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 ...

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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 $>1$ ...
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97 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 ...
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243 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 ...
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129 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 ...
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361 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 ...
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439 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 ...
3
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99 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. ...
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321 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 ...
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46 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
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453 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) ...
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271 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 ...
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204 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 ...
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70 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 ...
6
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249 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 ...
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133 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 ...
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2answers
163 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
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2answers
185 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 ...
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1answer
164 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 ...
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1answer
119 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
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1answer
134 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 ...
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0answers
94 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 ...
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160 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 $G$. ...
4
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1answer
152 views

Conductor CM abelian variety

This is probably well known but I am not an expert in the subject. Given an abelian variety $A$ of dimension $g$ with CM by $O_K$ where $K$ is a CM field of degree $2g$, let $N_A$ be the norm of the ...
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143 views

where do CM abelian varieties get good reduction?

Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ and assume that $A$ has complex multiplication by the ring of integers of a CM field $K$. Then $A$ has potentially good reduction, that is: ...
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661 views

Why study CM abelian varieties?

I know that abelian varieties of CM type have central importance in algebraic geomtry and number theory. There are many conjectures and concepts related to them like Andre-Oort, Coleman conjecture, ...
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171 views

Hodge structure of abelian surfaces

In my case, I have an abelian surface $A$ of (2,8)-polarization, and I have some finite group (may not be abelian group) $G$ acting on $A$ without fixed point. I want to understand when there is a ...
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1answer
154 views

the group of all biholomorphic group automorphisms of complex tori

My background is complex geometry, but when I confront complex tori, I feel it is easier to consider it as a compact connected complex Lie group although I just know the definition of Lie group. Let ...
3
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1answer
165 views

Covering of Abelian variety by product of elliptic curves

Let $A$ be an complex abelian variety of dimension $n$. Is it possible to find $n$ elliptic curves $E_1,\dots,E_n$ such that the product $E_1\times \dots \times E_n$ of the elliptic curves etale ...
5
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105 views

Effect of Hecke transform on the Mumford-Tate group

Let $Sh_{K}(G,X)$ be a Shimura variety and $Z\subset Sh_{K}(G,X)$ be a special subvariety. $Z$ is given by a Shimura sub-datum $(H,Y)$ with $H\subset G$ an algebraic subgroup which I call the ...
9
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0answers
148 views

Are automorphisms of abelian varieties detected by the formal group?

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$. Assume $k$ has characteristic $p$ and denote by $A(p)$ the $p$-divisible group of dimension $g$ associated with ...
11
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428 views

Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties. Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac ...
7
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1answer
374 views

What is the motivation for defining the conductor of an abelian variety?

Let $K$ be a $p$-adic field, and let $A$ be an abelian variety over $K$. The conductor of the abelian variety is often defined as $2u+t+\delta$, where $u$, $t$ and $\delta$ are invariants related to ...
4
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114 views

Are torsion points in a semi-abelian variety over $\mathbb C_p$ bounded?

Let $A$ be a semi-abelian variety defined over (a subfield of) $\mathbb C_p$. Consider its $p$-adic topology with some (non-canonical) metric. Can we bound the distance of torsion points to $0$ with ...
5
votes
1answer
242 views

Picard number of principally polarized abelian varieties

Let $A$ be an abelian variety of dimension $n$. Over $\mathbb{C}$, at least, it is known that the Picard number (that is, the rank of the Neron-Severi group of $A$) is less than or equal to $n^2$, ...
4
votes
1answer
308 views

Fourier-Mukai transform for abelian varieties

Let $A$ be an abelian variety over $\mathbb{C}$, $L$ be a very ample line bundle on $A$, then the dual abelian variety is $\hat{A} \cong A/K(L)$ with $K(L)$ the kernel of surjective morphism $A \to ...
5
votes
2answers
171 views

can all CM types be realized by Jacobians?

The question is kind of self contained, but let me develop a bit further. Assume K is a CM field of degree $2g$, that is, a quadratic imaginary extension of a totally real field. A CM type of $K$ is ...
3
votes
2answers
482 views

Faltings height of a CM abelian variety

Let A be a CM abelian variety, say simple of dimension g, with $End(A) = O_K$, where $K$ is a CM field of degree $2g$. Is there an upper bound for the Faltings height $h(A)$ in terms of the ...
3
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1answer
428 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 ...
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0answers
122 views

algebraicity of Néron-Tate canonical height for Abelian varieties over global function fields

(transcendence of canonical heights) Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always ...
1
vote
1answer
434 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 ...
0
votes
2answers
258 views

Picard number of abelian variety [closed]

I would like references or a result about the computation of the picard number of the jacobian of an algebraic curve. What about the special case when the picard number of the Jacobian is one (is ...
8
votes
1answer
302 views

What is the structure of the group of rational points of an abelian variety over a Laurent series field?

Let $K = \mathbb{F}_q((t))$, and let $A_{/K}$ be a nontrivial abelian variety. Then $A(K)$ is a compact $K$-adic Lie group. What can be said about its structure? By way of comparison, if ...
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1answer
140 views

Isogeny of abelian varieties

Suppose we have a curve $X$ (of genus $\geq 3$), and we know that $\{\phi_i : X \to E_i\ \textrm{ for } i = 1, ..., r\}$ are covers of degrees $d_i$ (with the $d_i$'s not necessarily all equal), ...
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119 views

stably birational abelian varieties are isomorphic

Can anybody help me to prove the following result: Proposition. Let $A$ and $B$ be abelian varieties over a field $k$ of characteristic zero. Assume that $A \times \mathbb{P}_k^n$ and $B \times ...
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1answer
140 views

Dual of a Complex 2-Torus

Is a complex torus $A$ of dimension 2 always isomorphic to its dual torus (i.e. the torus obtained by taking the dual lattice), or are there counterexamples to this?
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Kernels and cokernels for morphisms of abelian schemes up to isogenies

For $S$ a noetherian scheme, let $\mathcal{A}(S)$ be the additive category of abelian schemes over $S$ and $\mathcal{A}_{\mathbb{Q}}(S)$ be the category of abelian schemes up to isogenies, i.e. ...
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1answer
211 views

Could we construct the Jacobian variety of a smooth curve $C$ with genus $>2$ from its derived category $D(C)$?

Let's consider a smooth curve $C$ over $\mathbb{C}$. We know that the Jacobian variety $Jac(C)$ of $C$ is the moduli space of the degree $0$ line bundles on $C$. $Jac(C)$ is an abelian variety of ...
10
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0answers
192 views

Purity for abelian schemes up to $p$-isogenies

Let $S$ be a noetherian excellent regular scheme and $U\subset S$ an everywhere dense open of codimension $\geq 2$. For some fibered categories of geometric objects, it makes sense to ask whether the ...
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1answer
106 views

Degree of isogenies between (semi-)abelian schemes

Let $S$ be a connected noetherian normal scheme of dimension 0 or 1 (i.e. $S$ is a connected Dedekind scheme). Let $f:G\to G'$ be a morphism of semi-abelian schemes over $S$. In their book on Neron ...
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1answer
225 views

Example of non-modular elliptic surface?

In "On elliptic modular surfaces", Shioda proves some interesting theorems on smooth elliptic surfaces (admitting a section); he then focuses on "modular elliptic surfaces" and proves some more ...