Cavicchi, Mattia

On the Boundary and Intersection Motives of Genus 2 Hilbert-Siegel Varieties

Doc. Math. 24, 1033-1098 (2019)
DOI: 10.25537/dm.2019v24.1033-1098
Communicated by Thomas Geisser


We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties \(S_K\) corresponding to the group \(\mathrm{G}\mathrm{Sp}_{4,F}\) over a totally real field \(F\), along with the relative Chow motives \(^\lambda\mathcal{V}\) of abelian type over \(S_K\) obtained from irreducible representations \(V_\lambda\) of \(\mathrm{G}\mathrm{Sp}_{4,F}\). We analyse the weight filtration on the degeneration of such motives at the boundary of the Baily-Borel compactification and we find a criterion on the highest weight \(\lambda\), potentially generalisable to other families of Shimura varieties, which characterizes the absence of the \textit{middle weights} 0 and 1 in the corresponding degeneration. Thanks to Wildeshaus' theory, the absence of these weights allows us to construct Hecke-equivariant Chow motives over \(\mathbb{Q}\), whose realizations equal interior (or intersection) cohomology of \(S_K\) with \(V_{\lambda}\)-coefficients. We give applications to the construction of homological motives associated to automorphic representations.

Mathematics Subject Classification

14G35, 11F46, 11G18, 11F70


Shimura varieties, Hilbert-Siegel varieties, boundary motive, intersection motive, weight structures, motives for Hilbert-Siegel modular forms


