Abstract
We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or $\mathcal{T}$-)groups, and the normal zeta functions of $\mathcal{T}$-groups of class $2$. We deduce our theorems from a “blueprint result” on certain $p$-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to “linearise” the problems of counting subgroups and representations in $\mathcal{T}$-groups, respectively.
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[Berman/06] M. N. Berman, "Uniformity and functional equations for local zeta functions of $\mathfrak{K}$-split algebraic groups," , preprint , 2008.
@techreport{Berman/06,
author={Berman, M. N.},
TITLE={Uniformity and functional equations for local zeta functions of $\mathfrak{K}$-split algebraic groups},
TYPE={preprint},
YEAR={2008},
ARXIV={0802.4207},
} -
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@book {CurtisReiner-methods/81, MRKEY = {632548},
AUTHOR = {Curtis, Charles W. and Reiner, Irving},
TITLE = {Methods of Representation Theory, With Applications to Finite Groups and Orders},
VOLUME={I},
PUBLISHER = {John Wiley \& Sons},
ADDRESS = {New York},
YEAR = {1981},
PAGES = {xxi+819},
ISBN = {0-471-18994-4},
MRCLASS = {20-02 (10C30 12A57 20Cxx)},
MRNUMBER = {82i:20001},
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ZBLNUMBER = {0469.20001},
} -
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J. Denef, "On the degree of Igusa’s local zeta function," Amer. J. Math., vol. 109, iss. 6, pp. 991-1008, 1987.
@article {Denef/87, MRKEY = {919001},
AUTHOR = {Denef, Jan},
TITLE = {On the degree of {I}gusa's local zeta function},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {109},
YEAR = {1987},
NUMBER = {6},
PAGES = {991--1008},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {11S40},
MRNUMBER = {89d:11108},
MRREVIEWER = {Daniel Barsky},
DOI = {10.2307/2374583},
ZBLNUMBER = {0659.14017},
} -
[Denef/91] J. Denef, "Report on Igusa’s local zeta function," in Séminaire Bourbaki, Vol. $1990/91$, , 1992, vol. 201–203, pp. 359-386.
@incollection{Denef/91,
author = {Denef, Jan},
TITLE={Report on Igusa's local zeta function},
BOOKTITLE={Séminaire Bourbaki, {\rm Vol. $1990/91$}},
SERIES={Astérisque},
VOLUME={201--203},
PAGES={359--386},
YEAR={1992},
MRNUMBER={93g:11119},
ZBLNUMBER={0749.11054},
} -
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J. Denef and D. Meuser, "A functional equation of Igusa’s local zeta function," Amer. J. Math., vol. 113, iss. 6, pp. 1135-1152, 1991.
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AUTHOR = {Denef, Jan and Meuser, Diane},
TITLE = {A functional equation of {I}gusa's local zeta function},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {113},
YEAR = {1991},
NUMBER = {6},
PAGES = {1135--1152},
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} -
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M. du Sautoy, "A nilpotent group and its elliptic curve: non-uniformity of local zeta functions of groups," Israel J. Math., vol. 126, pp. 269-288, 2001.
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AUTHOR = {du Sautoy, Marcus},
TITLE = {A nilpotent group and its elliptic curve: non-uniformity of local zeta functions of groups},
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JOURNAL = {J. Reine Angew. Math.},
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MRCLASS = {11M41 (14G10 20F18)},
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DOI = {10.1515/crll.2002.063},
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} -
[duSG/00]
M. du Sautoy and F. Grunewald, "Analytic properties of zeta functions and subgroup growth," Ann. of Math., vol. 152, iss. 3, pp. 793-833, 2000.
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TITLE = {Analytic properties of zeta functions and subgroup growth},
JOURNAL = {Ann. of Math.},
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VOLUME = {152},
YEAR = {2000},
NUMBER = {3},
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MRCLASS = {11M41 (20E07)},
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ZBLNUMBER={1006.11051},
} -
[duSG/06] M. du Sautoy and F. Grunewald, "Zeta functions of groups and rings," in Proc. Internat. Congress of Mathematicians, Eur. Math. Soc., Zürich, 2006, vol. II, pp. 131-149.
