Questions tagged [abelian-varieties]
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 geometry and number theory.
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Fontaine-Mazur for GL_1
For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified ...
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Why polarization of abelian varieties?
Maybe this question is not suitable for here, but I don't think I would receive a satisfactory answer in Math StackExchange.
I could never understand the intuition behind polarization of abelian ...
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In which ways can the isogeny theorem fail for local fields?
Fix a field $K$ with absolute Galois group $G$. By an isogeny theorem over $K$, I mean the statement that the map $\operatorname{Hom}(A,B)\otimes\mathbb{Z}_l \to \operatorname{Hom}_G(T_l A, T_l B)$ is ...
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Why no abelian varieties over Z?
Motivation
I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form
the set $\{...
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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 ...
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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 Curves'...
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Why do people think that abelian varieties are the hardest case for the Hodge conjecture?
Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and hard enough) that ...
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Is every abelian variety a subvariety of a Jacobian?
Let $k$ be an infinite field and $A$ be an abelian variety over $k$. Can $A$ be embedded into a Jacobian variety $J$ over $k$?
In these notes by William Stein this is stated without proof in remark 1....
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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 ...
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When is "independence of l" known?
My question is for which varieties over local fields is "independence of l" known for
etale cohomology. Say $X/{\mathbb Q}_p$ is a complete non-singular variety and $W_l$ is the (complex) Weil-...
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Is hyperelliptic cryptography "practical"?
Previosly my impression on this subject was that hyperelliptic cryptography systems (as well as other possible cryptosystems based on abelian varieties of dimension $>1$) have no advantages over ...
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Which curves can be found on Abelian varieties?
We know that each genus 2 curve is embedded into its degree 1 Jacobian.
Under which conditions on $C$, $A$, $g$ and $n$ is it possible for a genus $g$ smooth curve $C$ to be embedded in an Abelian ...
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Crystalline cohomology of abelian varieties
I am trying to learn a little bit about crystalline cohomology (I am interested in applications to ordinariness). Whenever I try to read anything about it, I quickly encounter divided power ...
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non principally polarized complex abelian varieties
I've read in (abstracts of) papers that there are abelian varieties over fields of positive characteristic that admit no prinicipal polarization. Apparently its not the easiest thing to find an ...
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Are there Néron models over higher dimensional base schemes?
Are there Néron models for Abelian varieties over higher dimensional ($> 1$) base schemes $S$, let's say $S$ smooth, separated and of finite type over a field?
If not, under what additional ...
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Bhargava's work on the BSD conjecture
How much would Bhargava's results on BSD improve if finiteness of the Tate-Shafarevich group, or at least its $\ell$-primary torsion for every $\ell$, was known? Would they improve to the point of ...
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Are Jacobians principally polarized over non-algebraically closed fields?
How does one define the Torelli map $M_g \to A_g$ of moduli stacks? On geometric points a curve maps to its principally polarized Jacobian.
So what I am asking is: if I have a curve $C$ over a non-...
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Embedding abelian varieties into projective spaces of small dimension
Given a (complex) abelian variety $A$ of a fixed dimension $g$, let $d(A)$ be the dimension of the smallest complex projective space it embeds into.
Is $d(A)$ uniform over all abelian varieties of a ...
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Why were Abelian functions so important in the 19th century?
Felix Klein, when discussing how the popularity of areas in mathematics rises and falls, mentions that in his youth Abelian functions were at the summit of mathematics, and that later on their ...
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On Tate's "Endomorphisms of Abelian Varieties over Finite Fields", sketch of proof of main result?
Let $k$ be a field, $\overline{k}$ its algebraic closure, and $A$ an abelian variety defined over $k$, of dimension $g$. For each integer $m \ge 1$, let $A_m$ denote the group of elements $a \in A(\...
user97565
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Torsion points of abelian varieties in the perfect closure of a function field
The following is a problem, which was recently brought to my attention by H. Esnault and A. Langer.
Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ...
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Do all simple factors of jacobians of curves come from correspondences?
For this question I will let the overly ambiguous word curve mean: smooth projective and connected curve over $\mathbb C$ (or equivalently a smooth compact Riemann-Surface).
Let $C$ be a curve over ...
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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?
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Albanese variety over non-perfect fields
It is a result of Serre (Morphismes universels et varietes d'albanese) that the Albanese (abelian) variety, i.e. an initial object for morphisms to (torsors over) abelian varieties, exists for any ...
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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 1N]...
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Characterizations of complex Abelian varieties (especially 3-folds) among projective nonsingular varieties?
If $X$ is a complex Abelian variety of dimension $g$, then
The canonical sheaf is trivial
$\dim {\rm H}^i(X; \mathcal{O}_X) = \binom{g}{i}$.
When $g =1,2$, then any connected, projective ...
