In this paper we consider reduction maps $r_{v} : K_{2n+1}(F)/C_{F} \to K_{2n+1}(κ_{v})_{l}$ where $F$ is a number field and $C_{F}$ denotes the subgroup of $K_{2n+1}(F)$ generated by $l$-parts (for all primes $l$) of kernels of the Dwyer-Friedlander map and maps $r_{v} : A(F)\to A_{v}(κ_{v})_{l}$ where $A(F)$ is an abelian variety over a number field. We prove a generalization of the support problem of Schinzel for $K$-groups of number fields: Let $P_{1}, ..., P_{s}, Q_{1}, ..., Q_{s}\in K_{2n+1}(F)/C_{F}$ be the points of infinite order. Assume that for almost every prime $l$ the following condition holds: for every set of positive integers $m_{1}, ..., m_{s}$ and for almost every prime $v$ $$m_{1} r_{v}(P_{1})+... + m_{s} r_{v}(P_{s})=0 \mathrm{implies} m_{1} r_{v}(Q_{1})+... + m_{s}r_{v}(Q_{s})= 0. $$ Then there exist $α_{i}$, $β_{i}\in \mathbb{Z} \setminus \{0 \}$ such that $α_{i} P_{i}+β_{i} Q_{i}=0$ in $B(F)$ for every $i \in \{1, ... s\}$. We also get an analogues result for abelian varieties over number fields. The main technical result of the paper says that if $P_{1}, ..., P_{s}$ are nontorsion elements of $K_{2n+1}(F)/C_{F}$, which are linearly independent over $\mathbb{Z}$, then for any prime $l$, and for any set $\{k_{1},... ,k_{s}\}\subset \mathbb{N} \cup \{0\}$, there are infinitely many primes $v$, such that the image of the point $P_{t}$ via the map $r_{v}$ has order equal $l^{k_{t}}$ for every $t \in \{1, ..., s \}$.
The category of abelian varieties over $\mathbb{F}_q$ is shown to be anti-equivalent to a category of $\mathbb{Z}$-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over $\mathbb{F}_q$. On full subcategories cut out by a finite set $w$ of conjugacy classes of Weil $q$-numbers, the anti-equivalence is represented by what we call $w$-locally projective abelian varieties.
We study (weak) log abelian varieties with constant degeneration in the log flat topology. If the base is a log point, we further study the endomorphism algebras of log abelian varieties. In particular, we prove the dual short exact sequence for isogenies, Poincaré complete reducibility theorem for log abelian varieties, and the semi-simplicity of the endomorphism algebras of log abelian varieties.
In this paper we study super-isolated abelian varieties, that is, abelian varieties over finite fields whose isogeny class contains a single isomorphism class. The goal of this paper is to (1) characterize whether a product of super-isolated varieties is super-isolated, and (2) characterize which super-isolated abelian varieties admit principal polarizations, and how many up to polarized isomorphisms.
Analogy between Anderson T-motives and abelian varieties with multiplication by an imaginary quadratic field (MIQF) is a source of 2 results: 1. A description of abelian varieties with MIQF of dimension $r$ and signature $(n, r-n)$ in terms of "lattices" of dimension $r$ in $\Bbb C^n$; 2. A construction of exterior powers of abelian varieties with MIQF having $n=1$.
In two earlier articles, we proved that, if the Hodge conjecture is true for ALL CM abelian varieties over the complex numbers, then both the Tate conjecture and the standard conjectures are true for abelian varieties over finite fields. Here we rework the proofs so that they apply to a single abelian variety. As a consequence, we prove (unconditionally) that the Tate and standard conjectures are true for many abelian varieties over finite fields, including abelian varieties for which the algebra of Tate classes is not generated by divisor classes.
We prove that torsion codimension 2 algebraic cycles modulo rational equivalence on supersingular abelian varieties are algebraically equivalent to zero. As a consequence, we prove that homological equivalence coincides with algebraic equivalence for algebraic cycles of codimension 2 on supersingular abelian varieties over the algebraic closure of finite fields.
We prove a conjecture by Kawaguchi-Silverman on arithmetic and dynamical degrees, for self-morphisms of semi-abelian varieties. Moreover, we determine the set of the arithmetic degrees of orbits and the (first) dynamical degrees of self-morphisms of semi-abelian varieties.
