Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with abelian-varieties
Search options not deleted
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.
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 …
- 10.5k
10
votes
1
answer
537
views
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 …
- 10.5k
8
votes
Accepted
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
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
5
votes
Accepted
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 …
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 …
4
votes
Accepted
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
4
votes
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
Accepted
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
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 …
3
votes
Accepted
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 …
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 $C$ …
2
votes
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
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 …
- 10.5k