Abelian Type Research Articles

Weights and conservativity

The purpose of this article is to study conservativity in the context of triangulated categories equipped with a weight structure. As application, we establish (weight) conservativity for the restriction of the (generic) l-adic realization to the category of motives of Abelian type of characteristic zero.

Chow motives of abelian type over a base

Let S be a Noetherian scheme, let X be a smooth projective scheme over S, whose fibres are connected curves of genus g, and let J be the Jacobian scheme of the relative curve X over S. We generalise the theorem due to Rolph Schwarzenberger and prove that if S is integral and normal, and the structural morphism admits a section, then there exists a locally free sheaf on J, such that the relative symmetric power is isomorphic to the projective bundle over J, provided , and the ample divisor is mathrm{{Sym}}^{d-1}(X/S), embedded into by the section of the structural morphism from X to S. Then we use this result to generalise the theorem due to Shun-Ichi Kimura: if S is an integral regular scheme, separated and of finite type over a Dedekind domain, then all relative Chow motives of abelian type over S are finite-dimensional.

Tauberian theorems for the Stockwell transform of Lizorkin distributions

ABSTRACTWe prove the continuity of the Stockwell transform and the corresponding synthesis operator on the spaces of highly localized test functions over and , respectively. Using the obtained continuity results, we define and study the Stockwell transform on the space of Lizorkin distributions . We also provide Abelian and Tauberian type results relating the asymptotic behavior of distributions with the asymptotics of their Stockwell transform.

The solution space of the Einstein’s vacuum field equations for the case of five-dimensional Bianchi Type I (Type 4A1)

We consider the 4+1 Einstein’s field equations (EFE’s) in vacuum, simplified by the assumption that there is a 4D sub-manifold on which an isometry group of dimension four acts simply transitive. In particular, we consider the Abelian group Type 4A1; and thus the emerging homogeneous sub-space is flat. Through the use of coordinate transformations that preserve the sub-manifold’s manifest homogeneity, a coordinate system is chosen in which the shift vector is zero. The resulting equations remain form invariant under the action of the constant Automorphisms group. This group is used in order to simplify the equations and obtain their complete solution space which consists of seven families corresponding to 21 distinct solutions. Apart form the Kasner type all the other solutions found are, to the best of our knowledge, new. Some of them correspond to cosmological solutions, others seem to depend on some spatial coordinate and there are also pp-wave solutions.

Simple finite-dimensional right-alternative unital superalgebras with associative-commutative even part over a field of characteristic zero

We study simple finite-dimensional right-alternative unital superalgebras over a field of characteristic zero. We prove that if the even part of a superalgebra is associative and commutative, then the superalgebra is of Abelian type. The classification of such superalgebras is known.

Integral models of Shimura varieties with parahoric level structure

For a prime $p > 2$ , we construct integral models over $p$ for Shimura varieties with parahoric level structure, attached to Shimura data $(G,X)$ of abelian type, such that $G$ splits over a tamely ramified extension of ${\mathbf {Q}}_{\,p}$ . The local structure of these integral models is related to certain “local models”, which are defined group theoretically. Under some additional assumptions, we show that these integral models satisfy a conjecture of Kottwitz which gives an explicit description for the trace of Frobenius action on their sheaf of nearby cycles.

Annihilator ideals of two-generated metabelian p-groups

For certain metabelian [Formula: see text]-groups [Formula: see text] with two generators [Formula: see text] and [Formula: see text], the annihilator [Formula: see text] of the main commutator [Formula: see text] of [Formula: see text], as an ideal of bivariate polynomials with integer coefficients, is determined by means of a presentation for [Formula: see text]. It is proved that together with Schreier’s polynomials [Formula: see text], the annihilator [Formula: see text] identifies the group [Formula: see text] uniquely, and the Furtwängler isomorphism of the additive group underlying the residue class ring [Formula: see text] to the commutator subgroup [Formula: see text] of [Formula: see text] admits the calculation of the abelian type invariants of [Formula: see text]. The results are underpinned by class field theoretic realizations of the groups [Formula: see text] as Galois groups [Formula: see text] of second Hilbert [Formula: see text]-class fields [Formula: see text] over algebraic number fields [Formula: see text].

