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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3 votes
0 answers
91 views

Shafarevich conjecture for Abelian varieties over global function fields

Let $S$ be a finite set of places of a global function field $K$. Are there finitely many Abelian varieties over $K$ with good reduction outside $S$? What if we exclude isotrivial families?
0 votes
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77 views

Classification of principally polarized abelian surfaces - reference request [migrated]

I found in Encyclopedia of math https://encyclopediaofmath.org/wiki/Abelian_surface there is a claim that: "A principally polarized Abelian surface $(A,λ)$ is either the Jacobi variety $J(H)$ of ...
1 vote
0 answers
45 views

Abelian surface with CM by a prescribed quartic field

Given a quartic field $K$, is it possible to exhibit an explicit abelian surface $A$ defined over a number field with CM by the field $K$? For example, let's take the non-Galois CM-field $K=\mathbb Q(...
2 votes
2 answers
382 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 $S_i$, ...
2 votes
1 answer
349 views

Can an abelian surface be bielliptic

Is an abelian surface containing an elliptic curve a bielliptic surface? Suppose I have an abelian surface $A$ over the complex numbers that contains an elliptic curve $E$. Then $A \to A/E$ is an ...
4 votes
1 answer
688 views

Vanishing of $\text{Ext}^2$ sheaf from abelian variety to multiplicative group

Does anyone know a proof or reference for the following statement? Or if it's false (which seems unlikely to me), a counterexample? Let $k$ be a field (maybe we need it to be perfect) and $A$ an ...
2 votes
0 answers
164 views

Real structure(s) of a Shimura curve ("complex conjugation" of abelian surfaces)

For a complete lattice $L \subseteq \mathbb{C}^2$ let $A_L$ denote the complex abelian algebraic surface that is isomorphic (as a complex manifold) to the complex torus ${\mathbb{C}^2}/{L}$ (this ...
7 votes
1 answer
180 views

Explicit equations for the universal vector extension of an elliptic curve

The universal vector extension $E$ of an abelian variety $A$ is an algebraic group, an extension of $A$ by a vector group $0 \to V \to E \to A \to 0$, such that any other extension of $A$ by a vector ...
2 votes
1 answer
123 views

Endomorphism ring of the Jacobian of a generic smooth plane quartic

Let $C$ be an arbitrary smooth plane quartic defined over a number field $K$. Assume $C$ is not hyperelliptic, and denote by $J$ the Jacobian of $C$. How does $\text{End}(J)$ look like for a generic ...
1 vote
0 answers
67 views

Simplicity of abelian varieties and localization

Let $A$ be an abelian variety defined over a number field $K$. Let $v$ be a place of $K$ and denote by $K_v$ the $v$-adic completion of $K$ with respect to $||\cdot||_v$. Assume $A$ is simple, is it ...
2 votes
0 answers
166 views

Is the Weil restriction of an elliptic curve self-dual?

$\DeclareMathOperator\res{res}$Let $K=\mathbb{Q}(\sqrt{-3})$, and let $$p\equiv 1\pmod 3$$ be a prime split in $K$. Assume that $$p=\omega*\overline\omega,\quad\text{where}\quad\omega\equiv 1\pmod 3.$$...
1 vote
0 answers
153 views

Reference request showing that a very general Abelian variety $ A $ of genus $ g>1 $ has cyclic class group with ample generator

In Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM I asked for an example of a Cohen Macaulay, normal, ...
13 votes
3 answers
990 views

Faltings height in short exact sequences

Let $K$ be a number field and $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ a short exact sequence of abelian varieties over $K$. Let $h(A)$ denote the logarithmic Faltings height (...
0 votes
0 answers
69 views

Is there an $\mathbb{F}_{\!q}$-curve of geometric genus 3 and $\mathbb{F}_{\!q^3}$-cover to an elliptic $\mathbb{F}_{\!q}$-curve of $j$-invariant 0?

