**7**

votes

**0**answers

226 views

### Can an abelian variety/Q have no 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

288 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

**0**answers

66 views

### Theta divisor on the jacobian [migrated]

I have a classic questions!!(I can't find in the literature) so forgive me if it is so easy!
Given a curve $X$ of genus $g$, and let $\Theta$ be a theta divisor on the associated jacobian $J$.
What ...

**2**

votes

**1**answer

164 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 ...

**1**

vote

**0**answers

63 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

85 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

822 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

148 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

158 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

259 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 ...

**6**

votes

**1**answer

196 views

### Weil height of an Abelian Variety with everywhere (potentially) good reduction

Background: Suppose that $E$ is an elliptic curve over $\mathbb{Q}$ with everywhere (potentially) good reduction. there are many ways to define the height of $E$, and I will be concerned with the ...

**8**

votes

**1**answer

193 views

### Do all 0-dimensional Shimura Varieties show up (as CM points) in $\mathcal{A}_g$?

Question: Let $S$ be a 0-dimensional Shimura variety. Does $S$ necessarily admit a morphism (in the category of Shimura varieties) to $\mathcal{A}_g$ for some $g\geq 1$? Here $\mathcal{A}_g$ is the ...

**8**

votes

**2**answers

506 views

### Adjoining torsion points from abelian varieties

Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically ...

**11**

votes

**0**answers

238 views

### Am I missing something about this notion of Mirror Symmetry for abelian varieties?

This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s.
In the comments of the question, I was directed to the paper ...

**7**

votes

**0**answers

208 views

### Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s

It is well known (cf. Dolgachev) that there is a beautiful notion of mirror symmetry for lattice-polarized K3 surfaces. That is, if we are given a rank $r$ lattice $M$ of signature $(1, r - 1)$ and a ...

**6**

votes

**1**answer

218 views

### Do complex tori contain quasi-projective open subsets?

Complex tori are not associated to projective varieties in general.
But can one find an open $U$ inside a complex torus $\mathbb C^g/L$ such that $U$ is the analytification of a quasi-projective ...

**10**

votes

**2**answers

467 views

### On a proposition in Hartshorne's paper “Ample vector bundles on curves”

In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following:
Let $A$ be an abelian variety [over an alg. closed field ...

**2**

votes

**0**answers

63 views

### Abel-Prym map for Prym-Tyurin varieties

Let $(J,\Theta)$ be the Jacobian of a smooth projective curve $C$, and let $i:P\hookrightarrow J$ be an abelian subvariety of $J$ such that $i^*\Theta\equiv e\Xi$ for some principal polarization $\Xi$ ...

**10**

votes

**2**answers

488 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 ...

**1**

vote

**0**answers

139 views

### Marten's proof of torelli theorem

I am trying to read the proof of torelli theorem by Henrik H.Martens "A new proof of torelli's theorem" Annals of mathematics vol78 no. 1 .The proof seems to me like using mysterious combination of 3 ...

**4**

votes

**0**answers

138 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 ...

**0**

votes

**0**answers

28 views

### singular locus of $\mathcal{A}_3(2)^{hyp}$

The geometry of the Satake compactification of $\mathcal{A}_2(2)$ is very well known. It is the singular quartic 3-fold in $P^4$ known as $Igusa\ quartic$. I am looking for references (or ...

**2**

votes

**1**answer

125 views

### Endomorphism Ring of Simple Abelian Varieties

I know that if $A$ is a simple abelian variety over a number field $k$ with all endomorphisms defined over $K$ then $\mbox{End}(A_K)\otimes \mathbb{Q}$ is a division algebra with a positive ...

**1**

vote

**1**answer

79 views

### What is Rosati Form

I was reading a paper and they mentioned the Rosati form. Particularly, what they said was:
Let $A$ be an abelian surface defined over $k$ such that $ST_A^0$ (the connected component of the Sato-Tate ...

