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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4 votes
0 answers
268 views

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, $2\dim = \mbox{height}$.
4 votes
0 answers
361 views

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, ...
13 votes
4 answers
5k views

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 ...
3 votes
1 answer
527 views

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 ...
6 votes
0 answers
328 views

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 ...
6 votes
0 answers
182 views

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 ...
5 votes
0 answers
76 views

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 $\...
9 votes
1 answer
394 views

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 ...
7 votes
1 answer
418 views

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 ...
3 votes
0 answers
179 views

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 $\mathcal A$. ...
1 vote
0 answers
367 views

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, ...
2 votes
1 answer
281 views

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{...
1 vote
1 answer
229 views

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 \...
2 votes
0 answers
291 views

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. ...
8 votes
1 answer
2k views

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 ...
5 votes
2 answers
249 views

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 $\mathbb Z$, $N$ ...
7 votes
2 answers
509 views

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 ...
5 votes
1 answer
627 views

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 $p>2$. ...
1 vote
0 answers
486 views

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 ...
3 votes
0 answers
142 views

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 ...
3 votes
1 answer
377 views

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 ...
14 votes
1 answer
559 views

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 $\rho_1: \Gamma \rightarrow V_1$ ...
2 votes
0 answers
229 views

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 ...
2 votes
1 answer
166 views

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 ...
4 votes
0 answers
186 views

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^{-...
3 votes
1 answer
382 views

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, ...
21 votes
4 answers
3k views

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 ...
18 votes
1 answer
2k views

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 ...
3 votes
0 answers
119 views

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 $A/S$?
3 votes
0 answers
141 views

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 ...
6 votes
1 answer
530 views

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 $\infty$). ...
3 votes
0 answers
134 views

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,...
5 votes
0 answers
193 views

Intermediate Jacobian of abelian varieties

Is the intermediate Jacobian of an abelian variety again an abelian variety?
10 votes
1 answer
523 views

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 ...
4 votes
0 answers
115 views

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 $\...
1 vote
0 answers
124 views

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 $\#...
18 votes
1 answer
1k views

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 ...
1 vote
0 answers
77 views

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{...
2 votes
0 answers
79 views

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 ...
2 votes
1 answer
145 views

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)\...
3 votes
1 answer
152 views

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 ...
8 votes
1 answer
156 views

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]...
2 votes
0 answers
262 views

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 ...
14 votes
1 answer
563 views

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 + ...
7 votes
1 answer
455 views

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 ...
5 votes
0 answers
248 views

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 ...
10 votes
2 answers
591 views

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 ...
6 votes
0 answers
121 views

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 ...
12 votes
0 answers
564 views

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 ...
18 votes
1 answer
967 views

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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