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Results tagged with abelian-varieties
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user 5101
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.
10
votes
Accepted
Which abelian varieties over a local field can be globalized?
As phrased, this problem looks just as difficult as determining whether a given element of $\mathbb{Q}_p$ lies in $\mathbb{Q}$. There are real numbers which are unknown to be rational (e.g. $\pi + e$) …
- 21.4k
14
votes
Accepted
Albanese variety over non-perfect fields
The arguments of Serre can be in fact made to work over any separably closed field. The result in the general case can then be deduced using Galois descent. Details can be found in Section 2 and the a …
- 21.4k
9
votes
Accepted
Elements of arbitrary large order in the first Galois cohomology of an elliptic curve
Here is the kind of method I had in mind.
We have the elliptic curve Kummer sequence
$$0 \to E[n] \to E \to E \to 0,$$
Here I denote by $E[n]$ the $n$-torsion group scheme of $E$. Applying Galois coh …
- 21.4k
6
votes
Accepted
Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvabl...
In good cases, the ''isomorphism functor'' $\mathrm{Isom}(X,Y)$ is representable by a scheme. Hence, you are asking that if this scheme is non-empty (i.e. contains a geometric point), whether it conta …
- 21.4k
11
votes
Accepted
Defining isogenies over smaller fields
No: Consider the elliptic curve $E: y^2 = x^3 + x$ defined over $\mathbb{Q}$. Then the isogeny $y \mapsto iy$ and $x \mapsto -x$ is defined over $\mathbb{Q}(i)$ but obviously not over $\mathbb{Q}$.
I …
- 21.4k