Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Characteristic class and the $\varepsilon$-factor of an étale sheaf
HTML articles powered by AMS MathViewer

by Naoya Umezaki, Enlin Yang and Yigeng Zhao PDF
Trans. Amer. Math. Soc. 373 (2020), 6887-6927 Request permission

Abstract:

We prove a twist formula for the $\varepsilon$-factor of a constructible sheaf on a projective smooth variety over a finite field in terms of characteristic class of the sheaf. This formula is a modified version of the formula conjectured by Kato and Saito in [Ann. of Math. 168 (2008), pp. 33–96].

We give two applications of the twist formula. First, we prove that the characteristic classes of constructible étale sheaves on projective smooth varieties over a finite field are compatible with proper push-forward. Secondly, we show that the two Swan classes in the literature are the same on proper smooth surfaces over a finite field.

References
  • Tomoyuki Abe and Deepam Patel, On a localization formula of epsilon factors via microlocal geometry, Ann. K-Theory 3 (2018), no. 3, 461–490. MR 3830199, DOI 10.2140/akt.2018.3.461
  • Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354652
  • Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
  • A. Beilinson, Constructible sheaves are holonomic, Selecta Math. (N.S.) 22 (2016), no. 4, 1797–1819. MR 3573946, DOI 10.1007/s00029-016-0260-z
  • A. Beilinson, Topological $\scr E$-factors, Pure Appl. Math. Q. 3 (2007), no. 1, Special Issue: In honor of Robert D. MacPherson., 357–391. MR 2330165, DOI 10.4310/PAMQ.2007.v3.n1.a13
  • 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
  • P. Deligne and G. Henniart, Sur la variation, par torsion, des constantes locales d’équations fonctionnelles de fonctions $L$, Invent. Math. 64 (1981), no. 1, 89–118 (French). MR 621771, DOI 10.1007/BF01393935
  • Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520
  • P. Deligne, Les constantes des équations fonctionnelles des fonctions $L$, Séminaire à l’ I.H.E.S., 1980, notes de L. Illusie.
  • 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
  • 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 0354657
  • P. Deligne, Les constantes des équations fonctionnelles des fonctions $L$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 501–597 (French). MR 0349635
  • Lei Fu, Etale cohomology theory, Revised edition, Nankai Tracts in Mathematics, vol. 14, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. MR 3380806, DOI 10.1142/9569
  • William Fulton, Intersection theory, 2nd ed., 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. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
  • V. Ginsburg, Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), no. 2, 327–402. MR 833194, DOI 10.1007/BF01388811
  • A. Grothendieck et al., Cohomologie $\ell$-adique et fonctions L, Séminaire de Géométrie Algébrique du Bois-Marie 1965–1966 (SGA 5). dirigé par A. Grothendieck avec la collaboration de I. Bucur, C. Houzel, L. Illusie, J.-P. Jouanolou et J-P. Serre. Lecture Notes in Mathematics 589, Springer-Verlag, Berlin-Heidelberg-New York, (1977).
  • A. Grothendieck, Récoltes et Semailles, Réflexions et témoignages sur un passé de mathématicien, http://lipn.univ-paris13.fr/\~{}duchamp/Books\&more/Grothendieck/RS/pdf/RetS.pdf
  • Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
  • L. Illusie, Y. Lazslo, and F. Orgogozo, Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents, Astérisque, vol. 361-362, Soc. Math. France, 2014, Séminaire à l’École Polytechnique 2006-2008.
  • Luc Illusie, Théorie de Brauer et caractéristique d’Euler-Poincaré (d’après P. Deligne), The Euler-Poincaré characteristic (French), Astérisque, vol. 82, Soc. Math. France, Paris, 1981, pp. 161–172 (French). MR 629127
  • Kazuya Kato and Shuji Saito, Unramified class field theory of arithmetical surfaces, Ann. of Math. (2) 118 (1983), no. 2, 241–275. MR 717824, DOI 10.2307/2007029
  • Kazuya Kato and Takeshi Saito, Ramification theory for varieties over a perfect field, Ann. of Math. (2) 168 (2008), no. 1, 33–96. MR 2415398, DOI 10.4007/annals.2008.168.33
  • Nicholas M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR 955052, DOI 10.1515/9781400882120
  • Nicholas M. Katz, Travaux de Laumon, Astérisque 161-162 (1988), Exp. No. 691, 4, 105–132 (1989). Séminaire Bourbaki, Vol. 1987/88. MR 992205
  • Reinhardt Kiehl and Rainer Weissauer, Weil conjectures, perverse sheaves and $l$’adic Fourier transform, 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. 42, Springer-Verlag, Berlin, 2001. MR 1855066, DOI 10.1007/978-3-662-04576-3
