# Questions tagged [abelian-schemes]

The abelian-schemes tag has no usage guidance.

The abelian-schemes tag has no usage guidance.

48
questions

3
votes

1
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130
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Let $Y (N) $ be the moduli scheme of dimension two principally polarized Abelian schemes with level $N$. It is claimed in "G.Laumon - Fonctions zeta des variétés de Siegel" (Lemma 4.1) that ...

5
votes

0
answers

96
views

Let $\cal A$ be a smooth commutative group scheme over $S$, where $S$ is the spectrum of a discrete valuation ring with fraction field $K$ and residue field $k$. Suppose that $A:={\cal A}_K$ is an ...

1
vote

2
answers

267
views

The following came up when reading the definition of the moduli stack of principally polarized abelian varieties in [1].
Let $\pi_1:A_1 \to S_1$ and $\pi_2: A_2 \to S_2$ be abelian schemes over $S_i$, ...

2
votes

0
answers

167
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Let $C$ be a nice curve, i.e. $C$ is a smooth, projective, geometrically integral scheme of dimension $1$ over a field $k$. For example, (assuming the characteristic of $k$ is neither 2 or 3) an ...

4
votes

4
answers

743
views

Let $X$ be a smooth, projective ireducible scheme over an algebraically closed field $k$. I'm trying to understand when there exists an abelian variety $A$ such that $X$ is isomorphic to a prime ...

2
votes

0
answers

124
views

For an Abelian scheme over a ring of integers in a number field, Faltings has a theorem that describes how the Faltings' height changes through an isogeny. There are multiple references for this ...

5
votes

0
answers

257
views

Let $f \colon S \rightarrow C$ be a minimal elliptic surface and let $g \colon J \rightarrow C$ be its jacobian fibration. In this case, we know that the fibers of $g$ are better behaved that the ones ...

11
votes

1
answer

280
views

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 ...

2
votes

0
answers

215
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$\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\tors}{tors}$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 $...

6
votes

0
answers

238
views

Let $A$ be an abelian scheme over some base scheme $S$.
Let $A^\vee$ be the dual abelian scheme, defined as $\text{Pic}^0_{A/S}$ where $\text{Pic}_{A/S}(T)=\text{Pic}(A_T)/\text{Pic}(A)$. (maybe some ...

3
votes

1
answer

518
views

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 ...

2
votes

0
answers

271
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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.
...

4
votes

0
answers

162
views

Let $S$ be a scheme, $(\text{Sch}/S)_{\rm Ét}$ a big étale site, and $A$ a representable
(either in schemes or algebraic spaces over $S$) abelian sheaf on $(\text{Sch}/S)_{\rm Ét}$.
Suppose there is a ...

2
votes

1
answer

256
views

I would like to know the relation between the BSD conjecture for abelian schemes (as stated for example in T Keller's thesis) and the clasical BSD conjecture.
In particular, can one state the ...

2
votes

1
answer

412
views

Let $S$ be an integral scheme with function field $K = K(S)$. Let $\mathscr{A}, \mathscr{B}$ be Abelian schemes over $S$. Let $L/K$ be a separable field extension. Given $f_L \in \mathrm{Hom}(\mathscr{...

2
votes

1
answer

267
views

By a supersingular Abelian scheme, I mean an Abelian scheme which is fibrewise a supersingular Abelian variety, i.e. isogenous to a product of supersingular elliptic curves (F. Oort, Subvarieties of ...

14
votes

0
answers

476
views

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 ...

3
votes

1
answer

284
views

Let $k$ be a finite field, $\ell \neq \mathrm{char} k$ be prime, $X/k$ be a smooth projective geometrically integral variety of dimension $d$, and $\mathcal{A}/X$ be an Abelian scheme. Let $\eta \in H^...

3
votes

0
answers

192
views

It is well-known that every Abelian variety is isogenous to its dual, but what about Abelian schemes?

4
votes

1
answer

344
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It is known that for every abelian scheme $A$ over a ring $R$, there exists a subring $R_0$ of $R$ that is of finite type over $\mathbb{Z}$ and an abelian scheme $A_0$ over $R_0$ such that $A$ is ...

6
votes

0
answers

438
views

During my reading of Peter Scholze and Jared Weinstein's paper ``Moduli of $p$-divisible groups'' I found this assertion in the proof of Proposition 6.1.3. Consider the following situation. Let $k$ be ...

1
vote

1
answer

88
views

Let $G$ be a $\mathbb{Q}$-subgroup of $\mathrm{GSp}_{2g}$, reductive and defines a Shimura subdatum of $(\mathrm{GSp}_{2g},\mathfrak{H}_g)$. Let $V$ be the natural representation of $\mathrm{GSp}_{2g}$...

1
vote

1
answer

342
views

Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise ...

2
votes

1
answer

627
views

This question is a follow up to Why is the norm map dual to restriction under Tate local duality?
Let $A$ and $B$ be dual abelian schemes over a base scheme $S$. For an integer $n \ge 1$, consider ...

0
votes

1
answer

136
views

Let $k$ be an algebraic closure of a finite field, $\ell \neq \mathrm{Char}(k)$ be prime, $S/k$ a smooth projective geometrically connected surface and $C/k$ a smooth ample connected hypersurface ...

