Project A2: p-adic group algebras



Principal investigator

Participating scientists

Summary

The use of p-adic group algebras is a recent development, mostly in representation theory and arithmetic, but also in non-commutative algebra, algebraic topology, and functional analysis. On the one hand we will undertake the further systematic investigation of these algebras for their own sake. On the other hand, by actively pursuing quite diverse applications - notably the p-adic Langlands program, we hope to uncover unexpected interrelations as a fruitful source of new ideas.

During the second half of the last century Langlands has devised a vast program to relate algebraic invariants of arithmetic objects to analytic invariants like L-functions. In its local form over a p-adic field it sets into correspondence (in general still conjecturally) l-adic Galois representations (for some prime number l different from p) and complex representations of arbitrary reductive algebraic groups. Seemingly a technical point, the assumption about the prime number l in fact is a conceptual limitation of this program. But over at least the last decade it has become clear that p-adic Galois representations (i.e., the case l = p) are an indispensable tool for many central problems in number theory (e.g., Wiles' proof of Fermat). The present project therefore has the goal of conceiving an extension of the classical local Langlands program which will incorporate p-adic Galois representations. This encounters totally new phenomena and therefore requires conceptually new strategies. On the other hand Schneider together with Teitelbaum has developed from scratch over the last years the necessary new p-adic representation theory of general reductive groups. Technically this was done through the introduction and investigation of certain p-adic group rings. Breuil and Colmez have achieved a big step in the example of the group GL2(Qp) over the p-adic number field. In a joint paper with Breuil and in forthcoming work with Vigneras Schneider has formulated precise strategies to attack the general goal. This gives a sound basis from where to start the work in this project. Methodologically, the project comprises number theory, p-adic Hodge theory, p-adic analysis and functional analysis, p-adic Lie groups and Lie algebras, non-commutative algebra, and representation theory.
In fact, these very same p-adic group rings miraculously also appear in global number theory. The group G here is a compact p-adic Lie group which is realized as the Galois group of an infinite extension of global number fields. The ring in question can be algebraically viewed as the completed group ring of G which is the projective limit of the algebraic group rings of the finite quotients of G by open normal subgroups. Here the strategy is to view global arithmetic invariants (e. g. Selmer groups) as modules over this ring and to investigate them with methods of (non)commutative algebra. More specifically, the so called non-commutative Iwasawa theory attempts to construct p-adic L-functions of motives over number fields as elements in the algebraic K-group K1 of this completed group ring. Presently this Kgroup has been computed explicitly (by Kato and others) only in very few specific examples of groups G. It is the second goal of this project to develop systematic tools for understanding the algebraic K-theory of completed group rings.
There is another somewhat different class of p-adic group rings c0(G) which we want to study in this project. They are associated with countable discrete groups G and form a p-adic analog of L1-group algebras. In the case of the group G = Zn one obtains the well known Tate algebras from p-adic rigid analysis. Deninger introduced a p-adic entropy for G-actions and related it to a p-adic version of the Fuglede-Kadison determinant on finite von Neumann algebras. In this context it is important to determine the units and the K-theory of c0(G). More generally, the goal is to understand the structure of the rings c0(G). For residually finite groups G it would also be interesting to understand the relation of c0(G) with the completed group ring of the profinite completion of G.

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