In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. (Def: http://en.m.wikipedia.org/wiki/Abelian_variety)

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Existence of Shafarevich maps(theorem 3.6) on Kollar 's book

I have some problem when reading Theorem 3.6 of Kollar's book Shafarevich Maps and Automorphic Forms, page 41 (Corollary 3.5 of this article ), which states that Let $X$ be a normal variety, ...
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16 views

Linear system which gives $(m,n)$-polarization?

What is the dimension of $H^0(T,\mathcal{L})$, where $T$ is a complex torus of dimension $2$ and $\mathcal{L}$ is a line bundle which gives $T$ a $(m,n)$-polarization?
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49 views

Real Lie groups and elliptic curves

Let $f:A\to A'$ be a morphism of elliptic curves over the real numbers $\mathbb R$. It canonically induces a morphism $f(\mathbb R): A(\mathbb R)\to A'(\mathbb R)$ between the sets of real points, ...
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112 views

Can an elliptic curve have discrimant one?

Can the discriminant, $4a^3 +27b^2$ of an elliptic curve $$E: y^2=x^3+ax+b$$ be equal to 1. I believe that this should not be possible otherwise the curve would have good reduction at all primes $p$, ...
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36 views

Weil pairing, cyclotomic field in division field and determinant map for ell curve and abelian variety

Question 1: If $E/K$ is an elliptic curve defined over a number field $K$, then the Weil Pairing gives me that $K(\mu_n) \subseteq K(E[n](\bar{K}))$. If i identify $Gal(K(\mu_n)/K)$ with a subgroup ...
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21 views

Equation of the curve corresponding to a principal polarization

Let $\mathbb{C}^2/\Lambda$ be a principally polarized abelian surface. I think it is well-known how to write down the equation of the divisor (Riemann surface) corresponding to the polarization, in ...
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34 views

Are there invariants of formal group laws other than height?

By a theorem of Lazard, 1-d formal group laws over separably closed fields of char $p$ are classified up to isomorphism by their height. Are there invariants of formal group laws other than height ...
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59 views

Deforming line bundles on abelian varieties

Let $X$ ba an abelian variety over $\mathbb C$. I would like to understand how line bundles on $X$ deform. The obstructions to deform line bundles lie in $$\textrm{Ext}^2(L,L)=H^2(X,\mathscr O_X).$$ ...
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31 views

Is there a field k, not necessarily algebraically closed, for which variety V(y) in k^2 is reducible?

I am looking for a field $k$ such that the variety V in $k^2$ given by $V =\{(x,y)\in k^2|y=0\}$ is reducible. Thanks
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42 views

How to see whether a tori is an abelian variety or not?

Given an explicit lattice $\Lambda \cong \mathbb{Z}^{2n}$ in $\mathbb{C}^n$, how can one check whether the complex torus $\mathbb{C}^n/\Lambda$ is a projective or not?
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47 views

zeta function of abelian varieties and the exterior algebra

Let $X$ be a smooth projective variety over a finite field $k$. By Grothendieck, the zeta function of $X$ admits the cohomological expression $$ Z(X, t)=\prod_{j=0}^{2\dim X} \det (1-F t \ | \ ...
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31 views

Why is the $\mu_n$ representation rational?

In their paper "On the irregularity of cyclic coverings of algebraic surfaces" by F. Catanese and C. Ciliberto, the authors consider the following situation. Let $A = V/\Lambda$ be a $g$-dimensional ...
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104 views

Theta divisor on the jacobian

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 ...
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71 views

How Appell-Humbert theorem works in the simplest case of an elliptic curve

Line bundles on complex tori $V/\Lambda$ could be described by a pair $(H, \chi)$, where $H$ is a hermitian form on $V$ s.t. $\operatorname{Im} H(\Lambda, \Lambda) \subset \mathbb{Z}$, and $\chi$ is a ...
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63 views

Miranda's Exercise on the Jacobian of a complex torus

I have to prove that the Jacobian of a complex torus $X=\mathbb{C}/L$ is isomorphic to $X$ by explicity showing that the subgroups of periods $\Lambda \subset \mathbb{C}$ is a lattice which is ...
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1answer
41 views

Sheaf on abelian variety preserved by tensor product with a translation invariant line bundle

If $A$ is an abelian variety, $\mathscr{F}$ is a coherent sheaf on $A$, and $\mathscr{F}\otimes \mathscr{L}\cong \mathscr{F}$ for all translation invariant line bundles $\mathscr{L}$, why is the ...
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39 views

What are some elementary books which discuss projective lines on surfaces with examples?

