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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Application of the seesaw principle

The seesaw principle says in general the following. If $X,T$ are varieties with $X$ complete, and $\mathcal{L}$ a line bundle on $X\times T$, then $$ T_{1} = \{t\in T: \mathcal{L}_{X\times\{t\}} \...
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Isomorphism $(A\times B)^\vee\to A^\vee\times B^\vee$ for abelian varieties

Let $A$ and $B$ be abelian varieties, and consider the natural map $$f:(A\times B)^\vee\to A^\vee\times B^\vee$$ sending a line bundle on $A\times B$ to the restrictions to $A\times\{0\}$ and $\{0\}\...
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Size of torsion subgroup of an abelian variety

Fisrt, let $k$ be a number field, and let $A/k$ be an abelian variety. I know that by the Mordell–Weil theorem, $A(k)$ is finitely generated, but I saw here (in the introduction) that $A(k)_{tors}$ is ...
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What is a maximally isotropic subgroup of product of elliptic curves

From page 5 of https://mast.queensu.ca/~kani/papers/numgenl.pdf : Then $\psi$ is automatically an anti-isometry with respect to the $e_N$ -pairings on $E$ and on $E'$: $$e_N(\psi(x), \psi(y)) = e_N(x,...
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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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On Igusa Congruence Groups $\Gamma_g(n,2n)$ and moduli interpretation of $\mathcal{A}_g(n,2n)$

Consider the congruence subgroup $\Gamma_g(n)=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\in Sp_{2g}(\mathbb{Z}):\begin{bmatrix}a&b\\c&d\end{bmatrix}\equiv\begin{bmatrix}1_g&0\\0&...
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Pullback of a translation map of a divisor in Birkenhake-Lange's book "Complex Abelian Varieties"

I'm currently studying the book 'Complex Abelian Varieties' by Birkenhake and Lange. On page 74, after lemma 1.5, the authors make the following statement: 'Another observation, which will prove ...
William Gibson's user avatar
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Mumford Ch.17 on Very Ample Line Bundles

A famous result of Lefschetz says that if $L$ is an ample line bundle on an abelian variety then $L^{\otimes 3}$ is very ample. In Mumford's book on abelian varieities, this is mentioned in Chapter 17....
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U(n) is compact and algebraic, but not abelian—why not a contradiction?

the subgroup of unitary matrices $\text{U}(n) \subset GL(n, \mathbb{C})$ is compact and definitely algebraic, with an algebraic group law; on the other hand, it's not abelian. why is this not a ...
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Line bundles on complex tori

There is a line bundle-divisor correspondence. Namely, given a divisor $D$ which is a formal sum of codimension $1$ subvarieties, we can write $\mathcal{O}_X(D)$ to be the line bundle whose sections ...
daruma's user avatar
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Real Elliptic curves as compact abelian groups

It's well known that an elliptic curve of the form $y^3 = x^3 +ax +b$ admits a group structure, as long as a point at infinity $O$ is added to serve as identity. If we look at a real elliptic curve as ...
Pedro Lourenço's user avatar
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How to determine the type of a divisor $D$ on a product of elliptic curves?

Say $E_1, \dotsc, E_n$ are elliptic curves (over $\mathbb C$), and $$D \subset E_1 \times \dotsc \times E_n$$ is an (effective) divisor. Then the line bundle $\mathcal O(D)$ has a type $(d_1, \dotsc, ...
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Pullback of the Poincare bundle along $\varphi_{\mathcal L}$ is the Mumford bundle

I'm trying to understand why $(1\times\varphi_{\mathcal L})^*(\mathcal P_A) = \Lambda(\mathcal L)$ where $\Lambda(\mathcal L) = m^*\mathcal L\otimes\operatorname{pr}_1^*\mathcal L^{-1}\otimes\...
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Homomorphisms of (complex) abelian varieties with RM

I'm new to abelian varieties. Here's the question: Let $f: A\to B$ be a homomorphism of $n$-dimensional abelian varieties with real multiplication by $\mathcal{O}_L$, where $[L:\mathbb{Q}] = n$. If $f$...
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Action of complex conjugation on abelian surfaces with complex multiplication

Let $K$ be an imaginary quadratic field, and let $A$ be an abelian surface over $\mathbb{C}$ with complex multiplication by $K\times K$, meaning that $K\times K\subseteq End(A)\otimes\mathbb{Q}$. Then ...
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Why is the kernel of a separable isogeny Cartier dual to the kernel of the dual map?

Let $f:X\rightarrow Y$ be a separable isogeny of abelian varieties and $K$ be its kernel. Let $\widehat{f}:\widehat{Y}\rightarrow \widehat{X}$ be the dual map and $K'$ the kernel of this. It is known ...
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Criterion of very ampleness of $L^{\otimes 2}$ where $L$ is ample line bundle on abelian variety.

