
306 views

Component group of Neron model of a parametrized abelian variety

Let $A$ be an abelian variety of dimension $2$ over a $p$-adic field $K$ with (additive) valuation $v$. Assuming $A$ has multiplicative reduction, the theory of $p$-adic theta functions gives us an ...
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0answers
203 views

Chern class of line bundle inducing a principal polarisation

What can one say about the Chern class $c_1(\mathcal{L})$ of a line bundle $\mathcal{L}$ on an Abelian variety $A$ inducing a (principal) polarisation $A \to A^\vee$? Why am I asking this? My ...
3
votes
2answers
234 views

Induced maps of an automorphism of a curve on the tangent ot its jacobian and on its differential forms

Let $C$ be a smooth projective curve of genus $g$ over a field $k$ and $J$ be its jacobian (defined over $k$). Let $\sigma: C \rightarrow C$ be a $k$-automorphsm of $C$. This automorphism $\sigma$ ...
8
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0answers
317 views

Obstructions to deforming an abelian variety

Are abelian varieties unobstructed? That is, given an abelian scheme $X \to \mathrm{Spec}R $ for $R$ a local artinian ring and $\mathrm{Spec} R \to \mathrm{Spec} S $ a nilpotent thickening, can we ...
2
votes
1answer
823 views

books (or notes) on complex multiplication

This would be a vague question, but I still want to ask here. Do you have any recommended book on complex multiplicaton. I know only 2 books: Shimura's book ...
2
votes
1answer
276 views

Weil reciprocity on abelian varieties and biextensions?

I was once told, by someone who would likely be right about such things, that the version of Weil reciprocity for abelian varieties (as in Lang, Abelian Varieties) should come out of consideration of ...
2
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1answer
174 views

kernel of an isogeny and coker of its induced map on the Tate module

In a proof in Milne's note "Abelian Variety" (top on p.52), I saw an equality: $ \mathrm{Ker}(\beta)(l) = \mathrm{Coker}(T_{l}(\beta))$, here $\beta$ is an (separable) isogeny of an abelian variety ...
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0answers
198 views

Where can I find a copy of Serre's Cours au college de France 1985-1986?

Hi, I was wondering: where might I be able to find a copy of this work online? And are there any other resources for the proof of the open image theorem for abelian varieties with endomorphism ring ...