Papers by Grzegorz Banaszak
arXiv (Cornell University), Sep 4, 2012
In this paper we introduce new definition of Hodge structures and show that R-Hodge structures ar... more In this paper we introduce new definition of Hodge structures and show that R-Hodge structures are determined by R-linear operators that are annihilated by the Weierstrass σ-function 2010 Mathematics Subject Classification. 14D07.
arXiv (Cornell University), Apr 18, 2022
In this paper we give a complete characterization of the component group of the Sato-Tate group o... more In this paper we give a complete characterization of the component group of the Sato-Tate group of an abelian variety A of arbitrary dimension, defined over a number field K, in terms of the connectedness of the Lefschetz group associated to A.
EMS Press eBooks, Jul 14, 2004
In this paper we investigate the image of the l-adic representation attached to the Tate module o... more In this paper we investigate the image of the l-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre.
Acta Arithmetica, 2019
Using results of Greither-Popescu [19] on the Brumer-Stark conjecture we construct l-adic imprimi... more Using results of Greither-Popescu [19] on the Brumer-Stark conjecture we construct l-adic imprimitive versions of these characters, for primes l > 2. Further, the special values of these l-adic Hecke characters are used to construct G(F/K)-equivariant Stickelberger-splitting maps in the Quillen localization sequence for F , extending the results obtained in [1] for K = Q. We also apply the Stickelberger-splitting maps to construct special elements in K 2n (F) l and analyze the Galois module structure of the group D(n) l of divisible elements in K 2n (F) l. If n is odd, l n, and F = K is a fairly general totally real number field, we study the cyclicity of D(n) l in relation to the classical conjecture of Iwasawa on class groups of cyclotomic fields and its potential generalization to a wider class of number fields. Finally, if F is CM, special values of our l-adic Hecke characters are used to construct Euler systems in odd K-groups K 2n+1 (F, Z/l k). These are vast generalizations of Kolyvagin's Euler system of Gauss sums [33] and of the K-theoretic Euler systems constructed in [4] when K = Q.
Contemporary Mathematics, 2016
We make explicit Serre's generalization of the Sato-Tate conjecture for motives, by expressing th... more We make explicit Serre's generalization of the Sato-Tate conjecture for motives, by expressing the construction in terms of fiber functors from the motivic category of absolute Hodge cycles into a suitable category of Hodge structures of odd weight. This extends the case of abelian varietes, which we treated in a previous paper. That description was used by Fité-Kedlaya-Rotger-Sutherland to classify Sato-Tate groups of abelian surfaces; the present description is used by Fité-Kedlaya-Sutherland to make a similar classification for certain motives of weight 3. We also give conditions under which verification of the Sato-Tate conjecture reduces to the identity connected component of the corresponding Sato-Tate group.
On Galois cohomology of some 𝑝-adic representations and étale 𝐾-theory of curves
Contemporary Mathematics, 1999
The Stickelberger splitting map in the case of abelian extensions $F / \Q$ was defined in [Ba1, C... more The Stickelberger splitting map in the case of abelian extensions $F / \Q$ was defined in [Ba1, Chap. IV]. The construction used Stickelebrger's theorem. For abelian extensions $F / K$ with an arbitrary totally real base field $K$ the construction of \cite{Ba1} cannot be generalized since Brumer's conjecture (the analogue of Stickelberger's theorem) is not proved yet at that level
Indiana University Mathematics Journal, 2015
We make explicit a construction of Serre giving a definition of an algebraic Sato-Tate group asso... more We make explicit a construction of Serre giving a definition of an algebraic Sato-Tate group associated to an abelian variety over a number field, which is conjecturally linked to the distribution of normalized L-factors as in the usual Sato-Tate conjecture for elliptic curves. The connected part of the algebraic Sato-Tate group is closely related to the Mumford-Tate group, but the group of components carries additional arithmetic information. We then check that in many cases where the Mumford-Tate group is completely determined by the endomorphisms of the abelian variety, the algebraic Sato-Tate group can also be described explicitly in terms of endomorphisms. In particular, we cover all abelian varieties (not necessarily absolutely simple) of dimension at most 3; this result figures prominently in the analysis of Sato-Tate groups for abelian surfaces given recently by Fité, Kedlaya, Rotger, and Sutherland.
ContinuousK-theory
K-Theory, 1995
We study arithmetical properties of homotopy groups of thel-adic completion of Quillen&#3... more We study arithmetical properties of homotopy groups of thel-adic completion of Quillen'sK-theory space of number field, with a view on the Dwyer-Friedlander comparison map into étaleK-theory. The relation of these groups toK-theory is a complete analogy to the relation of continuous étale cohomology to étale cohomology. We identify the torsion subgroup of the resulting <img src="/fulltext-image.asp?format=htmlnonpaginated&src=GM254772G10Q7632_html\10977_2004_Article_BF00961470_TeX2GIFIE1.gif" border="0" alt=" $$\mathop {\underleftarrow
Journal of Pure and Applied Algebra, 1997
In this paper we prove that K-groups of the henselization of some local rings imbed into K-groups... more In this paper we prove that K-groups of the henselization of some local rings imbed into K-groups of the completion of these rings. One of the main tools we use is the Artin Approximation Theorem.
Journal of Pure and Applied Algebra, 2007
For a pair of rings A ⊂ B satisfying certain condition () we get general imbedding results for Ka... more For a pair of rings A ⊂ B satisfying certain condition () we get general imbedding results for Karoubi-Villamayor, Homotopy and Quillen K-theories. As a one of corollaries, namely Corollary 4, we get the generalization of the paper [BZ].
