# Questions tagged [abelian-varieties]

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.

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### Abelian sub-varieties for the product of complex complementary abelian varieties

Let $(A,\Theta)$ be a ppav such that $A=X\times Y$ for $X$ and $Y$ complementary abelian subvarieties. Suppose that $\mathrm{Hom}(X,Y)=\{0\}$. I'm trying to prove the following: Let $B\subset A$ be an ...
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### Is the Weil pairing on elliptic curves a perfect pairing

Let $E/K$ be an elliptic curve and $m$ be a positive integer such that char$(K)\nmid m$. I know that the Weil pairing defined on elliptic curves is a bilinear & non-degenerate pairing. However is ...
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### Endomorphisms of Lie group acting on cotangent space

Let $G$ be a (complex, compact, commutative) Lie group. Apparently the endomorphism ring $\textrm{End}(G)$ of $G$ (i.e., holomorphic group homomorphisms) acts on the cotangent space $T^\ast_eG$ at the ...
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### When are Kummer varieties isomorphic for abelian varieties?

Let $k$ be a field (if necessary algebraically closed), and $A,B$ are the same dimensional abelian varieties. Now we consider their Kummer varieties $K_A:=A/\{\pm 1\}$ and $K_B$ which are algebraic ...
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### Classification of principally polarized abelian surfaces - reference request

I found in Encyclopedia of math https://encyclopediaofmath.org/wiki/Abelian_surface there is a claim that: "A principally polarized Abelian surface $(A,λ)$ is either the Jacobi variety $J(H)$ of ...
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### CM Abelian surface with prescribed CM field

The number field $E=\mathbb Q(\sqrt{\sqrt{2}-3})$ is a CM-field since it is a totally imaginary extension of the totally real field $\mathbb Q(\sqrt 2)$. Is there a way to construct an abelian ...
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### Does an Isogeny induce an Injective map on $Pic^0$

Suppose that $\varphi \colon A_1 \to A_2$ is an isogeny between two abelian varieties (of the same dimension). Is the induced map $\varphi^* \colon Pic^\circ(A_2) \to Pic^\circ(A_1)$ always injective?
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### Definition of lattices being isotropic

I am reading Birkenhake-Lange's Complex Abelian Varieties. In Chapter 3 section 1, there is a notion of isotropic for lattices: Let $X=V/\Lambda$ be a complex abelian variety, where $V$ is a complex ...
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### Divisors of self-intersection 2 on an Abelian surface

I've came across a statement that says: If $A$ is an Abelian surface and $D$ is a positive divisor on $A$ with self-intersection number $2$, then either $D$ is a curve of genus $2$, or the sum of two ...
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### Weil Pairing on Product of Curves

I am reading the paper An efficient key recovery attack on SIDH and I am stuck on some things from page 5. Clearly I am lacking some background to understand this paper and any references would be ...
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### coordinate of abelian variety and formal group

Assume Elliptic curves over local field, E/k, has good ordinary reduction and $$E:y^2+axy+by=x^3+cx^2+dx+e$$. In this case, we know that with the coordinate system $z$ attached to the formal Lie group ...
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### If two smooth cubic threefolds are birationally equivalent, are they isomorphic?

Edit: here is a potential proof Let $X,Y$ be two smooth cubic threefolds over an algebraically closed field. Suppose that they are birationally equivalent. Then are they isomorphic? This seemingly ...
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### Tate conjecture for Abelian varieties without polarizations

I'm reading Levin's note to learn the Tate conjecture for Abelian varieties over number fields. It seems that Tate's original paper and this note both use a weak finiteness hypothesis There are ...
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### Restriction of translation on an abelian variety to a subvariety

Let $A/\mathbb{C}$ be an abelian variety, and let $X\subset A$ be a closed integral subvariety. Assume that $a\in A$ is such that $T_a(X)\subset X$, where $T_a$ is the translation on $A$ by $a$. Does ...
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### Searching book for abelian variety

Do you have interesting books presenting abelian variety (over an arbitrary field k) using the scheme point of view? Most of the lectures I know use the point of view presented in the first chapter of ...
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### Zeros of a finite support function on elliptic curve are torsion

Let $X$ be a elliptic curve over a field $k$ and $f$ be a holomorphic function on $X$, if $f$ only has two zeros $P$ and $Q$, then I'd like to know how to prove see that $P$ is a torsion point of $X$. ...
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### Let $E:y^2=x(x-a)(x-b)(a,b\in \Bbb{Q})$ be an elliptic curve. What is $\#E_d(\Bbb{Q})_{tor}$?

Let $E:y^2=x(x-a)(x-b)(a\neq b\in \Bbb{Q})$ be an elliptic curve　over a field of rational numbers. Let $d\in \Bbb{Z}$ be a square free integer and $E_d:dy^2=x(x-a)(x-b)$ be a quadratic twist of $E$. ...
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### Abelian Varieties over finite fields

I am interested in studying some algorithmic aspects of hyperelliptic curves on finite fields. For this I have studied the topics of Projective model of hyperelliptic curves, Local rings and ...
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### Books on abelian varieties [duplicate]

I was wondering if there were any particular recommendations for books on abelian varieties? Especially for the algebraic side. I’m following Mumford right now and he deals with the analytic side ...
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### Jacobian of hyper elliptic curve and $\Bbb{C}^g/Λ$

Let $C$ be an genus $g$ hyper elliptic curve. Let $J(C)$ be an Jacobi variety of $C$. Then, it is know that there is isomorphism $A \cong \Bbb{C}^g/Λ$, where $Λ$ is lattice. I have two questions about ...
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### Definition of degree of Frobenius of abelian variety and its dual

How can I define the degree of Frobenius map and its dual of abelian variety ? In algebraic curve case, we define degree of frobenius by corresponding field extension(For example, see Silverman's Prop....
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