Executive Summary : | The study of algebraic groups over local fields K has been extensively studied in the past 60 years. The first big step in this direction was the paper by N.Iwahori and H.Matsumoto [IM]. A little later, this question was taken up by F.Bruhat and J.Tits who considered the greatest possible generalization in their now well-known body of work, called Bruhat-Tits theory, beginning in the late 60’s and extending till 1985 [BT1], [BT2]. Bruhat-Tits theory develops the study of bounded subgroups of the group G (K ) where G is a connected reductive algebraic group under very general assumptions of being quasi-split over K . The main theme is to schematize these bounded groups and this is carried out in their papers in 1971 and 1984. Somewhat later in the mid-nineties, there was an approach to constructing the Bruhat-Tits buildings over higher dimensional local fields made by A. Parshin [Par] with possible motivation from questions in arithmetic. The present proposal has its origins in the work of Bruhat and Tits. In 2015, in a paper by the PI and C.S. Seshadri [BS], a somewhat different geometrical approach was observed in the setting when K = k((z)). This approach arose out of the study of a com- mon generalization of parabolic bundles on the one hand and bundles with structure group G on the other. One assumes for simplicity that G is almost simple, semisimple and simply connected. Later in the paper by the PI [Bal] a generalization of the Bruhat-Tits group schemes arose over two-dimensional regular local rings, unlike the classical situation when the base was always a discrete valuation rings or at best a Prufer ring. It served as a key technical tool to describe principal G-bundles on a family of smooth irreducible curves degenerating to a nodal curve with simple normal crossing singularities. The current project is aimed at a complete generalization of the classical theory of Bruhat-Tits group schemes over a base which is the spectrum of an arbitrary ring of the form A??z1,...,zn?? where A is a discrete valuation ring with possible tameness as- sumptions on the residue field characteristics. It opens several new directions both geometry and arithmetic. In geometry, the usual parahoric or loop groups have been intensely studied in the theory of principal G -bundles on curves. The knowledge of these group schemes entails a study of the stacks of torsors for these group scheme. These will be natural extensions of the theory of parabolic bundles over curves. |