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 ...

learn more… | top users | synonyms

16
votes
1answer
495 views

Is hyperelliptic cryptography “practical”?

Previosly my impression on this subject was that hyperelliptic cryptography systems (as well as other possible cryptosystems based on abelian varieties of dimension $>1$) have no advantages over ...
3
votes
0answers
70 views

How order of divisor with support at infinity is changed at reduction?

Jing Yu in his paper "On Arithmetic Of Hyperelliptic Curves" on page 5 asserts the following The most interesting case is certainly the case $k = \mathbb Q$ and $D \in \mathbb Z[t]$. To ...
3
votes
1answer
162 views

Is there a covering of Prym variety?

$\mathstrut$Hi, guys! Let $C$, $C^\prime$ be projective smooth irreducible algebraic curves over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$), $\phi : C$ $\to$ $C^\prime$ a ...
7
votes
0answers
205 views

Quadratic twists of 1-motives

Quadratic twists of elliptic curves (or, more generally, abelian varieties) are familiar objects in arithmetic geometry. I would like to extend that definition to the category of 1-motives over global ...
7
votes
1answer
142 views

Endomorphism algebras of abelian surfaces with real multiplication

Given an abelian variety $A$ over a field $F$, one may consider the ring of endomorphisms $End(A)$, the ring of $F$-rational maps $A \to A$ respecting the group structure on $A$. We may also consider ...
3
votes
1answer
185 views

Why is dual lattice a lattice, in the context of complex tori

I have a simple linear algebra question regarding the definition of dual of a lattice; it was asked by someone else here three months ago on mathstackexchange but got no answer and few views, so ...
0
votes
1answer
145 views

Will any two linearly equivalent ample divisors on an abelian variety intersect?

Let $X$ be an abelian variety of dimension $n>2$. Let $L$ be a very ample line bundle on $X$. Is it possible to find two divisors $D_1,D_2\in |L|$ which do not intersect or intersect in codimension ...
7
votes
1answer
1k views

Poincaré line bundle

I am being stuck by the proof of the existence of Poincaré line bundle of complex torus in Griffiths-Harris. Here is the question: Let $M$ be a complex torus and $M'$ be the complex torus dual to ...
6
votes
0answers
160 views

Simplicity of a rank 2 vector bundle over a principally polarized abelian surface

Let $A := \textrm{Jac}(C)$ be the Jacobian of a genus $2$ curve $C$, with principal polarization $\Theta$. Studying some branched covers of $A$, I was led to consider some rank $2$ holomorphic ...
2
votes
1answer
129 views

Points $\alpha_n$ of $A$ over the $m_n$-th layer in a $\mathbb{Z}_p$-ext. of $K$, where $A$ is an Ab. var. and $m_n$ is strictly increasing

I have the following setting: 1.) A Galois extension of number fields $K\hookrightarrow L$, with $\operatorname{Gal}(L/K)=\mathbb{Z}_{p}$. In my terminology, number field does not imply finiteness ...
1
vote
1answer
103 views

Derived equivalence of families of dual abelian varieties

Let $B$ be a smooth projective complex variety and $\pi:X\to B$ a smooth projective map whose fibres $X_b$ are abelian varieties. Let $\psi:Y\to B$ be the naturally associated bundle such that the ...
4
votes
0answers
243 views

Moduli of coherent sheaves on abelian varieties

Let $\mathcal{A}_g$ be the moduli space (stack?) of $g$-dimensional principally polarized abelian varieties. We have the universal family of abelian varieties $\chi_g\rightarrow \mathcal{A}_g$, where ...
7
votes
2answers
237 views

$p$-torsion of an abelian variety of $p$-rank $0$

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $A$ be an abelian variety over $k$ such that $A[p](k) = 0$, i.e., such that $A$ has $p$-rank $0$. If I am not mistaken, ...
12
votes
0answers
355 views

If $k$ is an algebraically closed field of any characteristic, then the fundamental group of $A$ is abelian

This is a followup to my earlier question, see here. I reproduce it as follows. Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using transcendental ...
3
votes
1answer
90 views

Is there a unique line bundle in the Kummer surface which pulls back to a totally symmetric line bundle?

Let $X=Jac(C)$ be an abelian surface over $\mathbb{C}$, the Jacobian of a genus 2 curve. Let $L$ be a symmetric line bundle. Let $Y$ be the Kummer surface, quotient of $X$ by the action of involution. ...
1
vote
1answer
141 views

Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvable extension?

Are there counterexamples to the following: Given two varieties $A$, $\tilde{A}$, both defined over $\mathbb{Q}$, one of which, say $A$, is a Shimura variety. Then, every isomorphism defined over ...
2
votes
1answer
112 views

Pullback of line bundles and divisors from $Kum(C)$ to $Jac(C)$

Let $C$ be a genus 2 curve over $\mathbb{C}$. Let $X=J(C)$. Consider the involution $i$ on $X$, $x\mapsto -x$. Let $Y=\frac{X}{(i)}$. This is a singular surface with 16 points of singularity - these ...
3
votes
1answer
258 views

How do I find a smooth curve in $J(C)$ through the 2-torsion points?

