AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Motivic Euler Products and Motivic Height Zeta Functions
About this Title
Margaret Bilu
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 282, Number 1396
ISBNs: 978-1-4704-6021-1 (print); 978-1-4704-7352-5 (online)
DOI: https://doi.org/10.1090/memo/1396
Published electronically: January 3, 2023
Keywords: Grothendieck rings of varieties,
Motivic integration,
Heights,
Poisson formula,
Manin’s conjecture,
Vanishing cycles,
Euler products
Table of Contents
Chapters
- 1. Introduction
- 2. Grothendieck Rings of Varieties and Motivic Vanishing Cycles
- 3. Motivic Euler Products
- 4. Mixed Hodge Modules and Convergence of Euler Products
- 5. The Motivic Poisson Formula
- 6. Motivic Height Zeta Functions
Abstract
A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the theory of weights in cohomology. We deduce from it the asymptotic behaviour, as the degree goes to infinity, of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the above moduli spaces: in particular, we get an estimate for their dimension and the number of components of maximal dimension. The main tools for this are a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, an extension of Hrushovski and Kazhdan’s motivic Poisson summation formula, and a motivic measure on the Grothendieck ring of varieties with exponentials constructed using Denef and Loeser’s motivic vanishing cycles.- Paul Balmer and Marco Schlichting, Idempotent completion of triangulated categories, J. Algebra 236 (2001), no. 2, 819–834. MR 1813503, DOI 10.1006/jabr.2000.8529
- V. V. Batyrev and Yu. I. Manin, Sur le nombre des points rationnels de hauteur borné des variétés algébriques, Math. Ann. 286 (1990), no. 1-3, 27–43 (French). MR 1032922, DOI 10.1007/BF01453564
- Victor V. Batyrev and Yuri Tschinkel, Rational points of bounded height on compactifications of anisotropic tori, Internat. Math. Res. Notices 12 (1995), 591–635. MR 1369408, DOI 10.1155/S1073792895000365
- Victor V. Batyrev and Yuri Tschinkel, Rational points on some Fano cubic bundles, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 1, 41–46 (English, with English and French summaries). MR 1401626
- Victor V. Batyrev and Yuri Tschinkel, Manin’s conjecture for toric varieties, J. Algebraic Geom. 7 (1998), no. 1, 15–53. MR 1620682
- Kai Behrend and Ajneet Dhillon, On the motivic class of the stack of bundles, Adv. Math. 212 (2007), no. 2, 617–644. MR 2329314, DOI 10.1016/j.aim.2006.11.003
- A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
- M. Bilu, Produits eulériens motiviques, PhD thesis, https://tel.archives-ouvertes.fr/tel-01662414, Université Paris-Saclay, 2017
- Margaret Bilu and Sean Howe, Motivic Euler products in motivic statistics, Algebra Number Theory 15 (2021), no. 9, 2195–2259. MR 4355473, DOI 10.2140/ant.2021.15.2195
- Franziska Bittner, The universal Euler characteristic for varieties of characteristic zero, Compos. Math. 140 (2004), no. 4, 1011–1032. MR 2059227, DOI 10.1112/S0010437X03000617
- Lev A. Borisov, The class of the affine line is a zero divisor in the Grothendieck ring, J. Algebraic Geom. 27 (2018), no. 2, 203–209. MR 3764275, DOI 10.1090/jag/701
- David Bourqui, Fonction zêta des hauteurs des surfaces de Hirzebruch dans le cas fonctionnel, J. Number Theory 94 (2002), no. 2, 343–358 (French, with English summary). MR 1916278, DOI 10.1006/jnth.2001.2739
- David Bourqui, Fonction zêta des hauteurs des variétés toriques déployées dans le cas fonctionnel, J. Reine Angew. Math. 562 (2003), 171–199 (French, with English summary). MR 2011335, DOI 10.1515/crll.2003.072
- David Bourqui, Produit eulérien motivique et courbes rationnelles sur les variétés toriques, Compos. Math. 145 (2009), no. 6, 1360–1400 (French, with English and French summaries). MR 2575087, DOI 10.1112/S0010437X09004291
- David Bourqui, Fonctions $L$ d’Artin et nombre de Tamagawa motiviques, New York J. Math. 16 (2010), 179–233 (French, with English and French summaries). MR 2657373
- David Bourqui, Fonction zêta des hauteurs des variétés toriques non déployées, Mem. Amer. Math. Soc. 211 (2011), no. 994, viii+151 (French, with English and French summaries). MR 2809202, DOI 10.1090/S0065-9266-2010-00609-4
- C. Brav, V. Bussi, D. Dupont, D. Joyce, and B. Szendrői, Symmetries and stabilization for sheaves of vanishing cycles, J. Singul. 11 (2015), 85–151. With an appendix by Jörg Schürmann. MR 3353002, DOI 10.5427/jsing.2015.11e
