Abstract
This paper gives explicit formulas for the Fourier expansion of general Eisenstein series and local Whittaker functions over SL2. They are used to compute both the value and derivatives of these functions at critical points.
Similar content being viewed by others
References
Abramowitz M, Stegun I. Pocketbook of Mathematical Functions. Abridged edition of Handbook of mathematical functions. Material selected by Michael Danos and Johann Rafelski. Thun: Verlag Harri Deutsch, 1984
Borcherds R E. Automorphic forms with singularities on Grassmannians. Invent Math, 1998, 132: 491–562
Bruinier J, Burgos Gil J, Kühn U. Borcherds products and arithmetic intersection theory on Hilbert modular surfaces. Duke Math J, 2007, 139: 1–88
Bruinier J H, Funke J. On two geometric theta lifts. Duke Math J, 2004, 125: 45–90
Bruinier J, Kudla S, Yang T H. Faltings heights of Big CM cycles and derivatives of L-functions. Preprint, 2010
Bruinier J, Kühn U. Integrals of automorphic Green’s functions associated to Heegner divisors. IMRN, 2003, 31: 1687–1729
Bruinier J, Yang T H. CM-values of Hilbert modular functions. Invent Math, 2006, 163: 229–288
Bruinier J, Yang T H. Faltings heights of CM cycles and derivatives of L-functions. Invent Math, 2009, 177: 631–681
Bruinier J, Yang T H. CM values of automorphic Green functions on orthogonal groups over totally real fields. Arithmetic Geometry and Automorphic Forms, to appear
Cassels J W S. Rational Quadratic Forms, London Mathematical Society Monographs, 13. London-New York: Academic Press, 1978.
Cohen H. Sums involving the values at negative integers of L-functions of quadratic characters. Math Ann, 1975, 217: 271–285
Errthum E. Singular moduli of Shimura curves. PhD Thesis, University of Maryland, 2007
Gross B, Zagier D. On singular moduli. J Reine Angew Math, 1985, 355: 191–220
Gross B, Zagier D. Heegner points and derivatives of L-series. Invent Math, 1986, 84: 225–320
Hirzebruch F, Zagier D. Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus. Invent Math, 1976, 36: 57–113
Kitaoka Y. Arithmetic of quadratic forms. In: Cambridge Tracts in Mathematics 106. Cambridge: Cambridge University Press, 1993
Kudla S. Central derivatives of Eisenstein series and height pairings. Ann of Math, 1997, 146: 545–646
Kudla S. Integrals of Borcherds forms. Compositio Math, 2003, 137: 293–349
Kudla S. Splitting metaplectic covers of dual reductive pairs. Israel J Math, 1994, 87: 361–401
Kudla S, Rallis S. On the Weil-Siegel formula. J Reine Angew Math, 1988, 387: 1–68
Kudla S, Rallis S. On the Weil-Siegel formula. II. The isotropic convergent case. J Reine Angew Math, 1988, 391: 65–84
Kudla S, Rapoport M, Yang T H. On the derivative of an Eisenstein series of weight one. IMRN, 1999, 7: 347–385
Kudla S, Rapoport M, Yang T H. Derivatives of Eisenstein series and Faltings heights. Compositio Math, 2004, 140: 887–951
Kudla S, Rapoport M, Yang T H. Modular forms and special cycles on Shimura curves. In: Annals of Math Studies Series, vol 161, Princeton University Press, 2006
Kudla S, Yang T H. On the pullback of an arithmetic theta function I. Preprint, 2010
Langlands R. On the Functional Equations Satisfied by Eisenstein Series. In: Lecture Notes in Mathematics, vol. 544. Berlin-New York: Springer-Verlag, 1976
Lebedev N N. Special Functions and Their Applications. New York: Dover, 1972
Rallis S. On the Howe duality conjecture, Compositio Math, 1984, 51: 333–399
Ranga Rao R. On some explicit formulas in the theory of Weil representation. Pacific J Math, 1993, 157: 335–371
Schofer J. Borcherds forms and generalizations of singular moduli. J Reine Angew Math, 2009, 629: 1–36
Siegel C L. Über die analytische Theorie der quadratischen Formen. II. Ann of Math, 1936, 37: 230–263
Weil A. Sur la formule de Siegel dans la théorie des groupes classiques. Acta Math, 1965, 113: 1–7
Yang T H. An explicit formula for local densities of quadratic forms. J Number Theory, 1998, 72: 309–356
Zagier D. Nombres de classes et formes modulaires de poids 3/2. C R Acad Sci Paris Sr A-B, 1975, 281: 883–886
Zhao L L. Period integral of automorphic Green functions. PhD Thesis, University of Wisconsin, 2011
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Wang Yuan on the Occasion of his 80th Birthday
Rights and permissions
About this article
Cite this article
Kudla, S.S., Yang, T. Eisenstein series for SL(2). Sci. China Math. 53, 2275–2316 (2010). https://doi.org/10.1007/s11425-010-4097-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-010-4097-1