Torzewski , Alexander (2018) Regulator constants of integral representations, together with relative motives over Shimura varieties. PhD thesis, University of Warwick.

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Abstract

This thesis is split into three largely independent chapters. The first concerns the representation theory of Zp[G]-lattices. Specifically, we investigate regulator constants, due to Dokchitser-Dokchitser, which are isomorphism invariants of lattices whose extension of scalars to Qp is self-dual. We first show that when G has cyclic Sylow p-subgroups then regulator constants are strong invariants of permutation modules in a way that can be made precise. Our main result is then that, subject to an additional technical hypothesis on G, this can be combined with existing work of Yakovlev to provide an explicit list of accessible invariants which completely determine, up to isomorphism, any Zp[G]-lattice whose extension to Qp is self-dual.

The second chapter is an application of this result in the context of number fields. Given a Galois extension of number fields K=F with Galois group G, the extension of scalars to Zp of the unit group of K modulo its torsion subgroup denes a Zp[G]-lattice. If we assume that G has cyclic Sylow p-subgroups and satisfies the aforementioned hypothesis, then the above result gives a list of invariants which determine the Galois module structure. The main result of this chapter is then that if p divides G at most once, we can explicate these invariants in terms of classical number theoretic objects. For example, in some cases this can be done in terms of capitulation of ideal classes and ramification information.

The final (unrelated) topic concerns relative motives over Shimura varieties. Given a Shimura datum (G; Ӿ) and neat open compact subgroup K ≤ G(Af ), denote the corresponding Shimura variety ShK(G;Ӿ) by S. The canonical construction described by Pink shows how to associate variations of Hodge structure on San to representations of G. It is expected that this should be motivic in nature, i.e. that there is a motive over S for every representation of G whose Hodge realisation is the variation of Hodge structure given by the canonical construction. Using mixed Shimura varieties, we show that this can be done functorially for representations with Hodge type {(-1; 0); (0,-1)} and that this is compatible with change of S. When (G;Ӿ) has a chosen PEL-datum, existing work of Ancona allows us to associate a motive over S to any representation of G. We then give results to show that in some cases this compatible with change of S and independent of the choice of PEL-datum.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Lattice theory, Galois modules (Algebra), Shimura varieties
Official Date:	September 2018
Dates:	Date Event September 2018 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Bartel, Alex
Sponsors:	Engineering and Physical Sciences Research Council
Extent:	v, 153 leaves
Language:	eng

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