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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125 views

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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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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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 ...
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101 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 $S_i$, ...
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229 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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$p$-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 ...
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Give 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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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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1answer
129 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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Abelian variety with CM defined over real numbers

Is there an abelian variety $A/\mathbb R$ of dimension $n$ such that $End_{\mathbb R}(A)\otimes \mathbb Q$ contains a field $K$ of degree $[K:\mathbb Q]=2n$? ($End_{\mathbb R}(A)$ is the ring of $\...
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Monodromy groups that are profinitely dense in Sp(2g,Z)

$\DeclareMathOperator\Sp{Sp}$Assume $g\geq 2$. It is known that there exist finitely generated subgroups of $\Sp(2g,\mathbb{Z})$ of infinite index that surject onto all finite quotients of $\Sp(2g,\...
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Ordinary abelian varieties and Frobenius eigenvalues

Say $A_0$ is an ordinary abelian variety over ${\mathbf{F}}_q$. Call $\mathcal{A}$ the canonical lift of $A_0$ over $R := W({\mathbf{F}}_q)$. It carries a lift of the $q$-th power map on $A_0$. We ...
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From a factor of automorphy on an abelian variety to a divisor

Given a complex abelian variety $A = V/\Gamma$ (for $\Gamma$ being a lattice in the complex vector space $V$), one knows how to describe a holomorphic line bundle in terms of factors of automorphy: By ...
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How often is the rank of J_0(p)^- zero

As mentioned in this answer there is a conjecture by Kimball Martin that, formulated slightly informally, has the following special case. Conjecture: On average $J_0(p)$ has 2 simple components when ...
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Elliptic fibrations on some Kummer surface in characteristic $2$

In the question I ask about one elliptic fibration on the surface $$ K\!: y^2 + x_1x_2y = (x_1x_2)^2(x_1 + x_2 + 1) + (x_1 + x_2)^2. $$ over a finite field $\mathbb{F}_q$ of characteristic $2$ such ...
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Richelot isogenies in characteristic $2$

I am interested in Richelot isogenies between ordinary abelian surfaces in characteristic $2$. If I am not mistaken, the corresponding theory is developed in Article "J.-B. Bost, J.-F. Mestre, ...
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Harmonic forms on a complex torus

Let $T=\mathbb{C}^3/\Lambda$ be a complex torus of our interest and $L$ be a holomorphic line bundle on $T$, I am interested in $H^{0,2}_{\bar\partial_L}(T,L)$, i.e., the $(0,2)$ harmonic forms taking ...
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Mirror partners of some Calabi-Yau threefolds

I don't have experience in mirror symmetry, hence I am not sure that my question is of research level. Sorry in advance. Let $k$ be an algebraically closed field of characteristic $\neq 2, 3$. ...
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What is the involution on the moduli space of genus 3 curves induced by the Torelli map

Let $M_g$ be the moduli space of genus $g$ curves, $A_g$ be the moduli space of principally polarized dimension $g$ abelian varieties. They have dimensions $3g-3,g(g+1)/2$ respectively. The Torelli ...
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Linear forms and the second Voronoi decomposition

This is not my area of expertise, so forgive me if the question is a bit naive. Given a collection of vectors $v_1,\ldots,v_d$ in $\mathbb{R}^n$ (with $d\geq n$), there is a corresponding set of ...
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Isomorphic endomorphism algebras implies isogenous (for abelian varieties over finite fields)?

$\newcommand{\F}{\mathbb{F}} \newcommand{\End}{\mathrm{End}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}}$ I would like to know if the following is true: Proposition A : Let $A_1, A_2$ ...
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272 views

Weil pairing on abelian varieties and etale Chern classes

Given a line bundle $L$ on an abelian variety $A/k$, there is an associated Weil pairing $e_L\colon\bigwedge^2V_pA\to\mathbb Q_p(1)$, where $p$ is a prime different from the residue characteristic of ...
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1answer
170 views

Is the Ueno fibration smooth?

Let $A$ be an abelian variety over $\mathbb{C}$ and let $X\subset A$ be a closed subvariety. Let $X\to Y$ be the Ueno fibration. (That is, $Y$ is of general type and a closed subvariety of $A/B$ where ...
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145 views

Isogeny and canonical isomorphism of global highest differential forms

Let $A$ and $B$ be Abelian Varieties of dimension $d$ over a local field $K$. Let $\phi : A \rightarrow B$ be an isogeny and $\phi^{\prime}$ its dual. Recall that one has a canonical isomorphism $...
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1answer
177 views

Finite quotients of the absolute Galois group of $\mathbb{Q}$ via torsion of elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be a prime. Then there is an action of the absolute Galois group of $\mathbb{Q}$ on $E[p]$ that factors through a finite quotient. Does any finite ...
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Jacobians $\mathbb{F}_q$-isogenous to the direct square of an ordinary elliptic $\mathbb{F}_q$-curve of $j$-invariant $0$

Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 + b$, of $j$-invariant $0$ over a finite field $\mathbb{F}_q$, such that $\sqrt{b} \not\in \mathbb{F}_q$. Question. What are some examples of ...
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107 views

Pullback of algebraic $K$-theory along the surjection of abelian varieties

Given a surjective homomorphism of abelian varieties $f:A\rightarrow B$ where $\text{dim}(A)>\text{dim}(B)$, does $f^*$ induce a rational injection of algebraic $K$-theory? According to the ...
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142 views

Generic rank of proper pushforward of the trivial line bundle

Given a proper surjective morphism $f:X\rightarrow Y$ where $X$ and $Y$ are smooth projective varieties. The proper pushforward $f_!$ is the homomorphism that sends the class of a coherent sheaf $M$ ...
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162 views

