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 ...
Manoel's user avatar
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9 votes
1 answer
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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 ...
Lukas's user avatar
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6 votes
0 answers
159 views

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 $...
Asvin's user avatar
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11 votes
1 answer
331 views

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 ...
Asvin's user avatar
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3 votes
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195 views

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 $...
Rodion N. Déev's user avatar
4 votes
0 answers
214 views

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 $\...
Joël's user avatar
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5 votes
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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 ...
Asvin's user avatar
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1 vote
1 answer
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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$...
Pierre's user avatar
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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 ...
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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 ...
user267839's user avatar
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7 votes
0 answers
149 views

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?
rj7k8's user avatar
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5 votes
2 answers
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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-...
Giulio Bresciani's user avatar
2 votes
0 answers
233 views

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, ...
A. GM's user avatar
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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 = ...
Dimitri Koshelev's user avatar
2 votes
1 answer
183 views

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....
Dimitri Koshelev's user avatar
3 votes
1 answer
211 views

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 ...
Dimitri Koshelev's user avatar
2 votes
1 answer
169 views

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 ...
Dimitri Koshelev's user avatar
1 vote
1 answer
174 views

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 $C\!: y^2 = (x^3 + b)(x^3-b)$, ...
Dimitri Koshelev's user avatar
2 votes
0 answers
260 views

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 ...
Asvin's user avatar
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-2 votes
1 answer
271 views

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.\...
PSUN's user avatar
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7 votes
1 answer
464 views

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 ...
Aaron Landesman's user avatar
3 votes
0 answers
175 views

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 ...
ybilu's user avatar
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0 answers
73 views

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 = ...
Dimitri Koshelev's user avatar
2 votes
0 answers
239 views

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 ...
Qirui Li's user avatar
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2 votes
0 answers
141 views

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 = ...
Dimitri Koshelev's user avatar
1 vote
0 answers
243 views

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 $G_0$ ...
Qirui Li's user avatar
  • 397
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, $2\dim = \mbox{height}$.
Qirui Li's user avatar
  • 397
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, ...
Harry's user avatar
  • 343
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 ...
57Jimmy's user avatar
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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 ...
Asvin's user avatar
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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 $\...
Z.A.Z.Z's user avatar
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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 $\mathcal A$. ...
kkgz's user avatar
  • 31
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 ...
sawdada's user avatar
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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, ...
MCS's user avatar
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3 votes
1 answer
585 views

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 ...
Z Wu's user avatar
  • 340
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 \...
Stanley Yao Xiao's user avatar
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. ...
user267839's user avatar
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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 ...
user267839's user avatar
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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{...
Soby's user avatar
  • 157
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 $\mathbb Z$, $N$ ...
sawdada's user avatar
  • 6,148
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 ...
Harry's user avatar
  • 343
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 ...
sawdada's user avatar
  • 6,148
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 $p>2$. ...
sawdada's user avatar
  • 6,148
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 ...
JHM's user avatar
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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 ...
Dimitri Koshelev's user avatar
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 ...
Rick Sanchez's user avatar
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 ...
Vincent's user avatar
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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 $\rho_1: \Gamma \rightarrow V_1$ ...
Zavosh's user avatar
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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^{-...
aytio's user avatar
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