- Jean-Luc Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque 140-141 (1986), 3–134, 251 (French, with English summary). Géométrie et analyse microlocales. MR 864073
- Sylvain E. Cappell, Laurentiu Maxim, Jörg Schürmann, Julius L. Shaneson, and Shoji Yokura, Characteristic classes of symmetric products of complex quasi-projective varieties, J. Reine Angew. Math. 728 (2017), 35–63. MR 3668990, DOI 10.1515/crelle-2014-0114
- Antoine Chambert-Loir and Yuri Tschinkel, On the distribution of points of bounded height on equivariant compactifications of vector groups, Invent. Math. 148 (2002), no. 2, 421–452. MR 1906155, DOI 10.1007/s002220100200
- Antoine Chambert-Loir and Yuri Tschinkel, Integral points of bounded height on partial equivariant compactifications of vector groups, Duke Math. J. 161 (2012), no. 15, 2799–2836. MR 2999313, DOI 10.1215/00127094-1813638
- Antoine Chambert-Loir and François Loeser, Motivic height zeta functions, Amer. J. Math. 138 (2016), no. 1, 1–59. MR 3462879, DOI 10.1353/ajm.2016.0002
- Antoine Chambert-Loir, Johannes Nicaise, and Julien Sebag, Motivic integration, Progress in Mathematics, vol. 325, Birkhäuser/Springer, New York, 2018. MR 3838446, DOI 10.1007/978-1-4939-7887-8
- Raf Cluckers and François Loeser, Constructible motivic functions and motivic integration, Invent. Math. 173 (2008), no. 1, 23–121. MR 2403394, DOI 10.1007/s00222-008-0114-1
- Raf Cluckers and François Loeser, Constructible exponential functions, motivic Fourier transform and transfer principle, Ann. of Math. (2) 171 (2010), no. 2, 1011–1065. MR 2630060, DOI 10.4007/annals.2010.171.1011
- Pierre Colmez and Jean-Pierre Serre (eds.), Correspondance Grothendieck-Serre, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 2, Société Mathématique de France, Paris, 2001 (French). MR 1942134
- Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 354657
- P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
- Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 498551
- 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 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 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 10.1016/S0040-9383(01)00016-7
- T. Ekedahl, The Grothendieck group of algebraic stacks, arXiv:
0903.3143v2 , 2009. - Benjamin Fayyazuddin Ljungberg, Moduli Spaces of Bundles Via Motivic Probabilities, ProQuest LLC, Ann Arbor, MI, 2019. Thesis (Ph.D.)–Stanford University. MR 4197621
- Jens Franke, Yuri I. Manin, and Yuri Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), no. 2, 421–435. MR 974910, DOI 10.1007/BF01393904
- 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 10.1215/S0012-7094-06-13232-5
- S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, A power structure over the Grothendieck ring of varieties, Math. Res. Lett. 11 (2004), no. 1, 49–57. MR 2046199, DOI 10.4310/MRL.2004.v11.n1.a6
- Brendan Hassett and Yuri Tschinkel, Geometry of equivariant compactifications of $\textbf {G}_a^n$, Internat. Math. Res. Notices 22 (1999), 1211–1230. MR 1731473, DOI 10.1155/S1073792899000665
- Sean Howe, Motivic random variables and representation stability II: Hypersurface sections, Adv. Math. 350 (2019), 1267–1313. MR 3949982, DOI 10.1016/j.aim.2019.04.030
- Ehud Hrushovski and David Kazhdan, Integration in valued fields, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 261–405. MR 2263194, DOI 10.1007/978-0-8176-4532-8_{4}
- Ehud Hrushovski and David Kazhdan, Motivic Poisson summation, Mosc. Math. J. 9 (2009), no. 3, 569–623, back matter (English, with English and Russian summaries). MR 2562794, DOI 10.17323/1609-4514-2009-9-3-569-623
- M. Kapranov, The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups, arXiv:
math/0001005 , 2000 - Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. MR 1074006, DOI 10.1007/978-3-662-02661-8
- János Kollár, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR 2289519
- King Fai Lai and Kit Ming Yeung, Rational points in flag varieties over function fields, J. Number Theory 95 (2002), no. 2, 142–149. MR 1924094, DOI 10.1016/S0022-314X(01)92757-X
- Michael Larsen and Valery A. Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3 (2003), no. 1, 85–95, 259 (English, with English and Russian summaries). MR 1996804, DOI 10.17323/1609-4514-2003-3-1-85-95
- Michael J. Larsen and Valery A. Lunts, Irrationality of motivic zeta functions, Duke Math. J. 169 (2020), no. 1, 1–30. MR 4047547, DOI 10.1215/00127094-2019-0035