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Etale endomorphisms of abelian varieties in positive characteristic
Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ${\bf F}_p$ (where $p>0$ is a prime number).
My question is : does there exist an abelian ...
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Torsion points in Abelian varieties over number fields
Hello,
Suppose $A$ is an Abelian variety of dimension $g$ over a number field $k$. Then using height functions one can show that there are non-torsion points in $A(\bar k)$. This looks like an ...
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Can we count isogeny classes of abelian varieties?
Let's fix a finite field F and consider abelian varieties of dimension g over F. Can we say how many isogeny classes there are? Is it even clear that there's more than one isogeny class? For g=1, ...
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Examples of rational families of abelian varieties.
I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1).
One can generate many examples as Jacobians of rational ...
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Elements of arbitrary large order in the first Galois cohomology of an elliptic curve
Let $E$ be an elliptic curve over $k=\mathbb{Q}$. Consider $H^1(k,E)$.
In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ...
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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 $S$, $...
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Zariski vs etale torsors over abelian varieties
Question. Let $A$ be an abelian variety (say, over the complex numbers), $G$ an algebraic group, $c$ a class in $H^1_{\rm et}(A, G)$. Denote the multiplication by $N$ map on A by $m_N:A\to A$. Does ...
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Counting abelian varieties over finite fields in a given isogeny class
Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the ...
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Does every Abelian variety have a finite resolution by Jacobians?
One knows that every Abelian variety is a quotient of a Jacobian. Does every Abelian variety have a finite resolution by Jacobians?
user19475
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Is there any rational curve on an Abelian variety?
Is that true that there is no rational curves contained in an Abelian variety? If it's true, is that because abelian varieties are not uniruled? How do I know whether an abelian variety is not ...
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Generalizations of Belyi's theorem
Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic ...
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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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Is the complement of an affine open in an abelian variety ample?
Let $U$ be an affine open subscheme of an abelian variety $A$ over $\mathbb{C}$. Is $A-U$ an ample divisor?
If $\dim A =1$ this is true.
If $\dim A = 2$, the complement is a divisor $D_1+\ldots + ...
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The Tate conjecture for abelian varieties
Let $k$ be a number field. Recall that Faltings proved the famous Tate conjecture, which states that for any abelian variety $X$ over $k$ and any prime $\ell$, the natural map
$$\mathrm{End}(X) \...
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Zeta function of Abelian variety over finite field
Let $A/\mathbf{F}_q$ be an Abelian variety of dimension $g$. Suppose one knows $|A(\mathbf{F}_{q^n})|$ for all $1 \leq n \leq g$. Does one know then $\zeta(A,s)$ (equivalently, $|A(\mathbf{F}_{q^n})|$ ...
user19475
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Weil pairing on abelian varieties and etale Chern classes
Given a line bundle $L$ on an abelian variety $A/k$, there is an associated Weil pairing $e_L\colon\bigwedge^2V_pA\to\mathbb Q_p(1)$, where $p$ is a prime different from the residue characteristic of ...
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If it quacks like an abelian variety over a finite field
Consider smooth projective varieties over a finite field. If a curve "looks like" an elliptic curve (i.e. has genus $1$) then it can be made into an elliptic curve.
Is there something ...
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Shimura's construction of an abelian variety from cusp forms of weight $2k$
Let $\Gamma \subset \mathrm{SL}_2(\mathbb{Z})$ be an arithmetic subgroup, and $S_{2k}(\Gamma)$ the space of holomorphic cusp forms of weight $2k$ for $\Gamma$.
Let $\rho_1: \Gamma \rightarrow V_1$ ...
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Characterizations of Abelian varieties (3-folds) in positive characteristic
From this question
Characterizations of complex Abelian varieties (especially 3-folds) among projective nonsingular varieties?
I learned that if $X$ is a smooth complex projective variety of dimension ...
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Geometry underlying a comparison of Dieudonné theories
Maybe these hypotheses aren't necessary, but for me $\mathbb G$ will be a smooth formal group of dimension 1 and finite height over a perfect field $k$.
There are several presentations of the ...
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Am I missing something about this notion of Mirror Symmetry for abelian varieties?
This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s.
In the comments of the question, I was directed to the paper http://arxiv.org/abs/...
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Lifting Abelian Varieties to p-adic fields
Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...
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"Albanese" schemes: When does an "initial abelian scheme" exist under a given scheme?
For a variety V, its Albenese variety Alb(V) is a variety with a map V → Alb(V) which factors uniquely into any map from V to an abelian variety. Can we say something similar for an arbitrary ...
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Complex torus, C^n/Λ versus (C*)^n
I'm having trouble distinguishing the various sorts of tori.
One definition of torus is the algebraic torus. Groups like $SU(2,\mathbb{C})$ and $SU(3,\mathbb{C})$ have important subgroups that are ...
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