  • [Anc15]. G. Ancona, Décomposition de motifs abéliens, Manuscripta Math. 146 (2015), 307-328. DOI 10.1007/s00229-014-0708-4; zbl 06418369; MR3312448; arxiv 1305.2874.
  • [Bon10]. M.V. Bondarko, Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory 6 (2010), 387-504. DOI 10.1017/is010012005jkt083; zbl 1303.18019; MR2746283; arxiv 0704.4003.
  • [BS73]. A. Borel, J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436-491. DOI 10.1007/BF02566134; zbl 0274.22011; MR0387495.
  • [BR16]. R. Brasca, G. Rosso, Eigenvarieties for non-cuspidal forms over certain PEL Shimura varieties, preprint (2016), version of June 19, 2018, 33 pages. arxiv 1605.05065.
  • [BW04]. J. I. Burgos, J. Wildeshaus, Hodge modules on Shimura varieties and their higher direct images in the Baily-Borel compactification, Ann. Sci. Ec. Norm. Sup. (4) 37, no. 3 (2004), 363-413. DOI 10.1016/j.ansens.2004.01.002; zbl 1073.14036; MR2060481; arxiv math/0209370.
  • [CD12]. D.-C. Cisinski, F. Deglise, Triangulated categories of mixed motives, preprint (December 2009), version of December 18, 2012, 279 pages. arxiv 0912.2110.
  • [CD16]. D.-C. Cisinski, F. Deglise, Etale motives, Compositio Math. 152 (2016), 556-666. DOI 10.1112/S0010437X15007459; zbl 06578150; MR3477640; arxiv 1305.5361.
  • [CH00]. A. Corti, M. Hanamura, Motivic decomposition and intersection Chow groups. I, Duke Math. J. 103 (3) (2000), 459-522. DOI 10.1215/S0012-7094-00-10334-1; zbl 1052.14504; MR1763656; arxiv math/9804123.
  • [Cloî17]. G. Cloitre, Sur le motif intérieur de certaines variétés de Shimura: le cas des variétés de Picard, Thèse, Université Paris 13, Sorbonne Paris Cité, June 2017, 68 pages.
  • [Cloz90]. L. Clozel, Motifs et formes automorphes: applications du principe de fonctorialité, in L. Clozel, J. Milne (eds.), Automorphic forms, Shimura varieties, and L-functions (Ann Arbor, MI, 1988), Vol. I, Perspect. Math., 10, Academic Press, Boston, MA (1990), 77-159. zbl 0705.11029; MR1044819.
  • [Del69]. P. Deligne, Formes modulaires et représentations \(\ell \)-adiques, Séminaire Bourbaki, Vol. 1968/69, Exp. No. 355, 139-172, Lecture Notes in Math. 175, Springer, Berlin (1971). zbl 0206.49901; MR3077124.
  • [Del71]. P. Deligne, Travaux de Shimura, Séminaire Bourbaki, Vol. 1970/71, Exp. No. 389, 123-165. Lecture Notes in Math. 244, Springer, Berlin (1971). DOI 10.1007/BFb0058700; zbl 0225.14007; MR0498581.
  • [DM91]. C. Deninger, J.P. Murre, Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math. 422 (1991), 201-219. zbl 0745.14003; MR1133323.
  • [Eke90]. T. Ekedahl, On the adic formalism, in P. Cartier et al. (eds.), The Grothendieck Festschrift, Volume II, Prog. in Math. 87, Birkhäuser Boston, MA (1990), 197-218. zbl 0821.14010; MR1106899.
  • [Fan16]. J. Fangzhou, Borel-Moore motivic homology and weight structure on mixed motives, Mathematische Zeitschrift 283 (2016), 1-35. DOI 10.1007/s00209-016-1636-7; zbl 1375.14023; MR3519998; arxiv 1502.03956.
  • [Fli05]. Y. Flicker, Automorphic forms and Shimura varieties of \(PGSp(2)\), World Scientific Publishing Co. Pte. Ltd. (2005), xii+325 pp. zbl 1103.11018; MR2170597.
  • [Fra98]. J. Franke, Harmonic analysis in weighted \(L^2\) spaces, Ann. Sci. Ec. Norm. Sup. (4) 31, no. 2 (1998), 181-279. DOI 10.1016/S0012-9593(98)80015-3; zbl 0938.11026; MR1603257.
  • [Fre90]. E. Freitag, Hilbert modular forms, Springer-Verlag (1990), viii+250 pp. zbl 0702.11029; MR1050763.
  • [Hard]. G. Harder, Cohomology of Arithmetic Groups, book in preparation, available at the address
  • [Harr91]. M. Harris, Hodge-de Rham structures and periods of automorphic forms, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc. (1994), pp. 573-624. zbl 0824.14015; MR1265564.
  • [HZ94]. M. Harris, S. Zucker, Boundary cohomology of Shimura varieties II. Hodge theory at the boundary, Invent. Math. 116 (1994), no. 1-3, 243-308. DOI 10.1007/BF01231562; zbl 0860.11031; MR1253194.
  • [HZ01]. M. Harris, S. Zucker, Boundary cohomology of Shimura varieties III. Coherent cohomology on higher-rank boundary strata and applications to Hodge theory, Mém. Soc. Math. Fr. (N.S.) no. 85 (2001), vi+116 pp. DOI 10.24033/msmf.398; zbl 1020.11042; MR1850830.
