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Questions tagged [abelian-schemes]

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55 questions
Newest Active Bountied Unanswered
2 votes
0 answers
114 views

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 ...
Razumikhin's user avatar
  • 123
2 votes
1 answer
320 views

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 $\...
Razumikhin's user avatar
  • 123
3 votes
0 answers
197 views

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 $\...
babu_babu's user avatar
  • 321
2 votes
1 answer
283 views

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 ...
TCiur's user avatar
  • 719
7 votes
0 answers
207 views

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. ...
Maarten Derickx's user avatar
  • 2,284
3 votes
0 answers
198 views

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 ...
Nathan Lowry's user avatar
  • 333
4 votes
1 answer
301 views

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 ...
Hans's user avatar
  • 3,151
3 votes
1 answer
284 views

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 ...
user avatar
5 votes
0 answers
156 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 ...
Damian Rössler's user avatar
  • 3,346
3 votes
2 answers
637 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 $S_i$, ...
red_trumpet's user avatar
  • 1,378
2 votes
0 answers
236 views

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 ...
Z Wu's user avatar
  • 632
4 votes
4 answers
989 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 ...
curious math guy's user avatar
  • 4,448
2 votes
0 answers
219 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 ...
Asvin's user avatar
  • 8,081
5 votes
0 answers
485 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 ...
Stefano's user avatar
  • 625
11 votes
1 answer
413 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 ...
Asvin's user avatar
  • 8,081

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