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Results tagged with abelian-varieties
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user 3485
Abelian varieties are projective algebraic varieties endowed with an Abelian group structure. Over the complex numbers, they can be described as quotients of a vector space by a lattice of full rank. They are analogs in higher dimensions of elliptic curves, and play an important role in algebraic geometry and number theory.
5
votes
Accepted
Can the simplicity of abelian varieities be implied by the reduction
Here is a slightly different proof. We have the following facts:
(1) If $B, C$ are abelian varieties over $K$, then the Néron model of $B\times C$ is the product of the Néron models (this is a sim …
- 11.1k
5
votes
Accepted
geometrical reducedness of the identity connected component (reference request)
You can find a proof here following hints of Raynaud. The main idea is that the prime-to-$p$ torsion of $G^0$ is dense in $G^0_\mathrm{red}$ and its Zariski closure (with reduced structure) is geometr …
- 11.1k
6
votes
Accepted
dual isogeny for abelian varieties over a general field
You can find the statement in the general case (any isogeny over arbitrary field) and the proof in van der Geer and B. Moonen's book (draft) on abelian varieties. More precisely it is in Chapter 5, Pr …
- 11.1k
16
votes
Accepted
Can we always find a curve which doesn't have semi-stable reduction
Yes. Take $f\in O_K$ a uniformizing element of some prime $\mathfrak p$. Consider the hyperelliptic curve defined by the equation
$$y^2=x^{2g+1}+f.$$
Then this curve doesn't have semi-stable reducti …
- 11.1k
5
votes
Torsion of an abelian variety under reduction.
To complete Pete and Milne's answers when A is not an abelian scheme (for example, when it is the Néron model of an abelian variety over ${\mathbb Q}_p$ with not necessary good reduction), then for an …
- 11.1k
11
votes
Are there Néron models over higher dimensional base schemes?
This is not answer to the OP, but to the remark of Pete on singular base scheme $S$. Here is an example explainning why one should assume regularity even in dimension $1$.
Let $T$ be a smooth curve …
- 11.1k