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Results tagged with abelian-varieties
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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
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Reference request. Finiteness of the Selmer group
Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with …
- 141
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answers
84
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Two pairings on the group $K(L)$ associated with a non-degenerate line bundle $L$ on an abel...
Let $A$ be an abelian variety over a field and let $L$ be a non-degenerate line bundle on $A$.
Then $L$ gives rise to a morphism $\lambda:A\to A^*$ from $A$ to its dual.
As usual, let $K(L):=\ker(\lam …
- 3,076
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answers
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Extension of a multiple of a rational point to an integral point of a semiabelian scheme
Let $\cal A$ be a smooth commutative group scheme over $S$, where $S$ is the spectrum of a discrete valuation ring with fraction field $K$ and residue field $k$. Suppose that $A:={\cal A}_K$ is an abe …
- 3,076
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Torsion points of abelian varieties in the perfect closure of a function field
The following is a problem, which was recently brought to my attention by H. Esnault and A. Langer.
Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field …
- 728
9
votes
1
answer
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Endomorphism ring of simple ordinary abelian variety
Is there an example of an ordinary and simple abelian variety $A$ over an algebraically closed field $K$ (of characteristic $p>0$) such that ${\rm End}(A)$ is not commutative? Note that the answer is …
- 790
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votes
1
answer
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Commutation of endomorphisms of abelian varieties
Let $A$ be an abelian variety over an algebraically closed field $k$.
Let $\phi:A\to A$ be an étale isogeny (over $k$). Suppose that the set $\cup_{r\geq 0}({\rm ker}\,\phi^{\circ r})(k)$ is
Zarisk …
- 28.8k
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votes
1
answer
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CM abelian varieties over the rationals
Let $K$ be a number field and let $A$ be an abelian variety of dimension $g$ over $K$. Let $L$ be a CM field and suppose that $[L:{\bf Q}]=2g$. Suppose that there exists an embedding $\iota:L\hookrigh …
- 2,616
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2
answers
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On a proposition in Hartshorne's paper "Ample vector bundles on curves"
In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following:
Let $A$ be an abelian variety [over an alg. closed field $ …
- 1,285
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0
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Ampleness of Hodge bundles over complex curves
Let $C$ be a smooth, proper and connected curve over
the complex numbers $\bf C$. Let ${\cal G}\to C$ be a smooth group scheme over $C$ and let $\epsilon_{\cal G}:C\to{\cal G}$ be its
zero-section. …
- 3,076
11
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1
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Mordell-Weil group of the universal abelian scheme
Let $n>2$ and let $k$ be either $\bf Q$ or a finite field whose characteristic is prime to $n$. Let $A_{g,n}$ be the moduli scheme, which represents the functor, which with every $k$-scheme $S$
assoc …
- 30.5k
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1
answer
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Etale endomorphisms of abelian varieties in positive characteristic
Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ${\bf F}_p$ (where $p>0$ is a prime number).
My question is : does there exist an abelian variety $ …
- 5,050