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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Isomorphic Jacobian Varieties Just Like Abelian Varieties — Torelli's Theorem
Torelli's theorem states:
Let $R$, $R'$ be compact Riemann surfaces of genus $g$, $J(R)$, $J(R')$ their Jacobian varieties, $\Theta$, $\Theta'$ their respective theta divisors. The Riemann surfaces ...
9
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1
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Tamagawa numbers
Let $K$ be a finite extension of $\mathbb{Q}_p$ with absolute Galois group $G_K$. Let $A$ be an abelian variety defined over $K$. The (geometric) Tamagawa number is defined as the order of the ...
6
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0
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What are the genus 4 curves with Jacobians that are 4-th powers?
Consider the moduli space of all genus $4$ curves $\overline{\mathscr M_4}$ of dimension $3\times 4 - 3 = 9$. Under the Torelli map, there is a map to $\overline{\mathscr A_4}$ (which has dimension $...
11
votes
1
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Lifting a splitting of an Abelian variety to characteristic 0
Let $R$ be the ring of integers in a (complete) algebraic closure of $\mathbb Q_p$ with maximal ideal $\mathfrak p$. Suppose I have an Abelian surface $\mathcal A/R$ such that over every $R/\mathfrak ...
3
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0
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Genus two curves on abelian surfaces
Considering a smooth genus two curve $C_2$, let $J(C_2)$ be its Jacobian surface, and take $p \in J(C_2)$ an $m$-torsion point. Let $A = J(C_2)/Z_m$, where $Z_m$ acts by $x \mapsto x+p$. The image of $...
4
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Mordell-Weil group of an abelian variety on the perfect closure of a finitely generated field
This question is closely related the question Over which fields does the Mordell-Weil theorem hold?
I consider the following question:
(1) Let $K$ be a finitely generated field extension of $\...
5
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0
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457
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A functor on Abelian varieties corresponding to this operation on Weil numbers
Let $A/\mathbb F_q$ be an abelian variety over a finite field with Weil numbers $q^{1/2}\alpha_1,\dots,q^{1/2}\alpha_n$.
Consider the numbers $q^{d/2}\alpha_1,\dots,q^{d/2}\alpha_n$. These are still ...
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1
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Intersection of a certain linear ideals of $K[[X_1,\ldots,X_{np}]]$ for ${\mathrm{ch}}(K) = p > 0$
Suppose ${\mathrm{ch}}(K) = p > 0$ and we consider the formal power series ring $K[[X_1,\ldots,X_{np}]]$ over $K$ in $np$ variables $X_1,\ldots, X_{np}$. Let $\Lambda$ be the set defined as follows$...
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0
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Application of Stein factorisation: rigidity lemma
Let $X,Y$ Noetherian schemes and $f:X \to Y$ proper map. The Stein factorisation factorizes $f$ as $X \xrightarrow{g} Spec \text{ } f_* \mathcal{O}_X \xrightarrow{h} Y$ with $h$ finite and $g$ has ...
3
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Abelian varieties by Moonen and van der Geer: proof of rigidity lemma
I try to understand a reduction step in the proof of rigidity lemma as proved in Moonen's and van der Geer's Abelian varieties (Lemma 1.11 on page 12- if the link not work the draft version is online ...
7
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149
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Conditions for $p$-divisible group to come from an abelian variety
Over $\mathbb{Z}_p$ or $\mathbb{Q}_p$, are there any known sufficient conditions on a $p$-divisible group for it to come from an abelian variety?
5
votes
2
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Isogeny components of Jacobians of étale covers
Let us work over $\mathbb{C}$. Fix $X$ a smooth projective curve of genus at least $2$. For every simple abelian variety $A$, it is easy to come up with a ramified covering $Y\to X$ with a non-...
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Standard application of Oort-Tate classification theorem
$\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\tors}{tors}$In Mazur's paper “Modular curves and the Eisenstein ideals”, on the bottom of page 159, it says that if $T$ is a open subscheme of $...
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abelian variety with all over non degenerate pairing
Suppose we have an abelian variety $A$ defined over the rational numbers $\mathbb{Q}$. It is known that the Weil pairing
$$
e_\ell: A[\ell]\times A[\ell] \rightarrow \mu_\ell
$$
is non degenerate, ...
3
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0
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The Weil restriction of an elliptic curve with respect to $\mathbb{F}_{p^2}/\mathbb{F}_{p}$
For a prime $p > 3$ consider the quadratic finite field extension $\mathbb{F}_{p^2}/\mathbb{F}_{p}$. Also, consider the elliptic curves
$$
E\!: y_0^2 = x_0^3 + ax_0 + b,\qquad
E^{(1)}\!: y_1^2 = ...
2
votes
1
answer
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Is there a way to find any $\mathbb{F}_2(t)$-point on the elliptic curve $\mathcal{E}$?
Consider the ordinary elliptic curves
$$
E\!:y_1^2 + x_1y_1 = x_1^3 + 1,\qquad E^\prime\!: y_2^2 + x_2y_2 = x_2^3 + x_2^2 + 1
$$
over the field $\mathbb{F}_2$. They are quadratic twists to each other....
