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

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.