Algebra & Number Theory

$L$-functions and periods of adjoint motives

Michael Harris

Full-text: Open access

Abstract

The article studies the compatibility of the refined Gross–Prasad (or Ichino–Ikeda) conjecture for unitary groups, due to Neal Harris, with Deligne’s conjecture on critical values of L-functions. When the automorphic representations are of motivic type, it is shown that the L-values that arise in the formula are critical in Deligne’s sense, and their Deligne periods can be written explicitly as products of Petersson norms of arithmetically normalized coherent cohomology classes. In some cases this can be used to verify Deligne’s conjecture for critical values of adjoint type (Asai) L-functions.

Article information

Source
Algebra Number Theory, Volume 7, Number 1 (2013), 117-155.

Dates
Received: 10 July 2011
Revised: 12 October 2011
Accepted: 20 February 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729931

Digital Object Identifier
doi:10.2140/ant.2013.7.117

Mathematical Reviews number (MathSciNet)
MR3037892

Zentralblatt MATH identifier
1319.11028

Subjects
Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18] 11G09: Drinfelʹd modules; higher-dimensional motives, etc. [See also 14L05]

Keywords
adjoint $L$-functions automorphic forms motives Ichino–Ikeda conjecture periods

Citation

Harris, Michael. $L$-functions and periods of adjoint motives. Algebra Number Theory 7 (2013), no. 1, 117--155. doi:10.2140/ant.2013.7.117. https://projecteuclid.org/euclid.ant/1513729931


Export citation

References

  • D. Blasius, M. Harris, and D. Ramakrishnan, “Coherent cohomology, limits of discrete series, and Galois conjugation”, Duke Math. J. 73:3 (1994), 647–685.
  • G. Chenevier and M. Harris, “Construction of automorphic Galois representations, II”, 2013. To appear in Cambridge J. Math.
  • L. Clozel, “Motifs et formes automorphes: applications du principe de fonctorialité”, pp. 77–159 in Automorphic forms, Shimura varieties, and $L$-functions (Ann Arbor, MI, 1988), vol. 1, edited by L. Clozel and J. S. Milne, Perspect. Math. 10, Academic Press, Boston, MA, 1990.
  • L. Clozel, M. Harris, and R. Taylor, “Automorphy for some $l$-adic lifts of automorphic mod $l$ Galois representations”, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181.
  • L. Clozel, M. Harris, and J.-P. Labesse, “Construction of automorphic Galois representations, I”, pp. 497–527 in Stabilization of the trace formula, Shimura varieties, and arithmetic applications, I: On the stabilization of the trace formula, edited by L. Clozel et al., International Press, 2011.
  • P. Deligne, “Valeurs de fonctions $L$ et périodes d'intégrales”, pp. 313–346 in Automorphic forms, representations and $L$-functions (Corvallis, OR, 1977), vol. 2, edited by A. Borel and W. Casselman, Proc. Sympos. Pure Math. 33, Amer. Math. Soc., Providence, R.I., 1979.
  • P. Deligne, “Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques”, pp. 247–289 in Automorphic forms, representations and $L$-functions (Corvallis, OR, 1977), vol. 2, edited by A. Borel and W. Casselman, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R.I., 1979.
  • F. Diamond, M. Flach, and L. Guo, “The Tamagawa number conjecture of adjoint motives of modular forms”, Ann. Sci. École Norm. Sup. $(4)$ 37:5 (2004), 663–727.
  • M. Dimitrov, “On Ihara's lemma for Hilbert modular varieties”, Compos. Math. 145:5 (2009), 1114–1146.
  • W. Fulton and J. Harris, Representation theory, Graduate Texts in Mathematics 129, Springer, New York, 1991.
  • W. T. Gan, B. Gross, and D. Prasad, “Symplectic local root numbers, central critical $L$-values, and restriction problems in the representation theory of classical groups”, pp. 1–110 in Sur les conjectures de Gross et Prasad, Astérisque 346, Société Mathématique de France, Paris, 2012.
  • W. T. Gan, B. Gross, and D. Prasad, “Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups”, pp. 111–-170 in Sur les conjectures de Gross et Prasad, Astérisque 346, Société Mathématique de France, Paris, 2012.
  • M. Harris, “$L$-functions and periods of polarized regular motives”, J. Reine Angew. Math. 483 (1997), 75–161.
  • M. Harris, “Cohomological automorphic forms on unitary groups, II: Period relations and values of $L$-functions”, pp. 89–149 in Harmonic analysis, group representations, automorphic forms and invariant theory, edited by J.-S. Li et al., Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 12, World Sci. Publ., Hackensack, NJ, 2007.
  • M. Harris, “A simple proof of rationality of Siegel–Weil Eisenstein series”, pp. 149–185 in Eisenstein series and applications, edited by W. T. Gan et al., Progr. Math. 258, Birkhäuser, Boston, MA, 2008.
  • M. Harris, “Beilinson–Bernstein localization over $\mathbb{Q}$ and periods of automorphic forms”, Int. Math. Res. Notices (2012).
  • M. Harris, J.-S. Li, and B. Sun, “Theta correspondences for close unitary groups”, pp. 265–307 in Arithmetic geometry and automorphic forms, edited by J. Cogdell et al., Adv. Lect. Math. (ALM) 19, International Press, 2011.
  • H. Hida, “Congruence of cusp forms and special values of their zeta functions”, Invent. Math. 63:2 (1981), 225–261.
  • A. Ichino and T. Ikeda, “On the periods of automorphic forms on special orthogonal groups and the Gross–Prasad conjecture”, Geom. Funct. Anal. 19:5 (2010), 1378–1425.
  • J.-P. Labesse, “Changement de base CM et séries discrètes”, pp. 429–470 in Stabilization of the trace formula, Shimura varieties, and arithmetic applications, I: On the stabilization of the trace formula, edited by L. Clozel et al., International Press, 2011.
  • A. Mínguez, “Unramified representations of unitary groups”, pp. 389–410 in Stabilization of the trace formula, Shimura varieties, and arithmetic applications, I: On the stabilization of the trace formula, edited by L. Clozel et al., International Press, 2011.
  • C. Mœglin, “Classification et changement de base pour les séries discrètes des groupes unitaires $p$-adiques”, Pacific J. of Mathematics 233:1 (2007), 159–204.
  • C. Moeglin and J.-L. Waldspurger, “La conjecture locale de Gross–Prasad pour les groupes spéciaux orthogonaux : le cas général”, in Sur les conjectures de Gross et Prasad, II, Astérisque 347, Société Mathématique de France, Paris, 2012.
  • N. Harris, A refined Gross–Prasad conjecture for unitary groups, thesis, University of California, San Diego, 2011.
  • J.-L. Waldspurger, “Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie”, Compositio Math. 54:2 (1985), 173–242.
  • P.-J. White, Le produit tensoriel automorphe et l'endoscopie sur le groupe unitaire, thesis, Université Paris Diderot- Paris 7, 2010.