Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

Splitting Quadratic Forms Over Integers of Global Fields

Abstract: We are interested in the decomposition of quadratic forms over the Hasse domains of global fields. By a global fleld F we mean either an algebraic number field or an algebraic function field in one variable over a finite constant field. Throughout assume that the characteristic of F is not 2. A Hasse domain o of F is a Dedekind domain which can be obtained as the intersection of almost all valuation rings on F. So o is the ring of integers corresponding to a set S of almost all the discrete spots on F. This is the accepted generalization of the situation in which F is the field Q of rational numbers and o is the ring Z of rational integers, or where F is an algebraic number field and o is the usual ring of integers contained therein. See O'Meara 2 [6], p. 79, and [5], Chapter X, or Weiss [9], pp. 189 ff. Now consider a regular quadratic space V over the global field F. With o a Hasse domain in F, we consider an o-lattice L on V; that is, L is a finitely generated o-module which spans V. We ask the following question: Is there some number no, depending only on o, such that L has a non-trivial orthogonal splitting L L,1 IL2 whenever rank L nO? This problem has been investigated in the case in which F== Q with Hasse domain o =Z. In fact, Erd6s and Ko [2] proved that in this situation for every integer n > 5 there exists an indecomposable definite quadratic form of rank n. This means that there does not exist a number no in the definite case! But the indefinite case was later investigated by G. L. Watson in his book Integral Quadratic Forms [8], where he proved that every indefinite Z-form of rank n ? 12 splits non-trivially, and in fact that with a slight restriction on the behavior at the archimedean spot, splitting occurs when n ? 8. Watson goes on to remark that "the general problem of deconlposition has not received the attention it deserves." It is our purpose in this

Free Fractions: An Invitation to (applied) Free Fields

Abstract: Long before we learn to construct the field of rational numbers (out of the ring of integers) at university, we learn how to calculate with fractions at school. When it comes to "numbers", we are used to a commutative multiplication, for example 2*3=6=3*2. On the other hand --even before we can write-- we learn to talk (in a language) using words, consisting of purely non-commuting "letters" (or symbols), for example "xy" is not equal to "yx" (with the concatenation as multiplication). Now, if we combine numbers (from a field) with words (from the free monoid of an alphabet) we get non-commutative polynomials which form a ring (with "natural" addition and multiplication), namely the free associative algebra. Adding or multiplying polynomials is easy, for example (2/3*xy+z)+1/3*xy=xy+z or 2*x(yx+3*z)=2*xyx+6*xz. Although the integers and the non-commutative polynomials look rather different, they share many properties, for example the unique number of irreducible factors: x(1-yx)=x-xyx=(1-xy)x. However, the construction of the universal field of fractions (aka "free field") of the free associative algebra is highly non-trivial (but really beautiful). Therefore we provide techniques (building on the work of Cohn and Reutenauer) to calculate with free fractions (representing elements in the free field or "skew field of non-commutative rational functions") to be able to explore a fascinating non-commutative world.

Integrality in the Steinberg module and the top-dimensional cohomology of SLn OK

Abstract: We prove a new structural result for the spherical Tits building attached to GLnK for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module Stn(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this integrality by proving that the complex of partial bases of O is Cohen–Macaulay. We apply this to prove new vanishing and nonvanishing results for H(GLnOK ;Q), where OK is the ring of integers in a number field and νn is the virtual cohomological dimension of GLnOK . The (non)vanishing depends on the (non)triviality of the class group of OK . We also obtain a vanishing theorem for the cohomology H(GLnOK ;V ) with twisted coefficients V . The same results hold for SLnOK as well.

On some geometric representations of GL(n,O)

Abstract: We study a family of complex representations of the group GL(n,O), where O is the ring of integers of a non-archimedean local field F. These representations occur in the restriction of the Grassmann representation of GL(n,F) to its maximal compact subgroup GL(n,O). We compute explicitly the transition matrix between a geometric basis of the Hecke algebra associated with the representation and an algebraic basis which consists of its minimal idempotents. The transition matrix involves combinatorial invariants of lattices of submodules of finite O-modules. The idempotents are p-adic analogs of the multivariable Jacobi polynomials.