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Results tagged with abelian-varieties
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user 9658
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
Endomorphism Ring of Simple Abelian Varieties
Albert's classification works/is good enough for (algebraically closed) fields of characteristic zero. For the complete list of all possibilities for a given $g$ (including the case of prime characte …
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4
votes
Accepted
Can we control the size of the intersection of two abelian subfactors of an abelian variety ?
Here is a (slightly more detailed) variant of what was pointed out by grp.
Let $E$ and $F$ be non-isogenous elliptic curves over $K$. Let $n$ be a positive integer. (If $p=char(K)>0$ and $p$ divides …
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8
votes
Accepted
Abelian varieties with given endomorphism algebra
Felipe, Albert is right: $r>1$. In particular, there are no complex abelian surfaces, whose endomorphism algebra is a definite quaternion algebra over the rationals. The same is true in any characteri …
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12
votes
Accepted
Tate conjecture for abelian varieties over a finitely generated extension of an algebraicall...
The answer is no: an easy (not interesting) counterexample is provided by "constant" abelian varieties, i.e., when $A$ and $B$ are defined over an algebraically closed field $k$ of characteristic zer …
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5
votes
Accepted
possible CM-types of abelian varieties
There are certain additional conditions on ``multiplicities" that are spelled out in the original Shimura's paper. Let me discuss a couple of interesting cases.
The case of CM type, i.e. when $n=g$. …
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6
votes
Etale endomorphisms of abelian varieties in positive characteristic
If $p-1$ is divisible by $24$ then there is an explicit example of an ordinary $7$-dimensional abelian variety $X$, whose endomorphism algebra is the imaginary quadratic field $Q(\sqrt{-3})$; in parti …
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3
votes
Accepted
Mumford-Tate groups of abelian surfaces
The list of of all possible Hodge (special Mumford-Tate) groups of complex abelian varieties up to dimension 4 (and for simple abelian varieties up to dimension 5) is contained in https://arxiv.org/pd …
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5
votes
Theta group representation
If I understand correctly, $K_1$ and $K_2$ are (mutually orthogonal maximal) isotropic subgroups of $K(L)$. Therefore they both can be lifted (non-canonically) to (finite) commutative (sic!) subgrou …
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11
votes
Accepted
Shimura datum of family of fake elliptic curves
Actually, fake elliptic curves are discussed in Chapter 9 of Lang's Introduction to algebraic and abelian functions.
In order to describe the map to $GSP(4,Q)$, recall that there is the standard ant …
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2
votes
lower bound for torsion of abelian varieties
Let $E$ be a supersingular elliptic curve over a finite field $K$ of characteristic $p$. If $K$ is sufficiently large then the generator (Frobenius automorphism) of the absolute Galois group of $K$ ac …
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3
votes
Accepted
lower bound for torsion of abelian varieties
Now let me address your last question. For the sake of simplicity, let us assume that $K$ is a global field of characteristic $p>2$ and the ring $End(A)$ of all endomorphisms of $A$ (over an algebraic …
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8
votes
Reference for a theorem of Tate on the endomorphism rings of AVs over finite fields
I guess $p=\operatorname{char}(k)$. For another (unified) proof of Tate's theorem (that works for primes $\ell\ne p$ and $\ell=p$) see arXiv:0711.1615 [math.AG]; MR2484084 (2010a:11117).
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9
votes
Accepted
Weil reciprocity on abelian varieties and biextensions?
Lang's reciprocity law and Poincar\'e-Mumford biextensions come together in the context of generalized N\'eron pairings
http://iopscience.iop.org/0025-5726/6/3/A03/pdf/0025-5726_6_3_A03.pdf
(Proof …
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16
votes
Accepted
n-th root of unity in n-th division field of abelian variety?
Actually, this is an exercise in Serre's Lectures on Mordell--Weil Theorem:
$K(A[n])$ always contains $\mu_n$ if $char(K)$ does not divide $n$ and $A$ is an abelian variety of positive dimension over …
- 5,050
2
votes
Generalization of $j(E) \in \overline { \Bbb{Z}}$ to abelian varieties of arbitrary dimension
The integrality of $j(E)$ means that an elliptic curve $E$ over a number field has potential good reduction everywhere (Deuring).
So, the following assertion may be viewed as a generalization of the …
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