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Questions tagged [abelian-varieties]

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions.

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Reference: Wikipedia.

Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.

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Some complex torus is not an abelian variety

Let $e_1,...,e_4\in\mathbb{C}^2$ whose coordinates are all algebraically independent. Let $\Lambda$ be the lattice spanned by these vectors. Why is $\mathbb{C}^2/\Lambda$ not an abelian variety?
Alan Lee
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Tate conjecture for Abelian varieties without polarizations

I'm reading Levin's note to learn the Tate conjecture for Abelian varieties over number fields. It seems that Tate's original paper and this note both use a weak finiteness hypothesis There are infinitely many $B_n=A/G_n$ which are isomorphic,…
Phanpu
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Searching book for abelian variety

Do you have interesting books presenting abelian variety (over an arbitrary field k) using the scheme point of view? Most of the lectures I know use the point of view presented in the first chapter of Hartshorne (so without scheme). I'm looking for…
Analyse300
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The group algebra decomposition for an abelian variety

I was reading this paper, and I want to understand the definition of $Fix_{G_k}{\mathcal{W}}$ given in $1.2$. I tried to use the formula on a specific example and it did not worked (I think that I did not understand the definition of Fix). Let…
Román
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What is the dimension of the coproduct of two abelian varieties?

Let's say I have two abelian varieties, $A$ and $B$, and I take their coproduct. My intuition says that $\dim(A+B) = \dim(A) + \dim(B)$. Is this accurate?
gradintherockies
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Definition of lattices being isotropic

I am reading Birkenhake-Lange's Complex Abelian Varieties. In Chapter 3 section 1, there is a notion of isotropic for lattices: Let $X=V/\Lambda$ be a complex abelian variety, where $V$ is a complex vector space and $\Lambda$ is a lattice. Let $L$…
finiteness
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Abelian Varieties over finite fields

I am interested in studying some algorithmic aspects of hyperelliptic curves on finite fields. For this I have studied the topics of Projective model of hyperelliptic curves, Local rings and Noetherian rings, Valuation at a point on a curve, Field…
George
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Pullback of the Poincare bundle along $\varphi_{\mathcal L}$ is the Mumford bundle

I'm trying to understand why $(1\times\varphi_{\mathcal L})^*(\mathcal P_A) = \Lambda(\mathcal L)$ where $\Lambda(\mathcal L) = m^*\mathcal L\otimes\operatorname{pr}_1^*\mathcal L^{-1}\otimes\operatorname{pr}_2^*\mathcal L^{-1}$ is the Mumford…
Nico
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division points of abelian varieties after reduction modulo a prime

I do not have much knowledge of this topic, so I would like also if you can give me some basic references regarding this, in addition to a possible answer to my question. Given an abelian variety $A$ defined over a field $K$ , and a positive integer…
A. GM
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