# Tagged Questions

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 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...