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.

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Difference between stabilizer and automorphism group

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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Elliptic curves are one dimensional abelian varieties

I am trying to prove that the group law on an elliptic curve induces abelian variety structure. Now I am aware of the following group structure of an elliptic curve, $$X(k)\longleftrightarrow \text{...
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"Jacobian variety" for surfaces

I heard that there isn't a functorial construction which associates an abelian variety to any 2-dimensional variety, equipped with an embedding of the surface inside the abelian variety. I find that a ...
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Question on isogenies of degree d.

I am trying to understand the following question (Proposition 5.12. in ABELIAN VARIETIES, Bas Edixhoven, Gerard van der Geer, and Ben Moonen) If $f: X \to Y$ is an isogeny of degree $d$ between ...
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Coherent sheaves of an Abelian variety

I am currently studying the very basics of Abelian varieties over a field $k$. Besides this, I am trying to understand Fourier-Mukai transforms in Algebraic Geometry. My current aim is to study the ...
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Why is a group scheme connected if its torsion is $p$-divisible for every prime $p$?

This arises from (4.12) of Mumford's Analytic construction of degenerating Abelian Varieties over Complete Rings. Why is a group scheme connected if its torsion is $p$-divisible for every prime $p$? I ...
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Preimage of a proper morphism

We know that proper  morphisms of  varieties take  closed subsets to closed subsets. What can I say of the preimage  of a proper morphism? For example, if I know  that $f(g(x))$  is an uncountable ...
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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 ...
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Definition of level covers in a moduli space.

I'm studying Samuel Grushevsky'paper, The Schottky Problem, and there is a definition that I do not understand, level covers, in the paper he defines two spaces $\mathcal{A}_g(l)$ and $\mathcal{A}_g(l,...
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Proof of the Theorem of the Cube

I am trying to understand the proof of the Theorem 2.5 here(Theorem of the Cube). It is required to prove that an invertible sheaf on $X\times Y\times Z$ is trivial when the restrictions to $\left \{ ...
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Equivalent definitions of CM abelian varieties

I am reading Milne's notes on CM (page 27) https://www.jmilne.org/math/CourseNotes/CM.pdf He defined a CM abelian variety $A$ to satisfy $$2\dim A=[\text{End}^{0}(A):\mathbb{Q}]_\text{red}.$$ Then ...
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Endomorphism group acting on the first homology group

I am using Milne's notes about abelian varieties. There is a proposition that says For any abelian variety $A$, $$2\dim A \geq [\text{End}^{0}A:\mathbb{Q}]_\text{red}.$$ The proof uses the fact that $\...
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Commutative subalgebras of a matrix algebra over a division algebra

Let $K$ be a field of characteristic $0$, $D$ a central semi-simple division algebra of dimension $d^2$ over $K$, and $n$ a positive integer. Let $R$ be a maximal subfield of $M_n(D)$, then is $\dim_K ...
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Finitely many elliptic curves isogenous to a given one (over number fields)

Let $K$ be a number field and $E/K$ be an elliptic curve (or an abelian variety). Is there an "easy" proof that there are only finitely many isomorphism classes of elliptic curves $E' / K$ ...
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Equivalence in the definitions of abelian variety

In the algebraic setting, an abelian variety $ X $ of dimension $ n $ over $ \mathbb{C} $ is defined as follows - $ X $ is a connected, projective algebraic group of dimension $ n $ over $ \mathbb{C} $...
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Which line bundle on $A$ is the pull-back of the Poincaré bundle via a given morphism $A \to A^*$?

Let $A$ be an Abelian variety (over an algebraically closed field of characteristic zero), and denote by $A^*$ its dual, which parameterizes degree zero line bundles on $A$. If $Q$ is a point of $A^*$,...
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What is contradiction of $\dim H^0(X, \pi^{*}({\cal O}_{X/F}(D_1)))> \dim H^0(X/F, {\cal O}_{X/F}(D_1))$

Sorry for my bad English. I'm confusing in Mumford's "Abelian varieties"new edition section 17 p.153. I have this circumstance; Let $X$ be an abelian variety, $\cal{L}$be an ample divisor,...
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Existence of very ample line bundle $M$ s.t. $M \otimes L_i$ is very ample

Sorry for my bad English. Let $X$ be projective variety over algebraically closed field $k$, and $L_1,\dots, L_n$ be any line bundles on $X$. Now can we construct very ample line bundle $M$ on $X$ ...
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proof of vanishing theorem of Abelian variety

Sorry for my bad English. In Mumford's "Abelian variety" section 16, there is vanishing theorem as follow; Let $X$ be an abelian variety of dimension $g$, and $L$ be an ample line bundle. ...
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Hyperalgebra of a group scheme

The following definitions are from Mumford's book on Abelian Varieties: A group scheme $G$ is a finite type scheme over an algebraically closed field $k$ with morphisms $m:G\times G\rightarrow G$, $e:...
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Can the type of a polarization of abelian varieties jump in a family?