I have the books: W. H. Blythe, On models of cubic surfaces (1905) and A. Henderson, The twenty-seven lines upon the cubic surface, and a couple more modern algebraic geometry books including I. R. ...
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31 views

Decomposition of abelian varieties up to isogeny

Let $A_1,A_2,B_1,B_2$ be simple abelian varieties over a number field $k$. Suppose that $A_1\times A_2$ is $k$-isogenous to $B_1\times B_2$. Can we deduce that (up to reordering the factors) $A_1$ is ...
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88 views

Differential of a morphism of abelian varieties

I am reading the lecture notes of J.S. Milne on Abelian varieties and I got stuck at some point. Let $\alpha,\beta\colon X\rightarrow Y$ be homomorphisms of abelian varieties $X$ and $Y$. Then for ...
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30 views

$X\to \textrm{End}(O_{X,e}/m_{X,e}^r)$ is a morphism

Suppose $X^n$ is a complete group variety over algebaically closed field $k$, then the group law can be shown to be commutative. In proving this, one step is to show $X\to ...
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76 views

Isogenies of abelian varieties

Let $\pi:X\to Y$ be a finite morphism of smooth projective curves over an algebraically closed field (of characteristic zero if necessary) which are both of genus $>1$. We have two "natural" maps ...
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26 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 ...
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40 views

What is $H_1(A_\mathbb{C}^{top},\mathbb{Q})$

Let $A$ be an abelian variety defined over a number field. I have seen in a few papers the singular homology $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ being used. I read up on the singular homology but it ...
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70 views

Examples of varieties with torsion in their integral Hodge structure

I am not so used to thinking about integral Hodge structures, so this question might be completely trivial. What are easy and interesting examples of smooth projective connected varieties $X$ with ...
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32 views

Is finiteness of rational points preserved by duality?

Sorry if this is obvious. I don't know much about Abelian varieties. Let $A/k$ be an abelian variety. Let's say $k$ has characteristic zero. Let $\widehat{A}$ be the dual abelian variety. Suppose ...
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18 views

is it possible to express the moduli of ppav's using torelli loci?

This is a probably vague question from an outsider: It is well known that it is possible to embed "the" moduli space of curves $M_g$ with fixed genus g into the moduli space of principally polarized ...
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1answer
30 views

Dual isogenies of complex tori in Birkenhake-Lange

Let $f: X\to Y$ be an isogeny of complex tori, of degree $n$. On page 13 of Complex Abelian Varieties, Birkenhake-Lange show that there is a dual isogeny $g: Y\to X$. Basically, they show ...
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36 views

The characteristic polynomial of Frobenius of an RM curve

Let $C$ be a genus two curve over $\mathbb{Q}$. We can reduce $C$ modulo a prime $p$ to obtain a curve $\bar{C}$ over $\mathbb{F}_p$. By counting points of $\bar{C}$ over $\mathbb{F}_p$ and ...
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1answer
48 views

Order of $A/2A$ for $A$ an Abelian variety

Let $A$ be an Abelian variety over $\mathbb R$ of dimension $g$. Then the size of $A(\mathbb R)/2A(\mathbb R)$ is $(\# A(\mathbb R)[2])/2^g$. I'm wondering how one might go about proving such a ...
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121 views

Subtori of a complex torus

I am beginning to read Birkenhake-Lange, Complex Abelian Varieties, where they define a complex torus as being a quotient $X=\mathbb{C}^g/\Lambda$, where $\Lambda$ is a lattice in $\mathbb{C}^g$. ...
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278 views

Picard group and jacobian of a singular curve

I found somewhere written that the jacobian of a reducible curve is the product of the jacobians, which seems quite reasonable to me. Where can I find some proof of it and a description of the Picard ...
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116 views

Deformation polarized abelian variety

In "$C$ is not algebraically equivalent to $C^{-}$ in its Jacobian" by Ceresa and then in "On the periods of certain rational integrals, II" by Griffiths (which is quoted as a reference in the first ...
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229 views

Prym variety associated to an étale cover of degree 2 of an hyperelliptic curve.

In view of this question, I have an additional question. The situation is as follows. Let $C$ be the hyperelliptic curve over $\mathbb{C}$, which is given on an affine by the equation $y^2 = x^5 +1 ...
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66 views

What is this cycle on the Jacobian of a curve?

