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Results tagged with abelian-varieties
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user 18060
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.
3
votes
Accepted
Definition field of isogeny between abelian varieties
Yes. The group of isogenies form a locally free module of rank $1$ over the endomorphism ring of $A$, hence are generated by a single isogeny of minimal degree $k$. So every isogeny is that isogeny co …
- 149k
5
votes
Accepted
Explicit period lattices for abelian surfaces
It seems to me that it's best to go back up to the quadratic extension and view it as a product of two elliptic curves $E \times E^\sigma$. If you can compute the Weierstrass equations for these ellip …
- 149k
9
votes
Accepted
The torsion point count in higher dimension
As far as I know, we expect that the image of the Galois group in $GL_{2g}(\mathbb A_\mathbb Q)$ is open in $G(\mathbb A_\mathbb Q)$ for $G$ the monodromy group of the Galois representation (which is …
- 149k
1
vote
Accepted
etale covers of line bundles on an abelian variety
For clarity, the best way to work with this is complex-analytically. I am sure there is a, probably more involved, algebraic proof.
Lemma: Let $M$ be a complex manifold and let $X$ be a $\mathbb C^\t …
- 149k
2
votes
Accepted
kernel of an isogeny and coker of its induced map on the Tate module
Let $x$ be an $l^n$-torsion point in the kernel of the isogeny. Take any point $y$ such that $l^ny=x$. $l^n\beta(y)=\beta(l^ny)=\beta(x)=0$, so $\beta(y)$ is an $l^n$-torsion point. This is well-defin …
- 149k
4
votes
Finite quotients of the absolute Galois group of $\mathbb{Q}$ via torsion of elliptic curves
For elliptic curves, the answer is no, as Chris Wuthrich and Aron Fehm point out in the comments. In fact I think every extension with Galois group a finite simple group not of the form $PSL_2(\mathbb …
- 149k
1
vote
Accepted
Are there any quadratic functions on an abelian variety not from the height machine?
The source has countable dimension over $\mathbb R$, since $A$ has countably many divisors defined over a finite extension of $K$, while the target, being the space of quadratic functions on a countab …
- 149k
2
votes
Accepted
Interpretation of $H_1(A_\mathbb{C}^{top},\mathbb{Q})$
$A_\mathbb C$ is not defined in the finite field case because you can't base change from a finite field to $\mathbb C$.
$H^1( A_\mathbb C^{top})$ is a vector space whose dimension is $2g$. One can de …
- 149k
4
votes
Accepted
Ordinary abelian varieties over a finite field
Because the abelian variety is ordinary, $\pi$ has the property that it's $p$-adic valuation at every place is either $0$ or $1$ (this follows easily from condition (IV) on the first page of Deligne's …
- 149k
5
votes
Semisimplicity of Frobenius on *integral* Tate module
Probably not.
Let $E_1$ and $E_2$ be two distinct elliptic curves with identical $\ell$-torsion representations, and let $A$ be $E_1 \times E_2$ mod the diagonal $\ell$-torsion representation.
Then …
- 149k
1
vote
Another question related to the isogeny theorem for elliptic curves
The class group of the endomorphism ring $\mathcal O_K$ is defined over $K$. But unless the class group of $\mathcal O_K$ is trivial, none of the CM curves are defined over $K$. Thus there is no bijec …
- 149k
6
votes
Moduli Spaces of Higher Dimensional Complex Tori
A complex torus of dimension $d$ can be written as a quotient $\mathbb C^d/\mathbb Z^{2d}$. Thus it is determined by a map $\mathbb Z^{2d} \to \mathbb C^{d} \cong \mathbb R^{2d}$. We can specify this …
- 149k
2
votes
Accepted
Non-degenerate points on a Jacobian surface
The product $(E \times C) / \sigma$, where $\sigma$ acts by inversion on $E$ and the hypereliptic involution on $C$, is an elliptic surface over $C/\sigma = \mathbb P^1$.
This surface has two sections …
- 149k
1
vote
Generic Mumford Tate group and algebraic points
It's possible to be more precise. The non-generic points form a countable family of subsets defined by algebraic equations of bounded degree.
To see this, we use Larsen's alternative - if the Mumford …
- 149k
10
votes
Accepted
Abelian variety with CM defined over real numbers
No.
Assume for contradiction that such an $A$ exists. First look at the singular cohomology $H^1(A_{\mathbb C}, \mathbb Q)$, which admits an action of $K$ and so is a $K$-vector space. It has dimensio …
- 149k