# 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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### Any kind of duality between differentials and Tate modules?

Let $X$ be a curve over some algebraically closed field $k$ and let $J$ be its Jacobian. I have read that one should think of the Tate module $T_lJ$ as being the first homology group of $X$ with ...
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### Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?

This is a cross-post! For the original post on SE (9 upvotes, no answer) see: https://math.stackexchange.com/questions/4475853/is-a-complex-algebraic-set-with-a-zariski-dense-subset-of-algebraic-...
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### Abelian subvarieties corresponding to vector subspaces

Let $S$ be a connected smooth projective surface. Let $C$ a smooth curve on $S$ In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following: Let \begin{equation*} r: ...
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### What's a right parameter space of abelian varieties over a non algebraically closed fields?

Let $k$ be a field of characteristic not 2 or 3. Then the set of elliptic curves over $k$ can be parametrized by the affine variety $S=D(4a^3+27b^2)\subset\mathbb{A}^2_k$ via the family $E\to S$ where ...
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### How to construct explicitly defining polynomials of an morphism between smooth irreducible curves?

Let $\phi\!: C_1 \to C_2$ be a separable morphism of smooth irreducible curves embedded as projectively normal models by invertible sheaves $\mathcal{L}_1$ and $\mathcal{L}_2$ respectively. Theorem 4....
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### Polarization induces alternating pairing on homology

Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$. We work up to isogeny (i.e., Hom sets are tensored with $\mathbb{Q}$). I am looking for a reference (and ideally, a short explanation) for ...
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### Do there exist elliptic curves over $H_K$ having everywhere good reduction and CM by $\mathcal{O}_K$?

For $K$ a number field, denote by $\mathcal{O}_K$ its ring of integers and by $H_K$ its Hilbert class field. For which imaginary quadratic field $K$ does there exist an elliptic curve $E$, defined ...
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### Galois cohomology of abelian varieties

Suppose $A$ is an abelian variety over a number field $K$ and call $M$ the maximal torsion free quotient of $A(\overline{K})$ equipped with its Galois action. For the first Galois cohomology of $M$, ...
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### Endomorphisms ring of complex abelian variety under isogenies

I’m trying to understand if over $\mathbb{C}$ two abelian varieties have the same complex multiplication if and only if they are isogenous. Is it true? If it is true this means that if I consider $A$ ...
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### Simultaneous rank jumping of elliptic curves over number fields

Here's something fun that I've been wondering about for a while out of curiosity. It has to do with the rank $\mathrm{rk}(E(K))$ of an elliptic curve $E$ over a number field $K$, and especially how it ...
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### What's the best reference for Abelian varieties?

I am curious about learning about Abelian varieties, specifically how they are in some ways generalizations of elliptic curves. I know of the two sources: https://www.jmilne.org/math/CourseNotes/AV....
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### About Definition 2 in Roĭtman's Paper

Let $A(X)$ be the Chow group of $0$-cycles on a nonsingular irreducible projective variety $X$ over an uncountable algebraically closed field of characteristic zero. In Definition 2 of Roĭtman's paper ...
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### Why is the Jacobian of a curve "irreducible" as a principally polarized abelian variety?

In J.P. Murre's "Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of mumford", in the proof of Theorem 3.11 he remarks that "the Jacobian of a ...
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### Product of Abelian varieties with complex multiplication

We take Abelian varieties $A_1, A_2,\dotsc,A_n$ over a number field. If $A_1, A_2,\dotsc,A_n$ have complex multiplication, then does the product $A_1\times A_2 \times \dotsb \times A_n$ have complex ...
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### Characterization of an Abelian surface

I have a smooth projective surface $X$, and two flat family of elliptic curves on it: $E_{1,t}$ and $E_{2,t}$, (I don't know what either $t$ runs through!) such that (1), for any i={1，2}, the closed ...
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### Toric degeneration of Kummer Surface

I am wondering if there are any explicit examples of a toric degeneration of a Kummer surface (e.g. as a family of projective varieties say), and what the central fibre can look like? (I am working ...
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### Do you know explicit examples of superelliptic curves $y^{\ell} = g(x)$ (for some prime $\ell > 3$) covering some elliptic curves?

For every elliptic curve $E$ Icart in $\S 2$ of the paper explicitly constructs a superelliptic curve $S\!: y^3 = f(x)$ and a cover $\varphi\!: S \to E$. Do you know explicit examples of superelliptic ...
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It is known that there are only finitely many isomorphism classes of abelian variety over a finite field. I am curious about the exact number of these isomorphism classes. Explicitly, fix $g$, let $\... • 443 5 votes 0 answers 96 views ### Extension of a multiple of a rational point to an integral point of a semiabelian scheme Let$\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$is an ... • 2,914 1 vote 2 answers 267 views ### What is the pull-back of a polarization of abelian schemes over different bases? The following came up when reading the definition of the moduli stack of principally polarized abelian varieties in [1]. Let$\pi_1:A_1 \to S_1$and$\pi_2: A_2 \to S_2$be abelian schemes over$S_i$, ... • 790 3 votes 2 answers 272 views ### Abelian varieties corresponding to Hodge substructures In an exercise of Voisin book, says: Let$j:C\rightarrow S$the inclusion of a smooth curve on a smooth connected projective surface. Set$H=ker(j_*:H^1(C,\mathbb{Z})\rightarrow H^3(S,\mathbb{Z}))$. ... • 439 1 vote 0 answers 78 views ###$p$-power torsion points of abelian varieties along$p$-adic Lie extensions Let$p$be a prime and$K$be a number field. Let$K_\infty$be a uniform$p$-adic Lie extension of dimension$l$over$K$with unique intermediate fields$K_n$of degree$p^{nl}$over$K$. We ... 2 votes 0 answers 167 views ### Examples of semi-abelian schemes over a curve Let$C$be a nice curve, i.e.$C$is a smooth, projective, geometrically integral scheme of dimension$1$over a field$k$. For example, (assuming the characteristic of$k$is neither 2 or 3) an ... • 291 2 votes 0 answers 106 views ### Splitting of prime and order of reduction of point of infinite order in an abelian variety I have already asked this question on stackexchange without much luck. I apologize if the question is too trivial to be asked here. Let$A$be an abelian variety defined over a number field$K$,$P \...
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Let $C$ be a smooth projective curve of genus $g$ with an involution $\iota: C \to C$. We have the quotient map $\pi: C \to C/\iota$, with $C/\iota$ a smooth curve of genus $h$. The pullback map \$\pi^...