Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T01:14:04.934Z Has data issue: false hasContentIssue false
  • English
  • Français

EULER SYSTEMS FOR HILBERT MODULAR SURFACES

Part of: Algebraic number theory: global fields Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  15 November 2018

ANTONIO LEI Open the ORCID record for ANTONIO LEI [Opens in a new window] ,
DAVID LOEFFLER Open the ORCID record for DAVID LOEFFLER [Opens in a new window]  and
SARAH LIVIA ZERBES Open the ORCID record for SARAH LIVIA ZERBES [Opens in a new window]
ANTONIO LEI
Affiliation:
Département de mathématiques et de statistique, Pavillon Alexandre-Vachon, Université Laval, Québec, QC, Canada G1V 0A6; [email protected]
DAVID LOEFFLER
Affiliation:
Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK; [email protected]
SARAH LIVIA ZERBES
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK; [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct an Euler system—a compatible family of global cohomology classes—for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is nontrivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.

MSC classification

Primary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F80: Galois representations 11R23: Iwasawa theory
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 6 , 2018 , e23
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2018

References

Ancona, G., ‘Décomposition de motifs abéliens’, Manuscripta Math. 146(3–4) (2015), 307–328.Google Scholar
Asai, T., ‘On certain Dirichlet series associated with Hilbert modular forms and Rankin’s method’, Math. Ann. 226(1) (1977), 81–94.Google Scholar
Beĭlinson, A., Higher Regulators and Values of L-Functions, Current Problems in Mathematics, 24 (Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984), 181–238.Google Scholar
Bertolini, M., Darmon, H. and Rotger, V., ‘Beilinson–Flach elements and Euler systems II: the Birch and Swinnerton–Dyer conjecture for Hasse–Weil–Artin L-functions’, J. Algebraic Geom. 24(3) (2015), 569–604.Google Scholar
Bloch, S. and Kato, K., ‘ L-functions and Tamagawa numbers of motives’, inThe Grothendieck Festschrift, Vol. I (ed. Cartier, P. et al. ) Progress in Mathematics, 86 (Birkhäuser, Boston, MA, 1990), 333–400.Google Scholar
Brauer, R. and Nesbitt, C., ‘On the modular characters of groups’, Ann. of Math. (2) 42 (1941), 556–590.Google Scholar
Brunault, F. and Chida, M., ‘Regulators for Rankin–Selberg products of modular forms’, Ann. Math. Québec 40(2) (2016), 221–249.Google Scholar
Brylinski, J.-L. and Labesse, J.-P., ‘Cohomologie d’intersection et fonctions L de certaines variétés de Shimura’, Ann. Sci. Éc. Norm. Supér. (4) 17(3) (1984), 361–412.Google Scholar
Darmon, H. and Rotger, V., ‘Diagonal cycles and Euler systems I: a p-adic Gross–Zagier formula’, Ann. Sci. Éc. Norm. Supér. (4) 47(4) (2014), 779–832.Google Scholar
Dembélé, L., Loeffler, D. and Pacetti, A., ‘Non-paritious Hilbert modular forms’, Preprint, 2016, arXiv:1612.06625.Google Scholar
Deninger, C. and Murre, J., ‘Motivic decomposition of abelian schemes and the Fourier transform’, J. Reine Angew. Math. 422 (1991), 201–219.Google Scholar
Dimitrov, M., ‘Galois representations modulo p and cohomology of Hilbert modular varieties’, Ann. Sci. Éc. Norm. Supér. (4) 38(4) (2005), 505–551.Google Scholar
Dimitrov, M., ‘On Ihara’s lemma for Hilbert modular varieties’, Compos. Math. 145(5) (2009), 1114–1146.Google Scholar
Grothendieck, A., ‘On the de Rham cohomology of algebraic varieties’, Publ. Math. Inst. Hautes Études Sci. 29 (1966), 95–103.Google Scholar
Huber, A. and Wildeshaus, J., ‘Classical motivic polylogarithm according to Beilinson and Deligne’, Doc. Math. 3 (1998), 27–133.Google Scholar
Im, J., ‘Twisted tensor L-functions attached to Hilbert modular forms’, inElliptic Curves and Related Topics, CRM Proceedings & Lecture Notes, 4 (American Mathematical Society, Providence, RI, 1994), 111–119.Google Scholar
