# Questions tagged [abelian-varieties]

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.

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### Classification of principally polarized abelian surfaces - reference request

I found in Encyclopedia of math https://encyclopediaofmath.org/wiki/Abelian_surface there is a claim that: "A principally polarized Abelian surface $(A,λ)$ is either the Jacobi variety $J(H)$ of ...
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### One unexpected observation related to algebraic curves and their Jacobians

Let $C$ be a projective irreducible algebraic curve over an algebraically closed field of characteristic $0$ (for simplicity). Assume that there is also a cover $\varphi\!: C \to E$ to an elliptic ...
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### Example of abelian variety over finite field which doesn't lift

What is an example of an abelian variety over a finite field $\mathbb{F}_p$ which doesn't lift to $\mathbb{Z}_p$? This question seems to hint that they should exist, but no example is given. Note that ...
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### How does the number of connected components of the Néron model change in a family of abelian varieties?

Given an elliptic curve $E/\mathbb{Q}_p$, it is known that the component group of the Néron model of $E$ is cyclic of order $-v(j(E))$ when $E$ has split multiplicative reduction, and in all other ...
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### Néron–Severi group of Abelian surfaces

Suppose that an Abelian surface $A$ is isogenous to the product of two elliptic curves $E \times E'$. When can we say that the Néron–Severi group is generated by the classes of these two elliptic ...
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### $p$-divisibility of Picard groups

Let $p$ be a prime number and let $k$ be a field with $char(k)\neq p$ such that all finite extensions have degree coprime to $p$. (For example, we can take $k=\mathbb{R}$ and $p\neq 2$ or let $k$ the ...
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### What does the Néron model of the dual abelian variety parametrize?

Let $K$ be a field which is complete with respect to a discrete valuation $v$ with ring of integers $R$ and residue field $k$. Let $A$ an abelian variety over $K$ and let $A^t$ be the dual abelian ...
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### Néron model, torsion and ramification

Let $B$ a discrete valuation ring, say for simplicity with residue field of characteristic $0$, and $K$ its quotient field. Assume that I have an abelian variety $A$ over $K$ and let $A'$ be its Néron ...
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### Automorphic classification of different types of abelian surfaces

For elliptic curves over $\mathbb{Q}$ the Mumford-Tate group is either $\mathrm{GL}_2$ or $\mathrm{Res}_\mathbb{Q}^F (\mathbb{G}_m)$ if it has CM with the imaginary quadratic field $F$. In this case ...
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### "Vanishing locus" of forms in the $h$-topology

Let $\Omega_{h}^p$ be the sheaf of $p$-forms in the $h$-topology defined as the sheafification for the $h$-topology of the presheaf, $$Y \mapsto \Omega^p_Y(Y)$$ Kebekus and Schnell show that when $X$...
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### Do rational maps to abelian varieties extend across rational singularities?

Let $X$ be a normal proper variety with only rational singularities and $A$ an abelian variety. Does a rational map $X \supset U \to A$ extend to a morphism $X \to A$? If not, what is a ...
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