Suppose $f: X \to Y$ is a projective family of abelian varieties, i.e. $f$ is a proper submersion of complex manifolds, which factors over $\mathbb P^N \times Y$, and each fiber $X_y = f^{-1}(y) \...
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How to proof $X/G\times X/G\cong (X\times X)/(G\times G)$ for scheme X and finite subgroup scheme $G\subset X$?

Sorry for my bad English. I'm having some trouble understanding the claim in Mumford's "Abelian Varieties" p.111. Let $k$ be algebraically closed field, $X$ be group scheme over $k$, and $...
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Splitting of prime and order of reduction of point of infinite order in an abelian variety

Let $A$ be an abelian variety defined over a number field $K$, $P \in A(K)$ a point of infinite order. Let $K_{\ell^n} = K(A[\ell^n])$ be the field of definition of the points in $A[\ell^n]$ and $K_{\...
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Line Bundle Calculations on Elliptic Curves

I'm working on Moonen's notes, Charpter 2. Corollary 2.10 shows that for fixed line bundle $L$ on Abelian variety $X$, $\varphi_L:X\rightarrow \mathrm{Pic}(X), x\mapsto [t_x^*L\otimes L^{-1}]$ is a ...
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Factorization of a morphism of varieties over an open dense subset extends over entire variety

Let $X, Y, Z$ three $k$-varieties. By "variety over field $k$" I mean a separated $k$-scheme of finite type which is geometrically integral. Suppose that $f: X \to Z, g: X \to Y$ and $h: Y \...
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Can we characterize a free action of a group scheme using $S$-points?

Let $k$ be an algebraically closed field. All schemes in this post will be separated and finite type over $k$. Let $G$ be a group scheme, $X$ be a scheme, and suppose $G$ acts on $X$. In Mumford's &...
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Genus of Abelian Varieties

The notes I was reading mentions that if $X$ is an abelian variety of dimension $g$, over a field $k$, then we have the following cohomology: $$H^i(X,\mathcal{O}_X)=k^{g \choose i},\ 0\le i\le g.$$ ...
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Surjective proper cover by ordinary varieties

Let $X$ be smooth proper variety over a finite field $k$ of positive characteristic $p$. Assume that $X$ is not ordinary, then my question is if there exists a smooth, projective ordinary $Y$ with a ...
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Translation of a divisor

Let $A$ be an abelian variety over a field $k$, and $D$ a divisor on $A$. if $a\in A(k)$, I am not sure how to define $D+a$, the translation of $D$? My guess is that, if $D=\Sigma_{i=1}^mn_iZ_i$ where ...
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Dual of subquotient of endomorphism of a Cartier dual

Let $A$ be an abelian variety and $V=A[p]$ its p-torsion subgroup, for some prime number $p$. Let $V^*$ its Cartier dual. If $V$ is isomorphic to a subquotient of $V^*$, may I conclude that $V^*$ is ...
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Definition of finite subgroup scheme

I was reading some notes on elliptic curves and abelian varieties and for an elliptic curve $E$ over a field $k$, I saw the following result: $$\operatorname{Isog}(E)\longleftrightarrow \{K\subset E \...
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Jacobians of curves

I'm only looking for some references. So I understand that if $C$ is a smooth curve of genus $g>1$, we have an injection $$C^{(2)}\rightarrow J(C),$$ $$(P,Q)\mapsto [P+Q-D_0]$$ where $C^{(2)}$ is a ...
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Torsion parts of elliptic curve

Given an elliptic curve $E$ over a field $k$, I know that that the kernel of the power map $[n]:E\rightarrow E$ (here $char(k)\not| n$) has the structure $ker[n]=\operatorname{Spec}(\bigsqcup_i K_i)$ ...
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Algebraicity of a map

The following result is a classical one: Let $A$ be an abelian variety over a field $k$ and let $D$ be a divisor on $A$, the following conditions are equivalent: i) The divisor $2D$ gives a finite ...
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Number of polarizations of a given degree is finite

