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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Deformation of abelian scheme
Let $A/k$ be an abelian variety over a characteristic $p>0$ perfect field $k$. Usually, we say $A$ admits a lift to $W_2(k)$ if there exists an abelian scheme $\mathcal{A}/W_2(k)$ such that $\...
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Poincaré reducibility isogeny degree
Suppose that $(X, H)$ is a polarized abelian variety. Suppose furthermore that $A \subset X$ is an abelian subvariety. Then by Poincaré reducibility there is a complementary abelian subvariety $B$ ...
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Lifting of abelian variety
Is any abelian variety over a perfect field of characteristic $p>0$ $W_2$-liftable?
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Understanding Nagata's 1959 proof of his conjecture for large square
In his 1959 paper On the 14-th problem of Hilbert, Nagata proved his conjecture when $r=s^2\geq16$. I am confused by the "obvious" Lemma 5 of the paper
Lemma 5. If $C$ is a plane curve of ...
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Comparison between Faltings height and naive height of a curve
Let $C$ be a smooth curve of genus $g\ge 2$ over a number field $K$, and let $J$ be its Jacobian variety. Suppose we have an embedding $C\to \mathbb{P}^2$ so that $C$ is defined by a homogeneous ...
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For a smooth projective curve $X$, is the map, $M(r,d)\rightarrow \mathrm{Pic}^d(X)\cong \mathrm{Pic}^0(X)$ an albanese morphism?
$\DeclareMathOperator\Pic{Pic}$Let $X$ be a smooth projective curve and $M(r,d)$ be the moduli space of semistable vector bundles of rank $r$ and degree $d$ over $X$. My aim is to show that, when we ...
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Does analytic rank always upper bound algebraic rank over function fields?
Let $A$ be an abelian variety over a global function field. Is the Mordell-Weil rank of $A$ known to be at most the analytic rank, i. e. the order at $s = 1$ of the $L$-function? This is stated in ...
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Rational points on Abelian variety with infinitely many zero reduction
Let $J$ be an abelian variety over a number field $K$, and $x \in J(K)$ be a $K$-ratinal point of $J$. Suppose that for infinitely many non-archimedean valuation $v$ of $K$, (where $J$ has good ...
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Independence of rational points of abelian varieties after localization
Let $K$ be a number field, $X$ a smooth projective curve over $K$ with genus $g$. Let $J$ be the jacobian of $X$ with $j: X \to J$, let $r$ be the rank of the $\mathbb Z$-module $J(K)$.
We may choose ...
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Motivation for Jacobian coordinates (elliptic curves)
In cryptography, it seems to be a common choice to use the so-called Jacobian coordinates to represent a point of an elliptic curve (see e.g. Elliptic Curves: Number Theory and Cryptography, L. C. ...
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Are abelian varieties over $\mathbb{F}_q[t]$ which have the same $L$-function isogenous?
Faltings' isogeny theorem implies that two abelian varieties over a number field $K$ which have the same characteristic polynomial of Frobenius at almost all primes are isogenous. If $K = \mathbb{Q}$ ...
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Action of complex conjugation on abelian varieties with complex multiplication
In the by now classical expository paper by Milne on the main theorem of complex multiplication
https://www.jmilne.org/math/articles/2007c.pdf
the author states in Th. 4.1 that if $A({\bf C})={\bf C}^...
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Reciprocity law for Siegel modular forms
Let $E$ be a CM number field of degree $2g$ over $\bf Q$, and let $I,J\subseteq{\cal O}_E$ be two ideals in the ring of integers of $E$. Let $\{\Phi_1,\dots,\Phi_g\}\subseteq{\rm Hom}(E,{\bf C})$ be a ...
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Polarizations and totally positive endomorphisms
Let $(A, \lambda)$ be a polarized Abelian variety.
Question: Are the polarizations of $A$ the same as isogenies $\lambda' \colon A \rightarrow A^{\vee}$ of the form $\lambda' = \lambda \circ \alpha$
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Confusion on theorem in paper on monodromy of $p$-rank strata of moduli of curves
Application 5.7 in "Monodromy of the $p$-rank strata of the moduli space of curves", by Achter and Pries,
seems to imply that for $g \ge 3$, there exists a supersingular smooth projective ...
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Vanishing of Weil–Châtelet group over solvably closed field
Let $A$ be a connected algebraic group defined over a solvably closed field $K$. Are there any results concerning the vanishing of the WC-group $H^1(G_K, A)$?
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If the motive of an abelian variety is defined over a subfield, is the abelian variety defined over that field?
Let $L / k$ be a finite extension of fields, let $M$ be a pure motive over $k$ (with rational coefficients), and suppose that the base change $M_L$ of $M$ to $L$ is isomorphic to $\mathfrak{h}^1(A)$. ...