  • [Héb11]. D. Hebert, Structures de poids à la Bondarko sur les motifs de Beilinson, Compositio Math. 147 (2011), 1447-1462. DOI 10.1112/S0010437X11005422; zbl 1233.14017; MR2834728; arxiv 1007.0219.
  • [KM74]. N. M. Katz, W. Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73-77. DOI 10.1007/BF01405203; zbl 0275.14011; MR0332791.
  • [Lan12]. K.-W. Lan, Toroidal compactifications of PEL-type Kuga families, Algebra Number Theory 6 (2012), no. 5, 885-966. DOI 10.2140/ant.2012.6.885; zbl 1302.11041; MR2968629.
  • [Lan13]. K.-W. Lan, Arithmetic compactifications of PEL-type Shimura varieties, London Mathematical Society Monographs Series 36, Princeton University Press (2013), xxvi+561 pp. zbl 1284.14004; MR3186092.
  • [Lem15]. F. Lemma, On higher regulators of Siegel threefolds I: the vanishing on the boundary, Asian J. Math. 19 (2015), 83-120. DOI 10.4310/AJM.2015.v19.n1.a4; zbl 1375.19010; MR3318014.
  • [LS04]. J.-S. Li, J. Schwermer, On the Eisenstein cohomology of arithmetic groups, Duke Math. J. 123, 1 (2004), 141-169. DOI 10.1215/S0012-7094-04-12315-2; zbl 1057.11031; MR2060025.
  • [LR91]. E. Looijenga, M. Rapoport, Weights in the local cohomology of a Baily-Borel compactification, in Complex geometry and Lie theory (Sundance, UT, 1989), Proc. Sympos. Pure Math. 53, Amer. Math. Soc. (1991), 223-260. zbl 0753.14016; MR1141203.
  • [M-SSYZ15]. S. Muller-Stach, M. Sheng, X. Ye, K. Zuo, On the cohomology groups of local systems over Hilbert modular varieties via Higgs bundles, Amer. J. Math. 137 (2015), no. 1, 1-35. DOI 10.1353/ajm.2015.0005; zbl 1357.14032; MR3318085; arxiv 1009.2011.
  • [Na13]. A. Nair, Mixed structures in Shimura varieties and automorphic forms, preprint (2013), 46 pages, available at the address
  • [Pin90]. R. Pink, Arithmetical compactifications of mixed Shimura varieties, Bonner Mathematische Schriften 209, Univ. Bonn (1990). MR1128753.
  • [Pin92]. R. Pink, On \(\ell \)-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification, Math. Ann. 292 (1992), 197-240. DOI 10.1007/BF01444618; zbl 0748.14008; MR1149032.
  • [Sap05]. L. Saper, \( \mathcal{L} \)-modules and the conjectures of Rapoport-Goresky-MacPherson, in J. Tilouine, H. Carayol, M. Harris, M.-F. Vignéras (eds.), Automorphic forms. I, Astérisque 298 (2005), 319-334. MR2141706; arxiv math/0112250.
  • [Sch90]. A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), 419-430. DOI 10.1007/BF01231194; zbl 0760.14002; MR1047142.
  • [Shi63]. H. Shimizu, On discontinuous groups operating on the product of the upper half planes, Ann. of Math. (2) 77 (1963), 33-71. DOI 10.2307/1970201; zbl 0218.10045; MR0145106.
  • [Vog81]. D. Vogan, Representations of real reductive groups, Prog. in Math. 15, Birkhäuser (1981), xvii+754 pp. zbl 0469.22012; MR0632407.
  • [Wei94]. C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press (1994), xiv+450 pp. zbl 0797.18001; MR1269324.
  • [Wil97]. J. Wildeshaus, Realizations of Polylogarithms, Lect. Notes Math. 1650, Springer-Verlag (1997), xii+343 pp. DOI 10.1007/BFb0093051; zbl 0877.11001; MR1482233.
  • [Wil09]. J. Wildeshaus, Chow motives without projectivity, Compositio Math. 145 (2009), 1196-1226. DOI 10.1112/S0010437X0900414X; zbl 1179.14021; MR2551994; arxiv 0806.3380.
  • [Wil12]. J. Wildeshaus, On the interior motive of certain Shimura varieties: the case of Hilbert-Blumenthal varieties, Int. Math. Res. Notices 2012 (2012), 2321-2355. DOI 10.1093/imrn/rnr109; zbl 1251.14014; MR2923168 arxiv 0906.4239.
  • [Wil15]. J. Wildeshaus, On the interior motive of certain Shimura varieties: the case of Picard surfaces, Manuscripta Math. 148 (2015), 351-377. DOI 10.1007/s00229-015-0747-5; zbl 1349.14106; MR3414481.
  • [Wil17]. J. Wildeshaus, Intermediate extensions of Chow motives of abelian type, Adv. Math. 305 (2017), 515-600. DOI 10.1016/j.aim.2016.09.032; zbl 1360.14078; MR3570143; arxiv 1211.5327.
  • [Wil19a]. J. Wildeshaus, Chow motives without projectivity, II, to appear in Int. Math. Res. Not. (2019). arxiv 1705.10502.
  • [Wil19b]. J. Wildeshaus, On the intersection motive of certain Shimura varieties: the case of Siegel threefolds, to appear in Annals of K-Theory (2019). arxiv 1706.02743.


Cavicchi, Mattia
LAGA, Institut Galilée, Université Paris 13, F-93430 Villetaneuse, France