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 geometry and number theory.

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0answers
44 views

Some basic questions on quotient of group schemes

Let $S$ be a fixed base scheme and $G, H$ be group schemes over $S$. Since I am mainly interested in commutative group schemes over fields, we may assume that $G,H$ are commutative and $S$ is a field ...
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2answers
341 views

Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor

It's known that the class number of $\mathbb{Q}(\sqrt{p})$ is $1$ for all primes $p<229$. Question: What would it be like for conceptual explanations of $h(\mathbb{Q}(\sqrt{p}))=1$ for the first ...
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1answer
179 views

For an abelian scheme, $R^pf_* \Omega^q$ is locally free and its formation is compatible with any base change

Let $k$ be a field, $\bar{R} \to R$ a local homomorphism of artinian local rings with the residue fields $k$, $I$ its kernel, $A/R$ an abelian scheme, and $\mathscr{T}$ its tangent sheaf. Let $A_0 = A ...
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1answer
218 views

$\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties

Reading M. Hindry and J. H. Silverman (Diophantine Geometry-An Introduction), I find the claim that $\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties. Mumford ...
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1answer
190 views

Surjectivity of the Abel-Prym map

It is well known that the Abel-Jacobi map restricted to $\text{Eff}_g(C)$ surjects onto the Jacobian $\text{Jac}(C)$, since every divisor of degree $g$ is effective. Is there an analogous statement ...
2
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1answer
170 views

Polarization of an abelian variety made by the sum of two divisors

Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$. In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...
6
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1answer
197 views

Homogeneous vector bundles on Abelian varieties

I recently encountered a result about vector bundles on Abelian varieties, which I found interesting. It characterizes homogeneous (translation invariant) vector bundles on Abelian varieties. More ...
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Isomorphism of certain irreducible representations over finite fields

We are given a faithful representation of a cyclic group of order 5 $\rho: C_5=G \rightarrow End_{\mathbb{F}_3}(V) $ with $dim_{\mathbb{F}_3}V=8$ as vector space. It is also known that $V=U\oplus W$ ...
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52 views

A map on Jacobians coming from a correspondence explicitly

From this question, we know that every map of the form $J(C) \to J(C)$ for a curve $C$ and it's jacobian $J(C)$ comes from a correspondence between $C\times C$ and in fact we can take this ...
6
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1answer
245 views

On the moduli stack of abelian varieties without polarization

(I am especially interested in abelian surfaces and characteristic 0). How bad is the moduli stack of abelian varieties (with no polarization or level structure)? Is it an Artin stack? DM (Deligne-...
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Sections of infinite order of elliptic surfaces

Let $X\to \mathbb{P}^1$ be a non-isotrivial elliptic surface over $\mathbb{C}$ with a section and with $X$ a smooth projective connected surface over $\mathbb{C}$. Let $\sigma:\mathbb{P}^1\to X$ be a ...
6
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For which (g,q) does there exist a supersingular curve?

We say a curve over a finite field $\mathbb F_q$ is supersingular if it's Frobenius eigenvalues (on $H^1(X,\mathbb Z_\ell)$) are of the form $q^{1/2}\alpha$ for $\alpha$ a root of unity. As far as I ...
3
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1answer
124 views

Surfaces of general type with $q=1$

Let $X$ be a smooth projective connected surface of general type over $\mathbb{C}$ with $q(X) = 1$, where $q(X) = \mathrm{h}^1(X,\mathcal{O}_X)$. Let $E$ be the Albanese variety of $X$, and let $X\to ...
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Proving the Immersion part of an Embedding

Trying to see the proof of embedding the Jacobian of a Compact Riemann Surface $X$ using Theta functions. So, using the Theta divisor we have the corresponding line bundle say $L$, we want to prove ...
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The profinite topology on the Mordell Weil group

In this lecture of Serre on his open image theorem, around 6 minutes, Serre mentions the following theorem of Tate: Let $A/k$ be an abelian variety over a number field and consider the Mordell-Weil ...
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A Lefschetz style formula for the $\ell^\infty$ torsion of an Abelian variety over a finite field

Let $A/\mathbb F_q$ be an abelian variety over a finite field. Define $A_\ell = A[\ell^\infty](\mathbb F_q)$, the $\ell^n$ ($n\geq 0)$ torsion points defined over the base field. I can assume $\ell \...
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110 views

Chow variety of 1-cycles on abelian surface

It is an easy exercise to show that on a K3 surface, a smooth genus $g$ curve moves in a $g$-dimensional linear system. Nearly the same exercise shows that on an abelian surface, the corresponding ...
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92 views

Structure of non-big divisors in an abelian variety

Let $A$ be an abelian variety over $\mathbb{C}$. If $A$ has an effective non-big divisor, then $A$ is not simple. (In a simple abelian variety, every non-zero effective divisor is ample.) What can ...
5
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113 views

Jacobian fibration of an abelian fibration

Let $f \colon S \rightarrow C$ be a minimal elliptic surface and let $g \colon J \rightarrow C$ be its jacobian fibration. In this case, we know that the fibers of $g$ are better behaved that the ones ...
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116 views

The Picard scheme of an ordinary singular curve

Let $k$ be an algebraically closed field, $C$ a proper reduced connected scheme over $k$ of dimension 1, whose singularity is at worse ordinary, $\pi : \tilde{C} \to C$ the normalization of $C$ and $...
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$A$ is a CM abelian variety over $\mathbb{C}$, $\sigma \in$ Aut$(\mathbb{C})$. How to define $\sigma A$?

