Questions tagged [abelian-schemes]
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55 questions
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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 [1].
Let $\pi_1:A_1 \to S_1$ and $\pi_2: A_2 \to S_2$ be abelian schemes over $S_i$, ...
CommunityBot
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Extending group structure
Let $f:X\rightarrow S$ be a proper smooth morphism admitting a section $s: S\rightarrow X$. Suppose $S$ is normal and $f$ is geometric fibrewisely abelian varieties. Will it follow that $X/S$ is an ...
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Deformation of abelian scheme
Let $A/k$ be an abelian variety over a characteristic $p>0$ perfect field $k$. Usually, we say $A$ admits a lift to $W_2(k)$ if there exists an abelian scheme $\mathcal{A}/W_2(k)$ such that $\...
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What are the higher $\mathrm{Ext}^i(A,\mathbf{G}_m)$'s, where $A$ is an abelian scheme?
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 ...
- 1,173
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Equivalent(?) definitions of relative $\mathrm{Pic}^0$ functor *for abelian scheme*
Let $\pi: X \to S$ be an abelian scheme. I know from the first chapter of Faltings--Chai that it is true at this level of generality that the dual abelian scheme exists, i.e., that the fppf sheaf $\...
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$l$-adic sheaf associated to an algebraic representation of $\mathrm{GSp}_{4}(\mathbb{Q})$
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 ...
CommunityBot
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Extending abelian schemes and their polarizations from an open subset
Let $S$ be a smooth quasi-projective scheme over $\mathbb{Z}$, and $A$ an abelian scheme over an open subset $U \subset S$. Suppose that $S\backslash U$ has codimension at least $2$ and that for every ...
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Failure of injectiveness of maps between cotangent spaces of abelian varieties
Let $p$ be a prime and $K$ a finite extension of $\mathbb Q_p$ with ramification index $e$. Let $\mathcal O_K$ be the ring of integers of $K$ and $k$ its residue field and the unique maximal ideal. ...
- 2,284
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How does the number of connected components of the Néron model change in a family of abelian varieties?
Given an elliptic curve $E/\mathbb{Q}_p$, it is known that the component group of the Néron model of $E$ is cyclic of order $-v(j(E))$ when $E$ has split multiplicative reduction, and in all other ...
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Néron model, torsion and ramification
Let $B$ a discrete valuation ring, say for simplicity with residue field of characteristic $0$, and $K$ its quotient field. Assume that I have an abelian variety $A$ over $K$ and let $A'$ be its Néron ...
- 49.1k
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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 ...
- 14.2k
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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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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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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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