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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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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How do we consider reduction of abelian variety?
Let $A/K$ be an abelian variety. $I$ be prime ideal of $R_K$($R_K$ is ring of integers of $K$).
I want to know what is the meaning of reduction of $A/K$ mod $I$.
Coeffieients of $A/K$ lies in $K$, not ...
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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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Lie algebra sheaf of abelian varieties 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....
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For abelian variety, can every isogeny decomposed into two certain isogenies?
For algebraic curve in positive characteristic,
every isogeny $f$ can be written as compositium of purely inseparable isogeny $φ$ and separable isogeny $λ$.
For general abelian variety $A$, is the ...
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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 $...
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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. ...
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The number of Lagrangian subgroup
(Sorry for my bad English.)
Let $p,\ell$ be distinct primes,
$(A,\lambda)$ be principal a polarized abelian varieties of dimension $g$ over $\overline{\mathbb{F}_p}$.
Then we have $\ell$-Weil paring $...
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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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Non-singularity of Group Variety
I'm preparing a presentation about Abelian Varieties(definition and proof of the commutativity of the group law) and I'm following Milne's notes.
At some point, he states that any group variety is non-...
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Reduction of the Degree of a Curve by a Substitution
Let $y^2=P_{2n}(x)$ be an (hyper)elliptic curve, where $P_{2n}$ is a polynomial of degree $2n.$
It is said that the substitution $$x=x_1^{-1}+\alpha\qquad \text{and} \qquad y=y_1x_1^{-n}$$ reduces the ...
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Filling in a Gap in the Proof of the Converse to Modularity
Here is the theorem:
Let $f = \sum_{n = 1}^\infty a_nq^n \in S_2(\Gamma_0(N))$ be a newform with rational coefficients. Then there is a (unique up to $\mathbb Q$-isogeny) elliptic curve $E/\mathbb Q$ ...
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Why does principally polarization on $E\times E$ for supersingular elliptic curve $E$ come from Jacobian of hyperelliptic superspecial curves?
Sorry for my bad English.
In p.13 of this pepar, we think principally polarized abelian surface $E\times E$ for supersingular elliptic curve $E$.
Then it is one of the next two case.
(i)product of E ...
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Extension of Action in Algebraic Groups
I am reading the proof in Borel's book, "Linear Algebraic Groups" of the fact if $G$ is connected affine group of dimension one, then it is either $\mathbb{G}_a$ or $\mathbb{G}_m$. In the ...
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Irreducible abelian variety in an algebraic set.
Suppose $X$ is an algebraic set (not irreducible) over an algebraically closed field $k$, and $X$ is a commutative group variety. If $dim(X)>0$, can we find an irreducible abelian subvariety of ...
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Relations between Galois representations and complex geometry
We know that the Hodge structures of rational singular cohomology of complex projective manifolds are semisimple. The Galois representations attached to etale cohomology of smooth projective varieties ...
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Every Hermitian scalar product in $V $ defines on $M=V /\Lambda$ a translation-invariant Kähler metric
I was reading the chapter on complex tori of Griffith Harris book "principles of Algebraic Geometry" and i read that every Hermitian scalar product on $V $ defines on the torus $M=\mathbb{C^...
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Reduction of an abelian variety modulo powers of a prime
Let $K$ be a local field and $A$ an abelian variety over $K$. We take it to be minimal in the sense of Néron. In the paper Abelian varieties over large algebraic fields by Frey and Jarden, there is ...
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Hodge-Tate decomposition for tori
I am reading the article "On the Order of the Reduction of a Point on an Abelian Variety", R.Pink (here is a link to the article).
I am stuck on the proof of Proposition 1.4. Here, the ...
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The cohomology of the Jacobian and its subvariety
I read the following in a paper:
Let $G\subset J$ a subgroup of the Jacobian $J$ which is a countable union of Zariski closed subsets in the abelian variety $J$, so the irredundant decomposition of ...
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Lifting the connected-étale sequence of a finite group scheme over a residue field
Let $R$ be a complete DVR with fraction field $K$, characteristic $0$ and algebraically closed residue field $k$ of characteristic $p>0$. Suppose $G_{0}$ is a finite flat group scheme over $k$ so ...
