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1
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1answer
106 views

Neron model: can number of components decrease after based change?

Suppose I have Neron model over some discrete valuation ring. Is there a result such that the number of components of the fiber over the closed point cannot decrease after some based change? In ...
6
votes
1answer
342 views

Understanding of Tamagawa numbers of hyperelliptic curve

One's can find following definition of tamagawa numbers in Dino Lorenzini paper "Torsion and Tamagawa numbers": Let $K$ be any discrete valuation field with ring of integers $O_K$ , uniformizer ...
7
votes
2answers
308 views

Calculate reduction of Jacobian of hyperelliptic curve

Suppose I have a hyperelliptic curve of genus $2$ over $\mathbb Q$. I want to get information about its Jacobian reduction at prime $p$ (especially, in case $p=2$). Also I'm interesting in the group ...
4
votes
1answer
229 views

Rational points on $X_0(15)$

The modular curve $X_0(15)$ has a canonical model over $\mathbf{Q}$, and it has genus $1$. As the cusp $\infty$ is rational, it is an elliptic curve. Roughly, my question is whether we can find all ...
2
votes
1answer
149 views

Specialization of sections in an elliptic fibration

Let $\pi: X \rightarrow S$ be the Neron model of an elliptic curve over a dedekind domain (but probably any minimal elliptic fibration will suffice). Let $\eta$ be the generic point of $S$, $K = ...
1
vote
0answers
197 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 $R$ ...
6
votes
1answer
216 views

Representability of sheaf of Ext^1 of a Néron model by $\mathbb{G}_m$

Let's work over a trait $S=\mathrm{Spec}R$, where $R$ is a dvr with fraction field $K$, residue field $k$. Given an abelian variety $A_K$ with semi-stable reduction, let $A$ over $S$ be its Néron ...
21
votes
1answer
886 views

Do all curves have Néron models

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$. Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$? By a Néron model, I mean ...
2
votes
1answer
373 views

reduction of elliptic curves

Let $X$ be an elliptic curve over a complete local field. The definition of semi-abelian reduction is: "the Neron model of $X$ is a semi-abelian scheme". On the other hand, the definition of ...
2
votes
1answer
313 views

Group of connected components of the global Néron-Raynaud model of a torus

Let $K = \mathbb{F}_q(C)$ be a global function field of an irreducible projective and smooth curve $C$ defined over a finite field of constants $\mathbb{F}_q$. Let $T$ be a $K$-torus. We choose one ...
1
vote
0answers
282 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 ...
2
votes
1answer
549 views

de jong's alteration theorem for families

What is the current status of de Jong's smooth alteration theorem for a family of schemes? His 1997 paper shows that given any family of curves $X/S$ with $S$ of finite type (and, say, local) over a ...
3
votes
1answer
425 views

Isomorphism on p-torsion of Neron models

Let $A$, $B$ be abelian varieties over $\mathbb{Q}$, with corresponding Neron models $\mathcal{A}$, $\mathcal{B}$ over $X=Spec{\mathbb{Z}}$. Let $p$ be an odd prime of good reduction for both $A$ and ...
6
votes
2answers
421 views

Global Sections of the Identity Component of Neron model

Let $A$ be an abelian variety over a number field $K$ and consider the Neron model $\mathcal{A}$ of $A$ over $X=Spec{\mathcal{O}_K}$. If $\mathcal{A}^0$ is the identity component of $\mathcal{A}$, ...
3
votes
1answer
310 views

Maps on the identity components of Neron models

Any map $A \to B$ of abelian varieties of the same dimension over a global field $K$ induces a map $\mathcal{A} \to \mathcal{B}$ on the corresponding Neron models over $X$ (where ...
4
votes
1answer
409 views

Special fiber of the Neron Model of an Abelian scheme in terms of Limit Hodge Structure

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 ...
10
votes
0answers
461 views

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 ...
15
votes
2answers
816 views

Are there Néron models over higher dimensional base schemes?

Are there Néron models for Abelian varieties over higher dimensional ($> 1$) base schemes $S$, let's say $S$ smooth, separated and of finite type over a field? If not, under what additional ...
9
votes
4answers
1k views

Torsion of an abelian variety under reduction.

Let $p$ be a prime. Suppose you have an Abelian scheme $A$ over $Spec\ \mathbb{Z}_p$. How do you prove that if $q$ is another prime, then the $q$-torsion of $A$ injects into the torsion of $A_p$, ...
6
votes
2answers
546 views

Néron theory for motives of arbitrary weight

SGA 7, tome 1, exp. IX, contains in its introduction and in section 13.4 remarks about ideas and conjectures of Deligne on a “théorie de Néron pour motifs de poids quelconque”. Would someone please ...