Abelian spiders and real cyclotomic integers

If Γ \Gamma is a finite graph, then the largest eigenvalue λ \lambda of the adjacency matrix of Γ \Gamma is a totally real algebraic integer ( λ \lambda is the Perron-Frobenius eigenvalue of Γ \Gamma ). We say that Γ \Gamma is abelian if the field generated by λ 2 \lambda ^2 is abelian. Given a fixed graph Γ \Gamma and a fixed set of vertices of Γ \Gamma , we define a spider graph to be a graph obtained by attaching to each of the chosen vertices of Γ \Gamma some 2 2 -valent trees of finite length. The main result is that only finitely many of the corresponding spider graphs are both abelian and not Dynkin diagrams, and that all such spiders can be effectively enumerated; this generalizes a previous result of Calegari, Morrison, and Snyder. The main theorem has applications to the classification of finite index subfactors. We also prove that the set of Salem numbers of “abelian type” is discrete.

Complex conjugation and Shimura varieties

In this paper we study the action of complex conjugation on Shimura varieties and the problem of descending these to the maximal totally real field of the reflex field. We prove the existence of such descent for many Shimura varieties whose associated adjoint group has certain factors of type A or D. This includes a large family of Shimura varieties of abelian type. Our considerations and constructions are carried out purely at the level of Shimura data and group theory.

Integral canonical models for automorphic vector bundles of abelian type

We define and construct integral canonical models for automorphic vector bundles over Shimura varieties of abelian type. More precisely, we first build on Kisin's work to construct integral canonical models over rings of integers of number fields with finitely many primes inverted for Shimura varieties of abelian type with hyperspecial level at all primes we do not invert, compatible with Kisin's construction. We then define a notion of an integral canonical model for the standard principal bundles lying over Shimura varieties and proceed to construct them in the abelian type case. With these in hand, one immediately also gets integral models for automorphic vector bundles.

Remarks on motives of abelian type

A motive over a field $k$ is of abelian type if it belongs to the thick and rigid subcategory of Chow motives spanned by the motives of abelian varieties over $k$. This paper contains three sections of independent interest. First, we show that a motive which becomes of abelian type after a base field extension of algebraically closed fields is of abelian type. Given a field extension $K/k$ and a motive $M$ over $k$, we also show that $M$ is finite-dimensional if and only if $M_K$ is finite-dimensional. As a corollary, we obtain Chow–Künneth decompositions for varieties that become isomorphic to an abelian variety after some field extension. Second, let $\varOmega$ be a universal domain containing $k$. We show that Murre's conjectures for motives of abelian type over $k$ reduce to Murre's conjecture (D) for products of curves over $\varOmega$. In particular, we show that Murre's conjecture (D) for products of curves over $\varOmega$ implies Beauville's vanishing conjecture on abelian varieties over $k$. Finally, we give criteria on Chow groups for a motive to be of abelian type. For instance, we show that $M$ is of abelian type if and only if the total Chow group of algebraically trivial cycles $\mathrm{CH}_*(M_\varOmega)_\mathrm{alg}$ is spanned, via the action of correspondences, by the Chow groups of products of curves. We also show that a morphism of motives $f: N \rightarrow M$, with $N$ finite-dimensional, which induces a surjection $f_* : \mathrm{CH}_*(N_\varOmega)_\mathrm{alg} \rightarrow \mathrm{CH}_*(M_\varOmega)_\mathrm{alg}$ also induces a surjection $f_* : \mathrm{CH}_*(N_\varOmega)_\mathrm{hom} \rightarrow \mathrm{CH}_*(M_\varOmega)_\mathrm{hom}$ on homologically trivial cycles.

On the second 2-class group [formula omitted] of some imaginary quartic cyclic number field K

Let H8 denote the quaternion group of order 8 and put G=Z/2Z×H8. Let K be some imaginary quartic cyclic number field whose 2-class group is of type (2,2,2). In this paper, we prove in particular that G is realizable over K, i.e., G≃Gal(K2(2)/K) where K2(2) is the second Hilbert 2-class field of K. Then we study the capitulation problem of the 2-ideal classes of K in the fourteen intermediate unramified extensions between K and its first Hilbert 2-class field. Additionally, these fourteen unramified extensions are constructed, and the abelian type invariants of their 2-class groups and the length of the 2-class field tower of K are given.

