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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How many elliptic curves have complex multiplication?

Let $K$ be a number field. Suppose we order elliptic curves over $K$ by naive height. What is the natural density of elliptic curves without complex multiplication? More generally, suppose we order $...
3
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1answer
80 views

Cohomology of the Munford line budle on an Abelian variety

Let $X$ be an Abelian variety over a field $k$; $L$ line bundle on $X$. I would like to calculate the cohomology of the Mumford line bundle $\Lambda(L)=m^*L\otimes p_1^*L^{-1}\otimes p_2^*L^{-1}$; ...
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33 views

elliptic k3 surface and Shioda Inose structure

We know that suppose given two elliptic curves $E$ and $E'$, there is a Kummer surface $km(E,E')$. And I'm curious suppose we know a $K3$ surface is kummer, how do we recover the pair $(E,E')$? For ...
4
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1answer
53 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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1answer
26 views

Endomorphism ring of an abelian variety and its reduction mod $\mathfrak{p}$

Let $A$ be an abelian variety defined over a number field $K$. Let $\mathfrak{p}$ be a prime of $K$ for which $A$ has good reduction and let $k=\mathcal{O}_{K,\mathfrak{p}}/\mathfrak{p}$. Let $\...
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1answer
55 views

Pullback of global sections?

Let $X$, $Y$ be abelian varieties and let $f:X\to Y$ be a morphism. They told me that we can define the pullback $f^*s$ of a global section $s\in\Gamma (L)$ where $L$ is an ample line bundle on $Y$, ...
0
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1answer
40 views

A question on very ample line bundle on Abelian Varities

I have a problem with some consideration that Mumford does about very ample line bundles in the prove of Riemann-Roch theorem. Namely, he says that if we consider a very ample line bundle $L=O(D)$ on ...
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37 views

Is any abelian subvariety of an abelian variety closed?

I am wondering because Milne here in Proposition 10.1, page 42, takes any abelian subvariety $B$ of an abelian variety $A$, with $0\neq B\neq A$, then he takes an ample line bundle $\mathcal{L}$ on $A$...
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82 views

Global sections and divisors

I'm trying to understand the proof of the Theorem at page 163 from Mumford, Abelian Varieties, and I have a question about one step. This is the situation: $X$ is an abelian variety (hence there's ...
0
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1answer
53 views

Definition of pullback of a Weil divisor on an abelian variety?

We are on an abelian variety, so Cartier divisors, Line bundles and Weil divisors are all equivalent. I would like to see the pullback of a Weil divisor. Is it true that, if $D=\sum n_i E_i$, then the ...
3
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1answer
43 views

The Mumford line bundle of $(-1)^* L$

Let $X$ be an abelian variety over a field $k$, $L$ a line bundle on $X$. Let $\varphi_L : X \to X^t$ be the morphism obtained by considering the Mumford line bundle $\Lambda (L) = m^*L \otimes p_1 ^...
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25 views

Lifting morphisms of $p$-divisible groups using Grothendieck Messing theory

During my reading of Peter Scholze and Jared Weinstein's paper ``Moduli of $p$-divisible groups'' I found this assertion in the proof of Proposition 6.1.3. Consider the following situation. Let $k$ be ...
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52 views

Why is $Supp D=Supp(\pi^*(\pi(Supp D)))$?

I'm trying to understand the proof of the theorem at page 163 of Mumford, Abelian Varieties. At some point we have the following situation: $X$ is an abelian variety, $D$ is an effective Weil divisor ...
2
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1answer
31 views

Understanding $X \to X^{tt}$ from the Poincare bundle

Let $X$ be an abelian variety over $k$, $X^t = \text{Pic} _{X/k}^0$ its dual, and $\mathscr{P}$ be the Poincare bundle on $X \times X^t$. View $\mathscr{P}$ as a family of line bundles on $X^t$ ...
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1answer
43 views

Line bundle and effective divisor?

$X$ is for me an abelian variety. I think I have seen this result somewhere but I can not find it anymore: For any line bundle $\mathcal{L}$, $\dim H^0(X,\mathcal{L})>0$ if an only if $\...
3
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35 views

Connected component of $0$: why is it an abelian variety?

With "Abelian variety" I mean a integral scheme $X$, proper over an algebraically closed field (complete variety) with a group law $m: X\times X \rightarrow X$ such that $m$ and the inverse map are ...
0
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42 views

Degree of étale morphism $\pi:X\rightarrow X/F$

Let $X$ be an abelian variety and let $F\subset X$ be a finite group. Then we get an étale morphism $\pi:X\rightarrow X/F$ (see construction of $X/F$ and $\pi$ in Mumford, Abelian varieties from page ...
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33 views

Why is it true that $Supp (T^*_xD)=SuppD-x$?

Let $X$ be an abelian variety, let $T_x: X\rightarrow X$, $z\mapsto z+x$ be the translation morphism, let $D=\sum n_iY_i$ be a Weil divisor. As I said here: Pullback of a Weil divisor? I am trying to ...
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0answers
62 views

Pullback of a Weil divisor?

How to define the pullback of a Weil divisor $D=\sum n_iY_i$? I'm particulary interested in $T^*_xD$ where $x\in X$, $X$ is an abelian variety and $T_x: X\rightarrow X$, $z\mapsto z+x$ is the ...
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0answers
45 views

Certain map of modules is iso [Mumford Abelian Varieties]

I have trouble showing the following in the proof of Prop. 2 in Abelian Varieties (pg.70 my edition, Chapter about quotients by finite groups): Suppose you have a Noetherian ring $B=A^G$ as ...
0
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1answer
57 views

How to apply Riemann-Roch Theorem and Kodaira Vanishing Theorem to an ample line bundle?

Let $X$ be an abelian variety. In "Mumford, Abelian Varieties" the Riemann-Roch Theorem has the following form: For all line bundles $\mathcal{L}$ on $X$, if $\mathcal{L}\cong\mathcal{O}_X(D)$, ...
2
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1answer
45 views

$A$ 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, ...
3
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48 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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0answers
33 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 varieties"...
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1answer
40 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 $$ {Hom}_{...
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32 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 $\Delta=g_2^3-27g_3^2$...
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1answer
60 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
37 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 ...
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1k 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 ...
2
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272 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, $...
3
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69 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 $...
1
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1answer
33 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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50 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 \mathbb ...
0
votes
2answers
53 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 ...
0
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0answers
56 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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22 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 ...
0
votes
1answer
94 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
175 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) \otimes_\...
3
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82 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 ...
2
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45 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 ...
0
votes
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 $...
3
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62 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 ...
3
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1answer
61 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
47 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 ...
2
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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
135 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
42 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 ...