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Siegel modular varieties and the Eisenstein cohomology of PGL2g+1

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We use the twisted topological trace formula developed in Weselmann (Compos. Math. 148:65–120, 2012) to understand liftings from symplectic to general linear groups. We analyse the lift from SP2g to PGL2g+1 over the ground field \({\mathbb{Q}}\) in further detail, and we get a description of the image of this lift of the L 2 cohomology of SP2g (which is related to the intersection cohomology of the Shimura variety attached to GSp2g ) in terms of the Eisenstein cohomology of the general linear group, whose building constituents are cuspidal representations of Levi groups. This description may be used to understand endoscopic and CAP-representations of the symplectic group.

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References

  1. Arthur, J.: The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, vol 61. American Mathematical Society Colloquium Publications, Providence, Rhode Island (2013). Preliminary version available on http://www.claymath.org/cw/arthur/pdf/arthur-book-2012b

  2. Ballmann, J., Weissauer, R., Weselmann, U.: Remarks on the fundamental lemma for stable twisted Endoscopy of classical groups. Manuskripte der Forschergruppe Arithmetik, vol. 7 (2002) Mannheim–Heidelberg. arXiv:1302.0034

  3. Bergström J., Faber C., van der Geer G.: Siegel modular forms of degree three and the cohomology of local systems. Selecta Mathematica 20, 83–124 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  4. Borel A.: Sur La Cohomologie des Espaces Fibres Principaux et des Espaces Homogenes de Groupes de Lie Compacts. Ann. Math. 57, 115–207 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  5. Borel, A., Wallach, N.: Continous Cohomology, Discrete Subgroups and Representations of Reductive Groups. Ann. Math. Stud. 94. Princeton University Press Princeton (1980)

  6. Cogdell J.W., Kim H., Piatetski-Shapiro I.I., Shahidi F.: On lifting from classical groups to GL N . Publ. Math. IHES 93, 5–30 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cogdell J.W., Kim H., Piatetski-Shapiro I.I., Shahidi F.: Functoriality for the classical groups. Publ. Math. IHES 99, 163–233 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cogdell, J.W., Piatetski-Shapiro, I.I., Shahidi, F.: Functoriality for the quasisplit classical groups. In: Arthur, J. (ed.) et al. On Certain L-functions. Conference in Honor of Freydoon Shahidi on the Occasion of his 60th Birthday, Clay Mathematics Proceedings vol. 13, pp. 117–140. (2011)

  9. Flicker Y.F.: On the symmetric square: total global comparison. J. Funct. Anal. 122, 255–278 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Franke J.: Harmonic analysis in weighted L 2-spaces. Ann. scient. École Norm. Sup. 31, 181–279 (1998)

    MATH  MathSciNet  Google Scholar 

  11. Franke J., Schwermer J.: A decomposition of spaces of automorphic forms and the Eisenstein cohomology of arithmetic groups. Math. Ann. 311, 765–790 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gelbart S., Jacquet H.: A relation between automorphic representations of GL(2) and GL(3). Ann. Sci. Éc. Norm. Supér. (4) 11, 471–542 (1978)

    MATH  MathSciNet  Google Scholar 

  13. Greub W., Halperin St., Vanstone R.: Connections, Curvature and Cohomology, Vol II: Cohomology of Principal Bundles and Homogeneous Spaces. Academic Press, New York, London (1973)

    Google Scholar 

  14. Greub W., Halperin St., Vanstone R.: Connections, Curvature and Cohomology, Vol III: Lie Groups, Principal Bundles and Characteristic Classes Academic Press. Academic Press, New York, London (1976)

    Google Scholar 

  15. Grobner, H., Raghuram, A.: On some arithmetic properties of automorphic forms of GL m over a division algebra. Int. J. Number Theory 10 (2014), doi:10.1142/S1793042114500110

  16. Ikeda T.: On the lifting of elliptic modular forms to Siegel cusp forms of degree 2n. Ann. Math. (2) 154, 641–681 (2001)

    Article  MATH  Google Scholar 

  17. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Grad. Texts in Math. 9, New York (1972)

  18. Jacquet H., Shalika J.A.: On Euler products and the classification of automorphic forms I. Am. J. Math. 103, 499–558 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jacquet H., Shalika J.A.: On Euler products and the classification of automorphic forms II. Am. J. Math. 103, 777–815 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  20. Knapp W.A.: Representation Theory of Semisimple Groups. Princeton University Press, Princeton (1986)

    MATH  Google Scholar 

  21. Kohnen W., Kojima H.: A maass space in higher genus. Compos. Math. 141, 313–323 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kostant B.: Lie algebra cohomology and the generalized Borel–Weil theorem. Ann. Math. 74, 329–387 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kottwitz, R., Shelstad, D.: Foundations of twisted endoscopys. Astérisque 255. Societe Mathematique de France, Paris (1999)

  24. Langlands, R.P.: On the classification of irreducible representations of real algebraic groups. Math. Surveys Monographs 31. American Mathematical Society, Providence, RI (1988)

  25. Mœglin C., Waldspurger J.-L.: Le spectre résiduel de GL(n). Ann. Sci. École Norm. Sup. (4) 22, 605–674 (1989)

    MATH  Google Scholar 

  26. Morel S.: Cohomologie d’intersection des variétés modulaires de Siegel, suite. Compos. Math. 147, 1671–1740 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Piatetski-Shapiro I.I.: On the Saito-Kurokawa lifting. Invent. math. 71, 309–338 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  28. Ramakrishnan D.: Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2). Ann. Math. 152, 45–111 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. Ramakrishnan D., Shahidi F.: Siegel modular forms of genus 2 attached to elliptic curves. Math. Res. Lett. 14, 315–332 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Schwermer J.: On arithmetic quotients of the Siegel upper half space of degree two. Compos. Math. 58, 233–258 (1986)

    MATH  MathSciNet  Google Scholar 

  31. Schwermer J.: On Euler products and residual Eisenstein cohomology classes for Siegel modular varieties. Forum Math. 7, 1–28 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  32. Soudry D.: The CAP representations of \({{\rm GSp}(4,\mathbb{A})}\). J. Reine Angew. Math. 383, 87–108 (1988)

    MATH  MathSciNet  Google Scholar 

  33. Soudry, D.: On Langlands functoriality from classical groups to GL n . In: Tilouine, J. (ed.) et al. Automorphic Forms (I) Proceedings of the Semester of the Émile Borel Center, Paris. Astérisque, vol. 298, pp. 335–390 (2005)

  34. Vogan D., Zuckerman G.: Unitary representations with nonzero cohomology. Compos. Math. 53, 51–90 (1984)

    MATH  MathSciNet  Google Scholar 

  35. Weissauer, R.: Existence of Whittaker models related to four dimensional symplectic Galois representations. In: Edixhoven, B. et al. (ed.) Modular forms on Schiermonnikoog, pp. 285–310. Cambridge University Press (2008)

  36. Weissauer, R.: Endoscopy for GSp(4) and the Cohomology of Siegel Modular Threefolds. Springer Lect. Notes Math. 1968. Berlin, Heidelberg (2009)

  37. Weselmann U.: A twisted topological trace formula for Hecke operators and liftings from symplectic to general linear groups. Compos. Math. 148, 65–120 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Mathematisches Institut, Im Neuenheimer Feld 288, 69121, Heidelberg, Germany

    Uwe Weselmann

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  1. Uwe Weselmann
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Weselmann, U. Siegel modular varieties and the Eisenstein cohomology of PGL2g+1 . manuscripta math. 145, 175–220 (2014). https://doi.org/10.1007/s00229-014-0676-8

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