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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Classification of principally polarized abelian surfaces - reference request

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 ...
finiteness's user avatar
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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(...
user413421's user avatar
2 votes
1 answer
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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 ...
Stormblessed's user avatar
7 votes
1 answer
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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 ...
Vik78's user avatar
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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 ...
DGrimm's user avatar
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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 ...
kindasorta's user avatar
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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 ...
kindasorta's user avatar
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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 $$p=\omega*\overline\omega,\quad\text{where}\quad\omega\equiv 1\pmod 3.$$...
yhb's user avatar
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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, ...
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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 ...
Dimitri Koshelev's user avatar
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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 ...
xir's user avatar
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4 votes
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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. ...
Paul Cusson's user avatar
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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 ...
Stormblessed's user avatar
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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 ...
Richard's user avatar
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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)...
Vik78's user avatar
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4 votes
2 answers
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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 ...
Vik78's user avatar
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1 answer
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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 $...
Dimitri Koshelev's user avatar
2 votes
1 answer
277 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 ...
Dimitri Koshelev's user avatar
2 votes
0 answers
144 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\...
Doug Liu's user avatar
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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 ...
Cornelius's user avatar
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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 ...
351910953's user avatar
  • 261
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160 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 ...
Stormblessed's user avatar
1 vote
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108 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 ...
Aitor Iribar Lopez's user avatar
3 votes
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251 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 ...
Paul Cusson's user avatar
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6 votes
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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. ...
Maarten Derickx's user avatar
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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 ...
Rayane B.'s user avatar
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176 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\...
Curious's user avatar
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11 votes
0 answers
341 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 ...
Daniel Loughran's user avatar
3 votes
0 answers
104 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 ...
Nathan Lowry's user avatar
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0 answers
186 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 ...
Stormblessed's user avatar
4 votes
2 answers
345 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 ...
Boaz Moerman's user avatar
6 votes
0 answers
204 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)=\{\...
Basil's user avatar
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4 votes
1 answer
264 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{...
Gabriel's user avatar
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148 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 $...
user135743's user avatar
3 votes
0 answers
184 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. ...
Doug Liu's user avatar
  • 423
3 votes
1 answer
171 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 ...
red_trumpet's user avatar
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1 vote
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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 ...
Francesco Polizzi's user avatar
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 $...
Dimitri Koshelev's user avatar
2 votes
0 answers
263 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\...
P. Usada's user avatar
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3 votes
0 answers
144 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 ...
Hans's user avatar
  • 2,863
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 ...
Hans's user avatar
  • 2,863
7 votes
1 answer
234 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 ...
Alireza Shavali's user avatar
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 $X$...
Ben C's user avatar
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4 votes
0 answers
199 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 ...
Ben C's user avatar
  • 3,039
1 vote
0 answers
83 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,...
divergent's user avatar
4 votes
1 answer
373 views

Étale group schemes and specialization

If $A$ is an abelian variety over a finite field $\mathbf{F}_q$, then $A(\mathbf{F}_q)$ (resp. $A(\overline{\mathbf{F}}_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}_q(t))$ is a ...
user avatar
0 votes
1 answer
320 views

Why an elliptic curve can be defined as an abelian variety of dimension 1?

Now we define an elliptic curve as "a smooth projective curve of genus one with a specified base point". (A little question by the way: Is the requirement "with a specified base point&...
HaomengXu's user avatar
6 votes
0 answers
322 views

How to explain the relationship between Tate–Shafarevich and Ideal Class Group, when all else fails?

In the short paper On the Tate–Shafarevich group of a number field of Sameer Kailasa, he reviews a curious phenomenon by which the class group of a number field $K$ appears as the exact kernel of the ...
Keith Millar's user avatar
  • 1,242
1 vote
0 answers
205 views

Moduli space of abelian surfaces

Let $K$ be a finite field with algebraic closure $\overline{K}$. The $j$-invariant gives a bijection between the the affine line $\mathbb{A}_K^1$ and $\overline{K}$-isomorphism classes of elliptic ...
Sebastian Monnet's user avatar
7 votes
0 answers
157 views

Is the universal object over a Hilbert scheme connected?

Hartshorne proved in his thesis that if $S$ is connected, then the Hilbert scheme $\operatorname{Hilb}^p=\operatorname{Hilb}^p(\mathbb{P}^n_S/S)$ is too (where $p\in \mathbb{Q}[z]$). Can the same be ...
Nathan Lowry's user avatar

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