**0**

votes

**1**answer

88 views

### Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector space

Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation.
I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of ...

**1**

vote

**0**answers

58 views

### symmetric theta structures and arithmetic subgroups

A symmetric theta structure is a theta structure that commutes with (a lift of) the natural involution $\imath: A \to A$ an an abelian variety. For simplicity I will assume that $A$ is a surface.
...

**2**

votes

**1**answer

101 views

### Degree of a smooth curve in an abelian variety

Let $A$ be an abelian variety, $g$ be a positive integer and $\mathcal{L}$ be an ample line bundle on $A$.
Question : Is there a real $r>0$ such that, for all smooth curve $C$ of genus $g$ in ...

**3**

votes

**0**answers

163 views

### Uniruled degenerations of abelian varieties

Suppose I have a smooth projective variety $X$ over $\mathbb{C}$ with $K_X$ semiample, and consider the fiber space $f:X\to Y$ given by $|\ell K_X|$, for some $\ell>0$ large, where $Y$ is a normal ...

**-1**

votes

**2**answers

180 views

### Interpretation of $H_1(A_\mathbb{C}^{top},\mathbb{Q})$

In the paper "Sato-Tate Distributions and Galois Endomorphism Modules in Genus 2" (arxiv: http://arxiv.org/abs/1110.6638), the authors use the singular homology $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ ...

**4**

votes

**0**answers

154 views

### Is a semiabelian algebraic space a scheme?

Let $S$ be a scheme and let $A$ be a commutative separated smooth $S$-group algebraic space of finite presentation each of whose geometric fibers is an extension of an abelian variety by a torus. Is ...

**1**

vote

**1**answer

135 views

### Divisors on an abelian surface

Let $A$ be an abelian surface given by the quotient of a product of two generic elliptic curves $E_1 \times E_2$ by the product $T_1 \times T_2$ of two translations by $2$-torsion points. Then $A$ ...

**2**

votes

**0**answers

163 views

### Is there an excplicit number field of definition for an Abelian Variety $A/\mathbb{C}$ with CM?

Consider a simple abelian variety $A/\mathbb{C}$ with sufficiently many CMs by $\mathcal{O}$, where $\mathcal{O}$ is an order in a CM field $K$. Specifically, $K$ is a CM field of degree $2g$, where ...

**0**

votes

**1**answer

187 views

### About Weil's proof of “Weil conjectures for curves and abelian varieties”

I know that the Weil's proof of the Weil conjectures for curves and abelian varieties is made under the lenguage of his "Foundation of algebraic geometry", however in "Polarizations and Grothendieck's ...

**4**

votes

**1**answer

149 views

### Reference or proof for the fact that $J(X_0(N))$ splits into abelian varieties with real multiplication

It´s known that $J_0(N) = J(X_0(N))= \bigoplus_f E(f)$ splits as a sum of abelian varieties parametrized by the Hecke eingenfunctions and that it´s an elliptic curve iff the Hecke eingenvalue is an ...

**1**

vote

**1**answer

156 views

### Why is the Tate local duality pairing compatible with the Cartier duality pairing?

This question is a follow up to Why is the norm map dual to restriction under Tate local duality?
Let $A$ and $B$ be dual abelian schemes over a base scheme $S$. For an integer $n \ge 1$, consider ...

**14**

votes

**2**answers

550 views

### Why polarization of abelian varieties?

Maybe this question is not suitable for here, but I don't think I would receive a satisfactory answer in Math StackExchange.
I could never understand the intuition behind polarization of abelian ...

**2**

votes

**2**answers

207 views

### Why is the norm map dual to restriction under Tate local duality?

Let $L/K$ be a finite Galois extension of nonarchimedean local fields, and let $A$ and $A^t$ be dual abelian varieties over $K$. Tate local duality tells us that $A^t(K)$ and $H^1(K, A)$ are ...

