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)


crystal; Maple
Full Text: DOI Euclid EuDML EMIS


[1] Cartan √Člie, Bull. Soc. Math. France 41 pp 1– (1913)
[2] Denef Jan, Amer. J. Math. 113 pp 1135– (1991) · Zbl 0749.11053
[3] Igusa Jun-ichi, J. reine angew. Math. 268 pp 110– (1974)
[4] Igusa Jun-ichi, J. reine angew. Math. 278 pp 307– (1975)
[5] Igusa Jun-ichi, Amer. J. Math. 111 pp 671– (1989) · Zbl 0707.14016
[6] Joyner David, ”A Maple package for the decomposition of certain tensor products of representations using crystal graphs” · Zbl 1097.17501
[7] Martin Roland, ”The universal p-adic zeta function associated to the adjoint group of SLl+1enlarged by the group of scalar multiples” (1992)
[8] Martin Roland, ”On generalized Igusa local zeta functions associated to simple Chevalley K-groups of type AlBlClDlE6E7E8F4and G2under the adjoint representation” (1992)
[9] Martin Roland, Electronic Research Announcements of the AMS 1 (3) (1996)
[10] Martin Roland, ”On the classification of Igusa local zeta functions associated to certain irreducible matrix groups”
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.