Euler reflexion formulas for motivic multiple zeta functions
Authors:
Lê Quy Thuong and Nguyen Hong Duc
Journal:
J. Algebraic Geom. 27 (2018), 91-120
DOI:
https://doi.org/10.1090/jag/689
Published electronically:
February 17, 2017
MathSciNet review:
3722691
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Abstract |
References |
Additional Information
Abstract:
We introduce a new notion of $\boxast$-product of two integrable series with coefficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability of and commuting with the limit of rational series. In the same context, we define a motivic multiple zeta function with respect to an ordered family of regular functions, which is integrable and connects closely to Denef-Loeser’s motivic zeta functions. We also show that the $\boxast$-product is associative in the class of motivic multiple zeta functions.
Furthermore, a version of the Euler reflexion formula for motivic zeta functions is nicely formulated to deal with the $\boxast$-product and motivic multiple zeta functions, and it is proved for both univariate and multivariate cases by using the theory of arc spaces. As an application, taking the limit for the motivic Euler reflexion formula we recover the well-known motivic Thom-Sebastiani theorem.
References
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- Jan Denef, Report on Igusa’s local zeta function, Astérisque 201-203 (1991), Exp. No. 741, 359–386 (1992). Séminaire Bourbaki, Vol. 1990/91. MR 1157848
- Jan Denef and François Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), no. 3, 505–537. MR 1618144
- Jan Denef and François Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201–232. MR 1664700, DOI https://doi.org/10.1007/s002220050284
- Jan Denef and François Loeser, Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J. 99 (1999), no. 2, 285–309. MR 1708026, DOI https://doi.org/10.1215/S0012-7094-99-09910-6
- Jan Denef and François Loeser, Geometry on arc spaces of algebraic varieties, European Congress of Mathematics, Vol. I (Barcelona, 2000) Progr. Math., vol. 201, Birkhäuser, Basel, 2001, pp. 327–348. MR 1905328
- Jan Denef and François Loeser, Lefschetz numbers of iterates of the monodromy and truncated arcs, Topology 41 (2002), no. 5, 1031–1040. MR 1923998, DOI https://doi.org/10.1016/S0040-9383%2801%2900016-7
- Jan Denef and Willem Veys, On the holomorphy conjecture for Igusa’s local zeta function, Proc. Amer. Math. Soc. 123 (1995), no. 10, 2981–2988. MR 1283546, DOI https://doi.org/10.1090/S0002-9939-1995-1283546-4
- Gil Guibert, Espaces d’arcs et invariants d’Alexander, Comment. Math. Helv. 77 (2002), no. 4, 783–820 (French, with English and French summaries). MR 1949114, DOI https://doi.org/10.1007/PL00012442
- Gil Guibert, François Loeser, and Michel Merle, Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink, Duke Math. J. 132 (2006), no. 3, 409–457. MR 2219263, DOI https://doi.org/10.1215/S0012-7094-06-13232-5
- Gil Guibert, François Loeser, and Michel Merle, Nearby cycles and composition with a nondegenerate polynomial, Int. Math. Res. Not. 31 (2005), 1873–1888. MR 2171196, DOI https://doi.org/10.1155/IMRN.2005.1873
- Ehud Hrushovski and François Loeser, Monodromy and the Lefschetz fixed point formula, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 2, 313–349 (English, with English and French summaries). MR 3346173, DOI https://doi.org/10.24033/asens.2246
- Quy Thuong Lê, The motivic Thom-Sebastiani theorem for regular and formal functions, to appear in J. Reine Angew. Math., DOI: 10.1515/crelle-2015-0022, arXiv:1405.7065.
- Quy Thuong Lê, A proof of the integral identity conjecture, II, preprint, arXiv:1508.00425.
