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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.
1
vote
Endomorphism ring of the Jacobian of a generic smooth plane quartic
See https://arxiv.org/abs/math/0405156 where Del Pezzo surfaces of degree 2 are used, in order to construct plane quartics, whose jacobians have endomorphism ring $\mathbb{Z}$.
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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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9
votes
Accepted
Isomorphic endomorphism algebras implies isogenous (for abelian varieties over finite fields)?
Here is a counterexample to Proposition A provided by 8-dimensional abelian varieties $A_1$ and $A_2$ over a finite field $F_{p^2}$ where $p$ is any prime that is congruent to $1$ modulo $17$ (e.g., …
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8
votes
Accepted
Weil pairing on abelian varieties and etale Chern classes
It seems that one of the pairings is the negative of the other (in char 0 this assertion is actually Lemma 2.6 of https://arxiv.org/pdf/1809.01440.pdf ).
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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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10
votes
Accepted
Abelian variety with prescribed endomorphism ring
The answer to your Question 3 is YES with the ground field $\mathbb{Q}$.
Here is a sketch of the proof. For each positive integer $q$ and a "parameter" $t$ (in char 0) consider the smooth projective …
- 14.2k
10
votes
Accepted
Does every Shimura variety contain a generic point defined over a number field?
The Answer to Question 1 is YES. It follows from Serre's variant of Hilbert's irreducibility theorem (for infinite Galois extensions) combined with the Tate conjecture on homomorphisms of abelian vari …
- 14.2k
7
votes
Abelian variety with prescribed endomorphism ring
The answer to your Question 1 is NO. See Th. 5.1 of my 2015 JPAA paper.
The answer to your Question 2 is YES. See 1963 Annals paper of Shimura "On analytic families ..".
- 14.2k
3
votes
Accepted
How to prove that $A$ is supersingular iff the Picard number $\rho(A)$ is equal to the secon...
Now let $k$ be a finite field. If $E$ is an elliptic curve over $k$ then $End(E)\otimes\mathbf{Q}$ is either an imaginary quadratic field or a definite quaternion algebra over $\mathbf{Q}$.
Warnin …
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4
votes
How to prove that $A$ is supersingular iff the Picard number $\rho(A)$ is equal to the secon...
If $A$ is not simple then it is isogenous to a product $E_1 \times E_2$ of elliptic curves $E_1$ and $E_2$, and the Picard numbers of $A$ and $E_1\times E_2$ do coincide. If $E_1$ is not isogenous to …
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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 …
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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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6
votes
Accepted
can all CM types be realized by Jacobians?
``Given $(K,\Phi)$ , there always exists a $g$ -dimensional abelian variety $A$ such that $End(A)\otimes Q=K$ and that $K$ acts on $H^0(A,\Omega^1)$ through $\Phi$. One easily constructs as a comp …
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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
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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