# Questions tagged [abelian-varieties]

Abelian varieties are projective algebraic varieties endowed with an Abelian group structure. Over the complex numbers, they can be described as quotients of a vector space by a lattice of full rank. They are analogs in higher dimensions of elliptic curves, and play an important role in algebraic ...

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### Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?

Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$?
Do there exist ...

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### Abelian variety over Q with many roots of unity

Given an abelian variety $A$ over the rational integers $\mathbb{Q}$, and a prime $p$, we know that $\mathbb{Q}(\zeta_p)$ is contained in $\mathbb{Q}(A[p])$, the $p$-division field of $A$, and where $\...

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### Relation between Faltings height and height on moduli space

Let $E$ be an elliptic curve over a number field $K$. The difference between the semistable Faltings height $h_F(E)$ of $E$ and the height $h(j_E)$ of the $j$-invariant of $E$ can be bounded in terms ...

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### Point Counts on $G$-torsors over Finite Fields

Let's assume we have a $G$-torsor $X_{1} \to X_{2}$, where $G$ is a finite abelian group, and both $X_{1}$ and $X_{2}$ are defined over $\text{Spec}(\mathbb{Z})$. Is there an easy way to compute $\#...

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### Roots of unity and coordinates of points in abelian varieties

We consider an abelian variety $A$ defined over the rational numbers $\mathbb{Q}$. For a torsion point $P\in A(\bar{\mathbb{Q}})$, consider the field $\mathbb{Q}(P)$ obtained by adjoining to $\mathbb{...

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### Weil pairings on abelian varieties restricted to subgroups of a given order

Let $A$ be an abelian variety of dimension $g$ defined over a number field $K$. Suppose $A$ has a principal polarization and $\ell$ is a prime number. We have a Weil pairing:
$$
e_\ell: A[\ell]\times ...

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### Pull-back of polarization

Let $(X, L)$ and $(Y, M)$ be two polarized abelian varieties .
According to Birkenhake C. and Lange H. in Complex Abelian Varieties a homomorphism of polarized abelian varieties $f:(Y, M)\...

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### why the division field of an abelian variety contains a cyclotomic field?

Given an abelian variety $A$ defined over $\mathbb{Q}$, for a positive integer (we can suppose prime) $\ell$, let $A[\ell]$ denote the group
of points of $A$ that are annihilated by $\ell$, the ...

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### Index of the endomorphism ring of an abelian surface

For an abelian surface $A/\mathbb{Q}$ such that $R:=\mathrm{End}_{\mathbb{Q}}(A)$ is an order in a real quadratic field $K$ (so a $\mathrm{GL}_2$-type surface), is there a bound on the index $[O_K : R]...

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### Polarization of the Jacobian in Torelli's theorem

I'm studying an example in book Yuji Shimizu and Kenji Ueno. Advances in Moduli Theory. Translations of Mathematical Monographs, vol. 206, that shows the importance of isomorphism as principally ...

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### Is the complement of an affine open in an abelian variety ample?

Let $U$ be an affine open subscheme of an abelian variety $A$ over $\mathbb{C}$. Is $A-U$ an ample divisor?
If $\dim A =1$ this is true.
If $\dim A = 2$, the complement is a divisor $D_1+\ldots + ...

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### Principally Polarized CM Abelian Variety

I am interested in considering examples of abelian varieties that are principally polarized with CM in dimension three. However, I am struggling to construct or find even a single instance.
In ...

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### analogue of Theorem of Mattuck for Abelian varieties over $\mathbf{F}_q(\!(t)\!)$

By a theorem of Mattuck [Abelian Varieties over $p$-Adic Ground Fields, Annals of Mathematics, Second Series, Vol. 62, No. 1 (Jul., 1955), pp. 92-119], for an Abelian variety $A$ of dimension $g$ over ...

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### Analogue of the original Birch–Swinnerton-Dyer conjecture for abelian varieties

$\newcommand{\Q}{\Bbb Q}
\newcommand{\N}{\Bbb N}
\newcommand{\R}{\Bbb R}
\newcommand{\Z}{\Bbb Z}
\newcommand{\C}{\Bbb C}
\newcommand{\F}{\Bbb F}
\newcommand{\p}{\mathfrak{p}}
$
Let $A$ be an abelian ...

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### Good reduction of abelian varieties over valuation rings via coverings

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$, and let $A$ be an abelian variety over $K$.
Suppose that there is a smooth proper scheme $\mathcal{X}$ over $\mathcal{O}_K$ whose ...

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### Do abelian varieties have Neron models over arbitrary valuation rings?

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$. Let $A$ be an abelian variety over $K$. Does $A$ have a Neron model?
If $\mathcal{O}_K$ is a discrete valuation ring, then this is ...

