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# 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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### 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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### 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 ...
1 vote
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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 $t$. ...
1 vote
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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 $C$ ...
1 vote
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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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### 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 ...
1 vote
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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 ...
1 vote
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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 ... 61 views

### 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 $A$. ...
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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 ...
1 vote
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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 ...
### 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)...