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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.

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Surjectivity of multiplication by $n$ on elliptic curves

Is there an abelian variety $A$ over a field $k$, such that $A(k^{\rm sep})$ is not a divisible group? The motivation of my question is the following : if $L$ is any algebraically closed field, then $...
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What is the Künneth formula for complete varieties?

I'm reading a part of Mumford's Abelian Varieties, and in the Chapter The theorem of the cube: II he claims that some "Künneth formula" tells us that if $L_1$ is a line bundle on a product $X \times ...
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Abelian Varieties

I have some questions about two statements from Bosch's "Algebraic Geometry and Commutative Algebra" about algebraic varieties (page 479): W fix an abelian variety $A$ over field $K$. Here the ...
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Abelian Variety Commutative

I have a question about a step in a proof from Lang's "Abelian Varieties" (page 20): By definition an abelian variety $A$ over field $k$ is a proper smooth $k$-group scheme that is irreducible. In ...
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Lifting of automorphism of rational surface to that on abelian variety

The paper I am referencing is "Normal Subgroups of the Cremona Group." https://arxiv.org/abs/1007.0895. In theorem 5.14, at the bottom of page 52, the author stated for the abelian surface $Y= \mathbb{...
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Does every closed subgroup of an abelian variety have a direct complement?

Consider A is an Abelian variety over $\mathbb{C}$ and B is a closed subgroup of A. I think there is an Abelian variety $C$ such that $A$ and $B\oplus C$ are isogenic. But I can't find a reference for ...
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Morphism of algebraic groups defined over real points

I am starting to study algebraic groups and I came across the following statement: let $S=\mathrm{Res}_{\mathbb{C}/\mathbb{R}}(\mathbb{G}_{m,\mathbb{C}})$ be the restriction of scalars of the ...
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What's the canonical definition of isogeny between semi-abelian schemes over base scheme S?

By the book, Degeneration of abelian varieties-[Faltings G , Chai C ], a semi-abelian scheme is a smooth separated commutative group scheme $\pi : G\rightarrow S$ with geometrically connected fibres, ...
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Finding a kernel generator of the dual isogeny

Let's say we have an isogeny $\phi:E\to E/\ker\phi$ between two elliptic curves over some finite field. Let's also assume we know $\ker\phi$ explicitly, or at least a generator of it, e.g. $\langle A\...
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Is every k-isogeny of abelian varieties given by polynomials over k?

Given an abelian variety $A$ over the rational integers $\mathbb{Q}$, for every finite group $G\subset A(\bar{\mathbb{Q}})$ consider the field $\mathbb{Q}(G)$ obtained by adjoining to $\mathbb{Q}$ the ...
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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 ...
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Smoothness of Picard scheme of an abelian variety as in Mumford's book

In Mumford's book, he shows $Pic^0$ of an abelian scheme is smooth by considering obstruction class: why is the homomorphism $\mu^*-p_1^*-p_2^*$ is injective on the second chomology? Mumford said in ...
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If isogenous elliptic curves have equal numbers of points, how can isogenies have non-trivial kernels?

Consider: Silverman, Ex V.5.4: Elliptic curves $E/\mathbb{F}_q$ and $E'/\mathbb{F}_q$ are isogenous if and only if $\#E(\mathbb{F}_q) = \# E'(\mathbb{F}_q)$. Silverman, Proposition 4.12: any finite ...
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Is there a “Weierstrass form” for isogenies?

Let $W_1$ and $W_2$ denote smooth subsets of $\mathbb{P}^2$ given by Weierstrass equations and suppose $\varphi : W_1 \rightarrow W_2$ is an isogeny. Is there a specific form that $\varphi$ can be ...
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Points of Scholze's Anticanonical Tower

In his paper "On torsion of locally symmetric spaces", Schole introduces a perfectoid space, which is the anticanonical tower. He claims that points coming from $\text{Spa}(C,C^+)$, where $C$ is a ...
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division points of abelian varieties after reduction modulo a prime

I do not have much knowledge of this topic, so I would like also if you can give me some basic references regarding this, in addition to a possible answer to my question. Given an abelian variety $A$ ...
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On an abelian variety, every element of $\Omega_0 \otimes k$ extends to a holomorphisc differential form

Let $X$ be an abelian variety over an algebraically closed field, $\alpha \in \Omega_{X,0} \otimes_{\mathscr{O}_{X,0} } k(0) $ (= the dual of $T_{X,0}$) Then does this $\alpha$ extend to an element of ...
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A natural isomorphism $\Omega_X \to T_{X,0}^* \otimes _k \mathscr{O}_X$ on an abelian variety $X$ (and the identification of vector fields)