This question may be a little silly: $A$ is a CM abelian variety over $\mathbb{C}$, $\sigma \in$ Aut$(\mathbb{C})$. How to define $\sigma A$? In my opinion, $\sigma$ can only act on $A$ when it’s ...
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When a CM abelian variety has complex multiplication by $\mathcal{O}_E$?

I'm reading Milne's note, Complex Multiplication. There are many properties, such as Shimura-Taniyama Formula provided that $A$ is an abelian variety with complex multiplication by $\mathcal{O}_E$. So ...
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151 views

Elliptic curves and archimedean place

here is my question : Let $K$ be any field, $ E \to spec(K)$. Let $ v \in M_K $ an archimedean place. We know that $ \overline{K_v} \simeq \mathbb{C}$ and there exists $\tau_v \in \mathbb{H}$ such ...
2
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0answers
134 views

Uniqueness of theta divisor

Let $A$ be an abelian variety (at least over $\mathbb{C}$). Suppose we have two theta divisors $\Theta_1$ and $\Theta_2$ on $A$, which give two principal polarizations on $A$. In general, are those ...
2
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120 views

Application of Galois descent

I not understand an assumption done at the beginning of the proof of Rigidity lemma in Moonens and van der Geers book about Abelian variaties (page 12). Here is it: Question: Why the assumption $k= \...
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124 views

Nef divisors on abelian varieties are pullbacks of ample ones

It is well known that for any polarization ( that is, ample line bundle) $L$ on an abelian variety $A$, there is an isogeny $\phi\colon A \to B$ to another abelian variety with a principal ...
3
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215 views

Example of a non log-canonical pair for an abelian variety with polarization of degree >2

Let $(A,L)$ be a polarized abelian variety of dimension $g$, with an indecomposable polarization of degree $\chi(L)=d$. There is a theorem of Debarre and Hacon about the singularities of pairs: ...
9
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150 views

Good reduction of finite etale covers of abelian varieties

Let $R$ be a dvr (whose residue characteristic is zero if it helps) with fraction field $K$. Let $A$ be an abelian variety over $K$ with good reduction over $R$. Let $X\to A$ be a finite etale ...
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1answer
208 views

Why does MAGMA claim that the automorphism group of a curve is trivial?

I have been trying to compute the Automorphism group of a curve using MAGMA with no success. This is what I have tried: I have tried to compute the Automorphism group of the curve $y^3=x^4-x$ and no ...
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2answers
227 views

Relation between elliptic curve and Fermat's last thereom

I am looking for a document (lecture note, book, blog) that elaborately explains how the elliptic curve $E (a, b) := y^2=x(x-a)(x-b)$ is associated with the solution of $a^n+b^n=c^n$. In 1969 ...
1
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0answers
113 views

Compatiblity of completion and fibre products. (Formal completion and formal groups)

Let $S$ be a scheme (not necessarily locally noetherian), $X$ a smooth separated group scheme over $S$, and $\hat{X}$ be the formal completion along with the identity section. Then does the group ...
9
votes
1answer
279 views

Isomorphic Jacobian Varieties Just Like Abelian Varieties — Torelli's Theorem

Torelli's theorem states: Let $R$, $R'$ be compact Riemann surfaces of genus $g$, $J(R)$, $J(R')$ their Jacobian varieties, $\Theta$, $\Theta'$ their respective theta divisors. The Riemann surfaces ...
9
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1answer
483 views

Tamagawa numbers

Let $K$ be a finite extension of $\mathbb{Q}_p$ with absolute Galois group $G_K$. Let $A$ be an abelian variety defined over $K$. The (geometric) Tamagawa number is defined as the order of the ...
6
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0answers
111 views

What are the genus 4 curves with Jacobians that are 4-th powers?

