Morrow, Matthew Thomas (2009) *Investigations in two-dimensional arithmetic geometry.* PhD thesis, University of Nottingham.

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## Abstract

This thesis explores a variety of topics in two-dimensional arithmetic geometry, including the further development of I. Fesenko's adelic analysis and its relations with ramification theory, model-theoretic integration on valued fields, and Grothendieck duality on arithmetic surfaces.

I. Fesenko's theories of integration and harmonic analysis for higher dimensional local fields are extended to an arbitrary valuation field F whose residue field is a local field; applications to local zeta integrals are considered.

The integral is extended to F^n, where a linear change of variables formula is proved, yielding a translation-invariant integral on GL_n(F).

Non-linear changes of variables and Fubini's theorem are then examined. An interesting example is presented in which imperfectness of a positive characteristic local field causes Fubini's theorem to unexpectedly fail.

It is explained how the motivic integration theory of E. Hrushovski and D. Kazhdan can be modified to provide a model-theoretic approach to integration on two-dimensional local fields. The possible unification of this work with A. Abbes and T. Saito's ramification theory is explored.

Relationships between Fubini's theorem, ramification theory, and Riemann-Hurwitz formulae are established in the setting of curves and surfaces over an algebraically closed field.

A theory of residues for arithmetic surfaces is developed, and the reciprocity law around a point is established. The residue maps are used to explicitly construct the dualising sheaf of the surface.

Item Type: | Thesis (PhD) |
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Supervisors: | Fesenko, I.B. |
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Faculties/Schools: | UK Campuses > Faculty of Science > School of Mathematical Sciences |
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ID Code: | 1016 |
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Deposited By: | Mr Matthew Morrow |
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Deposited On: | 26 Mar 2010 09:49 |
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Last Modified: | 26 Mar 2010 09:49 |
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