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Line bundle of complex tori
Let $\Lambda \subset V=\mathbb{C}^g$ be a lattice and $T_g=V/\Lambda$ be a $g$ dimensional complex torus.
According to the Appell-Humbert theorem, any line bundle $L$ of $T_g$ is isomorphic to the ...
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Understanding the Kummer extension of semi-abelian varieties
Currently, I am reading D. Bertrand's paper Galois representation and transcendental numbers (here is the link to the paper), and I have a question about Theorem 2 (on Kummer theory of semi-abelian ...
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Potential good reduction and characteristic polynomial of Frobenius
Let $R$ be a discrete valuation ring with fraction field $K$ (valuation being $v$) and finite residue field $k$, $l$ be a rational prime invertible in $R$, $D(\bar{v})$ be a decomposition subgroup of ...
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What is the union of picard variety?
This question comes from a defnition of FGA explained, the chapter of Picard scheme, wirtten by Professor Dr. Steven L. Kleiman.
Suppose $X/S$ is an abelian scheme over $S$, my question is how to ...
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Product of the dual is the dual of the product, abelian varieties
The question is simple. I want to show that if $X$ and $Y$ are abelian varieties, then $(X\times Y)^t \simeq X^t\times Y^t$.
Letting $i_X,i_Y:X,Y\to X\times Y$ be the inclusion maps whose duals are ...
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Expressing inverse beta regularized in terms of Riemann/Siegel theta
I am new to the Riemann/Siegel Theta function, but it represents many special cases of Inverse Beta Regularized $\text I^{-1}_s(a,b)$. The Riemann theta function can represent any Abelian function, ...
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Characteristic cycle of theta divisor
Let $A/\mathbb{C}$ be an abelian variety with $\dim A=n$ and $H\subset A$ be an irreducible closed hypersurface, which as a divisor defines a principal polarization of $A$. We call $H$ a theta divisor ...
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Hom scheme of abelian varieties is etale
Let $A$ and $B$ be abelian varieties over a field $k$. I want to show that the group scheme $H:=\underline{\operatorname{Hom}}_{\text{gp}}(A,B)$ is étale using rigidity for abelian schemes. Is the ...
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curves on abelian surface $E\times E$, part II
It is Chapter II Exercise 4.16.4 in Kollár, and we are working on $\mathbb{C}$. Let $E$ be a very general smooth elliptic curve and consider the abelian surface $X=E\times E$. Let $m,n$ be coprime ...
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Density of $p$-torsion of an abelian variety
Let $A$ be an abelian variety over a field $k$ with $\operatorname{char}k = p$. I have a couple of questions about the proof that the $p$-torsion of $A$, i.e. $A[p^\infty]$, is Zariski dense in $A$. ...
- 1,343
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Mordell's conjecture for curves over the field of algebraic numbers
Is there a conjecture/result about the number of rational solutions of curves with genus ≥ 2 over the algebraic closure of Q? Is this set thought to be also finite as in Faltings's theorem over number ...
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Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?
Edit: I now crossposted this question on MO: https://mathoverflow.net/questions/428384/is-a-complex-algebraic-set-with-a-zariski-dense-subset-of-algebraic-points-alrea
Let $X$ be a complex algebraic ...
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Converse to Proposition 2.23 in Darmon, Diamond, Taylor's FLT Notes
Can someone either prove or link me to a reference for Remark 2.24 (page 64) here? I am told that SGA7 covers this for general abelian varieties. I am wondering if a) anyone can pinpoint where in SGA7 ...
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Endomorphism of abelian variety which kills $p$-torsion is divisible by $p$
Let $A/\mathbb{Q}$ be an abelian variety, and suppose $T \in \operatorname{End}_{\mathbb{Q}}(A)$ is such that $T(A[p]) = 0$. I'd like to show that there is a $T' \in \operatorname{End}_{\mathbb{Q}}(A)...
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"Jacobian variety" for surfaces
I heard that there isn't a functorial construction which associates an abelian variety to any 2-dimensional variety, equipped with an embedding of the surface inside the abelian variety. I find that a ...
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