Kato, K., ‘ P-adic Hodge theory and values of zeta functions of modular forms’, Astérisque 295 (2004), 117–290. Cohomologies $p$ -adiques et applications arithmétiques. III.Google Scholar
Kings, G., ‘Higher regulators, Hilbert modular surfaces, and special values of L-functions’, Duke Math. J. 92(1) (1998), 61–127.Google Scholar
Kings, G., ‘Eisenstein classes, elliptic Soulé elements and the ℓ-adic elliptic polylogarithm’, inThe Bloch–Kato Conjecture for the Riemann Zeta Function (ed. Coates, J. et al. ) London Mathematical Society Lecture Note Series, 418 (Cambridge University Press, Cambridge, UK, 2015).Google Scholar
Kings, G., Loeffler, D. and Zerbes, S. L., ‘Rankin–Eisenstein classes for modular forms’, Amer. J. Math. (2015), to appear, arXiv:1501.03289.Google Scholar
Kings, G., Loeffler, D. and Zerbes, S. L., ‘Rankin–Eisenstein classes and explicit reciprocity laws’, Cambridge J. Math. 5(1) (2017), 1–122.Google Scholar
Lapid, E. and Rogawski, J., ‘On twists of cuspidal representations of GL(2)’, Forum Math. 10(2) (1998), 175–197.Google Scholar
Lei, A., Loeffler, D. and Zerbes, S. L., ‘Coleman maps and the p-adic regulator’, Algebra Number Theory 5(8) (2011), 1095–1131.Google Scholar
Lei, A., Loeffler, D. and Zerbes, S. L., ‘Euler systems for Rankin–Selberg convolutions of modular forms’, Ann. of Math. (2) 180(2) (2014), 653–771.Google Scholar
Lei, A., Loeffler, D. and Zerbes, S. L., ‘Euler systems for modular forms over imaginary quadratic fields’, Compos. Math. 151(9) (2015), 1585–1625.Google Scholar
Liu, Y., ‘Hirzebruch–Zagier cycles and twisted triple product Selmer groups’, Invent. Math. 205(3) (2016), 693–780.Google Scholar
Loeffler, D., ‘Images of adelic Galois representations for modular forms’, Glasgow Math. J. 59(1) (2017), 11–25.Google Scholar
Loeffler, D., Skinner, C. and Zerbes, S. L., ‘Syntomic regulators of Asai–Flach classes’, Preprint, 2016, arXiv:1608.06112.Google Scholar
Mazza, C., Voevodsky, V. and Weibel, C., Lecture Notes on Motivic Cohomology, Clay Mathematics Monographs, 2 (American Mathematical Society, Providence, RI, 2006).Google Scholar
Milne, J. S., ‘Canonical models of (mixed) Shimura varieties and automorphic vector bundles’, inAutomorphic Forms, Shimura Varieties, and L-Functions, Vol. I (Ann Arbor, MI, 1988), Perspectives in Mathematics, 10 (Academic Press, Boston, MA, 1990), 283–414.Google Scholar
Nekovář, J., ‘Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two’, Canad. J. Math. 64(3) (2012), 588–668.Google Scholar
Nekovář, J., ‘Eichler–Shimura relations and semi-simplicity of étale cohomology of quaternionic Shimura varieties’, Ann. Sci. E.N.S. 51 (2018), 1209–1283.Google Scholar
Nekovář, J. and Nizioł, W., ‘Syntomic cohomology and p-adic regulators for varieties over p-adic fields’, Algebra Number Theory 10(8) (2016), 1695–1790.Google Scholar
Rubin, K., Euler Systems, Annals of Mathematics Studies, 147 (Princeton University Press, Princeton, NJ, 2000).Google Scholar
Saito, M., ‘Mixed Hodge modules’, Proc. Japan Acad. Ser. A Math. Sci. 62(9) (1986), 360–363.Google Scholar
Saito, M., ‘Introduction to mixed Hodge modules’, Astérisque 179–180 (1989), 10, 145–162, Actes du Colloque de Théorie de Hodge (Luminy, 1987).Google Scholar
Shimura, G., ‘The special values of the zeta functions associated with Hilbert modular forms’, Duke Math. J. 45(3) (1978), 637–679.Google Scholar
Suslin, A., ‘Torsion in K 2 of fields’, K-Theory 1(1) (1987), 5–29.Google Scholar
Tian, Y. and Xiao, L., ‘ p-adic cohomology and classicality of overconvergent Hilbert modular forms’, Astérisque 382 (2016), 73–162.Google Scholar
Wildeshaus, J., ‘On the interior motive of certain Shimura varieties: the case of Hilbert–Blumenthal varieties’, Int. Math. Res. Not. IMRN 2012(10) (2012), 2321–2355.Google Scholar
Wiles, A., ‘On ordinary 𝜆-adic representations associated to modular forms’, Invent. Math. 94(3) (1988), 529–573.Google Scholar