Abstract
These notes were written to be distributed to the audience of the first author’s Takagi Lectures delivered June 23, 2018. These are based on a work-in-progress that is part of a collaborative project that also involves Akshay Venkatesh.
In this work-in-progress we give a new construction of some Eisenstein classes for GLN (Z) that were first considered by Nori [41] and Sczech [44]. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of SLN (Z) vector bundles and the explicit transgression of this Euler class by Bismut and Cheeger. Their proof indeed produces a universal form that can be thought of as a kernel for a regularized theta lift for the reductive dual pair (GLN, GL1). This suggests looking to reductive dual pairs (GLN, GLk) with k ≥ 1 for possible generalizations of the Eisenstein cocycle. This leads to fascinating lifts that relate the geometry/topology world of real arithmetic locally symmetric spaces to the arithmetic world of modular forms.
In these notes we do not deal with the most general cases and put a lot of emphasis on various examples that are often classical.
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References
M.F. Atiyah, H. Donnelly and I.M. Singer, Eta invariants, signature defects of cusps, and values of L-functions, Ann. of Math. (2), 118 (1983) 131–177.
A.A. Beĭlinson, Higher regulators and values of L-functions, In: Current Problems in Mathematics, 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181–238.
A.A. Beĭlinson, G. Kings and A. Levin, Topological polylogarithms and p-adic interpolation of L-values of totally real fields, Math. Ann., 371 (2018) 1449–1495.
A.A. Beĭlinson and A. Levin, The elliptic polylogarithm, In: Motives, Seattle, WA, 1991, Proc. Sympos. Pure Math., 55, Amer. Math. Soc., Providence, RI, 1994, pp. 123–190.
N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Operators, Grundlehren Text Ed., Springer-Verlag, 2004. Corrected reprint of the 1992 original.
J.-M. Bismut and J. Cheeger, Remarks on the index theorem for families of Dirac operators on manifolds with boundary, In: Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow, 1991, pp. 59–83.
J.-M. Bismut and J. Cheeger, Transgressed Euler classes of SL(2n, Z) vector bundles, adiabatic limits of eta invariants and special values of L-functions, Ann. Sci. École Norm. Sup. (4), 25 (1992) 335–391.
J.-M. Bismut, H. Gillet and C. Soulé, Analytic torsion and holomorphic determinant bundles. I. Bott—Chern forms and analytic torsion, Comm. Math. Phys., 115 (1988) 49–78.
D. Blasius, On the critical values of Hecke L-series, Ann. of Math. (2), 124 (1986) 23–63.
L.A. Borisov and P.E. Gunnells, Toric modular forms and nonvanishing of L-functions, J. Reine Angew. Math., 539 (2001) 149–165.
L.A. Borisov and P.E. Gunnells, Toric varieties and modular forms, Invent. Math., 144 (2001) 297–325.
R. Bott and L.W. Tu, Differential Forms in Algebraic Topology, Grad. Texts in Math., 82, Springer-Verlag, 1982.
F. Brunault, Beilinson—Kato elements in K2 of modular curves, Acta Arith., 134 (2008) 283–298.
P. Cassou-Noguès, Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques, Invent. Math., 51 (1979) 29–59.
P. Charollois and S. Dasgupta, Integral Eisenstein cocycles on GLn, I: Sczech’s cocycle and p-adic L-functions of totally real fields, Camb. J. Math., 2 (2014) 49–90.
P. Charollois, S. Dasgupta and M. Greenberg, Integral Eisenstein cocycles on GLn, II: Shintani’s method, Comment. Math. Helv., 90 (2015) 435–477.
P. Charollois and R. Sczech, Elliptic functions according to Eisenstein and Kronecker: an update, Eur. Math. Soc. Newsl., 101 (2016) 8–14.
S.-S. Chern, A simple intrinsic proof of the Gauss—Bonnet formula for closed Riemannian manifolds, Ann. of Math. (2), 45 (1944) 747–752.
P. Colmez, Algébricité de valeurs spéciales de fonctions L, Invent. Math., 95 (1989) 161–205.
R.M. Damerell, L-functions of elliptic curves with complex multiplication. I, Acta Arith., 17 (1970), 287–301.
P. Deligne, Valeurs de fonctions L et périodes d’intégrales. With an appendix by N. Koblitz and A. Ogus, In: Automorphic Forms, Representations and L-functions, Oregon State Univ., Corvallis, OR, 1977, Proc. Sympos. Pure Math., 33, Part 2, Amer. Math. Soc., Providence, RI, 1979, pp. 313–346.
P. Deligne and K.A. Ribet, Values of abelian L-functions at negative integers over totally real fields, Invent. Math., 59 (1980) 227–286.
G. Faltings, Arithmetic Eisenstein classes on the Siegel space: some computations, In: Number Fields and Function Fields—Two Parallel Worlds, Progr. Math., 239, Birkhäuser Boston, Boston, MA, 2005, pp. 133–166.
J. Flórez, C. Karabulut and T.A. Wong, Eisenstein cocycles over imaginary quadratic fields and special values of L-functions, J. Number Theory, 204 (2019) 497–531.
L.E. Garcia, Superconnections, theta series, and period domains, Adv. Math., 329 (2018) 555–589.
E. Getzler, The Thom class of Mathai and Quillen and probability theory, In: Stochastic Analysis and Applications, Lisbon, 1989, Progr. Probab., 26, Birkhäuser Boston, Boston, MA, 1991, pp. 111–122.