Journal of Number Theory, 1996
Euler systems introduced by A.Kolyvagin and K.Rubin may be used to construct interesting systems ... more Euler systems introduced by A.Kolyvagin and K.Rubin may be used to construct interesting systems of elements in algebraic K-theory. Let F/Q be an abelian field extension. We fix an odd prime p and a natural number m. Let S be the set of square free natural numbers
Journal of Number Theory, 2005
We consider the local to global principle for detecting linear dependence of points in groups of ... more We consider the local to global principle for detecting linear dependence of points in groups of the Mordell-Weil type. As applications of our general setting we obtain corresponding statements for Mordell-Weil groups of non-CM elliptic curves and some higher dimensional abelian varieties defined over number fields, and also for odd dimensional K-groups of number fields.
Journal of Number Theory, 2003
We consider the support problem of Erdös in the context of l-adic representations of the absolute... more We consider the support problem of Erdös in the context of l-adic representations of the absolute Galois group of a number field. Main applications of the results of the paper concern Galois cohomology of the Tate module of abelian varieties with real and complex multiplications, the algebraic K-theory groups of number fields and the integral homology of the general linear group of rings of integers. We answer the question of Corrales-Rodrigáñez and Schoof concerning the support problem for higher dimensional abelian varieties.
Journal of Number Theory, 2013
In this paper we study the divisibility and the wild kernels in algebraic K-theory of global fiel... more In this paper we study the divisibility and the wild kernels in algebraic K-theory of global fields F. We extend the notion of the wild kernel to all K-groups of global fields and prove that Quillen-Lichtenbaum conjecture for F is equivalent to the equality of wild kernels with corresponding groups of divisible elements in K-groups of F. We show that there exist generalized Moore exact sequences for even K-groups of global fields. Without appealing to the Quillen-Lichtenbaum conjecture we show that the group of divisible elements is isomorphic to the corresponding group of étale divisible elements and we apply this result for the proof of the lim 1 analogue of Quillen-Lichtenbaum conjecture. We also apply this isomorphism to investigate: the imbedding obstructions in homology of GL, the splitting obstructions for the Quillen localization sequence, the order of the group of divisible elements via special values of ζ F (s). Using the motivic cohomology results due to Bloch, Friedlander, Levine, Lichtenbaum, Morel, Rost, Suslin, Voevodsky and Weibel, which established the Quillen-Lichtenbaum conjecture, we conclude that wild kernels are equal to corresponding groups of divisible elements.
Journal of Number Theory, 2000
In this paper we de ne 2-adic cyclotomic elements in K-theory and etale cohomology of the integer... more In this paper we de ne 2-adic cyclotomic elements in K-theory and etale cohomology of the integers. We construct a comparison map which sends the 2-adic elements in K-theory onto 2-adic elements in cohomology. We also compute explicitly some of the product maps in K-theory of Z at the prime 2: 1. 2-adic cyclotomic elements in K 2n?1 (Z 1 2 ]) Z2 De nition 1.1. For a commutative ring with identity R we denote by K i (R; Z2) = lim ? K i (R; Z=2 m)
Journal of Number Theory, 2013
For a CM abelian extension F/K of an arbitrary totally real number field K, we construct the Stic... more For a CM abelian extension F/K of an arbitrary totally real number field K, we construct the Stickelberger splitting maps (in the sense of [1]) for both theétale and the Quillen K-theory of F and we use these maps to construct Euler systems in the even Quillen K-theory of F. The Stickelberger splitting maps give an immediate proof of the annihilation of the groups of divisible elements divK 2n (F) l of the even K-theory of the top field by higher Stickelberger elements, for all odd primes l. This generalizes the results of [1], which only deals with CM abelian extensions of Q. The techniques involved in constructing our Euler systems at this level of generality are quite different from those used in [3], where an Euler system in the odd K-theory with finite coefficients of abelian CM extensions of Q was given. We work under the assumption that the Iwasawa µ-invariant conjecture holds. This permits us to make use of the recent results of Greither-Popescu [16] on thé etale Coates-Sinnott conjecture for arbitrary abelian extensions of totally real number fields, which are conditional upon this assumption. In upcoming work, we will use the Euler systems constructed in this paper to obtain information on the groups of divisible elements divK 2n (F) l , for all n > 0 and odd l. It is known that the structure of these groups is intimately related to some of the deepest unsolved problems in algebraic number theory, e.g. the Kummer-Vandiver and Iwasawa conjectures on class groups of cyclotomic fields. We make these connections explicit in the introduction.
k-invariants for K-theory of curves over global fields
Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2009
We investigate the k-invariants for the K-theory and étale K-theory spaces of schemes. We give nu... more We investigate the k-invariants for the K-theory and étale K-theory spaces of schemes. We give numerical estimates of the orders of k-invariants for the K-theory and étale K-theory spaces of regular and proper models over of smooth, proper and geometrically irreducible curves defined over global fields F.
On the non-torsion elements in the algebraic K-theory of rings of integers
Journal für die reine und angewandte Mathematik (Crelles Journal), 1995
Homology, Homotopy and Applications, 2005
In this paper we investigate reduction of nontorsion elements in theétale K-theory of a curve X o... more In this paper we investigate reduction of nontorsion elements in theétale K-theory of a curve X over a global field F. We show that the reduction map can be well understood in terms of Galois cohomology of l-adic representations.
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Papers by Grzegorz Banaszak