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The zeta function of [sfr ][lfr ]2 and resolution of singularities

Published online by Cambridge University Press:  31 January 2002

MARCUS DU SAUTOY  and
GARETH TAYLOR
MARCUS DU SAUTOY
Affiliation:
The Mathematical Institute, University of Oxford, 24–29 St. Giles, Oxford OX1 3LB. e-mail: dusautoy@maths.ox.ac.uk
GARETH TAYLOR
Affiliation:
DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB. e-mail: G.L.Taylor@dpmms.cam.ac.uk
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Abstract

Let L be a ring additively isomorphic to ℤd. The zeta function of L is defined to be

where the sum is taken over all subalgebras H of finite index in L. This zeta function has a natural Euler product decomposition:

These functions were introduced in a paper of Grunewald, Segal and Smith [5] where the local factors ζL[otimes ]ℤp(s) were shown to always be rational functions in p−s. The proof depends on representing the local zeta function as a definable p-adic integral and then appealing to a general result of Denef’s [1] about the rationality of such integrals. The proof of Denef relies on Macintyre’s Quantifier Elimination for ℚp [8] followed by techniques developed by Igusa [6] which employ resolution of singularities.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 132 , Issue 1 , January 2002 , pp. 57 - 73
Copyright
2002 Cambridge Philosophical Society

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