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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$A$ abelian variety. Is the multiplication by $n_A$ surjective?

According with Mumford the answer is yes, but there are some obscure points in the proof. We know that there exists a very ample line bundle on $A$ since every abelian variety is projective. Hence, ...
2
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21 views

Why is the index $i(\mathcal{L})$ of an ample line bundle on an abelian variety equal to $0$?

I've seen that here https://www.math.uchicago.edu/~ngo/Shimura.pdf there's a theorem called Mumford's Vanishing Theorem (Theorem 2.2.2) which says: Let $\mathcal{L}$ be a line bundle on $X$ ...
2
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24 views

Dimension of an abelian subvariety in the proof of the Poincaré's Reducibility Theorem

I am trying to understand the proof of the Poincaré's Reducibility Theorem, that I'm reading from the book "Abelian varieties" of J.S. Milne (see Proposition 10.1) and from the book "Abelian ...
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1answer
29 views

Reference for the $\Bbb A^1_k$-rigidity of abelian $k$-varieties

Is there a reference that shows, for a field $k$, that abelian $k$-varieties are $\Bbb A^1_k$-rigid? A smooth variety $X$ over $k$ is $\Bbb A^1_k$-rigid if and only if the canonical map $$ ...
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39 views

Dimension of the zero-th cohomology for an ample line bundle [on hold]

Given an ample line bundle L on an abelian variety X, we have that $\dim H^0(X,L)>0$. Why?
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27 views

How can I “groupify” an elliptic curve over a non-field?

The book Primes of the form $x^2+ny^2$ by David A. Cox contains the following definitions regarding an elliptic curve (by which he means an equation $y^2=4x^3-g_2x-g_3$ such that ...
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1answer
54 views

What exactly is the $O_X$-module and the corresponding sheaf of modules?

I am very puzzled by the definition in the Wiki page. I understand that over a subset $U$ we can assign a sheaf of abelian groups, e.g. some analytic functions over $U$. So we consider that these ...
2
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1answer
30 views

Embedding Complex Tori in Projective Space

When we talk about projectively embedding complex tori $\mathbb{C}^{g}/\Lambda$ (i.e in Lefshetz Embedding Theorem), what exactly do we mean by an embedding. Is it in the differential geometry sense ...
2
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13 views

The bilinearity of the Cassels-Tate pairing

Let $K$ be a number field and let $A$ be an abelian variety over $K$ (I'm mostly interested in the case that $A$ is an elliptic curve). We use $v$ to denote places of $K$ and we write $H^i(k, A)$ for ...
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984 views

What is the Albanese map good for?

I am reading a textbook on complex manifolds and come across the Albanese map. For a compact Kähler manifold $X$, $$ T=H^0(X,\Omega_{X}^1)^*/H_1(M,\mathbb{Z}) $$ is a complex torus, called the ...
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244 views

Theta divisor on the jacobian

Given a curve $X$ of genus $g$, and let $\Theta$ be a theta divisor on the associated jacobian $J$. What is the degree of $\Theta$? and does $H^i(J,\Theta)=0$ for $i>0$? is two theta divisor are ...
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17 views

Translating from (short) Weierstrass form to a Lattice. [duplicate]

I know that given an elliptic curve in the form $\mathbb{C}/L$ for L some lattice, $\mathbb{Z}+\mathbb{Z}\tau$, then we can use the Weierstrass $\wp$ functions to turn it into short Weierstrass form, ...
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45 views

Cohomology of structure sheaf of abelian variety

Let $X$ be an abelian variety over $\mathbb{C}$ of dimension $n$. Consider the structure sheaf $O_X$. It's Euler characteristic is zero, because $\chi(O_X)= (O_X^n)/n!$. And the self intersection of ...
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1answer
31 views

Does adjoining a $p$-power divisor to an elliptic curve, where $p\neq 2$ is a prime, always result in a Galois group of order greater than $2$?

The question says it: Suppose I have a field $K$, whose characteristic is, for simplicity, zero, and an elliptic curve $E$ over $K$, and $x\in E(K)$. Suppose that $p$ is a prime different from $2$, ...
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43 views

A Complex Torus is generally simple!

Lets parametrize the set of lattices inside $\mathbb C^g$ with the open dense subset $U= GL_{2g}(\mathbb R)$ of $\mathbb R^{4g^2}$. Show that there exists a coutable family $(Z_n)_{n \in ...
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2answers
51 views

Does anyone know of any good sources on the algebraic theory of abelian varieties?