Consider the moduli space of all genus $4$ curves $\overline{\mathscr M_4}$ of dimension $3\times 4 - 3 = 9$. Under the Torelli map, there is a map to $\overline{\mathscr A_4}$ (which has dimension $...
11
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1answer
214 views

Lifting a splitting of an Abelian variety to characteristic 0

Let $R$ be the ring of integers in a (complete) algebraic closure of $\mathbb Q_p$ with maximal ideal $\mathfrak p$. Suppose I have an Abelian surface $\mathcal A/R$ such that over every $R/\mathfrak ...
3
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0answers
132 views

Genus two curves on abelian surfaces

Considering a smooth genus two curve $C_2$, let $J(C_2)$ be its Jacobian surface, and take $p \in J(C_2)$ an $m$-torsion point. Let $A = J(C_2)/Z_m$, where $Z_m$ acts by $x \mapsto x+p$. The image of $...
4
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0answers
153 views

Mordell-Weil group of an abelian variety on the perfect closure of a finitely generated field

This question is closely related the question Over which fields does the Mordell-Weil theorem hold? I consider the following question: (1) Let $K$ be a finitely generated field extension of $\...
5
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0answers
424 views

A functor on Abelian varieties corresponding to this operation on Weil numbers

Let $A/\mathbb F_q$ be an abelian variety over a finite field with Weil numbers $q^{1/2}\alpha_1,\dots,q^{1/2}\alpha_n$. Consider the numbers $q^{d/2}\alpha_1,\dots,q^{d/2}\alpha_n$. These are still ...
0
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0answers
126 views

Rigidity of universal abelian variety

When $\phi \colon A^{\mathrm{univ}} \to \overline{A_g}$ denotes the universal Abelian variety over a compactified Siegel moduli space $\overline{A_g}$, does there exist non-trivial automorphism $\...
1
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1answer
71 views

Intersection of a certain linear ideals of $K[[X_1,\ldots,X_{np}]]$ for ${\mathrm{ch}}(K) = p > 0$

Suppose ${\mathrm{ch}}(K) = p > 0$ and we consider the formal power series ring $K[[X_1,\ldots,X_{np}]]$ over $K$ in $np$ variables $X_1,\ldots, X_{np}$. Let $\Lambda$ be the set defined as follows$...
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0answers
96 views

Application of Stein factorisation: rigidity lemma

Let $X,Y$ Noetherian schemes and $f:X \to Y$ proper map. The Stein factorisation factorizes $f$ as $X \xrightarrow{g} Spec \text{ } f_* \mathcal{O}_X \xrightarrow{h} Y$ with $h$ finite and $g$ has ...
3
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0answers
165 views

Abelian varieties by Moonen and van der Geer: proof of rigidity lemma

I try to understand a reduction step in the proof of rigidity lemma as proved in Moonen's and van der Geer's Abelian varieties (Lemma 1.11 on page 12- if the link not work the draft version is online ...
7
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0answers
122 views

Conditions for $p$-divisible group to come from an abelian variety

Over $\mathbb{Z}_p$ or $\mathbb{Q}_p$, are there any known sufficient conditions on a $p$-divisible group for it to come from an abelian variety?
4
votes
1answer
184 views

Isogeny components of Jacobians of étale covers

Let us work over $\mathbb{C}$. Fix $X$ a smooth projective curve of genus at least $2$. For every simple abelian variety $A$, it is easy to come up with a ramified covering $Y\to X$ with a non-...
2
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0answers
142 views

Standard application of Oort-Tate classification theorem

$\DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\tors}{tors}$In Mazur's paper “Modular curves and the Eisenstein ideals”, on the bottom of page 159, it says that if $T$ is a open subscheme of $...
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0answers
50 views

abelian variety with all over non degenerate pairing

Suppose we have an abelian variety $A$ defined over the rational numbers $\mathbb{Q}$. It is known that the Weil pairing $$ e_\ell: A[\ell]\times A[\ell] \rightarrow \mu_\ell $$ is non degenerate, ...
2
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0answers
90 views

The Weil restriction of an elliptic curve with respect to $\mathbb{F}_{p^2}/\mathbb{F}_{p}$

For a prime $p > 3$ consider the quadratic finite field extension $\mathbb{F}_{p^2}/\mathbb{F}_{p}$. Also, consider the elliptic curves $$ E\!: y_0^2 = x_0^3 + ax_0 + b,\qquad E^{(1)}\!: y_1^2 = ...
2
votes
1answer
139 views

Is there a way to find any $\mathbb{F}_2(t)$-point on the elliptic curve $\mathcal{E}$?

Consider the ordinary elliptic curves $$ E\!:y_1^2 + x_1y_1 = x_1^3 + 1,\qquad E^\prime\!: y_2^2 + x_2y_2 = x_2^3 + x_2^2 + 1 $$ over the field $\mathbb{F}_2$. They are quadratic twists to each other....
3
votes
1answer
160 views

What is the geometric quotient of the abelian threefold?

Consider a finite field $\mathbb{F}_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$ and its element $\zeta \neq 1$, $\zeta^3 = 1$. Also, let $E\!: y^2 = x^3 + b$ be an elliptic curve of $j$-invariant ...
2
votes
1answer
135 views

What is the quotient $E \!\times\! E^\prime / G$?

Consider a finite field $\mathbb{F}_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$) $$ E\!:y_1^2 ...

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