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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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What is the Albanese map good for?

I am reading a textbook on complex manifolds and come across the Albanese map. For a compact Kähler manifold $X$, $$ T=H^0(X,\Omega_{X}^1)^*/H_1(M,\mathbb{Z}) $$ is a complex torus, called the Albanese torus of $X$. Fix a point $p\in X$, one obtains…
M. K.
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What is a Shimura variety and why should I care about them?

Shimura varieties have come up tangentially in talks with some of my advisors. My vague understanding is that they are "things that behave like moduli spaces of abelian varieties having some additional structure". I am familiar with the modular…
user0873
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How many elliptic curves have complex multiplication?

Let $K$ be a number field. Suppose we order elliptic curves over $K$ by naive height. What is the natural density of elliptic curves without complex multiplication? More generally, suppose we order $g$-dimensional abelian varieties over $K$ by…
Ashvin Swaminathan
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Why are elliptic curves important for elementary number theory?

Elliptic curves (or even Abelian varieties) are useful tools for many high-falutin' reasons They can be used to construct $\ell$-adic Galois representations One can find automorphic forms from an elliptic curve fairly easily There is a nice way to…
54321user
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What is the proper "addition formula" for $ \int \frac{\mathrm{d}x}{\sqrt{1-x^6}} $?

The followings are two well-known "addition formulas" (or "addition theorems"). $$ \int_{0}^{u} \frac{\mathrm{d}x}{\sqrt{1-x^2}} + \int_0^v \frac{\mathrm{d}x}{\sqrt{1-x^2}} = \int_{0}^{u\sqrt{1-v^2}+v\sqrt{1-u^2}} \frac{\mathrm{d}x}{\sqrt{1-x^2}}…
Tay Choi
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Elliptic curve over algebraically closed field of characteristic $0$ has a non-torsion point

Let $E/k$ be an elliptic curve over an algebraically closed field $k$ of characteristic $0$. Can one prove that the abelian group $E(k)$ is non-torsion? Better yet, can one prove that $E(k) \otimes_\mathbb Z \mathbb Q$ is an infinite-dimensional…
Bruno Joyal
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Why does Mumford want to avoid "reduction to Jacobians"?

In the introduction to his Abelian Varieties book, David Mumford writes: I don't believe the word "Jacobian" is ever used in this book. Rather stubbornly I wanted to prove that the theory of abelian varieties could be developed without the crutch…
Miles Lake
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Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?

Edit: I now crossposted this question on MO: https://mathoverflow.net/questions/428384/is-a-complex-algebraic-set-with-a-zariski-dense-subset-of-algebraic-points-alrea Let $X$ be a complex algebraic set, i.e. the (not necessarily irreducible)…
Luvath
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Divisors in an abelian surface

How to compute the Néron-Severi group of the abelian surface $Y = \mathbb{C}/\mathbb{Z}[i] \times \mathbb{C}/\mathbb{Z}[i]$. More generally, are there any result that compute the Néron-Severi group of product of curves? Suppose that surface $Y$ has…
rla
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An abelian variety not isogenous to a Jacobian

In the L-functions and Modular Forms Database is an isogeny class of an abelian variety of dimension $2$ over $\mathbb F_5$. They claim that it is principally polarizable but not the Jacobian of a curve over $\mathbb F_5$. The two ways they suggest…
Arkady
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Canonical divisor of an abelian variety

Let $A$ be an abelian variety over a field $k$ and let $K_A$ be its canonical divisor. Then I'm almost certain that $K_A$ is trivial, but I can't seem to prove it, nor find a counter example, nor find any reference on abelian varieties that even…
Hamish
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When $\phi_{\mathcal L}=0$ for $\mathcal L$ a line bundle over an abelian scheme $X/S$

Let $X\rightarrow S$ be a projective abelian scheme. To a line bundle $\mathcal L$ on $X$, we associate its Mumford line bundle $\Lambda(\mathcal L):= \mu^{\star}\mathcal L\otimes p_1^{\star}\mathcal L^{-1}\otimes p_2^{\star}\mathcal L^{-1}$ on…
Suzet
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What are applications of etale cohomology and Abelian varieties? (And what is arithmetic geometry?)

First, I apologize for my poor English. I like number theory such as "when can prime $p$ be written as $x^2 + y^2$?" and "find the integer solutions of this equation." Because I've heard that these problems can be solved by arithmetic geometry, I…
k.j.
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what are some examples of Abelian varieties?

I am looking at Milne's notes on Abelian varieties. Elliptic curves have an equation: $$ y^2 z = x^3 + a x z + b z^3 \text{ and }4a^3+27b^2 \neq 0$$ but also as complex manifold. $E(\mathbb{C}) = \mathbb{C}/\Lambda$. For Abelian varieties, there…
cactus314
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Miranda's Exercise on the Jacobian of a complex torus

I have to prove that the Jacobian of a complex torus $X=\mathbb{C}/L$ is isomorphic to $X$ by explicity showing that the subgroups of periods $\Lambda \subset \mathbb{C}$ is a lattice which is homotethic to the defining lattice $L$ for $X$, i.e.…
TheWanderer
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