**0**

votes

**0**answers

71 views

### Do principally polarized abelian varieties enjoy a genus expansion?

This is a vague question from an interested outsider:
It is well known that abelian varieties which arise as Jacobian of a curve (or a bit more general as Prym variety) are distinguished by the fact ...

**0**

votes

**1**answer

98 views

### rank of Abelian schemes under ample hypersurface section

Let $k$ be an algebraic closure of a finite field, $\ell \neq \mathrm{Char}(k)$ be prime, $S/k$ a smooth projective geometrically connected surface and $C/k$ a smooth ample connected hypersurface ...

**0**

votes

**0**answers

77 views

### the CM type of a CM abelian variety

Let $(A, F, i)$ be a CM abelian variety, by which I mean an abelian variety $A$ defined over $\overline{\mathbb{Q}}$, say of dimension $n$, a CM number field $K$ of degree $2n$ and an embedding $i: F ...

**2**

votes

**2**answers

401 views

### Elliptic curve E and Galois representation

Assume that an elliptic curve $E$ over $\Bbb Q$ has a reducible mod $p$ representation. Then
Q: Why is the semi-simplification of $E[p]$ the direct sum of ${\Bbb Z}/p{\Bbb Z}$ and $\mu_p$?
Next
...

**0**

votes

**2**answers

215 views

### Absolute Hodge implies Galois invariant?

Let $X$ be an Abelian variety defined over a number field $K$, suppose that it has a good reduction over a fine place $\mathfrak{p}$ of $K$. Let $G_{\mathfrak{p}}$ be the local Galois group for ...

**1**

vote

**1**answer

174 views

### Rigidity lemma over non-algebraically closed field

I would like to extend the rigidity lemma (as in Mumford's "Abelian varieties") to the case in which the base field $k$ is not algebraically closed.
I found a suitable proof in the draft of "Abelian ...

**1**

vote

**2**answers

134 views

### kernel of isogeny becomes constant after base change

Let $S = Spec(O_K)$ be the spectrum of the rings of integers of a number field $K$. Let $A/S \setminus T$ be an Abelian scheme over an open subscheme $S \setminus T \subseteq S$. Does the kernel of ...

**2**

votes

**1**answer

101 views

### Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication

Let $(A,a)$ be a principally polarised (with indecomposable polarisation) Abelian variety over $\mathbb C$. Assume that End(A) contains an order $R$ of a totally real number field of degree $>1$ ...

**2**

votes

**1**answer

85 views

### group structure on (subsets of) tropicalizations of Abelian varieties

In this paper Vigeland shows how one can define a group law on subset of a tropical elliptic curve, so that this group is homeomorphic to $S^1$. It is not clear to me what is the relationship between ...

**2**

votes

**1**answer

238 views

### Are Abelian varieties (sometimes) globally $F$-split?

As defined by Karen Smith here, beginning of section 3? If $E$ is an elliptic curve, then it is when $E$ is ordinary. I wonder about higher dimension cases. Any references would be greatly ...

**0**

votes

**1**answer

126 views

### Linear system on an abelian surface

On a K3 surface $S$, a linear system $|C|$ is said to be hyperelliptic if the corresponding map is of degree 2 and the image is of degree $g_a(C)-1$ in $\mathbb P^{g_a}$.
For $g_a(C) > 2$, if ...

**6**

votes

**3**answers

351 views

### Torsion group of the following elliptic curve

Let $p_1=2, p_2 = 3,\ldots,$ be the prime numbers, and define $n_i = \prod_{j=1}^i p_j$. Moreover, let $E_i $ be the elliptic curve defined by $y^2 = x^3 + n_i$.
Can one compute the torsion group ...

**5**

votes

**2**answers

402 views

### Modularity theorem for abelian varieties

There's alredy two posts on MO about the extension of modularity to elliptic curves over fields other than $\mathbb{Q}$ ([1], [2]), and another one about general algebraic varieties [3].
What is ...