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Results tagged with abelian-varieties
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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.
7
votes
Accepted
Global Sections of the Identity Component of Neron model
First, a remark: the free part of $A(K)$ is a quotient, not a sub, and so it is possible that a point of infinite order in $A(K)$ could have non-trivial image
in $\Phi_A(X)$. Probably what you mean …
- 19.4k
5
votes
Accepted
Maps on the identity components of Neron models
If I'm not mistaken, the map $Res^0$ is a bijection. Indeed, restriction induces maps
$$Hom_X(\mathcal A,\mathcal B) \to Hom_X(\mathcal A^0,\mathcal B^0)\to Hom_K(A,B).$$
The composite is a bijectio …
- 57.6k
35
votes
Why can projective varieties just have abelian group operations?
There are several different ways to see this. Here is one:
Let $G$ be our irreducible projective algebraic group variety over the field $k$, with identity element $e$. The group $G$ acts on itsel …
- 57.6k
10
votes
Accepted
Tate models for semistable algebraic varieties with mixed reduction over a local field
If $A$ has semi-abelian reduction, then $A$ is uniformized by a semi-abelian variety $G_A$,
namely there is an exact sequence
$$0 \to \Gamma_A \to G_A \to A \to 0,$$
where $\Gamma_A$ is free of finite …
- 57.6k
7
votes
Accepted
$p$-torsion in the Mordell-Weil group of Abelian varieties injecting in reduction
One has the finite flat group scheme $\mathbb Z/p$ over $\mathcal O_{K_{\mathfrak p}}$
(I write $K_{\mathfrak p}$ for the $\mathfrak p$-adic completion of $K$, and
$\mathcal O_{K_{\mathfrak p}}$ for …
- 57.6k
13
votes
Why do people think that abelian varieties are the hardest case for the Hodge conjecture?
Related to Jim Milne's answer, one might mention that Deligne proved that for abelian varieties, all Hodge cycles are "absolutely Hodge" (i.e. when you think of them embedded diagonally inside the pro …
- 57.6k
4
votes
Accepted
Galois orbits of newforms with prime power level
See Section 4 of my paper The Hecke algebra T_k has large index, joint with Frank Calegari, where this sort of argument is applied. Lemma 4.1 is a variant of the statement you prove; see also the las …
- 57.6k
39
votes
Accepted
Why no abelian varieties over Z?
It's a result related in spirit to Minkowski's theorem that $\mathbb Q$ admits no non-trivial unramified extensions. If $A$ is an abelian variety over $\mathbb Q$ with everywhere good reduction, then …
- 57.6k