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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0 answers
109 views

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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7 votes
1 answer
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Reference request. Finiteness of the Selmer group

Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with $...
3 votes
1 answer
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Are there any quadratic functions on an abelian variety not from the height machine?

Let $X$ be an abelian variety defined over a number field $K$. We know that the Neron--Tate height machine associates to a class in the Picard group of $X$ a unique quadratic function which is zero at ...
4 votes
0 answers
204 views

What does Colmez's conjecture tell us?

There is the well known conjecture of Colmez, which describes the logarithmic derivative of the $L$-function of a character via the periods of CM-abelian varieties. Equivalently it describes the ...
2 votes
1 answer
147 views

Intersection of translate of divisors on abelian variety

Setup. Let $K$ be an algebraically closed field of characteristic zero, and let $A/K$ be a simple abelian variety of dimension $n$. Let $\{ x_1,x_2,\dots,x_{m^{2n}}\}$ denote the $m$-torsion points of ...
2 votes
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247 views

Ampleness of the normal bundle to the Albanese image

Let $X$ be a projective surface of general type over $\mathbb{C}$, and assume that $\Omega_X$ is globally generated. Then the Albanese map $a \colon X \to \operatorname{Alb}(X)$ is a local embedding ...
3 votes
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Principally polarised supersingular abelian varieties

If $A/k$ is a supersingular abelian variety, then it is $k$-isogenous to $A'/k$ a superspecial abelian variety, see "A Note on supersingular abelian varieties" by Chia-Fu Yu. And if $A'/k$ ...
3 votes
1 answer
167 views

$p$-power torsion of semiabelian variety

Let $K$ be a finite extension field of $\mathbb{Q}_p$. Let us consider a semiabelian variety $G$ defined over $K$, i.e there exists an extension of an abelian variety $B$ and a torus $T$ defined over $...
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9 votes
1 answer
558 views

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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1 vote
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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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1 vote
1 answer
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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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1 vote
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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....
1 vote
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94 views

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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5 votes
1 answer
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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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0 votes
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Determine the CM type of a CM elliptic curve

I have something don't understand about the CM types of CM elliptic curves. I want to determine the CM type of certain elliptic curves. Let $K=\mathbb{Q}(\sqrt{-3})$ be the CM field and $\omega=\frac{-...
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Is Aut(X) the group of automorphisms of A which preserve X?

Let $X$ be a smooth closed subvariety of a complex abelian variety $A$. Assume $X$ is of general type and of codimension one with $\omega_X$ ample. Is the (finite) group $\mathrm{Aut}(X)$ the group ...
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3 votes
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Two pairings on the group $K(L)$ associated with a non-degenerate line bundle $L$ on an abelian variety

Let $A$ be an abelian variety over a field and let $L$ be a non-degenerate line bundle on $A$. Then $L$ gives rise to a morphism $\lambda:A\to A^*$ from $A$ to its dual. As usual, let $K(L):=\ker(\...
4 votes
1 answer
215 views

Difference between stabilizer and automorphism group of subvariety of an abelian variety

Let $X$ be a smooth closed subvariety of a complex abelian variety $A$. Assume $X$ is of general type and of codimension one with $\omega_X$ ample. Often, people speak about the stabilizer $\mathrm{...
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5 votes
1 answer
338 views

A constructive proof of the theorem of the cube

Do you know a constructive proof of the theorem of the cube ? More precisely, let $X$, $Y$, $Z$ be projective varieties (e.g., over an algebraically closed field $k$) with points $x$, $y$, $z$ ...
3 votes
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Galois invariant of Tate module

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $V$ be a de Rham representation of $G_K=\operatorname{Gal}(\overline{K}/K)$. By Corollary 3.8.4 of Bloch and Kato - L-functions and Tamagawa ...
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4 votes
1 answer
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Tri-homogenous polynomials of tridegree $(3,3,3)$ to add three points on an elliptic curve

Consider an elliptic curve $E \subset \mathbb{P}^2$ with the zero point $\mathcal{O}$. There are classical articles about complete systems of addition laws on $E$ (see Lange and Ruppert - Complete ...
3 votes
1 answer
172 views

