 # Hasse-Weil Zeta Functions for Linear Algebraic Groups

Turner, S. M (1996) Hasse-Weil Zeta Functions for Linear Algebraic Groups. PhD thesis, University of Glasgow.

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## Abstract

The aim of the project was to study two questions [unpublished] of R.W.K. Odoni, to discuss which we introduce the following terminology and notation. Let G be a linear algebraic group defined over an algebraic number field K. We will say that G has property (Z) if there exists a finite Galois extension l of K such that the Hasse-Weil zeta function zeta(G,K,s) of G is an alternating product of Artin L-functions for characters of Gal(l/K). Odoni's questions can then be formulated as follows. (Q1) Which G have property (Z) (and for which Galois extensions I of K)? (Q2) For which G does zeta(G, K, s) have a functional equation? While neither question was settled completely, the following progress was made. Main Result (A) If G is connected and solvable, it has property (Z) [6.2.2.1], (B) For each K-group G, there is a finite extension M of K such that the M-group G has property (Z) with Gal[l/M) trivial, and zeta(G,M,s) has a functional equation [6.3]. (C) If every connected almost K-simple K-group had property (Z), then all connected K-groups would too [6.2.2.1]. Among the former, those for which the Dynkin diagram has at most two components all have property (Z) [7.1.0.1]. (D) In particular, every almost simple K-group J has property (Z). Further, zeta(J, f, s) has a functional equation for an extension f of K of degree at most 2 [6.4]. In all cases, explicit expressions are given or can be easily reconstructed. Part (A) has almost certainly been known for a long time, though no statement of it was found. That there should seem to be so little in the literature about zeta functions for algebraic groups is quite surprising. The case in which G is a torus is dealt with in [Se59]; the connected solvable case (A) follows readily from this. In chapter (1) the necessary absolute algebraic geometry is introduced, until a definition of the notion of complete variety can be given [1.12]. In chapter (2), an account of the relative theory appears. While all of this material has been known for a long time, there is a paucity of convenient reference. The points of view of topology and algebra (especially Galois theory) are considered. The Weil restriction functor [2.8] is a key concept. Chapter (3) is an exposition of the standard theory of (linear) algebraic groups as far as that of connected groups [3.4]. In chapter (4) is expounded the theory of root systems [4.1] and that of connected reductive groups. Rationality questions are then treated, especially for connected semisimple groups. In chapter (5) will be found a resume of the algebraic number theory [5.1] and notions of zeta function [5.2] required. The heart of the chapter [5.3] is the notion of reduction (of a variety) modulo a prime ideal. In [5.4] will be found some results relating to preservation of properties under reduction modulo a prime. In the cases of some of these properties, an assertion of the result (though not a proof) was found in the literature. In other cases the result may be new. In chapters (6) and (7) the results announced above are obtained. In (6) will be found the proofs of parts (A), (B) and (D). The proof of (C) is deferred to (7) to avoid a very long chapter (6) which there seems no natural place to break.

Item Type: Thesis (PhD) Doctoral Mathematics 1996 Enlighten Team glathesis:1996-76474 Copyright of this thesis is held by the author. 19 Nov 2019 14:17 19 Nov 2019 14:17 http://theses.gla.ac.uk/id/eprint/76474 View Item