Questions tagged [neron-models]

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Lifting abelian varieties in (the closed fiber of) a fixed Neron model

Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
B. Cais's user avatar
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7 votes
0 answers
415 views

An abelian variety has good reduction $\iff$ the Neron model is proper

Let $R$ be a D.V.R. with the fraction field $K$, $A$ a $K$- abelian variety, $\mathfrak{A} \to \operatorname{Spec}R$ the Neron model of $A$. We say $A$ has good reduction if there exists a smooth ...
k.j.'s user avatar
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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
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6 votes
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121 views

Good reduction of abelian varieties over valuation rings via coverings

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$, and let $A$ be an abelian variety over $K$. Suppose that there is a smooth proper scheme $\mathcal{X}$ over $\mathcal{O}_K$ whose ...
Kriss's user avatar
  • 233
5 votes
0 answers
107 views

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 ...
Damian Rössler's user avatar
5 votes
0 answers
508 views

Reduction of torsion points on Neron Model

Let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $R$ and residue field $k$. Let $A/K$ be an abelian variety with Neron model $\mathcal{A}/R$. We denote by $\tilde{\mathcal{A}}/k$ the ...
Jędrzej Garnek's user avatar
4 votes
0 answers
314 views

$H^2(S, f_* \mathbb{G}_m)$ in the fppf versus etale topology for proper $f$

Let $f\colon X \rightarrow S$ be a proper morphism of schemes. Is the cohomology group $H^2(S, f_* \mathbb{G}_m)$ the same regardless of whether it is computed in the etale or the fppf topology? And ...
Question Mark's user avatar
4 votes
0 answers
217 views

An extension of group schemes admitting Neron models

Let $R$ be a discrete valuation ring, $K$ its field of fractions, and $$ 0 \rightarrow G_K' \rightarrow G_K \rightarrow G_K'' \rightarrow 0$$ a short exact sequence of smooth $K$-group schemes of ...
Question Mark's user avatar
4 votes
0 answers
159 views

Is this $S$-birational map an open immersion on its domain of definition?

My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...
Question Mark's user avatar
3 votes
0 answers
106 views

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 ...
Nathan Lowry's user avatar
3 votes
0 answers
145 views

What does the Néron model of the dual abelian variety parametrize?

Let $K$ be a field which is complete with respect to a discrete valuation $v$ with ring of integers $R$ and residue field $k$. Let $A$ an abelian variety over $K$ and let $A^t$ be the dual abelian ...
Hans's user avatar
  • 2,863
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
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2 votes
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158 views

A normal proper model of an abelian variety with geometrically integral special fiber smooth at the reduction of the origin

Let $A$ be an abelian variety over $\mathbb{Q}_p$. Does there exist a proper flat morphism $X\to \mathrm{Spec}\:\mathbb{Z}_p$ such that the generic fiber is isomorphic to $A$, the special fiber is ...
user avatar
2 votes
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158 views

Covering a finite set of points of height 1 by an affine open

Let $R$ be a Noetherian ring and let $X$ be a finite type, separated $R$-scheme that is normal and integral. Let $x_1, \dotsc, x_n \in X$ be points of height $1$. Does there exist an open affine $U \...
Question Mark's user avatar
1 vote
0 answers
187 views

Outline of the proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite

I have a question about proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite. Could you please tell (correct) me any strange parts about my understanding of the outline of the proof ? My ...
Duality's user avatar
  • 1,406
1 vote
0 answers
969 views

Is this essentially of finite type algebra actually of finite type?

Let $R$ be a discrete valuation ring with a uniformizer $\pi$ and $(A, \mathfrak{m}_A)$ a local $R$-algebra that is essentially of finite type (i.e., is a localization of a finite type $R$-algebra), ...
Question Mark's user avatar
1 vote
0 answers
256 views

How does the line bundles look like on a proper model (or Néron model) of an abelian variety?

How does the line bundles look like on a proper model (or Néron model) of an abelian variety? Who knows references about this? In particular, let us work over a trait $S=\mathrm{Spec} R$, where $R$ ...
Heer's user avatar
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1 vote
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507 views

Component group of Neron model of a parametrized abelian variety

Let $A$ be an abelian variety of dimension $2$ over a $p$-adic field $K$ with (additive) valuation $v$. Assuming $A$ has multiplicative reduction, the theory of $p$-adic theta functions gives us an ...
David Corwin's user avatar
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0 votes
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105 views

Extend line bundle on regular curve to it's regular model

Let $S$ be the spectrum of an excellent discrete valuation ring with field of fractions $K$ and $C$ be a proper integral regular curve over $K$. Assume, $C$ admits a proper regular flat model $\...
user267839's user avatar
  • 5,948
0 votes
0 answers
113 views

Does an isogeny between tori induce an isomorphism of the Lie algebras of their lft Néron models?

Let $f:T_1 \to T_2$ be an isogeny of tori over a number field $K$. Does $f$ induce an isomorphism of the Lie algebras of the lft Néron models of $T_1$ and $T_2$ ? Are there some interesting properties ...
Adrien MORIN's user avatar