Stabilizers in abelian varieties are also abelian? reference request

Let $K$ be a field of characteristic $0$ (number fields is a sufficient generality), $A/K$ an abelian variety, and $X\subseteq A$ a closed reduced subscheme. I am looking for a reference for the ...
1 vote
1 answer
84 views

The upper bound of number of the automorphism of principal polarization of abelian variety over algebraically closed field

I would like to find a upper bound of principal polarization of abelian variety in the following stiution: Suppose $A$ is an abelian variety over a $char=0$ algebraically closed field. And for any two ...
4 votes
0 answers
156 views

𝔾ₘ extensions vs line bundles over abelian varieties

Given a complex polarized abelian variety $V$, we can define a map $$\operatorname{Ext}^1\left(V, \mathbb{G}_m\right) \to \operatorname{Pic}\left(V\right)$$ by viewing an extension as a $\mathbb{G}_m$-...
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1 vote
1 answer
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Cohomology of the dual Abelian variety

I am interested in the (degree $1$) Betti cohomology of the dual $A^\vee$ of an Abelian variety $A$ (say, over $\overline{\mathbb{Q}}$). We can even assume $A$ to be an elliptic curve, if this makes ...
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2 votes
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143 views

Picard and Rosati for elliptic curves

I would like to ask for confirmation whether the following argument is correct. We work over an algebraically closed field $k$ of characteristic $0$. For an elliptic curve $E$, the Picard variety, or ...
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1 vote
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Different ways to construct the isogeny category of abelian varieties

Let $k$ be a field and let $\mathbf{AV}_{/k}$ be the category of abelian varieties over $k$. I'm interested in different definitions of the isogeny category of $\mathbf{AV}_{/k}$. Of course, the ...
4 votes
1 answer
204 views

Type vs degree of a polarized abelian variety

Let $(A,L)$ be a polarized abelian variety. I know that the degree of the polarization is the Euler characteristic of $L$, so that $d = \chi(L) = \dim H^0(A,L)$ since $L$ is ample. I've read in a lot ...
4 votes
1 answer
299 views

Tate-Shafarevich groups under finite Galois field extensions

Suppose $L/F$ is a finite Galois extension of number fields. Let $E$ be an elliptic curve over $F$ and $E_L$ its base change to $L$. Do we have that $\text{Sha}(E/F)$ is finite if and only if $\text{...
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3 votes
1 answer
304 views

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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3 votes
0 answers
145 views

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 $A$ ...
8 votes
0 answers
215 views

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 ...
1 vote
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Exterior power of Hodge structures

Let $V$ be a $\mathbb{Q}$-vector space and suppose there is a decomposition of $V_{\mathbb{C}}:=V \otimes_{\mathbb{Q}} \mathbb{C}$ into two $\mathbb{C}$-sub-vector spaces i.e., $V_{\mathbb{C}} \cong V^...
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4 votes
1 answer
173 views

Is Galois representation induced by semistable elliptic curve semistable?

$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Aut{Aut}$Let $E$ be a semi stable elliptic curve. Let $\overline{\rho_\ell}: \Gal(\overline{\Bbb Q}/\Bbb Q)\to \Aut (E[l]) $ be mod $\ell$ ...
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3 votes
1 answer
166 views

Universal covering of abelian variety

Let $A$ be an abelian variety over a field $K$. It is shown that its $p$-adic Tate module $T_p(A)= \varprojlim_{n} A[p^n](\overline{K}) \cong Hom(\mathbb{Q}_p/\mathbb{Z}_p,A(\overline{K})= \...
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1 vote
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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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3 votes
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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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3 votes
1 answer
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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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1 vote
1 answer
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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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3 votes
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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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5 votes
1 answer
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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 ...
3 votes
1 answer
250 views

The numbers of isomorphism classes of abelian variety over finite fields

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 $\...
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5 votes
0 answers
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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 ...
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 $S_i$, ...
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 $H=ker(j_*:H^1(C,\mathbb{Z})\rightarrow H^3(S,\mathbb{Z}))$. ...
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1 vote
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$p$-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 ...
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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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2 votes
1 answer
143 views

Induced action on Prym variety

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^...
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