- Eduard Looijenga, Motivic measures, Astérisque 276 (2002), 267–297. Séminaire Bourbaki, Vol. 1999/2000. MR 1886763
- Valery A. Lunts and Olaf M. Schnürer, Matrix factorizations and motivic measures, J. Noncommut. Geom. 10 (2016), no. 3, 981–1042. MR 3554841, DOI 10.4171/JNCG/253
- Valery A. Lunts and Olaf M. Schnürer, Motivic vanishing cycles as a motivic measure, Pure Appl. Math. Q. 12 (2016), no. 1, 33–74. MR 3613965, DOI 10.4310/PAMQ.2016.v12.n1.a2
- Serge Lang and André Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827. MR 65218, DOI 10.2307/2372655
- David B. Massey, The Sebastiani-Thom isomorphism in the derived category, Compositio Math. 125 (2001), no. 3, 353–362. MR 1818986, DOI 10.1023/A:1002608716514
- Laurentiu Maxim, Morihiko Saito, and Jörg Schürmann, Symmetric products of mixed Hodge modules, J. Math. Pures Appl. (9) 96 (2011), no. 5, 462–483 (English, with English and French summaries). MR 2843222, DOI 10.1016/j.matpur.2011.04.003
- J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 103–150. MR 861974
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
- M. Mustaţă, Zeta functions in algebraic geometry, Lecture notes available at http://www.math.lsa.umich.edu/~mmustata/zeta_book.pdf
- Emmanuel Peyre, Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J. 79 (1995), no. 1, 101–218 (French). MR 1340296, DOI 10.1215/S0012-7094-95-07904-6
- Emmanuel Peyre, Points de hauteur bornée et géométrie des variétés (d’après Y. Manin et al.), Astérisque 282 (2002), Exp. No. 891, ix, 323–344 (French, with French summary). Séminaire Bourbaki, Vol. 2000/2001. MR 1975184
- Emmanuel Peyre, Points de hauteur bornée sur les variétés de drapeaux en caractéristique finie, Acta Arith. 152 (2012), no. 2, 185–216 (French). MR 2864393, DOI 10.4064/aa152-2-5
- Chris A. M. Peters and Joseph H. M. Steenbrink, Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 52, Springer-Verlag, Berlin, 2008. MR 2393625
- Morihiko Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989) (French). MR 1000123, DOI 10.2977/prims/1195173930
- Morihiko Saito, Introduction to mixed Hodge modules, Astérisque 179-180 (1989), 10, 145–162. Actes du Colloque de Théorie de Hodge (Luminy, 1987). MR 1042805
- Morihiko Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333. MR 1047415, DOI 10.2977/prims/1195171082
- M. Saito, Thom-Sebastiani Theorem for Hodge Modules, preprint
- Stephen Hoel Schanuel, Heights in number fields, Bull. Soc. Math. France 107 (1979), no. 4, 433–449 (English, with French summary). MR 557080
- Ch. Schnell, An overview of Morihiko Saito’s theory of mixed Hodge modules, https://www.math.stonybrook.edu/~cschnell/pdf/notes/sanya.pdf
- M. Sebastiani and R. Thom, Un résultat sur la monodromie, Invent. Math. 13 (1971), 90–96 (French). MR 293122, DOI 10.1007/BF01390095
- Joseph Shalika, Ramin Takloo-Bighash, and Yuri Tschinkel, Rational points on compactifications of semi-simple groups of rank 1, Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002) Progr. Math., vol. 226, Birkhäuser Boston, Boston, MA, 2004, pp. 205–233. MR 2029871, DOI 10.1007/978-0-8176-8170-8_{1}3
- Joseph Shalika, Ramin Takloo-Bighash, and Yuri Tschinkel, Rational points on compactifications of semi-simple groups, J. Amer. Math. Soc. 20 (2007), no. 4, 1135–1186. MR 2328719, DOI 10.1090/S0894-0347-07-00572-3
- Joseph A. Shalika and Yuri Tschinkel, Height zeta functions of equivariant compactifications of the Heisenberg group, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 743–771. MR 2058627
- Joseph Shalika and Yuri Tschinkel, Height zeta functions of equivariant compactifications of unipotent groups, Comm. Pure Appl. Math. 69 (2016), no. 4, 693–733. MR 3465086, DOI 10.1002/cpa.21572
- Ramin Takloo-Bighash and Sho Tanimoto, Distribution of rational points of bounded height on equivariant compactifications of $\textrm {PGL}_2$ I, Res. Number Theory 2 (2016), Paper No. 6, 35. MR 3501019, DOI 10.1007/s40993-016-0037-7
- Sho Tanimoto and Yuri Tschinkel, Height zeta functions of equivariant compactifications of semi-direct products of algebraic groups, Zeta functions in algebra and geometry, Contemp. Math., vol. 566, Amer. Math. Soc., Providence, RI, 2012, pp. 119–157. MR 2858922, DOI 10.1090/conm/566/11218
- Ravi Vakil and Melanie Matchett Wood, Discriminants in the Grothendieck ring, Duke Math. J. 164 (2015), no. 6, 1139–1185. MR 3336842, DOI 10.1215/00127094-2877184
- Jean-Louis Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math. 36 (1976), 295–312 (French). MR 481096, DOI 10.1007/BF01390015