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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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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46 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 ...
2
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1answer
106 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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1answer
19 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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143 views

If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$

Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
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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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61 views

Is there a construction known for associating a K3 surface to a curve or cover of curves

Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian. Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
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123 views

Some complex torus is not an abelian variety

Let $e_1,...,e_4\in\mathbb{C}^2$ whose coordinates are all algebraically independent. Let $\Lambda$ be the lattice spanned by these vectors. Why is $\mathbb{C}^2/\Lambda$ not an abelian variety?
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119 views

Why is the trace map on an abelian variety continuous

Let $X$ be a (EDIT) variety with a group structure. For $a\in A$, let $t_a$ be the translation on $X$: $t_a(x) = a+x$. Why is the function $f:X\to \mathbf{C}$ given by $$f(a) = \sum_{i} (-1)^i ...
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22 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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60 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 ...
2
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49 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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69 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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34 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 ...
2
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37 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 ...
2
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64 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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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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45 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 $\mathscr{G}(L)$ ...
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33 views

Are there generalizations of Prym varieties to higher dimensions

Prym varieties are abelian varieties that are associated to a double cover of algebraic curves. Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way? ...
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125 views

Notation for a canonical quotient of an abelian variety in positive characteristic

This may be a somewhat silly question, but there it goes. Let $k$ be an algebraically closed field of characteristic $p>0$ and let $A=A_{/k}$ be an ordinary abelian variety of dimension $g\geq1$. ...
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38 views

Period matrix of abelian surface

Let's construct a complex torus as $(\mathbb{C}^\times)^2/\mathbb{Z}^2$, where the $\mathbb{Z}^2$-action is generated by $$ (z_1,z_2)\mapsto(az_1,bz_2), \ \ \ (z_1,z_2)\mapsto(cz_1,dz_2). $$ My ...
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41 views

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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45 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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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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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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119 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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21 views

the zeros of theta function?

Recall that the theta function with character $(a,b)\in \mathbb{R}^2$ is defined by $$ \vartheta_{a,b}(z, \tau) :=\sum^\infty_{n=-\infty} e^{\pi i (n + a)^{2} \tau + 2 \pi i(n + a)(z + b)},\quad a,b ...
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11 views

A $(d_1,\ldots,d_l)$-polarization of an abelian variety

I want to understand a $(d_1,\ldots,d_l)$-polarization of an abelian variety $A$ concretely. What does it mean to have such a polarization when $A$ is an elliptic curve? What kind of divisor does it ...
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117 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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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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33 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 ...