A.B. Goncharov, Regulators, In: Handbook of K-Theory. Vol. 1, 2, Springer-Verlag, 2005, pp. 295–349.
P. Graf, Polylogarithms for GL2 over totally real fields, preprint, ar**v:1604.04209.
G. Harder, Some results on the Eisenstein cohomology of arithmetic subgroups of GLn, In: Cohomology of Arithmetic Groups and Automorphic Forms, Luminy—Marseille, 1989, Lecture Notes in Math., 1447, Springer-Verlag, 1990, pp. 85–153.
G. Harder and N. Schappacher, Special values of Hecke L-functions and abelian integrals, In: Workshop Bonn 1984, Bonn, 1984, Lecture Notes in Math., 1111, Springer-Verlag, 1985, pp. 17–49.
G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers. Sixth ed., Revised by D.R. Heath-Brown and J.H. Silverman. With a foreword by A. Wiles, Oxford Univ. Press, Oxford, 2008.
R. Howe, θ-series and invariant theory, In: Automorphic Forms, Representations and L-Functions, Oregon State Univ., Corvallis, OR, 1977, Proc. Sympos. Pure Math., 33, Part 1, Amer. Math. Soc., Providence, RI, 1979, pp. 275–285.
T. Ishii and T. Oda, A short history on investigation of the special values of zeta and L-functions of totally real number fields, In: Automorphic Forms and Zeta Functions, World Sci. Publ., Hackensack, NJ, 2006, pp. 198–233.
H. Ito, A function on the upper half space which is analogous to the imaginary part of log η(z), J. Reine Angew. Math., 373 (1987) 148–165.
K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, In: Coho-mologies p-adiques et applications arithmétiques. III, Astérisque, 295, Soc. Math. France, Paris, 2004, pp. 117–290.
H. Klingen, Über die Werte der Dedekindschen Zetafunktion, Math. Ann., 145 (1962), 265–272.
S.S. Kudla, Seesaw dual reductive pairs, In: Automorphic Forms of Several Variables, Katata, 1983, Progr. Math., 46, Birkhäuser Boston, Boston, MA, 1984, pp. 244–268.
S.S. Kudla and J.J. Millson, Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, Inst. Hautes Études Sci. Publ. Math., 71 (1990) 121–172.
V. Mathai and D. Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology, 25 (1986) 85–110.
W. Müller, Signature defects of cusps of Hilbert modular varieties and values of L-series at s = 1, J. Differential Geom., 20 (1984) 55–119.
M.V. Nori, Some Eisenstein cohomology classes for the integral unimodular group, In: Proceedings of the International Congress of Mathematicians. Vol. 1, 2, Zürich, 1994, Birkhäuser, Basel, 1995, pp. 690–696.
R. Sczech, Dedekindsummen mit elliptischen Funktionen, Invent. Math., 76 (1984) 523–551.
R. Sczech, Eisenstein cocycles for GL2Q and values of L-functions in real quadratic fields, Comment. Math. Helv., 67 (1992) 363–382.
R. Sczech, Eisenstein group cocycles for GLn and values of L-functions, Invent. Math., 113 (1993) 581–616.
T. Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23 (1976) 393–417.
C.L. Siegel, Über die analytische Theorie der quadratischen Formen. III, Ann. of Math. (2), 38 (1937), 212–291.
C.L. Siegel, Advanced Analytic Number Theory. Second ed., Tata Inst. Fundam. Res. Stud. Math., 9, Tata Inst. Fundam. Res., Bombay, 1980.
C.L. Siegel, Gesammelte Abhandlungen. IV, (eds. K. Chandrasekharan and H. Maaß), Springer Collect. Works Math., Springer-Verlag, 2015. Reprint of the 1979 ed., Original publication incorrectly given as 1966 ed. on the title page.
D. Sullivan, La classe d’Euler réelle d’un fibré vectoriel à groupe structural SLn (Z) est nulle, C. R. Acad. Sci. Paris Sér. A-B, 281 (1975) Aii, A17–A18.
A. Weil, Elliptic Functions According to Eisenstein and Kronecker, Classics Math., Springer-Verlag, 1999. Reprint of the 1976 original.
F. Wielonsky, Séries d’Eisenstein, intégrales toroïdales et une formule de Hecke, Enseign. Math. (2), 31 (1985), 93–135.
Acknowledgements
N.B. would like to thank the Mathematical Society of Japan, the local organisers, and especially Professor Kobayashi, for their invitation and their kind hospitality in Kyoto. We all thank our collaborator Akshay Venkatesh as well as Javier Fresan for their comments and corrections on these notes. L.G. wishes to thank IHES for providing excellent conditions for research while this work was done. L.G. also acknowledges financial support from the ERC AAMOT Advanced Grant.
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Communicated by: Toshiyuki Kobayashi
This article is based on the 21st Takagi Lectures that the first author delivered at Research Institute for Mathematical Sciences, Kyoto University on June 23, 2018.
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Bergeron, N., Charollois, P. & Garcia, L.E. Transgressions of the Euler class and Eisenstein cohomology of GLN(Z). Jpn. J. Math. 15, 311–379 (2020). https://doi.org/10.1007/s11537-019-1822-6
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DOI: https://doi.org/10.1007/s11537-019-1822-6
Keywords and phrases
- cohomology of arithmetic groups
- characteristic classes
- Eisenstein series
- special values of automorphic L-series