The characteristic polynomials of abelian varieties over the finite field $\mathbb{F}_q$ with $q=p^n$ elements have a lot of arithmetic and geometric information. They have been explicitly described for abelian varieties up to dimension 4, but little is known in higher dimension. In this paper, among other things, we obtain the following three results on the characteristic polynomial of abelian varieties. First, we prove a relation between $n$ and $e$, where $e$ is a certain multiplicity associated with a simple abelian variety of arbitrary dimension over $\mathbb{F}_q$. Second, we explicitly …
In this article, we prove that the Zilber-Pink conjecture for abelian varieties over an arbitrary field of characteristic $0$ is implied by the same statement for abelian varieties over the algebraic numbers. More precisely, the conjecture holds for subvarieties of dimension at most $m$ in the abelian variety $A$ if it holds for subvarieties of dimension at most $m$ in the largest abelian subvariety of $A$ that is isomorphic to an abelian variety defined over $\bar{ \mathbb{Q}}$.
We prove that for any two integers $g\geq 4$ and $g'\leq 2g-1$, there exist abelian varieties over $\overline{\mathbb{Q}}$ which are not quotients of a Jacobian of dimension $g'$. Our method in fact proves that most Abelian varieties satisfy this property, when counting by height relative to a fixed finite map to projective space.
We prove, following Deligne and André, that the Hodge classes on abelian varieties of CM-type can be expressed in terms of divisor classes and split Weil classes, and we describe some consequences. In particular, we show that Grothendieck's standard conjecture of Lefschetz type implies the Hodge conjecture for abelian varieties (Abdulali, André, ...).
We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and $A$-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and $A$-motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication.
Recent developments on the uniformity of the number of rational points on curves and subvarieties in a moving abelian variety rely on the geometric concept of the degeneracy locus. The first-named author investigated the degeneracy locus in certain mixed Shimura varieties. In this expository note we revisit some of these results while minimizing the use of mixed Shimura varieties while working in a family of principally polarized abelian varieties. We also explain their relevance for applications in diophantine geometry.
In this paper, we define and construct canonical filtered $F$-crystals with $G$-structure over the integral models for Shimura varieties of abelian type at hyperspecial level defined by Kisin. We check that these are related by $p$-adic comparison theorems to the usual lisse sheaves, and as an application we also use this to show that the Galois representations generated from the $p$-adic étale cohomology of Shimura varieties with nontrivial coefficient sheaves are crystalline, at least in the case of proper abelian type Shimura varieties.
We count the number of isomorphism classes of degree $d$-twists of some polarized abelian varieties over finite fields of odd prime dimension. This can be seen as a higher dimensional analogue of the counting problem for elliptic curves case.
In this paper we prove the mixed Ax-Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular we present an application to studying the generic rank of the Betti map.
We prove a non-archimedean analogue of the fact that a closed subvariety of a semi-abelian variety is hyperbolic modulo its special locus, and thereby generalize a result of Cherry.
We give a uniform bound on the degree of the maximal torsion cosets for subvarieties of an abelian variety. The proof combines algebraic interpolation and a theorem of Serre on homotheties in the Galois representation associated to the torsion subgroup of an abelian variety.
We prove that the automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fields are cyclic, and give a complete list of finite groups that can be realized as such automorphism groups.
We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field $k$. We also compute the Grothendieck group of the category of $A$-isotypic abelian varieties, for any simple abelian variety $A$, assuming $k$ has characteristic 0, and for any elliptic curve $A$ in any characteristic.
In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.
We study probability measures corresponding to families of abelian varieties over a finite field. These measures play an important role in the Tsfasman- Vladuts theory of asymptotic zeta-functions defining completely the limit zeta-function of the family. J.-P.Serre, using results of R.M.Robinson on conjugate algebraic integers, described the possible set of measures than can correspond to families of abelian varieties over a finite field. The problem whether all such measures actually occur was left open. Moreover, Serre supposed that not all such measures correspond to abelian varieties (for…
We give a formula for the class number of an arbitrary CM algebraic torus over $\mathbb{Q}$. This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.
In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasi-projective varieties. Then, in terms of Prym varieties associated to the cyclic covers, we prove a structure theorem on their Mordell-Weil group. Our results give an explicit method for construction of elliptic curves, hyper- and super-elliptic Jacobians that have large ranks over function fields of certain varieties.
We provide a characterization of almost ordinary abelian varieties over finite fields, and use this characterization to provide lower bounds for the sizes of some almost ordinary isogeny classes.
We study a certain class of simple abelian varieties of type $\mathrm{IV}$ (in Albert's classification) over number fields with Mumford-Tate groups of type $A$. In particular, we show that such abelian varieties have ordinary reduction away from a set of places of Dirichlet density zero, thus confirming a special case of a broader conjecture of Serre's. We also study the splitting types and Newton polygons of the reductions of the abelian varieties of this type with small dimension (nine).
We prove that Shimura varieties of abelian type with infinite level at $p$ are perfectoid. As a corollary, the moduli spaces of polarized K3 surfaces with infinite level at $p$ are also perfectoid.
