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A counterexample in the theory of local zeta functions. (English) Zbl 0869.11095
This paper is a report on a discovery of a counterexample for Igusa’s conjecture on a \(p\)-adic integral of a complex power function over the integral points. Such integrals are called generalized Igusa local zeta functions. The author of this paper proved that the generalized Igusa local zeta function associated to the representation \(\rho\) of \(\text{SL}_7\), where \(\rho\) is the Cartan product of the first, third and fifth fundamental representations of \(\text{SL}_7\), is explicitly computable and shown not to satisfy the functional equation expected by Igusa’s conjecture.
Reviewer: M.Muro (Yanagido)
11S40 Zeta functions and \(L\)-functions
11M41 Other Dirichlet series and zeta functions
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
Maple; crystal
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