# Questions tagged [abelian-schemes]

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### Extension of a multiple of a rational point to an integral point of a semiabelian scheme

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 ...
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### What is the pull-back of a polarization of abelian schemes over different bases?

The following came up when reading the definition of the moduli stack of principally polarized abelian varieties in . Let $\pi_1:A_1 \to S_1$ and $\pi_2: A_2 \to S_2$ be abelian schemes over $S_i$, ...
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### Give examples of semi-abelian schemes over a curve

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 ...
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### Which schemes are divisors of an abelian variety?

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 ...
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### Faltings' height theorem for isogenies over finite fields

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 ...
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### Jacobian fibration of an abelian fibration

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 ...
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### Dual Abelian scheme (relative Picard functor) vs Ext sheaf

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 ...
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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 ...
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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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### On representable abelian sheaves vs abelian sheaves

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 ...
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### BSD conjecture for abelian schemes and the classical version

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 ...
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### Abelian scheme isogenous to dual

It is well-known that every Abelian variety is isogenous to its dual, but what about Abelian schemes?
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### Elimination of noetherian hypothesis for abelian schemes

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 ...
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### Lifting morphisms of $p$-divisible groups using Grothendieck-Messing theory

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 ...
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### Tangent spaces of an indecomposable family of abelian varieties (parametrized by a Hodge type Shimura variety)

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 $\mathrm{GSp}_{2g}$...
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### Schematic image of a relative Cartier divisor of a fiberwise dense open

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 ...
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### Why is the Tate local duality pairing compatible with the Cartier duality pairing?

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 ...
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### rank of Abelian schemes under ample hypersurface section

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 ...
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### Ampleness of Hodge bundles over complex curves

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. ...
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### Dieudonné modules over rings of charateristic zero

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 ...
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### Mordell-Weil group of the universal abelian scheme

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 $S$ ...
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### Level n-structure as defined by Mumford in GIT

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 ...
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### every abelian scheme quotient of a Picard scheme?

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.
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### étale cohomology with values in an abelian scheme is torsion?

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 ...
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### vanishing of étale cohomology groups with small support with values in an abelian scheme

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, $H^i_Z(S, A) = 0$ ...
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### CM liftings of abelian varieties and liftings of Frobenius

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 ...
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### Extending abelian schemes

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 ...
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### finite flat commutative group schemes arising from Abelian varieties

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 $\ell^n$?
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### references for abelian schemes

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 ...
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### A silly question: is the number of points on a Jacobian (of a curve, over a finite field) known?

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 ...
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 ...