Let $E$ be an elliptic curve $y^2 = x^3 + b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$ such that $3 \mid (q-1)$. Is there an absolutely irreducible $\mathbb{F}_{\!q}$-curve $C$ of ...
2 votes
0 answers
75 views

Is there any work on the intersection loci of the universal theta divisor with torsion sections?

Let $Y$ be a Siegel modular variety of some non-stacky level and genus $g$, carrying over it a universal principally polarized family of dimension-$g$ abelian varieties $A\to Y$. Inside $A$, with fine ...
4 votes
1 answer
218 views

Homogeneous polynomials cutting out complex abelian varieties

This is an update to a previous question of mine. The more clarified questions, results and definitions make me feel like this warrants a separate post instead of a large edit of the original one. ...
2 votes
0 answers
102 views

From rational to integral generators of Néron–Severi group

Suppose I've found rational generators for the Néron–Severi group $\mathrm{NS}(A)$ for an abelian variety over $\mathbb{C}$. How would I check if they are integral generators for $\mathrm{NS}(A)$. Are ...
1 vote
0 answers
132 views

Calculation of de Rham cohomology of abelian varieties/ jacobian varieties

It's known that for a elliptic curve like $E:y^2=x(x-1)(x-t)$ we have a basis $\frac{dx}{y}, \frac{xdx}{y}$ for $H_{dR}^1(E)$. But find such a basis is not an easy thing. I wonder for a general ...
4 votes
2 answers
297 views

Is the set of points on an abelian surface which project to rational points on the Kummer surface a subgroup?

Let $C$ be a hyperelliptic curve of genus 2 defined over $\mathbb{Q}$, let $J$ be its Jacobian, and let $X$ be the Kummer surface associated to $J$ (i. e. $X$ is the singular Kummer surface which ...
2 votes
1 answer
214 views

Looking for an example of a point $P$ on an abelian variety $X$ such that no curve on $X$ contains all multiples of $P$

Is there an example of an abelian variety $X$ defined over a number field $K$, with $\dim X > 1$, and a $K$-rational point $P$ on $X$, such that no curve $C$ on $X$ (say defined over a number field)...
1 vote
1 answer
206 views

The ideal in $\mathbb{Z}[x]$ of all vanishing polynomials of a curve automorphism

Let $C$ be a projective irreducible algebraic curve of genus $g$ over an algebraically closed field of characteristic $0$ (for simplicity). Given an automorphism $\alpha \in \mathrm{Aut}(C)$ of order $...
2 votes
1 answer
279 views

One unexpected observation related to algebraic curves and their Jacobians

Let $C$ be a projective irreducible algebraic curve over an algebraically closed field of characteristic $0$ (for simplicity). Assume that there is also a cover $\varphi\!: C \to E$ to an elliptic ...
2 votes
0 answers
150 views

Product subvariety of a simple abelian variety

Terminology: A subvariety $V$ of an abelian variety $A/\mathbb{C}$ is called a product if there are integral closed subvarieties $U,W\subset A$ such that $\dim U,\dim W>0$ and the sum morphism $U\...
1 vote
1 answer
163 views

Is multiplication by $d$ on the Jacobian of a nodal curve étale?

Let $k$ be an alg.closed of char$k=0$ and let $A$ be an abelian variety over $k$. This Lemma on stacks project states that $[d]\colon A\to A$ is étale. In particular, when $A$ is the Jacobian of a ...
3 votes
0 answers
157 views

Grothendieck-Messing in characteristic 0?