- 1,135
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$L$-function form of Tate conjecture for divisors on abelian varieties
Let $k$ be the function field of a $d$-dimensional regular integral finite-type scheme $Y$ over $\mathbb{Z}$. Conjecture 2 in Tate's paper in the Woods Hole proceedings predicts (among other things) ...
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How big can the degree of the field of definition of a morphism of abelian variety be?
Excuse me if some parts of the questions are well known, I am still a PhD student and have a lot to learn.
This question came to me while doing research on the Coleman's conjecture (as presented in a ...
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Canonical descent of Serre-Tate canonical lift
Deligne seems to say on page 239 here that the Serre-Tate canonical lift of an ordinary abelian variety over the algebraic closure $\overline{k}$ of a finite field of characteristic $p$ canonically ...
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What is the Dirichlet density of the reducible locus in the moduli of polarized abelian varieties in positive characteristic?
By section 2 of https://swc-math.github.io/aws/2024/2024KaremakerNotes.pdf, for $q = p^r$ a prime power, any $g, d \ge 1$, and any $n \ge 3$ prime to $q$, there exists a stratified fine moduli space $...
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Neron model of dual abelian variety
Let $A/K$ be an abelian variety where $K = \operatorname{Frac}(R)$ for some discrete valuation ring $R$. Let $\mathcal{A}$ be its Neron model. Assume that $A$ has potential good reduction, so that the ...
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Distance to boundary of $A_{g,n}$
Let $A$ be an principally polarized abelian variety of dimension $g$ over $\mathbb{C}$ and let $\mathcal{A}_{g}$ be the coarse moduli space of PPAVs of dimension $g$ so that $A$ defines a point $x$ on ...
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First cohomology of a variety and of its Albanese variety
Let $X$ be a smooth proper variety over an algebraically closed field $k$, $A_X$ its Albanese variety and $\alpha:X\to A_X$ the "canonical" map. It is known that for both étale and ...
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Existence of an irreducible polynomial of Frobenius
Suppose that $A$ is an abelian variety defined over a number field $K$, and that $\operatorname{End}(A)=\operatorname{End}(A_{\bar K})$. Suppose further that the ring of endomorphisms is $\mathcal O$, ...
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It seems natural to try to construct a non-degenerate pairing generalizing the Weil pairing
It is well known that a principally-polarized abelian variety $A$ has a non-degenerate (e.g., Weil) pairing of the form $A[n]\times A[n] \to \mu_n$, where $n \in \mathbb{N}$. However, I have never ...
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Does an Albanese variety have minimal dimension if this condition holds?
Let k be an algebraic closed field (if it matters, I'm interested in the case when this is the closure of a finite field).
Let $f(x,y)$ be a polynomial (or more generally a rational function) that is ...
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Equivalent(?) definitions of relative $\mathrm{Pic}^0$ functor *for abelian scheme*
Let $\pi: X \to S$ be an abelian scheme. I know from the first chapter of Faltings--Chai that it is true at this level of generality that the dual abelian scheme exists, i.e., that the fppf sheaf $\...
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Period lattice of CM modular forms on $\Gamma_0(N)$ and $\Gamma_1(N)$
Let K be an imaginary quadratic field $Q(\sqrt{-3})$. Let $p$ be a prime which split in $K$ and Let $\varpi$ be one of the factors of $p$ in $K$. Let $E$ be an elliptic curve $y^2=x^3+\varpi^2/4$ with ...
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Non-trivial maps over $\mathbb C$ and $\mathbb Q^{\mathrm{alg}}$?
$\newcommand{\alg}{\mathrm{alg}}\DeclareMathOperator\Hom{Hom}$Let $X$, $Y$ be two smooth projective varieties over $\mathbb Q^{\alg}$. Assume that there exists a non-constant map $f_{\mathbb C}: X_\...
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Is there infinite number of torsion points of an abelian variety with coordinates in the unit disk $\mathfrak{m}_{\mathbb{C}_p}$?
Let $\mathbb{C}_p$ denote the completion of the algebraic closure of the p-adic numbers $\mathbb{Q}_p$. Consider the unit disk $\mathfrak{m}_{\mathbb{C}_p}$.
Let $A$ be an $n$-dimensional abelian ...
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What concept is the line bundle associated with in the Riemann era?
I read Felix Klein's book, ON RIEMANN’S THEORY OF ALGEBRAIC FUNCTIONS AND THEIR INTEGRALS
In this connection we must
remember that it was chiefly the multiplicity of value of the integrals which for ...
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Degree of a map induced by divisors with weighted coefficients
Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $4$, and view $C$ as a subvariety of its Jacobian $A := \operatorname{Jac}(C)$ via the Abel–Jacobi map. Consider the map
$$f:C\times C \...