  • G. Laumon, Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 131–210 (French). MR 908218
  • Masayoshi Nagata, A generalization of the imbedding problem of an abstract variety in a complete variety, J. Math. Kyoto Univ. 3 (1963), 89–102. MR 158892, DOI 10.1215/kjm/1250524859
  • Deepam Patel, de Rham ${\scr E}$-factors, Invent. Math. 190 (2012), no. 2, 299–355. MR 2981817, DOI 10.1007/s00222-012-0381-8
  • D. Patel, K-theory of algebraic microdifferential operators, preprint.
  • Wayne Raskind, Abelian class field theory of arithmetic schemes, $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 85–187. MR 1327282
  • Shuji Saito, Functional equations of $L$-functions of varieties over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), no. 2, 287–296. MR 763423
  • T. Saito, Characteristic cycles and the conductor of direct image, arXiv:1704.04832.
  • Takeshi Saito, On the proper push-forward of the characteristic cycle of a constructible sheaf, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 485–494. MR 3821182
  • Takeshi Saito, The characteristic cycle and the singular support of a constructible sheaf, Invent. Math. 207 (2017), no. 2, 597–695. MR 3595935, DOI 10.1007/s00222-016-0675-3
  • Takeshi Saito and Yuri Yatagawa, Wild ramification determines the characteristic cycle, Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), no. 4, 1065–1079 (English, with English and French summaries). MR 3679621, DOI 10.24033/asens.2339
  • Takeshi Saito, $\epsilon$-factor of a tamely ramified sheaf on a variety, Invent. Math. 113 (1993), no. 2, 389–417. MR 1228131, DOI 10.1007/BF01244312
  • Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
  • Jean-Pierre Serre, Groupes de Grothendieck des schémas en groupes réductifs déployés, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 37–52 (French). MR 231831
  • J. Tate, Number theoretic background, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607
  • J. T. Tate, Fourier analysis in number fields, and Hecke’s zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 305–347. MR 0217026
  • N. Umezaki, E. Yang, and Y. Zhao, A blow up formula for Gysin pull-back, https://yangenlin.files.wordpress.com/2018/08/buf.pdf.
  • Isabelle Vidal, Formule du conducteur pour un caractère $l$-adique, Compos. Math. 145 (2009), no. 3, 687–717 (French, with English and French summaries). MR 2507745, DOI 10.1112/S0010437X08003850
  • Isabelle Vidal, Formule de torsion pour le facteur epsilon d’un caractère sur une surface, Manuscripta Math. 130 (2009), no. 1, 21–44 (French, with English and French summaries). MR 2533765, DOI 10.1007/s00229-009-0267-2
  • Weizhe Zheng, Six operations and Lefschetz-Verdier formula for Deligne-Mumford stacks, Sci. China Math. 58 (2015), no. 3, 565–632. MR 3319927, DOI 10.1007/s11425-015-4970-z
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14F20, 11G25, 11S40
  • Retrieve articles in all journals with MSC (2010): 14F20, 11G25, 11S40
Additional Information
  • Naoya Umezaki
  • Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
  • MR Author ID: 1171997
  • Email: umezaki@ms.u-tokyo.ac.jp, umezakinaoya@gmail.com
  • Enlin Yang
  • Affiliation: School of Mathematical Sciences, Peking University, No.5 Yiheyuan Road Haidian District., Beijing, 100871, People’s Republic of China
  • Email: yangenlin@math.pku.edu.cn, enlin.yang@mathematik.uni-regensburg.de
  • Yigeng Zhao
  • Affiliation: School of Sciences, Westlake University, 310024 Hangzhou, People’s Republic of China; and Institute of Sciences, Westlake Institute for Advanced Study, 310024 Hangzhou, People’s Republic of China
  • MR Author ID: 1278775
  • Email: zhaoyigeng@westlake.edu.cn, yigeng.zhao@mathematik.uni-regensburg.de
  • Received by editor(s): November 7, 2018
  • Received by editor(s) in revised form: June 14, 2019, and September 5, 2019
  • Published electronically: August 6, 2020
  • Additional Notes: Enlin Yang is the corresponding author.
    The second author was partially supported by NSFC grants 11901008, and Alexander von Humboldt Foundation for his research at Universität Regensburg and Freie Universität Berlin.
    Both the second and the third authors were partially supported by the DFG through CRC 1085 Higher Invariants (Universität Regensburg). The authors are grateful to these institutions.
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 6887-6927
  • MSC (2010): Primary 14F20; Secondary 11G25, 11S40
  • DOI: https://doi.org/10.1090/tran/8187
  • MathSciNet review: 4155195