Let $X$ be an abelian variety over a field, $\Omega_X$ the differential sheaf, $\mathscr{T}_X$ the tangent sheaf, i.e., $\mathscr{Hom}(\Omega_X, \mathscr{O}_X)$, and $T_{X,0}$ be the tangent space at $...
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Tate module of an Abelian Scheme

It is well known that if $A/k$ is an Abelian Variety over (the spectrum of) a field, a very important object to consider is its Tate module $T(A):=\underset{\underset{n}{\longleftarrow}}{\lim}A[p^n](\...
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What is the (geometic) genus of the image of curve under quotient map of its jacobian

I guess I need to explain my question. Let $C$ be a smooth genus $2$ curve, and let $J$ be its Jacobian. Fix an immersion $\varphi: C\to J$ and we identify $C$ with its image $\varphi(C)$. Now ...
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For an isogeny of abelian varieties $f : X \to Y$, is $Y = X/ \operatorname{ker}f$?

Let $k$ be a field, $f : X \to Y$ be an isogeny of $k$-abelian varieties. Then there exists the canonical separable isogeny $\pi : X \to X/\operatorname{ker}f$, such that $X/\operatorname{ker}f$ is a $...
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Points of elliptic curves over different fields

Let $f : E \to E'$ be an isogeny between elliptic curves (or abelian varieties), defined over a field $k$. Let $X$ be the kernel of $f$. Let $L \supset k$ be an algebraically closed field. Is it ...
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Is there one canonical principal polarization of a Jacobian per nonisomorphic curve?

Why is there only one canonical principal polarization per Jacobian? I don't yet see why it is true, but I have seen "the canonical polarization" stated many times. This is perhaps a naive question, ...
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Making rigorous Mumford's argument about the sheaf of differentials on an Abelian Variety

Mumford in his book on Abelian Varieties gives the following argument to compute the sheaf of differentials on an Abelian Variety: Let $X/k$ be an Abelian Variety over a field with identity $0$. Let $...
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Quotient by the canonical subgroup of an elliptic curve

This question bothers me! Consider a $p$-adically complete and separated ring $R$, and an elliptic curve $E$ over $R$. It is clear that, if $E$ has ordinary reduction, then there exists a subgroup ...
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factorisation of morphisms of abelian varieties

Corollary 4.11 of Silverman's book The Arithmetic of elliptic curves (p. 73) says Let $$ \phi:E_1 \rightarrow E_2 \text{ and } \psi:E_1 \rightarrow E_3 $$ be nonconstant isogenies, and assume that $\...
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A generic 4-torus$=\frac{C^2}{\Lambda}$ is simple for $\Lambda$ a rank $4$ lattice in $C^2$.

Let $\Lambda$ be a generic rank 4 lattice in $C^2$ where $C$ is the complex number. Then $A=\frac{C^2}{\Lambda}$ is generically non-isogenous to a product of elliptic curves. $\textbf{Q:}$ How do I ...
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Let $E/F$ be a finite field extension of number field $F$. Can one complete integral basis of $F/Q$ to integral basis of $E/Q$?

Let $E/F$ be a finite field extension of number field $F$. $\textbf{Q:}$Can one complete integral basis of $F/Q$ to integral basis of $E/Q$? I doubt this works. Maybe one wants $E$ as a composite ...
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stabilizer of action of a group scheme on a scheme

Let $G$ be a group scheme over a basis $S$, and $X$ be a scheme over $S$. Let $\rho: G \times_SX \to X$ be an action of $G$ on $S$. If $T$ is an $S$-scheme and $x \in X(T)$, then the definition of the ...
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Is intersection with the power of a polarization injective on the Chow Ring?