Let $X$ be the Jacobian of a genus 2 curve over $\mathbb{C}$. Let $L=\mathcal{O}(nC)$, where n is an even number. Is it possible to find a smooth curve from $|L|$ which is fixed by the involution ...
11
votes
0answers
195 views

Are Hecke eigenvalues on the cohomology of the Newton polygon strata automorphic?

Fix a genus $g$, a prime $p$, and a Newton polygon $\Delta$ of an abelian variety of genus $g$. Let $\mathcal A_{g, \overline{\mathbb F}_p, \Delta}$ be the moduli stack of abelian varieties of genus ...
2
votes
1answer
143 views

A curve in an abelian surface and its image in the Kummer surface

This is related to a question that I already asked Curve through the 16 singular points of a Kummer surface. I am new to Abelian varieties and I am trying to understand more about them. Let $X=J(C)$ ...
3
votes
2answers
223 views

Curve through the 16 singular points of a Kummer surface

Let $X$ be an abelian surface over $\mathbb{C}$. Consider the Kummer surface $K$ associated to $X$, that is the quotient of $X$ by the action of involution on $X$, $x\mapsto -x$. Kummer surface is a ...
11
votes
0answers
185 views

Do all simple factors of jacobians of curves come from correspondences?

For this question I will let the overly ambiguous word curve mean: smooth projective and connected curve over $\mathbb C$ (or equivalently a smooth compact Riemann-Surface). Let $C$ be a curve over ...
10
votes
1answer
211 views

What can we say about tropical maps $\mathbb{P}^1 \to A$ for an Abelian variety $A$?

It's well known that maps $\mathbb{P}^1_\mathbb{C} \to A$ are constant for any Abelian variety $A$ (in fact, for any complex torus). Is there any similar statement in the tropical case? Naively, the ...
11
votes
1answer
928 views

Quotient of abelian variety by an abelian subvariety

Let $k$ be a field and $A$ an abelian variety over $k$. Suppose that $B$ is an abelian subvariety of $A$. Consider the following fact: There exists an abelian variety $C$ over $k$ and a surjective ...
2
votes
0answers
136 views

Is the Jacobian of curve self-dual?

Given $C$ an algebraic curve, its Jacobian is isomorphic to its Albanese variety by Abel-Jacobi Theorem. But generally Jacobian and Albanese varieties are dual abelian varieties, does this imply that ...
12
votes
0answers
265 views

Counting abelian varieties over finite fields in a given isogeny class

Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the ...
9
votes
0answers
277 views

Does bounded-degree base extension yield Zariski-dense Mordell-Weil group?

If $d$ and $n$ are positive integers, does there exist a constant $B=B(d,n)$ with the following property? For any $n$-dimensional abelian variety $A$ over a degree-$d$ number field $K$, there is an ...
7
votes
0answers
226 views

Is the compositum of all quadratic extensions of the rationals an ample field?

Let $K$ be the compositum of all quadratic extensions of $\mathbb{Q}$, that is $$K = \mathbb{Q}(\sqrt{d} \ : \ d \in \mathbb{Q}).$$ Is there a (geometrically irreducible) smooth variety ...
4
votes
0answers
284 views

formal group laws of Abelian varieties in positive characteristic

Let $G$ be an algebraic group defined over an (algebraically closed) field $k$. Then one can obtain a formal group law by completing the multiplication map $m: G \times G \to G$ at the unit of $G$. ...
0
votes
1answer
128 views

Complex plane mod lattice to elliptic curve correspondence generalization

If we observe the correspondence $$\mathbb{C}/\Lambda \rightarrow E: Y^{2} = X^{3} - \frac{g_{2}(\Lambda)}{4}X - \frac{g_{3}(\Lambda)}{4},$$ we see the relationship between weight 4 and weight 6 ...
0
votes
0answers
74 views

Functor of order $n$ in Mumford's abelian variety

Let $T$ be a contravariant functor on the category of complete varieties into the Category $\underline{\mathrm{Ab}}$ of abelian groups. Let $X_0,\ldots,X_n$ be any system of complete varieties, ...
4
votes
2answers
283 views

Definition field of isogeny between abelian varieties

Let $K$ be a number field. Let $A$ and $B$ be abelian varieties over $K$. Assume that $A$ and $B$ are isogenous over $\bar{K}$, the algebraic closure of $K$. We further assume that the endomorphism ...
6
votes
1answer
206 views

A Siegel modular form related to the product of two eta functions

I am looking for a Siegel modular form of genus $2$ (living on the Siegel modular 3-fold $A_2=\mathrm{Sp}(4,\mathbb{Z})\backslash \mathfrak H_2$) which becomes "roughly" the product of two eta ...
3
votes
1answer
147 views

CM abelian varieties over the rationals

Let $K$ be a number field and let $A$ be an abelian variety of dimension $g$ over $K$. Let $L$ be a CM field and suppose that $[L:{\bf Q}]=2g$. Suppose that there exists an embedding ...
8
votes
0answers
173 views

Distribution of Mordell–Weil ranks of higher genus curves

By "nice curve", I mean a smooth, projective, geometrically integral curve over $\newcommand{\Q}{\mathbb{Q}}\newcommand{\Jac}{\operatorname{Jac}}\Q$ with at least one $\Q$-rational point. The ...
3
votes
2answers
270 views

N-th root of unity in N-th division field of abelian variety?

