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Results tagged with abelian-varieties
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user 519
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
$p$-divisibility of Picard groups
$\newcommand{\wt}{\widetilde}$
$\newcommand{\mr}{\mathrm}$
The question has a positive answer, in fact, regularity of $C$ is not needed. The proof as written below works under the assumption that $C$ …
- 10.5k
10
votes
1
answer
537
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Orders of reductions of rational points on elliptic curves
I am looking for references where the following (or similar questions) have been studied:
Let $K$ be a number field or a function field in one variable over a finite field and let $E$ be an elliptic …
- 13.8k
4
votes
Accepted
Essential dimension and the moduli space of abelian varieties
The two notions are related using Theorems 4.1 and 6.1 of the paper of Brosnan, Reichstein and Vistoli:
Theorem 6.1 reduces the computation of the essential dimension of the stack to that of the gen …
- 10.5k
1
vote
Examples of rational families of abelian varieties.
One can construct some families over rational bases which are not Jacobians by taking quotients:
For example, let $A$ be a fixed abelian variety of dimension $> 1$ and let $S$ be the space of all smo …
5
votes
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Nef divisors on abelian varieties
I think it is probably easier to prove that $L$ is ample when all the inequalities are strict:
The assumption for $i=n$ implies that $K(L)$ is finite, i.e., $L$ is non-degenerate, by the second stat …
- 10.5k
3
votes
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Intersection multiplicity in abelian varieties
The following answer expands on my comment.
We use Fulton's definition of the intersection product. Consider the diagonal embedding $\Delta$ of $A$ in $A \times A$ (which is regular) and intersect t …
- 10.5k
4
votes
Accepted
Nef classes on abelian varieties in positive characteristic
Here is a sketch of a purely algebraic proof based on the theory developed in Chapter 3 of Mumford's "Abelian Varieties".
Let $L$ be a nef line bundle on the abelian variety $A$ of dimension $g$. If …
- 10.5k
4
votes
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Mumford-Tate groups of abelian varieties with potentially good reduction everywhere
An elliptic curve with integral $j$-invariant has potential good reduction everywhere. If it does not have CM then its Mumford-Tate group is $GL_{2,\mathbb{Q}}$ which is not anisotropic modulo its cen …
- 10.5k
4
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CM abelian varieties and potential good reduction
Potential good reduction everywhere is quite far from having complex multiplication.
For elliptic curves, the condition is equivalent to the $j$-invariant being an algebraic integer. For $F = \mathbb …
- 10.5k
4
votes
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Kuga-Satake with p-adic methods
A good place to look is the paper "Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic". J. Reine Angew. Math. 648 (2010), 13–67, by Jordan Rizov. There are also related results by Yv …
- 10.5k
8
votes
Canonical liftings of endomorphisms of ordinary abelian varieties
A reference which proves a more general result is Theorem 1 of the appendix to the paper by Mehta and Srinivas "Varieties in positive characteristic with trivial tangent bundle."
With an appendix by S …
- 10.5k
8
votes
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Reference for a theorem of Tate on the endomorphism rings of AVs over finite fields
I think the result appears in:
Waterhouse, W. C.; Milne, J. S.: Abelian varieties over finite fields. 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Broo …
- 10.5k
2
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Algebraic cycles of dimension 2 on the square of a generic abelian surface
As far as I know, there is no smooth projective variety over $\mathbb{C}$ of dimension $n>2$ with all possible Hodge numbers nonzero (i.e. $h^{p,q} \neq 0$ for all $p+q = n$) for which the Griffiths g …
- 10.5k
15
votes
Which curves can be found on Abelian varieties?
For any $g >0$ there exists an abelian surface $A$ containing a smooth curve $C$ of genus $g$; the surface can be assumed to be simple if $g > 1$:
For $g=1$, one can just consider $C \times C$. For …
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