Let $C$ be a curve and $J$ it's Jacobian. There is the standard Abel-Jacobi map $a:C\rightarrow J$ which is given by $Q\rightarrow Q-P$ for some fixed point $P$ (here I am regarding $J$ as the degree ...
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87 views

Etale subgroup-scheme

Let $A$ and $B$ be two abelian varieties over a field $k$ and let $l$ be a prime number not dividing the caracteristic of $k$. Let $\phi : A \to B \in Hom(A,B)$ be such that $\phi$ is zero on ...
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72 views

Abelian Varieties over C vs. Abelian Varieties over C_p

Suppose $A$ is a complex abelian variety. Then $A$ is a complex torus $\mathbb C^g/\Lambda$ where $\Lambda$ is a lattice. On the other hand abelian varieties over $\mathbb C_p$ can have good ...
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1answer
59 views

Group schemes appearing in the $p$-torsion of an abelian scheme

This question is motivated by this one. In that question, there is stated without reference the fact that for an abelian variety over a field of characteristic $p$ (which I assume means the base field ...
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103 views

calculate the group of all biholomorphic group automorphisms of complex tori

My backgrand 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 ...
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142 views

Tate module of product of abelian varieties

Is there a way to relate the Tate module $T_{l}(A \times B)$ of the product of two abelian varieties $A$ and $B$ over a field $k$ (where $l \neq \text{char}(k)$), to the Tate modules $T_{l}(A)$ and ...
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63 views

Trivial divisor on elliptic curve

Suppose $E$ is an elliptic curve over $k$, and $(E,+)$ is an abelian group(suppose we fix some closed point as identity). Let $[p]$ denote the Weil divisor corresponding to the closed point $p \in ...
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114 views

Pullback of a vector bundle on Abelian variety via $(-1)$

Let $A$ be an abelian variety over some field $k$ and $(-1) : A \to A$ is the inverse map of $A$ as an algebraic group. If $V$ is a vector bundle over $A$ what is $(-1)^* V$? In other words, is there ...
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49 views

Specifying the real multiplication on a curve

Suppose we have a curve $C$ of genus two over an algebraically closed field of characteristic zero with a given Weierstrass point $p_0$, furthermore we assume this curve to have real multiplication by ...
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133 views

“Néron-Ogg-Shafarevich criterion” in positive characteristic

The Néron-Ogg-Shafarevich theorem usually seems to be cited to say that an abelian variety over a finite extension $K/{Q}_p$ has good reduction at $\ell \neq p$ if and only if the associated ...
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Why does Mumford want to avoid “reduction to Jacobians”?

In the introduction to his Abelian Varieties book, David Mumford writes: I don't believe the word "Jacobian" is ever used in this book. Rather stubbornly I wanted to prove that the theory of ...
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Divisors in an abelian surface

How to compute the Néron-Severi group of the abelian surface $Y = \mathbb{C}/\mathbb{Z}[i] \times \mathbb{C}/\mathbb{Z}[i]$. More generally, are there any result that compute the Néron-Severi group of ...
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100 views

Triviality of the tangent space of an abelian variety

The tangent bundle of an abelian variety $A/K$ is trivial. Mumford's proof of this fact goes something like this: given the (non-zero) value of a section in a fiber, one can determine the value in all ...
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200 views

Translation of a Paper of Tate

I'm just wondering if anyone knows if the paper "Classes d'isogenie des varietes abeliennes sur un corps fini" by John Tate has been translated into English. My French is not that good and I found it ...
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110 views

Does the Euler characteristic increase if I add an effective divisor

Let $D$ be an effective divisor on a smooth projective connected complex algebraic variety $X$. Suppose that $D\leq E$. Is it true that $$\chi(X,\mathcal{O}_X(D)) \leq \chi(X,\mathcal{O}_X(E))?$$
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148 views

Linear system of divisors on complete variety

I am currently reading Mumford's abelian varieties and Milne's notes on them and I have a problem understanding the proof that they are projective. Both of them use that a complete linear system of ...
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44 views

Theta group representation

Let $(X,L)$ be a polarized abelian variety over $k=\overline{k}$, and let $K(L)$ be the kernel of the isogeny $X\to X^\vee$ that sends $x$ to $t_x^*L\otimes L^{-1}$. The theta group $\mathscr{G}(L)$ ...