**-1**

votes

**0**answers

188 views

### Faltings height on pair $(\mathcal X,\mathcal D)$

Let $(\mathcal X,\mathcal L)$ be a semi-stable Abelian variety over number field $K$ and possessing a Neron differential $\omega\in \operatorname{H}^0(X,\Omega_X^{\text{dim}X})$, then the Faltings ...

**16**

votes

**1**answer

503 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

**0**answers

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

**1**answer

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 ...

**3**

votes

**1**answer

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

**1**answer

146 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

**0**answers

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 ...

**2**

votes

**1**answer

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

**1**answer

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

**0**answers

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 ...

**12**

votes

**0**answers

356 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 ...

**7**

votes

**2**answers

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, ...

**3**

votes

**1**answer

92 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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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 ...

**2**

votes

**0**answers

137 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

**0**answers

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

**0**answers

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 ...

**6**

votes

**0**answers

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 ...

**7**

votes

**0**answers

227 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 ...

**0**

votes

**1**answer

129 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

**0**answers

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

**2**answers

285 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 ...

**3**

votes

**1**answer

148 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

**0**answers

174 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

**2**answers

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 ...

**6**

votes

**1**answer

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 ...

**7**

votes

**1**answer

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

**0**answers

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

**2**answers

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 ...

**2**

votes

**2**answers

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 ...

**1**

vote

**1**answer

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 ...

**7**

votes

**1**answer

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

**0**answers

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

**1**answer

159 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 ...

**11**

votes

**0**answers

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 ...

**2**

votes

**1**answer

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

**1**answer

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

**0**answers

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 ...

**1**

vote

**0**answers

93 views

### Need information about particular kind of quotients of semisimple algebraic groups by free abelian discrete subgroups

Let me start with the simplest version of the question since already there I don't know anything.
For a complex number $q$, consider the quotient space $X_q:=\mathrm{SL}_2(\mathbb ...

**19**

votes

**1**answer

908 views

### When is “independence of l” known?

My question is for which varieties over local fields is "independence of l" known for
etale cohomology. Say $X/{\mathbb Q}_p$ is a complete non-singular variety and $W_l$ is the (complex) ...

**2**

votes

**0**answers

182 views

### The uniform boundedness of rational torsion for traceless abelian surfaces over a function field

The function field analog of the theorem of Mazur-Kamienny-Merel (giving a universal bound in $[K:\mathbb{Q}]$ for the size of the $K$-rational torsion of an elliptic curve) is immediate just from the ...

**3**

votes

**1**answer

167 views

### Pathological behavior of Lie algebra under a map of abelian schemes

I am trying to understand Example 7.5/9 from the book "Neron models". There one has a discrete valuation ring $R'$ that is the localization of $\mathbb{Z}[\zeta_p]$ at $p$, so that the absolute ...

**3**

votes

**1**answer

321 views

### Coherent cohomology of an abelian scheme and base change

Let $f\colon A \rightarrow S$ be an abelian scheme of dimension $d$. I would like a reference or an argument for the fact that $R^1f_* \mathcal{O}_A$ is locally free of dimension $d$ and that its ...