Let K be a number field and A/K an abelian variety over it. Can it be that K(A[n]) does not contain a primitive n-th rooth of unity? If the answer is yes is it always possible to bound the bad n ...
3
votes
0answers
128 views

Varieties acted upon faithfully by an abelian variety

Let $X$ be a smooth projective variety over the complex numbers. Suppose that some positive-dimensional abelian variety $A$ acts faithfully on $X$. Examples of such varieties $X$ are provided by ...
6
votes
2answers
243 views

Extending the Abel-Jacobi map over the DM-compactification $\overline{\mathcal{M}}_2$?

Let $\mathcal{M}_2$ be the moduli space of genus two curves and $\mathcal{A}_2$ the moduli space of principally polarized abelian surfaces. Then the Abel-Jacobi map gives an open embedding ...
7
votes
1answer
230 views

Abelian varieties with good reduction everywhere over function fields

There is a famous theorem due to J.-M. Fontaine, Il n'y a pas de variété abélienne sur Z (and independently by V.A. Abrashkin) that there are no abelian varieties over Z. I was wondering whether ...
1
vote
1answer
180 views

a problem about ideals of polynomial rings

Let $\{f_n\}_{n=1}^\infty\in \mathbb{C}[x,y]$ be a sequence of polynomials given by the following expressions $$ f_n(x,y)=\sum_{i=0}^{[\dfrac{n}{2}]}(-1)^{n-i}{{n-i}\choose i}x^{n-2i}y^i. $$ Let ...
2
votes
2answers
110 views

Reduction of tangent space of abelian variety

Let $A$ be an abelian variety defined over a number field $K$ with good reduction everywhere. Let $\mathcal P$ be a prime of $K$. Does reduction of $A$ modulo $\mathcal P$ induce a well-defined ...
5
votes
1answer
256 views

On the cohomology ring of the Hilbert scheme of points on k3 or abelian surfaces

There are many results on the cohomology of the Hilbert scheme of points of a surface. Gottsche calcaluted the Betti numbers and Nakajima got the generators of the cohomology. Also there are results ...
11
votes
0answers
367 views

Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial? This follows from the conjecture that the maximal ...
1
vote
0answers
77 views

Local duality for abelian varieties

Let $A$ be an abelian variety over a p-adic field $K$. Let $I$ be the inertia group of $K$. There is a Yoneda pairing $$H^n(\hat{\mathbb{Z}},A^I) \times Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z}) ...
2
votes
1answer
158 views

integral basis for the Lie algebra of the Neron model of an abelian variety

Let $A$ be an abelian variety over a number field $K$. Let $\mathcal{A}$ be the Neron model of $A$ over $O_K$. Let $\Omega_{\mathcal{A}/O_K}$ be the sheaf of invariant differential forms on ...
2
votes
1answer
336 views

Exactness on rational points of algebraic groups

Let $k$ be a finite extension of the p-adic number field $Q_p$ and G be a connected algebraic (not affine) group over $k$. It is well-known (see e.g. [1] Proposition 3.1) that G decomposes as ...
2
votes
1answer
218 views

An explicit formula for Weil pairing on a complex torus

I begin by defining the Weil pairing in general (as in Oda's 1969 paper). My question is about an explicit formula for this pairing in the case of an elliptic curve over complex numbers. Let ...
2
votes
0answers
94 views

degenerate abelian surfaces

I am wondering if the family of degenerate abelian surfaces constructed by K. Hulek, C. Kahn and S.H. Weintraub in "Moduli spaces of Abelian Surfaces: Compactification, Degenarations, and Theta ...
4
votes
1answer
433 views

Connected cycles of Shimura curves in $A_{g}$ not contained in larger Shimura subvarieties

Is there always a finite family of Shimura curves $(C_{i})$ in $A_{g}$ the moduli space of principally polarized abelian varieties of dimension $g(\geq 2)$, such that the union $\cup C_{i}$ is ...
1
vote
1answer
148 views

Hasse-Weil L-Functions of CM Abelian Varieties

In Shimura's paper "On the Zeta Function of an Abelian Variety With Complex Multiplication", in his terminology, the `one-dimensional part' of the zeta function is identified with a Hecke $L$-function ...