**0**

votes

**1**answer

321 views

### Minimal polynomial of symmetric endomorphism on abelian variety

Let $(A,\Theta)$ be a principally polarized abelian variety over an algebraically closed field $k$, and let $f$ be a symmetric endomorphism of $A$ (that is, $f^\dagger=f$ where $\dagger$ denotes the ...

**1**

vote

**0**answers

45 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

95 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 ...

**6**

votes

**1**answer

561 views

### Rank 2 vector bundle on a product of elliptic curves

Let $E$, $F$ be two complex elliptic curves, and $A=E \times F$. Let us denote by
$\pi_E \colon A \to E, \quad \pi_F \colon A \to F$
the natural projections. For all $p \in F$ let us write $E_p$ ...

**3**

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**0**answers

143 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**

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**2**answers

174 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**

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**0**answers

128 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**

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**1**answer

126 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

**1**answer

301 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 ...

**2**

votes

**0**answers

137 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

173 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

144 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**

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**2**answers

145 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 ...

**1**

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**1**answer

118 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

454 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 ...

**0**

votes

**0**answers

53 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 ...

**2**

votes

**2**answers

354 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

**1**answer

94 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

71 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 ...

**0**

votes

**2**answers

196 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 ...

**7**

votes

**5**answers

2k views

### Generalizations of Belyi's theorem

Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic ...

**1**

vote

**1**answer

159 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 ...

**5**

votes

**2**answers

337 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 ...

**1**

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**2**answers

119 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

71 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

225 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 ...

**1**

vote

**1**answer

56 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 ...

**0**

votes

**1**answer

124 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

327 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 ...

**9**

votes

**1**answer

262 views

### Motives over finite field not generated by hyperelliptic curves

So the question is that, over a finite field, does there exist an abelian variety $A$ for which there does not exist a generically one-to-one morphism from a hyperelliptic curve $C$ to $A$.
p.s. A ...

**3**

votes

**0**answers

90 views

### Ampleness of Hodge bundles over complex curves

Let $C$ be a smooth, proper and connected curve over
the complex numbers $\bf C$. Let ${\cal G}\to C$ be a smooth group scheme over $C$ and let $\epsilon_{\cal G}:C\to{\cal G}$ be its
zero-section. ...

**6**

votes

**1**answer

218 views

### Are there perverse sheaves on abelian varieties with small Euler characteristic?

Let $A$ be a simple abelian variety of dimension $g$. Let $K$ be an irreducible perverse sheaf on $A$. We know that $\chi(A,K)\geq 0$. (Corollary 1.4 of Franecki and Kapranov.) How small can ...

**6**

votes

**1**answer

512 views

### Why are torsion points dense in an abelian variety?

Hi everyone,
let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$ of characteristic $p\geqslant 0$. I'm trying to prove that the subgroup $A'$ which is the union of ...

**3**

votes

**0**answers

35 views

### Abelian varieties/$p$-divisible groups are an integral category

A preabelian category is called integral if epimorphisms are stable
under pullbacks and monomorphisms are stable under pushouts.
A major property of integral category is that by inverting bimorphisms ...

**7**

votes

**1**answer

392 views

### The Tate conjecture for abelian varieties

Let $k$ be a number field. Recall that Faltings proved the famous Tate conjecture, which states that for any abelian variety $X$ over $k$ and any prime $\ell$, the natural map
$$\mathrm{End}(X) ...

**10**

votes

**0**answers

226 views

### Lifting Abelian Varieties to p-adic fields

Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...

**7**

votes

**1**answer

195 views

### showing that abelian varieties are de Rham *without* showing that they are crystalline

If $X$ is a smooth projective variety over a $p$-adic field $K$, then Faltings' Theorem says that the etale cohomology of $X_{\overline{K}}$ is crystalline.
There have been various steps towards this ...

**2**

votes

**1**answer

184 views

### Canonical lifts from $\mathbb F_q$ and CM-theory

One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...

**3**

votes

**2**answers

257 views

### Defining isogenies over smaller fields

I'm having some issues with abelian varieties and fields of definition. This already became clear in my previous question on Jacobians. Here's another question. If somebody can explain some nice facts ...

**1**

vote

**0**answers

51 views

### Ampleness of the Canonical Bundle for Siegel Modular Varieties

Background
Throughout I only work with varieties over $\mathbb{C}$.
For $p$ a prime number, Let $Y(p)$ denote the modular curve parametrizing elliptic curves together with full $p$-torsion ...

**4**

votes

**1**answer

195 views

### isomorphism of extensions by abelian varieties

Let $A$ and $G$ be abelian varieties over $\mathbb{C}$. An element $P$ of $\text{Ext}(A, G)$ is an exact sequence
$0 \to G \to P \to A \to 0$,
here one can give $P$ the structure of an abelian ...

**6**

votes

**7**answers

1k views

### Is there any rational curve on an Abelian variety?

Is that true that there is no rational curves contained in an Abelian variety? If it's true, is that because abelian varieties are not uniruled? How do I know whether an abelian variety is not ...

**13**

votes

**7**answers

2k views

### Examples of rational families of abelian varieties.

I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1).
One can generate many examples as Jacobians of rational ...

**2**

votes

**0**answers

123 views

### Must the coordinates of a polynomial iteration have about the same size?

Original post. The following statements seem plausible (not to say intuitively obvious), but I do not see how to prove them.
Let us say that a polynomial mapping of $\mathbb{C}^2$ is reducible if ...

**0**

votes

**2**answers

121 views

### Poincaré bundle and Weil pairing for Abelian schemes

In which situations is there a Poincaré bundle for Abelian schemes? In [Mumford, Abelian varieties] only the case of Abelian varieties is treated.
The same question for the Weil pairing ...

**3**

votes

**2**answers

178 views

### abelian varieties with the same CM type are isogenous

Does anybody have a reference for the following fact?
All abelian varieties with complex multiplication and same CM type are isogenous over $\overline{\mathbb{Q}}$?
Here abelian variety with ...

**1**

vote

**1**answer

147 views

### Algebraic Hodge decomposition of CM abelian varieties

On p. 205 of Katz's paper entitled "p-adic L-functions for CM fields" Katz says that
"Shimura's algebraicity theorem, in our context, is an easy consequence of the fact that Hodge decomposition of ...

**1**

vote

**0**answers

51 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 ...

**2**

votes

**1**answer

115 views

### Potential good reduction of abelian varieties

In Corollary 3 on page 498 of the article "Good reduction of abelian varieties" it says that, under some specified conditions, the minimal subextension $L/K$ of $\overline{K}/K$ over which an abelian ...

**1**

vote

**0**answers

79 views

### A question on Kähler differentials and cotangent spaces on schemes

I have the following question (should be easy for those who know something about the field):
On page 92 (97 of the old edition) of Mumford's book "Abelian varieties", the author talks about an ...