I have a copy of Mumford's book, but as a final year undergraduate I am finding it to be a little too dense as a starting text. Something lighter would be appreciated to get an intuition before ...
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50 views

Pushforward of structure sheaf by multiplication map

Let $A$ be an abelian variety (over $\mathbf C$ for simplicity), of dimension $d$. For any integer $n$, denote by $f_n: A \rightarrow A$ the multiplication-by-$n$ map. This is an etale map of ...
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19 views

The induced map $\phi: \mathbb{C}^n \to \mathbb{C}^m$ in the construction of toric varieties

Let $\Sigma(1)$ denote the set of one dimensional cones in a fan $\Sigma$. The corresponding vectors in the lattice are denoted $(v_1, \ldots, v_n)$ and to each $v_i$ we associate a homogeneous ...
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1answer
83 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 ...
12
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1answer
169 views

Elliptic curve over algebraically closed field of characteristic $0$ has a non-torsion point

Let $E/k$ be an elliptic curve over an algebraically closed field $k$ of characteristic $0$. Can one prove that the abelian group $E(k)$ is non-torsion? Better yet, can one prove that $E(k) ...
3
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68 views

Does roots of L-polynomial of curve being roots of unity imply the curve is supersingular?

If $C$ is a supersingular curve over $\mathbb F_q$, then $\frac1{\sqrt q}$ of roots of $L$ function are roots of unity. What about converse? If $\frac{1}{\sqrt q}$ of roots of $L$ polynomial are ...
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43 views

Asking for some exercises to help me understanding abelian varieties better?

I want to study Mumford's Abelian Varieties in the coming winter break. I tried to study it before, but I didn't find my self really understanding(or memorizing) too much. I guess a better and more ...
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1answer
27 views

Sum of abelian subvarieties

Let $A$ be an abelian variety over a field $K$. Let $B,C\subseteq A$ be abelian subvarieties. 1) When people write $B+C$, do they mean something like the smallest abelian subvariety containing both ...
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60 views

Where does the terminology “fake/false elliptic curve” come from?

In describing, say, the moduli of Shimura curves, people often refer to "fake" or "false elliptic curves" ("les fausses courbes elliptiques"), which are abelian surfaces whose endomorphism ring is an ...
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1answer
60 views

Why is $\mathcal{O}_F \otimes \mathbb{Z}_p$ a Dedekind ring?

In the paper Compactifications de l'espace de modules de Hilbert-Blumenthal (Compositio. '78), M. Rapoport relates some assertions from a letter of Deligne to Serre, dated 1972. At the very beginning, ...
2
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1answer
45 views

Ribbons of abelian varieties

Let $X$ be a scheme over $\mathbb C$, such that $A=X_{\textrm{red}}$ is an abelian variety. Suppose the tangent spaces to $X$ are all of dimension $\dim X+1$. Then $X$ is non-reduced everywhere, but I ...
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20 views

Explicit models for abelian surfaces

Is there a nice descriprion of birational models for abelian surfaces of given polarization degree?
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34 views

A curve and its translate are linearly equivalent in a complex torus?

Let $X=\frac{V}{\Lambda}$ be a complex torus of dimension $2$. Suppose $C$ is a curve in $X$. Consider $C+a=C_1$, the curve which is a translate of $C$ by $a\in X$. Are $C$ and $C_1$ linearly ...
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33 views

Abelian hypersurfaces

Is there any way to tell from its defining equation when an affine hypersurface has a group law? I am aware that all such groups are matrix groups; in that case I might want to ask when a variety over ...
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1answer
121 views

References about moduli space of abelian varieties with level structure

In the course of one of my research project, I have been advised to try to have a look to "Moduli Space of Abelian Varieties with Level Structure". I am interested in references where this topic is ...
2
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1answer
40 views

What is the ideal sheaf of a translated subvariety of an abelian variety?

Let $A$ be an abelian variety (over an algebraically closed field $k$) and $X\subset A$ a nonsingular subvariety with ideal sheaf $\mathscr I_X\subset \mathscr O_A$. Let $\tau_a:A\to A$ be the ...
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12 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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45 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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100 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 ...
3
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1answer
33 views

Find normal affine subgroup $N$ such that $G/N $ is an abelian variety ( $G=\mathbb{A}^1 \setminus 0 $)

I'm reading Shafarevich Basic Algebraic Geometry. I read Chevalley Theorem. It asserts that every algebraic group $G$ has a normal subgroup $N$ such that $N$ is affine and $G/N$ is an abelian variety. ...
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44 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, ...
3
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1answer
73 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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1answer
20 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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1answer
68 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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2answers
120 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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34 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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78 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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53 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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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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1answer
50 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 ...
3
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1answer
134 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 ...
2
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104 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 ...
4
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2answers
282 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 ...
2
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1answer
48 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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45 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. ...