I was reading a set of notes by Zijian Yao titled "Mordell's conjecture after Faltings and Lawrence-Venkatesh". I am unable to find a source or the proof of the following statement which ...
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Matrix equation $iA = AB$

Let $A \in \mathrm{M}_{n, 2n}(\mathbb{C})$ be a matrix which rows are linearly independent over $\mathbb{R}$. Let $$ B := \begin{pmatrix} A \\ \overline{A} \end{pmatrix}^{-1} \begin{pmatrix} i 1_n &...
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Equivalence of definition of hyperalgebra

sorry for my bad English. I am reading Mumford's "Abelian Variety", and in this book he defined hyperalgebra $\mathbb{H}$ in p98, as follows. Let $k$ be algebraic closed field, and $G$ be ...
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Definitions of CM abelian varieties

I was reading through a presentation by Oort (https://www2.math.upenn.edu/~chai/UPenn2013-beamer.pdf) and noticed something which disturbed me: he defines (slide 38) a simple CM abelian variety to be ...
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Constant over some curve on $X$, if $X\to \mathbb{P}^n$ is not finite.

Sorry for my bad English. I confuse about Mumford’s abelian variety p58. Let $X$ be complete irreducible algebraic variety over algebraic closed field $k$. And $f:X\to \mathbb{P}^n$ is not finite ...
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Definition of Abeliants in G.W. Anderson's paper

I'm currently reading the paper "Abeliants and their application to an elementary construction of Jacobians" by G. Anderson. But I am having problem understanding the definition of the ...
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Genus and jacobian of a curve "parametrized by two elliptic curve"

Let $C : \text{"}y^2=f(x), z^2 = h(x)\text{"}$, $\text{deg}(f) = \text{deg}(g) = 3$, $f$ and $g$ without common roots, be a (affine) curve (with coordinates $[x:y:z:t]$ in the associated smooth ...
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Resolution of coherent sheaves on abelian varieties.

If $A$ is a commutative unital ring and $E$ is a finite rank projective $A$-module there is a surjective $A$-linear map $\phi: A^n \rightarrow E$, with kernel $F:=ker(\phi)$ and $F\oplus E \cong A^n$ ...
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Finding a subvariety that is finite over the target.

Given a surjective homomorphism of abelian varieties like $f:A\rightarrow B$ where $\text{dim}(A)>\text{dim}(B)$, is it possible to find a subvariety $Z$ of $A$ such that restriction of $f$ to $Z$ ...
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Mumford representation of points on the Jacobian of (real) hyperelliptic curves

I'm having trouble understanding why do we have a bijection between points on Jac(C) and divisors in Mumford representation, where C is a hyperelliptic curve. So I found here https://en.wikipedia.org/...
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monodromy on kernel of polarization for abelian surfaces

Let $(A,{\mathcal L})$ be a $(1,n)$-polarized complex abelian surface. By general theory, the polarization corresponds to a map $A \to A^\vee$, and let $K({\mathcal L})$ be its kernel. Then $K({\...
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torsion points on complex torus

I’m not sure how to convince myself that "set of torsion points is dense on complex torus". Is this obvious? I could not find a reference or a proof. Could anyone help me clarify or verify ...
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Endomorphism ring of supersingular elliptic curve

Let $E$ be a supersingular elliptic curve over $F_q$ where $q=p^n$, then $\operatorname{End}(E)$ is an order in quaterion algebra, hence a non-commutative ring. Question: Is there an endomorphism $\...
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$E(\mathbb{F}_q)$ is a torsion group where $E$ is an elliptic curve?

Let $E$ be an elliptic curve defined over $\mathbb{F}_q$ with $q=p^n$, then how to deduce $E(\mathbb{F}_q)$ is a torsion group? In other words, for any $\mathbb{F}_q$-rational point $P$, why does ...
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Non-trivial $2$-torsion point on elliptic curves

If $E$ is an elliptic curve over $\mathbb{F}_p$ where $p\geq5$ and $\#E(\mathbb{F}_p)$ is even, then does $E$ have a non-trivial $2$-torsion point defined over $E(\mathbb{F}_p)$? In other words, if we ...
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Homogeneous space of elliptic curve in Silverman's AEC

I have a question in X.3 proposition 3.2 of Silverman's book AEC. Let $E/K$ be an elliptic curve and $C/K$ be a homogeneous space for $E/K$. Fix a point $p_0\in C$ and define a map $\theta: E\...
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