3
votes
1
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What is the geometric quotient of the abelian threefold?
Consider a finite field $\mathbb{F}_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$ and its element $\zeta \neq 1$, $\zeta^3 = 1$.
Also, let $E\!: y^2 = x^3 + b$ be an elliptic curve of $j$-invariant ...
2
votes
1
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What is the quotient $E \!\times\! E^\prime / G$?
Consider a finite field $\mathbb{F}_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$)
$$
E\!:y_1^2 ...
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1
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Is the Jacobian isogenous over $\mathbb{F}_p$ to the direct product of the elliptic curves?
Let $\mathbb{F}_p$ be a finite field such that $p \equiv 1 \ (\mathrm{mod} \ 3)$ and $p \equiv 3 \ (\mathrm{mod} \ 4)$. Consider the Jacobian of the hyperelliptic curve $C\!: y^2 = (x^3 + b)(x^3-b)$, ...
2
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0
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Which endomorphisms of the Tate module of an abelian variety are "algebraic"?
For an abelian variety $A$ over a field $k$ with characteristic different from $\ell$ and Galois group $G = Gal(\overline k/k)$, there is always an injective map of the form:
$$\mathbb Q_\ell\otimes ...
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Existence of divisor in the Jacobian of smooth curve of genus two whose intersection with theta divisor is 1
Let $C$ be a smooth projective curve of genus $2$ and $J$ denotes the Jacobian of $C$. Let $\theta$ be the image of $C$ under the abel Jacobi map.
Is there exist a divisor $D$ in $J$ such that $D.\...
7
votes
1
answer
464
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Does the compactified Torelli map extend to a proper map of stacks?
Let $M_g^{ct}$ denote the moduli stack of compact type genus $g$ stable curves and $A_g$ the moduli stack of principally polarized $g$-dimensional abelian varieties.
Can someone provide a reference ...
3
votes
0
answers
175
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Subquotients of abelian varieties with good reduction
Let $B$ be an abelian variety over a DVR with good reduction, and let $A$
be a subquotient of $B$. Then $A$ has good reduction.
I know a proof of this statement using Neron-Ogg-Shafarevich. Is ...
1
vote
0
answers
73
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Is there a non-singular irreducible genus $2$ curve $C/\mathbb{F}_p$ and two $\mathbb{F}_p$-coverings $C \to E$, $C \to E^{(1)}$ of some degree $n$?
Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$), $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$)
$$
E\!:y_1^2 = ...
2
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0
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Vector extension for p-divisible group
Background:
I am trying to understand a proof of Messing's book at Page 120. My goal it to understand the universal vector extention.
Reference:
Messing, The crystals associated to Barsotti-Tate ...
2
votes
0
answers
141
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Is there a way to explicitly find any rational $\mathbb{F}_p$-curve on the Kummer surface?
Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$), $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$)
$$
E\!:y_1^2 = ...
1
vote
0
answers
243
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Deformation of $p$-divisible group
To try to understand the deformation of $p$-divisible group more explicit, I am thinking given a connected $p$-divisible group $G_0$ on $\overline{\mathbb{F}_q}$, Choose a deformation $G$ of $G_0$ ...
4
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The $p$-divisible groups which arise from an abelian variety over characteristic $p$?
Can you classify all conditions for $p$-divisible groups arising from an abelian variety? For example, $2\dim = \mbox{height}$.
4
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0
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Is $h^1(X,O_X)$ always equal to the dimension of the Albanese?
Let $X$ be a projective integral scheme over $\mathbb{C}$.
If $X$ is smooth, then $\mathrm{h}^1(X,\mathcal{O}_X)$ is the dimension of the Albanese variety of $X$. Probably, even if $X$ is normal, ...
3
votes
1
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Uniqueness of presentation for semi-abelian varieties
Let $k$ be any field and $G$ a semi-abelian variety over $k$, i.e., an algebraic group that fits into an exact sequence
$$ 1 \to T \to G \to A \to 1$$
of algebraic groups, where $T$ is an algebraic ...
6
votes
0
answers
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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 ...
5
votes
0
answers
76
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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 $\...
3
votes
0
answers
179
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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 $\mathcal A$.
...
7
votes
1
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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 ...
1
vote
0
answers
367
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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, ...
3
votes
1
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Given a semi-abelian scheme, is the set of points such that the fibres are abelian varities open?
Let $\pi:\mathcal{A}\rightarrow C$ be a semi-abelian scheme, i.e. $\mathcal{A}$ is a smooth separated commutative group scheme over $C$ via $\pi$ with geometrically connected fibres, such that each ...
1
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1
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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 \...
2
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0
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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.
...
8
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1
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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 ...
2
votes
1
answer
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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{...
5
votes
2
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249
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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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2
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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 ...
6
votes
0
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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 ...
5
votes
1
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627
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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 $p>2$. ...
1
vote
0
answers
486
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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 ...
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 ...
2
votes
0
answers
229
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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 ...
14
votes
1
answer
559
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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 $\rho_1: \Gamma \rightarrow V_1$ ...
4
votes
0
answers
186
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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^{-...