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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The $p$-divisible groups which arise from an abelian variety over characteristic $p$?
Can you classify all conditions for $p$-divisible groups arising from an abelian variety? For example, $2\dim = \mbox{height}$.
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Is $h^1(X,O_X)$ always equal to the dimension of the Albanese?
Let $X$ be a projective integral scheme over $\mathbb{C}$.
If $X$ is smooth, then $\mathrm{h}^1(X,\mathcal{O}_X)$ is the dimension of the Albanese variety of $X$. Probably, even if $X$ is normal, ...
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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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Uniqueness of presentation for semi-abelian varieties
Let $k$ be any field and $G$ a semi-abelian variety over $k$, i.e., an algebraic group that fits into an exact sequence
$$ 1 \to T \to G \to A \to 1$$
of algebraic groups, where $T$ is an algebraic ...
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How to decide whether the isogeny between Neron models is etale?
Let there be an isogeny $f:A_1 \rightarrow A_2$ between two abelian varieties over a $p$-adic field $F$ and assume $f$ has degree $p^n$. By the universal property we get a moprhism $f_0: \mathcal{A}_1 ...
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Abelian varieties with rank 0 over each global field
For each global field $K$, can we always find an Abelian variety $A$ with $A(K)$ rank $0$?
By Lang-Neron, Mordell Weil is also true for finite type fields (fields finitely generated over their prime ...
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Polarization type of the complement abelian subvariety
Assume that $P$ is a Prym variety of a ramified double cover (hence not principally polarized). Let $A,B\subset P$ a complementary pair. Assume that the type of the polarization of $A$ is given by $\...
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Analogue of the original Birch–Swinnerton-Dyer conjecture for abelian varieties
$\newcommand{\Q}{\Bbb Q}
\newcommand{\N}{\Bbb N}
\newcommand{\R}{\Bbb R}
\newcommand{\Z}{\Bbb Z}
\newcommand{\C}{\Bbb C}
\newcommand{\F}{\Bbb F}
\newcommand{\p}{\mathfrak{p}}
$
Let $A$ be an abelian ...
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Is $\operatorname{dim} H^1$ of an abelian variety the same for any Weil cohomology?
Let $A$ be an abelian variety over a field $k$ of dimension $g$, and $H$ be a Weil cohomology theory for smooth projective varieties over $k$ with characteristic $0$ coefficient field $E$.
Is it ...
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component group of Neron models
Let $A$ be an abelian variety over a discretely valued field $K$ and $\mathcal A$ its Neron model over $R$ (the ring of integers of $K$) and $\mathcal A^0$ the identity component of $\mathcal A$.
...
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Riemann Surfaces of Infinite Genus and Transcendental Curves
I'm a graduate student struggling to find a topic for his doctoral dissertation which hasn't already been explored. At present, my hope was to see if classical results about Jacobian Varieties, ...
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Lifting of automorphism of rational surface to that on abelian variety
The paper I am referencing is "Normal Subgroups of the Cremona Group." https://arxiv.org/abs/1007.0895. In theorem 5.14, at the bottom of page 52, the author stated for the abelian surface $Y= \mathbb{...
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On a refinement of Mordell's conjecture for curves
Let $C$ be an algebraic curve of genus $g \geq 2$, defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. It is then defined over a finite extension $K$ of $\mathbb{Q}$. We assume that $C(K) \ne \...
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Property of Complete Variety
I have a question about a step in the proof from Lang's "Abelian Varieties" (page 20):
By definition an abelian variety $A$ over field $k$ is a proper smooth $k$-group scheme that is irreducible.
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Why is, for a group scheme of finite type, "smooth" (resp. irreducible) equivalent to "geometrically reduced" (resp. geometrically irreducible)?
I have some questions about two statements from Bosch's "Algebraic Geometry and Commutative Algebra" about algebraic varieties (page 479). Since I still don't have the permission to add images I quote ...
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What integer value can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$?
Fix a positive integer $g$. What positive integer $N$ can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$ ?
For example, as there is no abelian varieties over $\mathbb Z$, $N$ ...
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Is the symmetric product of an abelian variety a CY variety?
Let $n>1$ be a positive integer and let $A$ be an abelian variety over $\mathbb{C}$. Then the symmetric product $S^n(A)$ is a normal projective variety over $\mathbb{C}$ with Kodaira dimension zero ...
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rank of Jacobian of Fermat curve and Chabauty-Coleman method
Consider the fermat curve $F(p)$ over $\mathbb Q$ which is the projective closure of $X^p+Y^p=1$ inside projective plane, where $p$ is a prime number and without loss of generality we assume $p>2$. ...
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List of Automorphism groups of Abelian Varieties for Dummies
(%Edited after abx comment%)
I seek explicit linear integral representations $\rho: Aut(X,\omega) \to Sp_{2g}(\mathbb{Z})$, when $(X,\omega)$ is complex $g$-dimensional PPAV. I prefer explicit ...
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Are there three ordinary elliptic curves $E$, $E_1$, $E_2$ such that $E^2 \cong E_1 \!\times\! E_2$?
Consider the elliptic curve $E\!: y^2 = x^3 + 1$ of $j$-invariant $0$ over an algebraically closed field $k$ of characteristics $p$. Let me remind that $E$ is ordinary (i.e., non-supersingular) iff $p ...
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Purely inseparable isogeny
How to prove purely inseparable isogeny between two abelian varieties is radical (universally injective)? Purely inseparable morphism means the extension between the two function fields is purely ...
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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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field of definition of CM abelian varieties
When $A$ is a CM abelian variety of dimension $1$ (i.e., an elliptic curve), then we have a result that if it has CM by a maximal order then it has a model over a number field $F$ where $F$ is the ...