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Abel-Jacobi map and Riemann-Roch
It's well known fact that if $E$ is elliptic curve, then $E \simeq Pic^0 (E)$ via Abel-Jacobi map $P \mapsto (P) - (O)$. The proof of that in Silverman's The Arithmetic of Elliptic Curves (Chapter III,...
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Theta height of abelian variety
Let $(A, L)$ be a principally polarized Abelian variety defined over $\overline{\mathbb{Q}}$, of dimension $g$, with $L$ ample and symmetric (Line bundle correspond to the theta - $\Theta$ divisor). ...
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A letter from David Masser to Daniel Bertrand, November 1986
In the paper Fonctions thêta et points de torsion des variétés abéliennes by Sinnou David, one of the references is a letter from David Masser to Daniel Bertrand from November 1986, that is [Ma3]- D. ...
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Albanese in tensor triangular geometry
I am new to algebraic geometry. My question is the following :
We know the notion of Albanese variety and chow group for in algebraic geometry. P. Balmer defined the tensor triangular chow group for ...
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Does this elliptic curve over a cyclotomic tower have finitely many integral points?
$\mathbb{Q}(\zeta_{p^\infty})$, also written as $\mathbb{Q}(\mu_{p^\infty})$ or $\mathbb{Q}(p^\infty)$, denotes $\mathbb{Q}$ adjoined with the $p^{n}$th roots of unity for all $n$. It's the union of a ...
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Why is $a(\mathbb{D}(G))=g-\dim V^2(\mathbb{D}(G))$ and not $g-\dim V(\mathbb{D}(G))$?
In A short guide to $p$-torsion of abelian varieties in
characteristic $p$, Pries writes that the $a$-number of a Dieudonné module $\mathbb{D}(G)$ is $g-\dim V^2\mathbb{D}(G)$.
If $G$ is the $p$-...
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Which abelian varieties over a local field can be globalized?
As the title says, if $\mathcal{A}$ is an abelian variety over $\mathbb{Q}_p$, is there a criterion as to if I should expect there to exist $A$ over $\mathbb{Q}$ such that
$$\mathcal{A}\cong A\times_{\...
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Questions about elliptic curves with level-$n$ structure
Let $n$ be a positive integer, which is $4$ or a prime number $l$.
Let $E$ be an elliptic curve defined over a number field $K$.
Assume that all the $n$-torsion points of $E$ are defined over $K$, i.e....
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How can complex abelian varieties degenerate to tropical abelian varieties
There is a similar interesting question here
which has not been answered. I therefore ask this question in the hope to get an answer. I wonder how a family of complex abelian varieties can exactly ...
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Néron-Tate height on abelian varieties and PDEs
Let $A$ be an abelian variety over some number field $K$. We know the $\mathbb{C}$-points of $A$ form a complex analytic manifold, so $A(\mathbb{C})$ is a smooth manifold in fact. It then makes sense ...
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Weil pairing for an abelian variety uniformized as torus modulo lattice over a non-archimedean field
Suppose I have an abelian variety $A$ with split toric reduction over a non-archimedean field $K$, so that $A$ can be realized as $T / \Lambda$ where $\Lambda$ is a lattice in the torus $T$. Letting $...
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Constructing non-split $\mathbf{G}_m$-extensions of elliptic curves
I would like to find examples of abelian surfaces $A$ over a DVR $R$ with residue field $k$, so that the special fibre of $A$ is a non-split $\mathbf{G}_{m/k}$-extension of an elliptic curve over $k$.
...
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The cyclic twist of elliptic curve is a principally polarized abelian variety
Let $L/K$ is a cyclic extension of degree $p$, and let $E/K$ be an elliptic curve.
Let $E^L$ be the kernel of the map $Res^L_{K}(E) \rightarrow E$, where $Res^L_{K}(E)$ is the Weil-restriction.
Is the ...
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Why does a line bundle on an abelian variety give a group extension only if it is algebraically trivial?
If $X$ is an abelian variety and $L$ is a line bundle, deleting the zero-section one obtains a diagram
$$
0 \to \mathbb G_m \to Y \stackrel{\phi}{\to} X \to 0
$$
where $\mathbb G_m = \phi^{-1}(0)$,
...
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477
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Moduli space of complex and anti-complex tori?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SL{SL}$By Will Sawin's answer to Moduli Spaces of Higher Dimensional Complex Tori the moduli space of complex $d$-tori is $X ...
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p-adic uniformization pairing
I am studying the paper of R. Greenberg and G. Stevens, $p$-adic $L$-functions and $p$-adic periods of modular forms.
I am currently trying to understand the definition of $\mathcal{L}$-invariant for ...
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Reference request: generalized Jacobian variety for higher dimensional variety
Let $X\subset \mathbf{P}^n$ be a hypersurface such that the singular locus of $X$ consists of a single ordinary double point. I'm trying to find a reference to the "generalized" intermediate ...
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