Xuejun Guo

The K 1 Group of Tiled Orders

In this article, we study the K 1 of tiled rings over local rings. Our results generalize some existing ones obtained by Keating and Xi.

HIGHER CLASS GROUPS OF GENERALIZED EICHLER ORDERS

ABSTRACT In this paper we study the possible torsion in even-dimensional higher class groups Cl 2n (Λ)(n ≥ 1) of an order Λ in a semisimple algebra A over a number field F with a ring of integers 𝒪 F . We show that for certain orders, called generalized Eichler orders, bip-torsion in Cl 2n (Λ) can only occur for primes p dividing prime ideals ℘ of 𝒪 F , at which Λ is not maximal. In particular, the results apply to Eichler orders in quaternion algebras and to hereditary orders.

Higher Class Groups of Locally Triangular Orders over Number Fields

In this paper, we study the K-theory of triangular rings. As an application, we show that for a locally triangular order Λ, the p-torsion in the higher class group Cl2n(Λ) can only occur for primes p which lie under the prime ideals ℘ of

Computing the Tame Kernel of ℚ(ζ8)

{\cal O}_F

ON K 2 OF DIVISION ALGEBRAS

, at which Λ is not maximal.

Imaginary Quadratic Fields with Ono Number 3

Abstract In this paper, it is proved that the tame kernel of ℚ(ζ8) is trivial.

An embedding theorem for Eichler orders

ABSTRACT In this paper, it is proved that if F is a global field, then for any integer n > 3, there is an extension field E over F of degree n such that K 2 E is not generated by the Steinberg symbols {a, b} with a ∈ F*, b, ∈ E*. If however, F is a number field and D is a finite-dimensional central division F-algebra with square free index, then K 2 D is always generated by the Steinberg symbols {a, b} with a ∈ F*, b ∈ D*. Finally, the tame kernels of central division algebras over F are expressed explicitly.

On the representation numbers of ternary quadratic forms and modular forms of weight 3/2☆

In this article, we prove that an imaginary quadratic field F has the ideal class group isomorphic to ℤ/2ℤ ⊕ ℤ/2ℤ if and only if the Ono number of F is 3 and F has exactly 3 ramified primes under the Extended Riemann Hypothesis (ERH). In addition, we give the list of all imaginary quadratic fields with Ono number 3.