Let $A$ be abelian variety, and $L$ be an ample line bundle on $A$. Then by Lefschetz's theorem $L^{\otimes 3}$ is very ample. I want to know good criterion if $L^{\otimes 2}$ is very ample.
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de Rham cohomology of dual abelian variety

Suppose $X$ is an abelian variety over a field $k$. Consider the de Rham cohomology of $X^{\vee}$ (denoted by $H_{dR}^i(X^{\vee})$). I want to ask is $H_{dR}^i(X^{\vee})$ naturally dual to $H_{dR}^i(X)...
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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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Why $[K:\Bbb{Q}]\leq2\dim A$ holds for CM abelian variety?

Let $A$ be an abelian variety. Let $K$ be a CM field. ${\rm End}^0(A):={\rm End}(A)\otimes_{\Bbb{Z}}\Bbb{Q} $. $A$ is said to have CM by $K$ if only if ${\rm End}^0(A)$ contains $K$. Then, why $[K:\...
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Definition $L$ function of elliptic curve

Let $E$ be an elliptic curve. My book reads, we denote '$L$ series of $E$ regrading $l$ adit representation of $Gal(\overline{\Bbb{Q}}/\Bbb{Q})$' as $L(E/\Bbb{Q}):= \prod_{p}\frac{1}{1-a_pp^{-s}+...
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Is there a known criteria for a complex torus to be isomorphic to its dual?

It is known that a principally polarized complex torus $T$ is isomorphic to its dual. Indeed, if $L$ is a line bundle over $T$, s.t. $c_1(E)$ is a principal polarization on $T$, then the mapping $x \...
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4 votes
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Lie algebra sheaf of an abelian variety and the derived pushforward of its structure sheaf

Let $A$ be an abelian variety over a base scheme $S$, write $\pi: A \rightarrow S$, equipped with the zero section $e: S \rightarrow A$. Let $A^{\vee}$ be the dual variety of $A$. I am hoping to ...
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To prove given elliptic curve is Jacobi variety of given genus $1$ curve

Let $E$ be an elliptic curve $X^3+Y^3+60Z^3=0$ and $C$ be a genus $1$ curve given by $C:3X^3+4Y^3+5Z^3=0$.  I want to prove elliptic curve $E$ is Jacobi variety of $C$. What I should to prove is that $...
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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....
Pont's user avatar
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7 votes
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Can you get a $\mathbb{C}$-basis of $\mathbb{C}^{n}$ from an $\mathbb{R}$-basis by picking one vector out of each of $n$ pairs?

Let $T = \{v_1, \dots, v_{2n}\} \subseteq \mathbb{C}^n$ be a $\mathbb{R}$-linearly independent set of vectors. Now consider the $2^n$ subsets $S \subseteq T$ of size $n$ which contain exactly one of $...
Carlos Esparza's user avatar
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Number of rational points of $C:y^5=-x^2+x$ over $ \Bbb{F}_5$

Let $C:y^5=-x^2+x$ be a super ellipticcurve over $\Bbb{Q}$. I counted up the point of $ \sharp C( \Bbb{F}_5)$ and obtained $C( \Bbb{F}_5)=6$. I want to know what $ \sharp C( \Bbb{F}_{25})$ is. I guess ...
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Definition of 'Abelian variety has CM by $K$'

Let $K$ be a number field, let $L⊆K$ be CM field. $A/K$ be an Abelian variety over $K$. $A/K$ is said to have CM by $L$ if there is embedding $L⊆End_K(A) \otimes \Bbb{Q}$. But if we adapt this ...
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Holomorphic $1$ form on hyperelliptic curve

Let $C$ be an hyper elliptic curve. It is known that $H^0(C,Ω_C/ \Bbb{Q})$, space of holomorphic $1$ form on $C$ has basis(sometimes called 'Hermite basis') and can be expanded by local parameter $t$. ...
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Definition of $Λ^*: End(Ω(J))→End(Ω(C))$ from Abel Jacobi map $Λ$?

Let $C$ be a hyper elliptic curve of genus $2$. Let $J(C)$ be its Jacobean. There is Abel Jacobi map $Λ:C→J(C)$. Let $Ω(J)$ and $Ω(C)$ be space of holomorphic differential form on $J$ and $C$ ...
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Homomorphisms of abelian varieties constitute a finitely generated abelian group

Let $X, Y$ be abelian varieties over $k$. Let $l$ be a prime not equal to the characteristic of $k$. Then one shows that $\text{Hom}_k(X, Y)\to \text{Hom}_{\mathbb{Z}_l}(T_l X, T_l Y)$ is injective. ...
Fabio Neugebauer's user avatar
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1 answer
774 views

What is a Shimura variety and why should I care about them?

Shimura varieties have come up tangentially in talks with some of my advisors. My vague understanding is that they are "things that behave like moduli spaces of abelian varieties having some ...
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