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156 views

### Why are modular curves non-trivial covers of the $j$-line

This is a very soft question.
Let $n\geq 1$ and let $Y(n)$ be the (open) modular curve associated to $\Gamma(n)\subset SL_2(\mathbb{Z})$. Interpreted correctly, $Y(n)\to Y(d)$ is finite etale, ...

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### What is known about abelian varieties with several principal polarizations?

Let ``the simple case" be when the polarized abelian variety does not break up into a product of polarized abelian varieties.
I am trying to get an idea of what is known about abelian varieties with ...

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### isomorphic abelian varieties

Let $A$ and $B$ be isogenous abelian varieties defined over a field $k$.
Suppose $A(L)\cong B(L)$ for all finite extensions $L$ of $k$. Does this imply that $A\cong B$?
It would be different if we ...

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### abelian variety over a regular extension of a field

I want to read Manin proof of Mordell Conjecture over function fields.I understand most of the article but I have problems with "kernel theorem"and it's proof:
consider $A$ is an abelian variety over ...

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### Parallel transport for variety over finite field

I was wondering: Given a variety over a finite field, say the projective plane or sphere over $\mathbb{F}_q$. Then I can try to define parallel transport along (geodesic) curves. In particular, I can ...

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### Is every abelian variety a subvariety of a Jacobian?

Let $k$ be an infinite field and $A$ be an abelian variety over $k$. Can $A$ be embedded into a Jacobian variety $J$ over $k$?
In these notes by William Stein this is stated without proof in remark 1....

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### What are the possible bad reductions for an abelian variety of dimension $g$ and a maximal endomorphism ring?

Perhaps the most basic fact about abelian varieties with CM is they have an everywhere potential good reduction (Serre-Tate). On the face of it it might appear that there isn't much more to be added ...

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### An example of curves with the same Jacobian, but different Jacobian automorphism groups (wrt their respective canonical principal polarizations)?

I am trying to understand examples of differing curves with the same Jacobian, and the quirks of the Jacobian.
Here is my question: What is an example where $Aut(Jac(X, a))$ and $Aut(Jac(X', a'))$ ...

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### Numerical equivalent positive non-degenerate divisor induced projective embedding involves Veronese map?

This is a part of material I do not understand from "Analytic Theory of Abelian Varieties" by Swinnerton-Dyer.
Let $A=\mathbb{C}^n/\Lambda$ be an abelian variety with positive-definite Hermitian form ...

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### 1-dimensional p-divisible groups, level structures and Cartier divisors

I am confused about the 1-dimensionality of $p$-divisible groups and its role in defining level structures.
Here's how I view/understand/not understand things:
If a $p$-divisible group arises from a ...

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### Largest ranks achieved by abelian varieties of fixed dimension

This is a follow-up to this earlier question on elliptic curves: Largest rank assumed by infinitely many elliptic curves
Let $g \geq 1$ be an integer. For each $g$, what is known about the largest ...

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### Semisimplicity of the p-adic étale Tate module over $F_p(t)$

Let $k$ be a finitely generated field of positive characteristic p.
Let $A$ be an abelian variety over $k$ and write $T_p(A)$ for the $p$-adic étale Tate module of $A$. Is it known if the natural ...

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### Singularities of quotient of a vector bundle by a lattice

Let $V$ be a complex vector bundle of rank $g$ over an open unit disc $\Delta$ and $H$ be a integral local system of rank $2g$ over $\Delta$ i.e., for every $t \in \Delta$, $H_t \cong \mathbb{Z}^{2g}$....

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### Derived categories of coherent sheaves and degenerations of abelian varieties

By the work of Burban-Drozd (https://projecteuclid.org/euclid.dmj/1076621984), we know what happens to the derived category of coherent sheaves when an elliptic curve degenerates into a nodal curve or ...

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### Singular abelian surfaces that can be defined over $\mathbb Q$

An abelian surface $A$ is called singular if it has maximal Picard number $\rho(A) = 4$.
By work of Shioda-Mitani, any singular abelian surface $A$ is the product $A = E_1 \times E_2$ of two ...

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### Factors of the Jacobian of modular curves

Let $J_1(p)$ be the Jacobian of the modular curve $X_1(p)$ for p an odd prime. We know that $J_1(p)$ is isogenous to a direct sum of abelian varieties $\oplus_{f}A_f$ where the sum runs over Hecke/...

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### Intersections with a Power of an Ample Divisor on an Abelian Variety

Let $A$ be a $g$-dimensional, complex abelian variety, let $H$ be an ample divisor, let $D\in Pic^0(A)$, and let $0\leq k\leq g$.
Question 1: Does $D^k\neq 0\in CH^k(A,\mathbb{Q})$ imply that $H^{g-k}...