Let $A$ is an abelian scheme over a base scheme $S$. Let $S \rightarrow S'$ be a thickening defined by an ideal of square zero (for example). If $p$ is locally nilpotent on $S$, then Serre-Tate and ...
2 votes
0 answers
162 views

Tangent space to the moduli space of abelian varieties

Letting $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties. I saw without reference that the tangent space of $\mathcal{A}_g$ at a point $t$ could be canonically identified ...
3 votes
0 answers
254 views

Explicit family of polynomials describing embedded torus in complex projective space

This question is cross-posted (with modifications) from MSE. The original question is probably unfit for MathOverflow (although a professor I asked said that this is very nontrivial), but I'm hoping ...
1 vote
0 answers
112 views

Abelian varieties with endomorphism structure

Let me stick to principally polarised abelian varieties $X$ over $\mathbb C$. I have seen several definitions of what it means for $X$ to have real multiplication by a totally real field $F$: There ...
6 votes
0 answers
170 views

Failure of injectiveness of maps between cotangent spaces of abelian varieties

Let $p$ be a prime and $K$ a finite extension of $\mathbb Q_p$ with ramification index $e$. Let $\mathcal O_K$ be the ring of integers of $K$ and $k$ its residue field and the unique maximal ideal. ...
4 votes
0 answers
121 views

Road map for learning about the computational/general theory of modular curves/isogenies of abelian varieties for cryptography

I am a graduate math/crypto student. So I've had some free time last year and I heard about elliptic curves in cryptography and how a resilient cryptosystem got demolished by a spectacular attack ...
1 vote
0 answers
149 views

Vanishing of pushforwards under Fourier-Mukai

In the book Fourier-Mukai and Nahm transforms in geometry and mathematical physics page 85 corollary 3.5 there is the following claim: First some notation. Let $X$ be an abelian variety of dimension $...
3 votes
0 answers
184 views

Tate isogeny theorem over varieties?

Let $X$ be a nice scheme, $\pi:E\to X$ an elliptic curve, and $\ell$ a prime invertible on $K$. Then we can consider the "Tate module" $(R^1\pi_*\mathbb{Z}_{\ell})^\vee=\hbox{''}\varprojlim\...
11 votes
0 answers
346 views

Example of abelian variety over finite field which doesn't lift

What is an example of an abelian variety over a finite field $\mathbb{F}_p$ which doesn't lift to $\mathbb{Z}_p$? This question seems to hint that they should exist, but no example is given. Note that ...
3 votes
0 answers
106 views

How does the number of connected components of the Néron model change in a family of abelian varieties?

Given an elliptic curve $E/\mathbb{Q}_p$, it is known that the component group of the Néron model of $E$ is cyclic of order $-v(j(E))$ when $E$ has split multiplicative reduction, and in all other ...
0 votes
0 answers
194 views

Néron–Severi group of Abelian surfaces

Suppose that an Abelian surface $A$ is isogenous to the product of two elliptic curves $E \times E'$. When can we say that the Néron–Severi group is generated by the classes of these two elliptic ...
4 votes
2 answers
350 views

$p$-divisibility of Picard groups

Let $p$ be a prime number and let $k$ be a field with $char(k)\neq p$ such that all finite extensions have degree coprime to $p$. (For example, we can take $k=\mathbb{R}$ and $p\neq 2$ or let $k$ the ...
4 votes
1 answer
267 views

If $A$ is an abelian variety over $k$ and $S$ is a $k$-scheme, what's $A'(S)$ geometrically?

Let $A$ be an abelian variety over a field $k$ with group operation $m\colon A\times A\to A$, and let $A'$ be the dual abelian variety. I know that $A'(k)$ is isomorphic to the subgroup $\operatorname{...
6 votes
0 answers
206 views

Mumford's definition of an abelian variety's $Pic^0$

I'm not sure whether this is a research-level question, but upon skimming through Mumford book of Abelian Varieties I noticed he gives this definition $$ \begin{equation} \label{eq} \text{Pic}^0(A)=\{\...
3 votes
0 answers
85 views

Can you find a Darboux basis for any skew integral form on a full-rank lattice in $ℂ^n$ so that the first $n$ vectors are $ℂ$-linearly independent?