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What is Rosati Form
I was reading a paper and they mentioned the Rosati form. Particularly, what they said was:
Let $A$ be an abelian surface defined over $k$ such that $ST_A^0$ (the connected component of the Sato-Tate ...
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lemma II.2.4 in Harris-Taylor (about drinfeld-katz-mazur level structure on 1-dimensional $p$-divisible groups)
Lemma II.2.4 on page 82 in Harris and Taylor's "The Geometry and Cohomology of Some Simple Shimura Varieties" (or lemma 3.2 here), says that given a Drinfeld(-Katz-Mazur) level structure $\alpha:(p^{-...
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How to produce low genus curves on abelian surfaces?
I would like to find "simple" complex algebraic curves (i.e. low genus) on a complex abelian surface $A$ (which are not just abelian subvarieties or translates of them). For example, a genus 2 curve, ...
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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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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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dual of quotient abelian variety
Let $A$ be an complex abelian variety and $S\subset A$ a finite subgroup.
How to calculate the dual abelian variety of $A/S$?
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Variety Isomorphism Problem for Abelian Surfaces
This is a special case of this question, where it is asked whether there exists an algorithm to determine whether two varieties are isomorphic. There, an answer by Bjorn Poonen explains how to solve ...
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Quaternion algebra actions on $\ell$-adic cohomology
Let $E$ be a supersingular elliptic curve over $\mathbf{F}_p$, and $H$ its endomorphism algebra $\text{End}(E)\otimes_{\mathbf{Z}}\mathbf{Q}$, a quaternion algebra (non split at $p$ and $\infty$).
...
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Is the generalized Kummer threefold rational in characteristics 3?
Let $E_i\!: y_i^2 = x_i^3 - x_i$, $i = 1, 2, 3$ be three copies of the supersingular elliptic curve in characteristics $3$. Consider on $E_i$ the following automorphism of order $3$:
$$
\sigma(x_i,...
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Intermediate Jacobian of abelian varieties
Is the intermediate Jacobian of an abelian variety again an abelian variety?
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Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?
Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$?
Do there exist ...
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Abelian variety over Q with many roots of unity
Given an abelian variety $A$ over the rational integers $\mathbb{Q}$, and a prime $p$, we know that $\mathbb{Q}(\zeta_p)$ is contained in $\mathbb{Q}(A[p])$, the $p$-division field of $A$, and where $\...
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Point Counts on $G$-torsors over Finite Fields
Let's assume we have a $G$-torsor $X_{1} \to X_{2}$, where $G$ is a finite abelian group, and both $X_{1}$ and $X_{2}$ are defined over $\text{Spec}(\mathbb{Z})$. Is there an easy way to compute $\#...
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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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Roots of unity and coordinates of points in abelian varieties
We consider an abelian variety $A$ defined over the rational numbers $\mathbb{Q}$. For a torsion point $P\in A(\bar{\mathbb{Q}})$, consider the field $\mathbb{Q}(P)$ obtained by adjoining to $\mathbb{...
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Weil pairings on abelian varieties restricted to subgroups of a given order
Let $A$ be an abelian variety of dimension $g$ defined over a number field $K$. Suppose $A$ has a principal polarization and $\ell$ is a prime number. We have a Weil pairing:
$$
e_\ell: A[\ell]\times ...
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Pull-back of polarization
Let $(X, L)$ and $(Y, M)$ be two polarized abelian varieties .
According to Birkenhake C. and Lange H. in Complex Abelian Varieties a homomorphism of polarized abelian varieties $f:(Y, M)\...
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why the division field of an abelian variety contains a cyclotomic field?
Given an abelian variety $A$ defined over $\mathbb{Q}$, for a positive integer (we can suppose prime) $\ell$, let $A[\ell]$ denote the group
of points of $A$ that are annihilated by $\ell$, the ...
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Index of the endomorphism ring of an abelian surface
For an abelian surface $A/\mathbb{Q}$ such that $R:=\mathrm{End}_{\mathbb{Q}}(A)$ is an order in a real quadratic field $K$ (so a $\mathrm{GL}_2$-type surface), is there a bound on the index $[O_K : R]...
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Polarization of the Jacobian in Torelli's theorem
I'm studying an example in book Yuji Shimizu and Kenji Ueno. Advances in Moduli Theory. Translations of Mathematical Monographs, vol. 206, that shows the importance of isomorphism as principally ...
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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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analogue of Theorem of Mattuck for Abelian varieties over $\mathbf{F}_q(\!(t)\!)$
By a theorem of Mattuck [Abelian Varieties over $p$-Adic Ground Fields, Annals of Mathematics, Second Series, Vol. 62, No. 1 (Jul., 1955), pp. 92-119], for an Abelian variety $A$ of dimension $g$ over ...
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Principally Polarized CM Abelian Variety
I am interested in considering examples of abelian varieties that are principally polarized with CM in dimension three. However, I am struggling to construct or find even a single instance.
In ...
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Do abelian varieties have Neron models over arbitrary valuation rings?
Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$. Let $A$ be an abelian variety over $K$. Does $A$ have a Neron model?
If $\mathcal{O}_K$ is a discrete valuation ring, then this is ...
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Good reduction of abelian varieties over valuation rings via coverings
Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$, and let $A$ be an abelian variety over $K$.
Suppose that there is a smooth proper scheme $\mathcal{X}$ over $\mathcal{O}_K$ whose ...
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What is known about abelian varieties with several principal polarizations?
Let ``the simple case" be when the polarized abelian variety does not break up into a product of polarized abelian varieties.
I am trying to get an idea of what is known about abelian varieties with ...
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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 ...