Simple finite-dimensional right-alternative superalgebras with semisimple strongly associative even part

It is shown that a simple finite-dimensional right-alternative unital superalgebra with semisimple strongly associative even part over a field of characteristic is either non-associative, or a superalgebra of Abelian type. A classification of such superalgebras over an algebraically closed field is given. Bibliography: 21 titles.

Mod $p$ points on Shimura varieties of abelian type

We show that the mod p points on a Shimura variety of abelian type with hyperspecial level, have the form predicted by the conjectures of Kottwitz and Langlands-Rapoport. Along the way we show that the isogeny class of a mod p point contains the reduction of a special point.

Successive Approximation of p-Class Towers

Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are determined by abelian type invariants of p-class groups C1pE of unramified extensions E/F with degree [E : F] = pn-1. Illustrated by the most extensive numerical results available currently, the transfer kernels (TE, F) of the p-class extensions TE, F : C1pF → C1pE from F to unramified cyclic degree-p extensions E/F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups S G of all 3-groups G with coclass cc(G) = 1, and establishing a general theorem on the connection between the p-class towers of a number field F and of an unramified abelian p-extension E/F, we are able to provide a theoretical proof of the realization of certain 3-groups S with maximal class by 3-tower groups of dihedral fields E with degree 6, which could not be realized up to now.

Criteria for Three-Stage Towers of <i>p</i>-Class Fields

Let p be a prime and K be a number field with non-trivial p-class group ClpK. A crucial step in identifying the Galois group G∞p of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G2pk, that is the second derived quotient MsG/Gn. The family τ1K of abelian type invariants of the p-class groups ClpL of all unramified cyclic extensions L/K of degree p is called the index- abelianization data (IPAD) of K. It is able to specify a finite batch of contestants for the second p-class group M of K. In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD (τ1Kτ(2)K) includes data on unramified abelian extensions L/K of degree p2 and enables sharper bounds for the order of M in the case Clpks(p,p,p), where current im-plementations of the p-group generation algorithm fail to produce explicit contestants for M , due to memory limitations. The iterated IPAD of second order τ(2)K contains information on non-abelian unramified extensions L/K of degree p2, or even p3, and admits the identification of the p-class tower group G for various infinite series of quadratic fields K=Q(√d) with ClpKs(p,p) possessing a p-class field tower of exact length lpK=3 as a striking novelty.

Simple right alternative superalgebras of Abelian type whose even part is a field

We study central simple unital right alternative superalgebras of Abelian type of arbitrary dimension whose even part is a field. We prove that every such superalgebra , except for the superalgebra , is a double, that is, the odd part can be represented in the form for a suitable . If the generating element commutes with the even part , then is isomorphic to a twisted superalgebra of vector type introduced by Shestakov [1], [2]. But if commutes with the odd part , then is isomorphic to a superalgebra introduced in [3] and called an -double. We prove that if the ground field is algebraically closed, then is isomorphic to one of the superalgebras , , .

Galois-generic points on Shimura varieties

We discuss existence and abundance of Galois-generic points for adelic representations attached to Shimura varieties. First, we show that, for Shimura varieties of abelian type, l-Galois-generic points are Galois-generic; in particular, adelic representations attached to such Shimura varieties admit (“lots of”) closed Galois-generic points. Next, we investigate further the distribution of Galois-generic points and show the Andre–Pink conjecture for them: if S is a connected Shimura variety associated to a Q-simple reductive group, then every infinite subset of the generalized Hecke orbit of a Galois-generic point is Zariski-dense in S. Our proof follows the approach of Pink for Siegel Shimura varieties. Our main contribution consists in showing that there are only finitely many Hecke operators of bounded degree on (adelic and connected) Shimura varieties. Compared with other approaches of this result, our proof, which relies on Bruhat–Tits theory, is effective and works for arbitrary Shimura varieties.

Intermediate extension of Chow motives of Abelian type

In this article, we give an unconditional construction of a motivic analogue of the intermediate extension in the context of Chow motives of Abelian type. Our main application concerns intermediate extensions of Chow motives associated to Kuga families to the Baily–Borel compactification of a Shimura variety.

Perfectoid Shimura Varieties of Abelian Type

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.