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On higher regulators of Siegel threefolds II: the connection to the special value

Part of: Arithmetic algebraic geometry $K$-theory in number theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  27 March 2017

Francesco Lemma
Francesco Lemma*
Affiliation:
Institut mathématique de Jussieu-Paris Rive Gauche, UMR 7586, Bâtiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France email [email protected]
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Abstract

We establish a connection between motivic cohomology classes over the Siegel threefold and non-critical special values of the degree-four $L$-function of some cuspidal automorphic representations of $\text{GSp}(4)$. Our computation relies on our previous work [On higher regulators of Siegel threefolds I: the vanishing on the boundary, Asian J. Math. 19 (2015), 83–120] and on an integral representation of the $L$-function due to Piatetski-Shapiro.

Keywords

Beilinson conjectureSiegel threefoldspinor L-function

MSC classification

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 19F27: Étale cohomology, higher regulators, zeta and $L$-functions
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 5 , May 2017 , pp. 889 - 946
Copyright
© The Author 2017 

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References

Ancona, G., Décomposition de motifs abéliens , Manuscripta Math. 146 (2015), 307–328.CrossRefGoogle Scholar
Arthur, J., Automorphic representations of GSp(4) , in Contributions to automorphic forms, geometry and number theory (John Hopkins University Press, Baltimore, MD, 2004), 65–81.Google Scholar
Asgari, M. and Schmidt, R., Siegel modular forms and representations , Manuscripta Math. 104 (2001), 173–200.Google Scholar
Beals, R. and Szmigielski, J., Meijer G-functions: a gentle introduction , Notices Amer. Math. Soc. 60 (2013), 866–872.Google Scholar
Beilinson, A. A., Higher regulators and values of L-functions , J. Soviet Math. 30 (1985), 2036–2070.CrossRefGoogle Scholar
Beilinson, A. A., Notes on absolute Hodge cohomology , in Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II, Contemporary Mathematics, vol. 55 (American Mathematical Society, Providence, RI, 1986), 35–68.CrossRefGoogle Scholar
Beilinson, A. A., Higher regulators of modular curves , in Applications of algebraic K-theory to algebraic geometry and number theory, Contemporary Mathematics, vol. 55 (American Mathematical Society, Providence, RI, 1988), 1–34.Google Scholar
Blasius, D., Harris, M. and Ramakrishnan, D., Coherent cohomology, limits of discrete series and Galois conjugation , Duke Math. J. 73 (1994), 647–685.Google Scholar
Borel, A., Stable real cohomology of arithmetic groups II , in Manifolds and Lie groups: papers in honor of Y. Matsushima, Progress in Mathematics, vol. 14 (Birkhäuser, Boston, MA, 1981), 21–55.CrossRefGoogle Scholar
Borel, A., Regularization theorems in Lie algebra cohomology. Applications , Duke Math. J. 50 (1983), 605–624.Google Scholar
Borel, A. and Wallach, N. R., Continous cohomology, discrete subgroups and representations of reductive groups, Annals of Mathematical Studies, vol. 94 (Princeton University Press, Princeton, NJ, 1980).Google Scholar
Bump, D., Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55 (Cambridge University Press, Cambridge, 1997), 574.Google Scholar
Bump, D., Friedberg, S. and Furusawa, M., Explicit formulas for the Waldspurger and Bessel models , Israel J. Math. 102 (1997), 125–177.Google Scholar
Casselman, W., The unramified principal series of p-adic groups I: the spherical function , Compositio Math. 40 (1980), 387–406.Google Scholar
Cisinski, D.-C. and Déglise, F., Triangulated categories of mixed motives, Preprint (2009),arXiv:0912.2110.Google Scholar
Deligne, P., Théorie de Hodge II , Publ. Math. Inst. Hautes Études Sci. 40 (1974), 5–77.Google Scholar
Deligne, P., Valeurs de fonctions L et périodes d’intégrales, Proceedings of Symposia in Pure Mathematics, vol. 33, Part 2 (American Mathematical Society, Providence, RI, 1979), 313–346.Google Scholar
Deninger, C., Higher regulators and Hecke L-series of imaginary quadratic fields I , Invent. Math. 96 (1989), 1–70.Google Scholar
Deninger, C., Higher regulators and Hecke L-series of imaginary quadratic fields II , Ann. of Math. (2) 132 (1990), 131–158.Google Scholar
Deninger, C. and Scholl, A. J., The Beilinson conjectures , in L-functions and arithmetic (Durham, 1989), London Mathematical Society Lecture Note Series, vol. 153 (Cambridge University Press, Cambridge, 1991), 173–209.Google Scholar
Esnault, H. and Viehweg, E., Deligne–Beilinson cohomology , in Beilinson’s conjectures on special values of L-functions, Perspectives in Mathematics, vol. 4 (Academic Press, Boston, MA, 1988), 43–91.Google Scholar