Any skew bilinear form $\omega$ on $\mathbb{Z}^{2n}$ can be brought into the form \begin{equation} \begin{pmatrix} 0 & \Delta \\ -\Delta & 0 \end{pmatrix}, \quad ...
3 votes
0 answers
196 views

What is a twisted D-module?

Let $X/\mathbb{C}$ be an abelian variety, $Y$ be the dual abelian variety, and $P$ be the Poincaré bundle on $X\times Y$. On p.207, Correction to “Sheaves with connection on abelian varieties” (by M. ...
3 votes
1 answer
177 views

How to determine the type of a divisor on a product of elliptic curves?

I already asked this on Math.SE, but didn't receive an answer yet. Say $E_1, \dotsc, E_n$ are elliptic curves (everything over $\mathbb C$), and $D \subset E_1 \times \dotsc \times E_n$ is an ...
1 vote
0 answers
133 views

Smooth symmetric divisors in abelian varieties without points of order $2$

Let $X=V/\Lambda$ be a complex abelian variety of dimension $g$, endowed with a polarization $M$ of type $(d_1, \ldots, d_g)$. A divisor $D \in |M|$ is called symmetric if $(-1)_X^*D=D$, namely if it ...
1 vote
0 answers
89 views

Is it possible to lift a pair of points on an elliptic $\mathbb{F}_{\!q}$-curve to a pair of short points on an elliptic $\mathbb{F}_{\!q}(t)$-curve?

Let $E$ be an (ordinary) elliptic curve over a finite field $\mathbb{F}_{\!q}$ of a (quite large) characteristic. For simplicity, suppose that $E(\mathbb{F}_{\!q})$ is a prime group. In addition, let $...
2 votes
0 answers
267 views

A gap in a proof of Orlov’s result on the group of autoequivalences of the derived category of an abelian variety

Let $X$ be an abelian variety over an algebraically closed field of charactristic $0$. In this paper, Orlov showed that there is a short exact sequence $$0\to \mathbb Z \oplus X \times \hat X \to\...
3 votes
0 answers
145 views

What does the Néron model of the dual abelian variety parametrize?

Let $K$ be a field which is complete with respect to a discrete valuation $v$ with ring of integers $R$ and residue field $k$. Let $A$ an abelian variety over $K$ and let $A^t$ be the dual abelian ...
4 votes
1 answer
230 views

Néron model, torsion and ramification

Let $B$ a discrete valuation ring, say for simplicity with residue field of characteristic $0$, and $K$ its quotient field. Assume that I have an abelian variety $A$ over $K$ and let $A'$ be its Néron ...
7 votes
1 answer
236 views

Automorphic classification of different types of abelian surfaces

For elliptic curves over $\mathbb{Q}$ the Mumford-Tate group is either $\mathrm{GL}_2$ or $\mathrm{Res}_\mathbb{Q}^F (\mathbb{G}_m)$ if it has CM with the imaginary quadratic field $F$. In this case ...
4 votes
0 answers
207 views

Do rational maps to abelian varieties extend across rational singularities?

Let $X$ be a normal proper variety with only rational singularities and $A$ an abelian variety. Does a rational map $X \supset U \to A$ extend to a morphism $X \to A$? If not, what is a ...
2 votes
0 answers
120 views

"Vanishing locus" of forms in the $h$-topology

Let $\Omega_{h}^p$ be the sheaf of $p$-forms in the $h$-topology defined as the sheafification for the $h$-topology of the presheaf, $$ Y \mapsto \Omega^p_Y(Y) $$ Kebekus and Schnell show that when $X$...
2 votes
0 answers
85 views

Does Albanese construction yield a morphism to moduli of abelian varieties?

Let $M_h$ be the (coarse) moduli space of polarized manifolds with Hilbert function $h$. I would like to know if the albanese $Alb(X)$ of a polarized manifold $X$ gives rise to a morphism $M_h\to A_{g,...

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