×

The plectic weight filtration on cohomology of Shimura varieties and partial Frobenius. (English) Zbl 1474.14042

Summary: We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of J. Nekovář and A. J. Scholl [“Plectic Hodge theory I”, Preprint, https://www.repository.cam.ac.uk/bitstream/handle/1810/292018/plechdgI.pdf?sequence=1]. This is achieved with the help of S. Morel’s work [Compos. Math. 147, No. 6, 1671–1740 (2011; Zbl 1248.11042)] on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.

MSC:

14G35 Modular and Shimura varieties
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11G18 Arithmetic aspects of modular and Shimura varieties
16W10 Rings with involution; Lie, Jordan and other nonassociative structures

Citations:

Zbl 1248.11042
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ayoub, Joseph and Zucker, Steven, ‘Relative Artin motives and the reductive Borel-Serre compactification of a locally symmetric variety’, Invent. Math., 188(2) (2012), 277-427. · Zbl 1242.14016
[2] Beĭlinson, A. A., Bernstein, J. and Deligne, P., ‘Faisceaux pervers’, in Analysis and Topology on Singular Spaces, I (Luminy, 1981), Vol. 100 of Astérisque (Société Mathématique de France, Paris, 1982), 5-171. · Zbl 0536.14011
[3] Blasius, Don, ‘Hilbert modular forms and the Ramanujan conjecture’, in Noncommutative Geometry and Number Theory (Springer, Vieweg2006), 35-56. · Zbl 1183.11023
[4] Blasius, Don and Rogawski, Jonathan D., ‘Motives for Hilbert modular forms’, Invent. Math., 114(1) (1993), 55-87. · Zbl 0829.11028
[5] Burgos, José I. and Wildeshaus, Jörg, ‘Hodge modules on Shimura varieties and their higher direct images in the Baily-Borel compactification’, in Annales Scientifiques de l’Ecole Normale Supérieure, Vol. 37 (Elsevier, 2004), 363-413. · Zbl 1073.14036
[6] Deligne, P., Cohomologie étale, Vol. 569 of Lecture Notes in Mathematics (Springer, Berlin, 1977). · Zbl 0345.00010
[7] Deligne, Pierre and De Shimura, Travaux, Lecture Notes in Math., 244 (1971), 123-165.
[8] Deligne, Pierre. Théorie de Hodge: II. Publications Mathématiques de l’IHÉS, Tome40 (1971), pp. 5-57. · Zbl 0219.14007
[9] Deligne, Pierre. La conjecture de Weil: II. Publications Mathématiques de l’IHÉS, Tome52 (1980), pp. 137-252. · Zbl 0456.14014
[10] Dimitrov, Mladen, ‘Compactifications arithmétiques des variétés de Hilbert et formes modulaires de Hilbert pour \({\varGamma}_1\left(c,n\right)\)’, in Geometric Aspects of Dwork Theory, Vols. I, II (Walter de Gruyter, Berlin, 2004), 527-554. · Zbl 1076.14029
[11] Faltings, Gerd and Chai, Ching-LiDegeneration of abelian varieties With an appendix by David Mumford of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] (Springer-Verlag, Berlin, 1990), xii+316. · Zbl 0744.14031
[12] Freitag, Eberhard, Hilbert Modular Forms (Springer, Berlin, 1990). · Zbl 0702.11029
[13] Grothendieck, Alexander, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (North-Holland Publishing Co., Amsterdam; Masson & Cie, Éditeur, Paris, 1968). · Zbl 0197.47202
[14] Hida, Haruzo, \(p\)-Adic Automorphic Forms on Shimura Varieties. Springer Monographs in Mathematics (Springer, New York, 2004). · Zbl 1055.11032
[15] Illusie, Luc, Laszlo, Yves and Orgogozo, Fabrice, eds., Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents (Société Mathématique de France, Paris, 2014), 363-364. With the collaboration of Déglise, Frédéric, Moreau, Alban, Pilloni, Vincent, Raynaud, Michel, Riou, Joël, Stroh, Benoît, Temkin, Michael and Zheng, Weizhe.
[16] Imai, Naoki and Mieda, Yoichi, ‘Toroidal compactifications of Shimura varieties of PEL type and its applications’, in Algebraic Number Theory and Related Topics 2011Research Institute for Mathematical Sciences, Kyoto, 2013), 3-24. · Zbl 1361.11042
[17] Kottwitz, Robert E., ‘Points on some Shimura varieties over finite fields’ J. Amer. Math. Soc., 5(2) (1992), 373-444. · Zbl 0796.14014
[18] Lan, Kai-Wen, Arithmetic Compactifications of PEL-Type Shimura Varieties, Number 36 (Princeton University Press, Princeton, NJ, 2013). · Zbl 1284.14004
[19] Lan, Kai-Wen and Stroh, Benoît, ‘Nearby cycles of automorphic étale sheaves’, Compos. Math., 154(1) (2018), 80-119. · Zbl 1432.11062
[20] Morel, Sophie, ‘Cohomologie d’intersection des variétés modulaires de Siegel suite’, Compos. Math., 147(6) (2011), 1671-1740. · Zbl 1248.11042
[21] Morel, Sophie, ‘Mixed \(\ell \)-adic complexes for schemes over number fields’, (2019), Preprint.
[22] Mumford, David, Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics, No. 5 (Oxford University Press, London, 1970). · Zbl 0223.14022
[23] Nair, Arvind N., ‘Mixed structures in Shimura varieties and automorphic forms, Preprint.
[24] Nekovář, Jan, ‘Eichler-Shimura relations and semisimplicity of étale cohomology of quaternionic Shimura varieties’, Ann. Sci. de l’École Norm. Supérieure. Quatrième Série, 51(5) (2018), 1179-1252. · Zbl 1458.11100
[25] Nekovář, J. and Scholl, A. J., ‘Introduction to plectic cohomology’, in Advances in the Theory of Automorphic Forms and Their \(L\)-Functions, Vol. 664 of Contemp. Math., (American Mathematical Society, Providence, RI, 2016) 321-337. · Zbl 1402.11092
[26] Nekovář, J. and Scholl, A. J., Plectic Hodge Theory I (2017), Preprint.
[27] Pink, Richard, Arithmetical Compactification of Mixed Shimura Varieties, Vol. 209 of Bonner Mathematische Schriften [Bonn Mathematical Publications] (Universität Bonn, Mathematisches Institut, Bonn, Germany, 1990). · Zbl 0748.14007
[28] Pink, Richard, ‘On \(l\)-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification’, Math. Ann., 292(2) (1992), 197-240. · Zbl 0748.14008
[29] Tian, Yichao and Xiao, Liang, ‘\(p\)-Adic cohomology and classicality of overconvergent Hilbert modular forms’, Astérisque, 382 (2016), 73-162. The Stacks Project, Available at http://stacks.math.columbia.edu. · Zbl 1423.11091
[30] Wildeshaus, Jörg, ‘On the interior motive of certain Shimura varieties: the case of Hilbert-Blumenthal varieties’, Int. Math. Res. Not. IMRN (10), 2012 (2012), 2321-2355. · Zbl 1251.14014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.