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Results tagged with abelian-varieties
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user 5015
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.
2
votes
kernel of isogeny becomes constant after base change
In less fancy language, you are asking for the ramification of the extension of $K$ where you adjoin the coordinates of the $n$-torsion points of $A$. Under your assumption of everywhere good reductio …
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2
votes
Why is the norm map dual to restriction under Tate local duality?
If $m$ is an integer coprime to the characteristic of $k$, then the pairing between $A^t(K)/m A^t(K)$ and the $m$-torsion part of $H^1(K,A)$ is compatible with the cup pairing
$$H^1(K,A^t[m]) \times …
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4
votes
Accepted
Tamagawa numbers of abelian varieties and torsion.
There is no general relation between the local $p$-primary torsion and the Tamagawa numbers. I believe one can have $p$-torsion points that map to non-trivial or to the trivial element in the group of …
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8
votes
Accepted
Tate–Shafarevich group of Jacobian of Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$
$\DeclareMathOperator{\sha}{Ш}$
I am not sure that the proof that Sha has order 9 is anywhere spelled out in full. Here the ideas how to do it.
First, that the order of $C$ is three in the $\sha$ is j …
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5
votes
Accepted
Questions about elliptic curves with level-$n$ structure
Let $K$ be a finite extension of $\mathbb{Q}_p$ with $p\neq 2$. Suppose that $E/K$ is an elliptic curve with additive reduction and such that $E$ has full $4$-torsion over $K$. By the Kodaira classif …
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2
votes
Accepted
integral basis for the Lie algebra of the Neron model of an abelian variety
The same problem already appears in the formulation of the Birch and Swinnerton-Dyer conjecture for say an elliptic curve over a number field. When there is no longer a global minimal Weierstrass mode …
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5
votes
Accepted
Tate-Shafarevich groups under finite Galois field extensions
The remark added to the question shows that the kernel of $Ш(E/F) \to Ш(E/L)^G$ is finite where $G$ is the finite Galois group of $L/F$.
$\DeclareMathOperator{\coker}{coker}$
Here is an argument why t …
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