Grzegorz Banaszak

Support problem for the intermediate Jacobians of l-adic representations

Abstract We consider the support problem of Erdos 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-Rodriganez and Schoof concerning the support problem for higher dimensional abelian varieties.

On Galois representations for abelian varieties with complex and real multiplications

Abstract In this paper, we study the image of l-adic representations coming from Tate module of an abelian variety defined over a number field. We treat abelian varieties with complex and real multiplications. We verify the Mumford–Tate conjecture for a new class of abelian varieties with real multiplication.

Galois Actions on the Eigenproblem of the Heisenberg Heptagon

We analyse the exact solution of the eigenproblem for the Heisenberg Hamiltonian of magnetic heptagon, i.e. the ring of N=7 nodes, each with spin 1/2, within the XXX model with nearest neighbour interactions, from the point of view of finite extensions of the field

A support problem for K -theory of number fields☆

\mathbb{Q}

Galois Symmetries of Bethe Parameters for the Heisenberg Pentagon

of rationals. We point out, as the main result, that the associated arithmetic structure of these extensions makes natural an introduction of some Galois qubits. They are two-dimensional subspaces of the Hilbert space of the model, which admit a quantum informatic interpretation as elementary memory units for a (hypothetical) computer, based on their distinctive properties with respect to the action of related Galois group for indecomposable factors of the secular determinant. These Galois qubits are nested on the lattice of subfields which involves several minimal fields for determination of eigenstates (the complex Heisenberg field), spectrum (the real Heisenberg field), and Fourier transforms of magnetic configurations (the cyclotomic field, based on the simple 7th root of unity). The structure of the corresponding lattice of Galois groups is presented in terms of Kummer theory, and its physical interpretation is indicated in terms of appropriate permutations of eigenstates, energies, and density matrices.

A Note on the Quillen–Lichtenbaum Conjecture and the Arithmetic of Square Rings

In this paper we consider a support problem for the reduction map in the odd K -theory of number fields.

On the map between K-groups of henselization and completion of some local rings

In this paper the field generated by the Bethe parameters related to the XXX model for the Heisenberg pentagon is considered. For the interior of the Brillouin zone, the Galois group of the Bethe number field over the rationals is determined. This Galois group is recognized as the group of arithmetic symmetries of the Bethe parameters.

Arithmetic of Partial Fibres in Relative Position Space of Bethe Ansatz

v W(Fv) → W (F ) → 0, where v runs over finite places of F. In [B] the group K 2n(O)l was called the l-part of the wild kernel for K2n(F ). We expect that the groups WK2n(F )l and K w 2n(O)l coincide. It is unknown in the moment, since we do not know the K-theory with coefficients in Z/l of a local field in the case when l is divisible by the residue characteristic, except for the case of Ql which follows by results of Bökstedt and Madsen [BM].

Hodge structures in topological quantum mechanics

Abstract 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.

On the map between homology of henselization and completion of some local rings

We consider fibration of the relative position space for the system of r Bethe pseudoparticles on a ring of N nodes. We define primary and secondary configurations in the relative position space. We study cardinalities of the set of primary configurations and of the partial fibres of the fibration.