@incollection {duSG/06, MRKEY = {2275592},
AUTHOR = {du Sautoy, Marcus and Grunewald, Fritz},
TITLE = {Zeta functions of groups and rings},
BOOKTITLE = {Proc. Internat. {C}ongress of {M}athematicians},
VENUE={Madrid, August 22--30, 2006},
VOLUME={II},
PAGES = {131--149},
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YEAR = {2006},
MRCLASS = {11M41 (11M45 20E07 20F18)},
MRNUMBER = {2008d:11102},
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} -
[duSLubotzky/96]
M. du Sautoy and A. Lubotzky, "Functional equations and uniformity for local zeta functions of nilpotent groups," Amer. J. Math., vol. 118, iss. 1, pp. 39-90, 1996.
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AUTHOR = {du Sautoy, Marcus and Lubotzky, Alexander},
TITLE = {Functional equations and uniformity for local zeta functions of nilpotent groups},
JOURNAL = {Amer. J. Math.},
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NUMBER = {1},
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ISSN = {0002-9327},
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MRNUMBER = {97g:11137},
URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v118/118.1du_sautoy.pdf},
ZBLNUMBER={0864.20020},
} -
[duSSegal/00] M. du Sautoy and D. Segal, "Zeta functions of groups," in New Horizons in Pro-$p$ Groups, Boston, MA: Birkhäuser, 2000, vol. 184, pp. 249-286.
@incollection {duSSegal/00,
author={du Sautoy, Marcus and Segal, Dan},
TITLE={Zeta functions of groups},
MRKEY = {1765115},
BOOKTITLE = {New Horizons in Pro-{$p$} Groups},
SERIES = {Progr. Math.},
VOLUME = {184},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {2000},
PAGES = {249--286},
ISBN = {0-8176-4171-8},
MRCLASS = {20-06 (20E18)},
MRNUMBER = {2001b:20001},
ZBLNUMBER={1188.11047},
} -
[duSTaylor/02]
M. du Sautoy and G. Taylor, "The zeta function of $\mathfrak{sl}_2$ and resolution of singularities," Math. Proc. Cambridge Philos. Soc., vol. 132, iss. 1, pp. 57-73, 2002.
@article {duSTaylor/02, MRKEY = {1866324},
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JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
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YEAR = {2002},
NUMBER = {1},
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ISSN = {0305-0041},
CODEN = {MPCPCO},
MRCLASS = {11M41 (11S40 14B05 17B20)},
MRNUMBER = {2002j:11100},
MRREVIEWER = {Margaret M. Robinson},
DOI = {10.1017/S0305004101005369},
ZBLNUMBER={1020.11074},
} -
[duSautoyWoodward/06]
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@book {duSautoyWoodward/06, MRKEY = {2371185},
AUTHOR = {du Sautoy, Marcus and Woodward, Luke},
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SERIES = {Lecture Notes in Math.},
VOLUME = {1925},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2008},
PAGES = {xii+208},
ISBN = {978-3-540-74701-7},
MRCLASS = {20E07 (11M41)},
MRNUMBER = {2009d:20053},
MRREVIEWER = {Alexander Lubotzky},
DOI = {10.1007/978-3-540-74776-5},
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} -
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[Howe-compact/77]
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NOTE = {(with an appendix by A. Beauville)},
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MRREVIEWER = {Driss Essouabri},
DOI = {10.1112/S0024609306018881},
ZBLNUMBER={1120.11040},
} -
[Wlodarczyk/05]
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@article {Wlodarczyk/05, MRKEY = {2163383},
AUTHOR = {W{\l}odarczyk, Jaros{\l}aw},
TITLE = {Simple {H}ironaka resolution in characteristic zero},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {18},
YEAR = {2005},
NUMBER = {4},
PAGES = {779--822},
ISSN = {0894-0347},
MRCLASS = {14E15},
MRNUMBER = {2006f:14014},
MRREVIEWER = {Santiago Encinas},
DOI = {10.1090/S0894-0347-05-00493-5},
ZBLNUMBER={1084.14018},
} -
[Woodward/05] L. Woodward, "Zeta functions of groups: computer calculations and functional equations," PhD Thesis , University of Oxford, 2005.
@phdthesis{Woodward/05,
author={Woodward, L.},
TITLE={Zeta functions of groups: computer calculations and functional equations},
SCHOOL={University of Oxford},
YEAR={2005},
} -
[X] R. I. Liu, "Counting subrings of $\mathbb{Z}^n$ of index $k$," J. Combinatorial Theory, vol. 114, pp. 278-299, 2007.
@article{X,
author={Liu, R. I.},
TITLE={ Counting subrings of $\mathbb{Z}^n$ of index $k$},
JOURNAL={J. Combinatorial Theory},
VOLUME={114},
YEAR={2007},
PAGES={278--299},
}