Fulton, W. and Harris, J., Representation theory: a first course, Graduate Texts in Mathematics, vol. 129 (Springer, New York, 1991).Google Scholar
Harris, M., Automorphic forms and the cohomology of vector bundles on Shimura varieties , in Automorphic forms, Shimura varieties and L-functions II, Perspectives in Mathematics, vol. 11 (Academic Press, Boston, MA, 1990), 41–91.Google Scholar
Harris, M., Hodge–de Rham structures and periods of automorphic forms , in Motives, Proceedings of Symposia in Pure Mathematics, vol. 55, Part 2 (American Mathematical Society, Providence, RI, 1994), 573–624.Google Scholar
Harris, M., L-functions and periods of polarized regular motives , J. Reine Angew. Math. 483 (1997), 75–161.Google Scholar
Harris, M., Occult period invariants and critical values of the degree four L-function of GSp(4) , in Contributions to automorphic forms, geometry and number theory (John Hopkins University Press, Baltimore, MD, 2004), 331–354.Google Scholar
Harris, M. and Zucker, S., Boundary cohomology of Shimura varieties III. Coherent cohomology on higher-rank boundary strata and applications to Hodge theory , in Mémoires de la Société Mathématique de France (Société Mathématique de France, Paris, 1999).Google Scholar
Huber, A. and Wildeshaus, J., Classical motivic polylogarithm according to Beilinson and Deligne , Doc. Math. 3 (1998), 27–133.Google Scholar
Huber, A. and Wildeshaus, J., Correction to the paper: ‘Classical motivic polylogarithm according to Beilinson and Deligne’ , Doc. Math. 3 (1998), 297–299.Google Scholar
Jannsen, U., Deligne homology, Hodge-D-conjecture and motives , in Beilinson’s conjectures on special values of L-functions, Perspectives in Mathematics, vol. 4 (Academic press, Boston, MA, 1988), 305–372.CrossRefGoogle Scholar
Kato, K., p-adic Hodge theory and values of zeta functions of modular forms , Astérisque 295 (2004), 117–290.Google Scholar
Kings, G., Higher regulators, Hilbert modular surfaces and special values of L-functions , Duke Math. J. 92 (1998), 61–127.Google Scholar
Knapp, A. W., Representation theory of semisimple Lie groups: an overview based on examples, Princeton Mathematical Series, vol. 36 (Princeton University Press, Princeton, NJ, 1986), 1–774.CrossRefGoogle Scholar
Lemma, F., Régulateurs supérieurs, périodes et valeurs spéciales de la fonction L de degré 4 de GSp(4) , C. R. Math. Acad. Sci. Paris 346 (2008), 1023–1028.Google Scholar
Lemma, F., On higher regulators of Siegel threefolds I: the vanishing on the boundary , Asian J. Math. 19 (2015), 83–120.Google Scholar
Milne, J. S. and Shih, K.-y., The action of complex conjugation on a Shimura variety , Ann. of Math. (2) 113 (1981), 569–599.Google Scholar
Mokrane, F. and Tilouine, J., Cohomology of Siegel varieties , Astérisque 280 (2002), 1–135.Google Scholar
Molev, A., Yangians and classical Lie algebras, Mathematical Surveys and Monographs, vol. 142 (American Mathematical Society, Providence, RI, 2007), 1–400.Google Scholar
Moriyama, T., Entireness of the spinor L-functions for certain generic cusp forms on GSp(2) , Amer. J. Math. 126 (2004), 899–920.CrossRefGoogle Scholar
Moriyama, T., Generalized Whittaker functions on GSp(2, ℝ) associated with indefinite quadratic forms , J. Math. Soc. Japan 63 (2011), 1203–1262.Google Scholar
Nekovar, J., Beilinson’s conjectures , in Motives, Proceedings of Symposia in Pure Mathematics, vol. 55, Part 1 (American Mathematical Society, Providence, RI, 1994), 537–570.CrossRefGoogle Scholar
Pépin Lehalleur, S., The motivic $t$ -structure for relative $1$ -motives, Preprint (2015),arXiv:1512.00266.Google Scholar
Piatetski-Shapiro, I. I., L-functions for GSp4 , O. Taussky-Todd: in memoriam , Pacific J. Math. 181(3) (1997), 259–275.Google Scholar
Pink, R., Arithmetical compactifications of mixed Shimura varieties , Bonner Math. Schriften 209 (1990), available at http://www.math.ethz.ch/∼pink/dissertation.html.Google Scholar
Scholbach, J., Arakelov motivic cohomology II , J. Algebr. Geom. 24 (2015), 755–786.CrossRefGoogle Scholar
Takloo-Bighash, R., L-functions for the p-adic group GSp(4) , Amer. J. Math. 122 (2000), 1085–1120.CrossRefGoogle Scholar
Tate, J. T., Fourier analysis in number fields and Hecke’s zeta function , in Algebraic number theory, eds Cassels, J. W. S. and Frölich, A. (Academic Press, New York, 1967), 305–347.Google Scholar
Taylor, R., On the l-adic cohomology of Siegel threefolds , Invent. Math. 114 (1993), 289–310.Google Scholar
Voevodsky, V., Suslin, A. and Friedlander, E. M., Cycles, transfers and motivic homology theories, Annals of Mathematic Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000).Google Scholar
Vogan, D. and Zuckermann, G., Unitary representations with non-zero cohomology , Compositio Math. 53 (1984), 51–90.Google Scholar
Weissauer, R., Four dimensional Galois representations , Astérisque 302 (2005), 67–150.Google Scholar