- Eduard Looijenga, Motivic measures, Astérisque 276 (2002), 267–297. Séminaire Bourbaki, Vol. 1999/2000. MR 1886763
- D. Zagier, Evaluation of the multiple zeta values $\zeta (2,\ldots ,2,3,2,\ldots ,2)$, Ann. of Math. (2) 175 (2012), no. 2, 977–1000. MR 2993756, DOI https://doi.org/10.4007/annals.2012.175.2.11
References
- Raf Cluckers and François Loeser, Constructible motivic functions and motivic integration, Invent. Math. 173 (2008), no. 1, 23–121. MR 2403394, DOI https://doi.org/10.1007/s00222-008-0114-1
- Jan Denef, Report on Igusa’s local zeta function, Astérisque 201-203 (1991), Exp. No. 741, 359–386 (1992). Séminaire Bourbaki, Vol. 1990/91. MR 1157848
- Jan Denef and François Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), no. 3, 505–537. MR 1618144
- Jan Denef and François Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201–232. MR 1664700, DOI https://doi.org/10.1007/s002220050284
- Jan Denef and François Loeser, Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J. 99 (1999), no. 2, 285–309. MR 1708026, DOI https://doi.org/10.1215/S0012-7094-99-09910-6
- Jan Denef and François Loeser, Geometry on arc spaces of algebraic varieties, European Congress of Mathematics, Vol. I (Barcelona, 2000) Progr. Math., vol. 201, Birkhäuser, Basel, 2001, pp. 327–348. MR 1905328
- Jan Denef and François Loeser, Lefschetz numbers of iterates of the monodromy and truncated arcs, Topology 41 (2002), no. 5, 1031–1040. MR 1923998, DOI https://doi.org/10.1016/S0040-9383%2801%2900016-7
- Jan Denef and Willem Veys, On the holomorphy conjecture for Igusa’s local zeta function, Proc. Amer. Math. Soc. 123 (1995), no. 10, 2981–2988. MR 1283546, DOI https://doi.org/10.2307/2160651
- Gil Guibert, Espaces d’arcs et invariants d’Alexander, Comment. Math. Helv. 77 (2002), no. 4, 783–820 (French, with English and French summaries). MR 1949114, DOI https://doi.org/10.1007/PL00012442
- Gil Guibert, François Loeser, and Michel Merle, Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink, Duke Math. J. 132 (2006), no. 3, 409–457. MR 2219263, DOI https://doi.org/10.1215/S0012-7094-06-13232-5
- Gil Guibert, François Loeser, and Michel Merle, Nearby cycles and composition with a nondegenerate polynomial, Int. Math. Res. Not. 31 (2005), 1873–1888. MR 2171196, DOI https://doi.org/10.1155/IMRN.2005.1873
- Ehud Hrushovski and François Loeser, Monodromy and the Lefschetz fixed point formula, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 2, 313–349 (English, with English and French summaries). MR 3346173
- Quy Thuong Lê, The motivic Thom-Sebastiani theorem for regular and formal functions, to appear in J. Reine Angew. Math., DOI: 10.1515/crelle-2015-0022, arXiv:1405.7065.
- Quy Thuong Lê, A proof of the integral identity conjecture, II, preprint, arXiv:1508.00425.
- Eduard Looijenga, Motivic measures, Astérisque 276 (2002), 267–297. Séminaire Bourbaki, Vol. 1999/2000. MR 1886763
- Don Zagier, Evaluation of the multiple zeta values $\zeta (2,\ldots ,2,3,2,\ldots ,2)$, Ann. of Math. (2) 175 (2012), no. 2, 977–1000. MR 2993756, DOI https://doi.org/10.4007/annals.2012.175.2.11
Additional Information
Lê Quy Thuong
Affiliation:
Department of Mathematics, Vietnam National University, 334 Nguyen Trai Street, Thanh Xuan District, Hanoi, Vietnam
Email:
leqthuong@gmail.com
Nguyen Hong Duc
Affiliation:
Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi City, Quang Binh, Vietnam
Address at time of publication:
BCAM – Basque Center for Applied Mathematics, Mazarredo, 14, 48009 Bilbao, Basque Country, Spain
MR Author ID:
842994
Email:
hnguyen@bcamath.org
Received by editor(s):
November 25, 2015
Received by editor(s) in revised form:
May 2, 2016
Published electronically:
February 17, 2017
Additional Notes:
The first author’s research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant No. FWO.101.2015.02. The second author was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant No. 101.04-2014.23. This research was also supported by ERCEA Consolidator Grant 615655 - NMST and by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323.
Article copyright:
© Copyright 2017
University Press, Inc.