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### Can an abelian variety dominate a variety of general type?

Let $X$ be a projective (not necessarily smooth) normal variety of general type over $\mathbb{C}$. Let $A$ be an abelian variety and let $A\to X$ be a surjective morphism.
Is $X$ zero-...

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### The quotient of a superspecial abelian surface by the involution

Let $E_i\!: y_i^2 = f(x_i)$ be two copies of a supersingular elliptic curve over a field of odd characteristics. Consider the involution
$$
i\!: E_1\times E_2 \to E_1\times E_2,\qquad (x_1, y_1, x_2, ...

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### Non algebraizable formal abelian schemes

I'd like to collect a good number of examples of formal schemes over $\text{Spf}(\mathbf{Z}_p)$, whose special fiber is a projective variety over $\mathbf{F}_p$, but that are not algebraizable.
If ...

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### Under what conditions are superspecial abelian surfaces isomorphic over a finite field?

Let $E_1$, $E_2$, $E_3$, $E_4$ be supersingular elliptic curves over a finite field $\mathbb{F}_{p^2}$, where $p$ is an odd prime. There is a well known theorem stating that over the algebraic closure ...

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### Generic Mumford Tate group and algebraic points

I will stick with a concrete example for this question, but it should probably be cast in a more general framework.
Let $Sym_g(\mathbf{C})$ be the space of symmetric matrices of order $g$ with ...

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### Endomorphisms of abelian varieties with real multiplication

Let us work over $\mathbb{C}$ to make life easier.
I've came across to the following definition. Let $F$ be a totally real number field of degree $g$, with ring of integers $\mathcal{O}_F$. An ...

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### Translates of a line bundle on a complex $n$-torus

Suppose $\mathbb T:=V/\Gamma$ is a complex $n$-torus (i.e., $V$ is an $n$-dimensional $\mathbb C$-vector space and $\Gamma$ is a rank $2n$ lattice in $V$). Fix a holomorphic line bundle $L\in\text{Pic}...

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### Endomorphism ring of simple ordinary abelian variety

Is there an example of an ordinary and simple abelian variety $A$ over an algebraically closed field $K$ (of characteristic $p>0$) such that ${\rm End}(A)$ is not commutative? Note that the answer ...

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### schemes vs varieties in abelian varieties and maximal subscheme where line bundle is trivial

This is a very elementary question as I am just learning about abelian varieties by reading Mumford's book,
Let $X$ be a complete variety (irreducible and reducible over an algebraically closed field)...

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### Bhargava's work on the BSD conjecture

How much would Bhargava's results on BSD improve if finiteness of the Tate-Shafarevich group, or at least its $\ell$-primary torsion for every $\ell$, was known? Would they improve to the point of ...

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### Chow groups modulo homological equivalence for abelian varieties

Let $X$ be an abelian variety over a field $k$.
Let $A^p_{\rm hom}(X)$ be the $p$-th Chow group of cycles modulo homological equivalence ($\ell$-adic, if $k$ is of char $p$).
Do we have $$A^p_{\rm ...

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### Is the isogeny class 1109.a of abelian surfaces in the LMFDB complete?

The LMFDB lists the Jacobian of the genus-2 curve 1109.a.1109.1 (http://www.lmfdb.org/Genus2Curve/Q/1109/a/1109/1) as being isolated in its rational-isogeny class. However, the LMFDB does not purport ...

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### Descent via an explicit isogeny (genus 2)

This question is related to a previous question posted by me here answered by Prof. M. Stoll. 5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians.
Here I ask some technicalities of a ...

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### Points of abelian varieties over purely transcendental extensions

I heard about the result in the theory of abelian varieties which says the following: given an abelian variety $X$ defined over a field $k$ and a purely transcendental extension $k\subset L\subset L'$ ...

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### Tate-Shafarevich group over number fields

Let $A$ be an abelian variety over a number field $K$, $\text{Sha}(A/K)$ its Tate-Shafarevich group, $\ell$ a prime.
Is it known that the $\ell$-primary torsion subgroup $\text{Sha}(A/K)\{\ell\}$ is ...

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### Why do “large” opens of abelian surfaces have “small” canonical bundle?

Let $A$ be an abelian surface, and let $S$ be a finite set of points. Let $U=A\setminus S$. Note that $U$ is a "large" open of $A$.
Let $B\to A$ be a proper birational surjective morphism with $B$ ...

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### How to prove that a certain curve does not lie in a proper abelian subvariety of abelian variety

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ by the equation
$$E: y^2=x^3-Ax+B=:f(x).$$
Consider the abelian variety $E^3:=E \times E \times E \subset \mathbb{P}^2 \times \mathbb{P}^2 \...