Let $A$ be an abelian variety of dimension g, and $x\in A$. Let $\theta$ be a polarization. Let $\theta_x$ denote its translate by $x$. Let $0\leq k\leq g$. If we have $$ \theta^{g-k}(\theta_x-\theta)...
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Endomorphism of frobenius scheme

Firstly, let $k$ be an algebraically closed field of characteristic $p>0$. Consider the local-local group scheme $$\alpha_p=\operatorname{Spec} k[\alpha]/(\alpha^p)\,,$$ then the claim in Oort's ...
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Krull-Schmidt for abelian varieties

Let $k$ be a field. If $A$ is an abelian variety over $k$, does $A$ have a unique decomposition as a direct product of directly indecomposable abelian varieties, up to rearrangement and isomorphism ...
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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 ...
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Morphism from complete variety to an affine variety

I am working on Mumfords Abelian Varieties and am not 100% familiar with varieties yet. In some proof it is used that any morphism from a complete variety to an affine variety is constant. I could not ...
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Commutativity of Abelian schemes

$S$ is a scheme, by an Abelian scheme over $S$ I mean a group scheme $X/S$ such that the structure morphism $X \rightarrow S$ is proper, flat and finite presentation. How to prove that $X$ is ...
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why is multiplication by n surjective on an abelian variety [duplicate]

I've just read the proof of $\S4$ (iv) in Mumford's "Abelian Varieties", on page 42-43, which says that multiplication by $n$ not divisible by $p=\text{char}(k)$ on an abelian variety $X$ is ...
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Relative Abelian Varieties

If $A$ is an abelian variety, we have an addition map $\mu:A\times A\to A$. Now, suppose we have a relative abelian variety $\mathcal{A}\to B$, i.e. the morphism is flat and proper and for any $b\in ...
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Mordell-Weil for Abelian Varieties and Jacobian of Curves

I've read in some references (some of them important, like page $328$ from Heights in Diophantine Geometry, by Bombieri and Gubler) that André Weil proved in his PhD thesis that the rank of an abelian ...
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Dimension of the image equal to dimension of space implies surjectivity in complete varieties

I do not know much on abelian varities but I am working on Mumford's Abelian Varities and in the second chapter the arose some questions to me. Actually, it seems to me as I am lacking of a lot of ...
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Pushforward of structure sheaf under an isogeny splits

In Edixhoven, van der Geer, abelian varieties, the proof of the statement 9.18.(iii) includes the claim that for any isogeny $f : X \to Y$ of degree $k$ there is a so-called "trace map" $f_{\ast}\...
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How to find a generator of a torsion subgroup of an elliptic curve with specific order?

Let $E(\mathbb F_q)$ bei any elliptic curve over a finite field with characteristic > 3. Is there any mathematical way, or even algorithm, to find a r-torsion subgroup or a generator of, when r is ...
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Showing that a certain residue field is finite

Let $E^{\prime}$ be an elliptic curve defined over $\mathbb{Q}$. Let $p$ be a prime and suppose we have a map $\rho_{E^\prime}: G_{\mathbb{Q}}\rightarrow PGL_2(\mathbb{Z}_p)$. Let us further assume $E^...
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group action on topological space induces group action on cohomology

I have two questions about group actions on topological spaces. Suppose we have a free, properly discontinuous left group action, $G$ acts on a Hausdorff space $X$. We want to show that there is a ...
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Automorphisms induced by a curve

Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $g \ge 2$ with Jacobian $J(C)$. By the Torelli theorem, we have that $\text{Aut }J(C)$ is either isomorphic to $\text{Aut }(C)$ or $\...
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What is an example of a formal group law that does not come from an abelian variety?

I am curious to find an example of a 1-d formal group law that does not come from a splitting of the formal group law of an abelian variety. I am aware that we can craft logarithms from the formal ...
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Converting from one form of period matrix to another

There are two possible definitions of the phrase ''period matrix'' discussed in the following post: http://www.martinorr.name/blog/2015/10/06/periods-of-abelian-varieties/ Is there a concrete example ...
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Classification of Principally polarized abelian surface.

We know that a principally polarized abelian surface is either the Jacobian of a smooth curve of genus 2 or the canonically polarized product of two elliptic curves. Can anyone suggest me proof of ...
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Isogenies between elliptic curves and their endomorphism rings.

Question 1: Let $E_1,E_2$ be two elliptic curves over a field $k$ of characteristic zero with complex multiplication. Let $R_1,R_2$ be the endomorphism rings of the elliptic curves and suppose that ...
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Questions about the Neron-Ogg-Shafarevich criterion

One version of the Neron-Ogg-Shafarevich criterion for abelian varieties says that for a local field $K$ with valuation ring $R$ and perfect residue field $k$ and an abelian variety over $A$, $A$ has ...
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Twisted cubic as abelian variety

I am trying to define a group structure on the twisted cubic $C$ as follows $$\begin{array}{ccc} C\times C&\longrightarrow C\\ ([p_0,..,p3],[q_0,...,q3])&\longmapsto & \begin{cases} [p0q0,...