3
votes

0
answers

226
views

Let $C$ be a smooth, proper and connected curve over
the complex numbers $\bf C$. Let ${\cal G}\to C$ be a smooth group scheme over $C$ and let $\epsilon_{\cal G}:C\to{\cal G}$ be its
zero-section. ...

0
votes

2
answers

500
views

In which situations is there a Poincaré bundle for Abelian schemes? In [Mumford, Abelian varieties] only the case of Abelian varieties is treated.
The same question for the Weil pairing $\mathscr{A}[...

9
votes

0
answers

506
views

For $S$ a noetherian scheme, let $\mathcal{A}(S)$ be the additive category of abelian schemes over $S$ and $\mathcal{A}_{\mathbb{Q}}(S)$ be the category of abelian schemes up to isogenies, i.e. ...

11
votes

0
answers

328
views

Let $S$ be a noetherian excellent regular scheme and $U\subset S$ an everywhere dense open of codimension $\geq 2$. For some fibered categories of geometric objects, it makes sense to ask whether the ...

3
votes

0
answers

508
views

Under what conditions on the base $X$ are Abelian schemes $\mathcal{A}/X$ projective, and projective in which sense?

4
votes

1
answer

766
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Q: Is there a simple proof of the fact that the Weil restriction of an abelian scheme along a finite étale morphism is an abelian scheme ?
Details: Let $S$ be a scheme and $f:S'\rightarrow S$ a ...

15
votes

1
answer

1k
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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 $S$, $...

4
votes

0
answers

316
views

Dear Colleagues,
would appreciate if you could recommend references, if such a theory exits, for the following question.
Let $A$ be an Abelian scheme over $\text{Spec}(R)$, where $R$ is a subring of ...

10
votes

1
answer

690
views

Let $n>2$ and let $k$ be either $\bf Q$ or a finite field whose characteristic is prime to $n$. Let $A_{g,n}$ be the moduli scheme, which represents the functor, which with every $k$-scheme $S$
...

2
votes

0
answers

214
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In Mumford's GIT, the definition of level $n$ structure ($n \geq 2)$ is $2g$ sections $\{\sigma_1, \dots, \sigma_{2g}\} : S \rightarrow A$ such that two conditions hold: (i) For geometric points the ...

15
votes

1
answer

1k
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Is every abelian scheme $\mathcal{A}/X$ under suitable conditions on $X$ a quotient of a Picard scheme of a curve $\mathcal{C}/X$? I need it for $X/\mathbf{F}_q$ smooth projective.

6
votes

3
answers

808
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Let $A/X$ be an abelian scheme. Is $H^n(X,A)$ torsion for $n > 0$?
Perhaps this can be proved analogously as Proposition IV.2.7 of Milne's Étale cohomology (where it is proved that the ...

3
votes

1
answer

407
views

Let $S/k$ be a smooth variety and $A/S$ be an abelian scheme. Let $Z \hookrightarrow S$ be a reduced closed subscheme of codimension $\geq 2$.
I want to show that in this situation, $H^i_Z(S, A) = 0$ ...

1
vote

1
answer

592
views

It is well-known that if $A$ is an ordinary abelian variety over a finite perfect field $ k$ of characteristic $ p>0$ and $ W=W(k)$ is the ring of Witt vectors over $ k$, then the canonical ...

4
votes

1
answer

408
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Let $R$ be a regular local ring of dimension at least 2, and let $U$ be the complement of the closed point in $\mathrm{Spec} R$. Given a polarized abelian scheme over $U$, under what hypotheses can ...

9
votes

2
answers

1k
views

How are the finite flat group schemes $\mathcal{A}[\ell^n]$ arising from an Abelian scheme $\mathcal{A}/S$ singled out among other finite flat commutative group schemes of exponent $\ell^n$?

1
vote

1
answer

551
views

Hi,
I have a very basic question.
I am looking for references explaining how to construct explicitily a Jacobian starting from a curve or examples of projective equations for an abelian scheme. I ...

4
votes

1
answer

600
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In a cryptography book I read that people does not known how to compute the number of points on a Jacobian of a hyperelliptic curve $C$ over a finite field $F_q$? Is this true? It seems easy to ...

4
votes

1
answer

470
views

Let $\mathcal{A}$ be an Abelian scheme over a smooth curve $S^*\subset S$ and let $\mathcal{A_S}$ be the Neron model of $\mathcal{A}$ over $S$. Is it possible to describe the special fiber of the ...

2
votes

1
answer

391
views

Let $S$ be a locally noetherian scheme, $Y$ a locally noetherien $S$-scheme and $X$ an abelian scheme over $S$. It is known that the map between groups $Hom(Y,X) \to Hom(Pic(X/S),Pic(Y/S)), f \mapsto ...

1
vote

3
answers

562
views

I know that the fibre of $A_{g,n}$ over $\mathbf{F}_p$ is quasi-projective (of what dimension?). Can one exhibit some smooth projective subvarieties of high dimension in it? What are references for ...

2
votes

1
answer

394
views

Let $A$ be an abelian variety of dimension g and a polarization $L$ of type $(d_1,.....,d_g)$ (let alone the case $d_i=d_j,$ $\forall i, j$). What is the degree of the generators of the homogeneous ...

13
votes

1
answer

1k
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Let $S$ be a base scheme, let $A/S$ be an abelian scheme, and let $\mathbf{G}_m/S$ be the multiplicative group; consider $A$ and $\mathbf{G}_m$ as objects in the abelian category of sheaves of abelian ...