Ramification behavior of field given by adjoining $p$-torsion point on formal group of abelian variety

Setup. Let $p > 2$ be a prime, let $K$ be the completion of the maximal unramified extension of $\mathbb{Q}_p$, and fix an algebraic closure $\overline{K}$ of $K$. Let $A/K$ be an abelian variety ...
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1answer
179 views

Semisimplicity of the étale cohomology mod $p$

Let $X$ be a smooth projective variety over a field $k$. Then if $\ell\neq \text{char} k$, $k$ is finite, and $X$ is an abelian variety it was shown by Weil that the $\ell$-adic cohomology of $X_{k^{...
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152 views

Covering abelian varieties over finite fields with the product of curves

Question. Given an $n$-dimensional abelian variety $X$ over a finite field, is it possible to find smooth projective curves $C_1,\ldots, C_n$ such that there exists a finite regular map $C_1\times \...
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356 views

Canonical lift of the deformation of an ordinary abelian variety

If $A/k$ is a principally polarised ordinary abelian variety ($k$ a perfect field of characteristic $p$, we may assume it is finite for simplicity), we have a canonical lift $\hat{A}/W(k)$. Now if I ...
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The intersection form on a Jacobian

$\DeclareMathOperator{\End}{End}$ Let $J=\mathop{Jac}(C)$ be a Jacobian, $\Theta$ the theta divisor. Since it is principal, it induces a bijection between the Néron-Severri group $NS(J)$ and the ...
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Non-degenerate points on a Jacobian surface

Let $C$ be a (hyperelliptic) genus $2$ curve over a number field $K$ with a $K$-rational Weierstrass point $\infty$. We embed $C$ in its Jacobian $J$ via $\infty$. Question: Is there a quadratic ...
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The image of a curve under the multiplication endomorphism of its Jacobian

Let $X$ be a complex smooth projective curve of genus $g\geq 2$. Embed $X$ in its Jacobian ${\rm{J}}(X)\cong{\rm{Div_0}}(X)/{\rm{Div_p}}(X)$ where ${\rm{Div_0}}(X)$ is the group of degree zero ...
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Simple abelian varieties of $\mathrm{GL}_2$ type with positive rank and large dimension

$\DeclareMathOperator\GL{GL}$I would like know if there are known constructions of simple abelian varieties of $\GL_2$ type of arbitrarily large dimension and positive Mordell-Weil rank, whose rank is ...
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Computing homology class of curve in product of elliptic curves

I have a smooth, projective curve $X/\mathbb{C}$ of genus $g$, embedded in a product of elliptic curves $A = \prod_{i=1}^g E_i$. Since $H_*(A; \mathbb{Z})$ with the Pontryagin product is isomorphic to ...
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177 views

Kernel of dual isogeny of elliptic curve

Let $E$ be an elliptic curve defined over an algebraic number field $K$ and $X$ be a cyclic subgroup of $E(K)$ of order $p^n$, then we have the isogeny $E\rightarrow E/X$ and the dual isogeny $ E/X\...
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Homomorphisms of abelian varieties and Tate modules

Let $A$ and $B$ be abelian varieties over a field $k$ and $\ell$ be a prime different from the characteristic of $k$, we have an injection $Hom(A,B) \otimes_\mathbb{Z} \mathbb{Z}_\ell \to Hom (T_\ell ...
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163 views

Integer solutions of Diophantine equation $y^2= 1+4n^{\underline k} $

I am looking for the integer solutions for the diophantine equation $y^2 =4n(n-1)(n-2)\cdots (n-k+1)+1$ for a given $k$ where $n>k+1>5$. In other words, $$y^2=1+4n^{\underline k},\tag{I}$$ where ...
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308 views

Polarizations in algebraic and symplectic geometry

In context of Abelian varieties there are a couple of equivalent ways to introduce the polarization of a algebraic variety. One way is to choose a line bundle $\mathcal{L}$ which satisfies certain ...
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Polarization of Prym varieites

I'm trying to understand polarization and rational Hodge structure of spectral curves and Prym varieties. Excuse me that this is similar to my previous question. I want to prove the following, Let $X$...
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95 views

Using principal polarisation to "cancel" Jacobian summands in isomorphism

I'm working through the sketch proof of irrationality of cubic threefolds in Huybrechts' The geometry of cubic hypersurfaces. Let $J(X)$ denote the intermediate Jacobian of a cubic threefold $X \...
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378 views

A question about $p$-adic monodromy of abelian varieties

Let $S_0$ be a smooth (projective?) and (geometrically) connected scheme over a finite field of characteristic $p$ and let $S$ be its base change to an algebraic closure of the finite field. Let $\pi:...
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473 views

What is the correct notion of representation for abelian varieties?

Zeroth question - am I right that in the "ordinary" sense an abelian variety does not possess any representations at all? More precisely, a representation of an algebraic group $G$ (over an ...
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2answers
440 views

A geometric definition of the addition law on abelian surfaces

Most people will have see a geometric "explanation" of the addition law on elliptic curves given by embedding it as a cubic in the projective plane and cutting it with lines. Is there a ...
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1answer
937 views

If it quacks like an abelian variety over a finite field

Consider smooth projective varieties over a finite field. If a curve "looks like" an elliptic curve (i.e. has genus $1$) then it can be made into an elliptic curve. Is there something ...
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679 views

Which schemes are divisors of an abelian variety?

Let $X$ be a smooth, projective ireducible scheme over an algebraically closed field $k$. I'm trying to understand when there exists an abelian variety $A$ such that $X$ is isomorphic to a prime ...

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