Kodaira–Spencer isomorphisms and degeneracy maps on Iwahori-level Hilbert modular varieties: the saving trace

Diamond, Fred 

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e50

Résumé

We consider integral models of Hilbert modular varieties with Iwahori level structure at primes over p, first proving a Kodaira–Spencer isomorphism that gives a concise description of their dualizing sheaves. We then analyze fibres of the degeneracy maps to Hilbert modular varieties of level prime to p and deduce the vanishing of higher direct images of structure and dualizing sheaves, generalizing prior work with Kassaei and Sasaki (for p unramified in the totally real field F). We apply the vanishing results to prove flatness of the finite morphisms in the resulting Stein factorizations, and combine them with the Kodaira–Spencer isomorphism to simplify and generalize the construction of Hecke operators at primes over p on Hilbert modular forms (integrally and mod p).

Détail

DOI : 10.1017/fms.2025.13

Bibliographie

[BBM82] Berthelot, P., Breen, L. and Messing, W., Théorie de Dieudonné Cristalline. II (Lecture Notes in Mathematics) vol. 930 (Springer, Berlin, 1982). Google Scholar

[BM90] Berthelot, P. and Messing, W., ‘Théorie de Dieudonné cristalline. III. Théorèmes d’équivalence et de pleine fidélité’, in The Grothendieck Festschrift , Vol. I (Progr. Math.) vol. 86 (Birkhäuser Boston, Boston, MA, 1990), 173–247. Google Scholar

[Cal20] Calegari, F., ‘Motives and -functions’, Current Developments in Mathematics 2018 (International Press, Somerville, MA, 2020), 57–123. Google Scholar

[Con00] Conrad, Brian, Grothendieck Duality and Base Change (Lecture Notes in Mathematics) vol. 1750 (Springer, Berlin, 2000). Google Scholar | DOI

[Con07] Conrad, B., ‘Arithmetic moduli of generalized elliptic curves’, J. Inst. Math. Jussieu 6(2) (2007), 209–278. Google Scholar | DOI

[DR73] Deligne, P. and Rapoport, M., ‘Les schémas de modules de courbes elliptiques’, in Modular Functions of One Variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (Lecture Notes in Mathematics) vol. 349 (Springer, Berlin, 1973), 143–316. Google Scholar | DOI

[DDW24] Deo, S. V., Dimitrov, M. and Wiese, G., ‘Unramifiedness of weight 1 Hilbert Hecke algebras’, Algebra Number Theory 18(8) (2024), 1465–1496. Google Scholar | DOI

[Dia23] Diamond, F., ‘Geometric weight-shifting operators on Hilbert modular forms in characteristic ’, J. Inst. Math. Jussieu 22(4) (2023), 1871–1930. Google Scholar | DOI

[Dia24] Diamond, F., ‘Compactifications of Iwahori-level Hilbert modular varieties’, J. Number Theory 263 (2024), 255–296. Google Scholar | DOI

[DK23] Diamond, F. and Kassaei, P. L., ‘The cone of minimal weights for mod Hilbert modular forms’, Int. Math. Res. Not. IMRN 2023(14) (2023), 12148–12171. Google Scholar | DOI

[DKS23] Diamond, F., Kassaei, P. and Sasaki, S., ‘A Jacquet-Langlands relation and Serre filtration via the geometry of Hilbert modular varieties: splicing and dicing’, Astérisque 439 (2023), 1–111. Google Scholar

[DS23] Diamond, F. and Sasaki, S., ‘A Serre weight conjecture for geometric Hilbert modular forms in characteristic ’, J. Eur. Math. Soc. (JEMS) 25(9) (2023), 3453–3536. Google Scholar | DOI

[DW20] Dimitrov, M. and Wiese, G., ‘Unramifiedness of Galois representations attached to Hilbert modular forms of weight ’, J. Inst. Math. Jussieu 19(2) (2020), 281–306. Google Scholar | DOI

[ERX17a] Emerton, M., Reduzzi, D. and Xiao, L., ‘Unramifiedness of Galois representations arising from Hilbert modular surfaces’, Forum Math. Sigma 5 (2017), Paper No. e29, 70 pp. Google Scholar | DOI

[ERX17b] Emerton, M., Reduzzi, D. A. and Xiao, L., ‘Galois representations and torsion in the coherent cohomology of Hilbert modular varieties’, J. Reine Angew. Math. 726 (2017), 93–127. Google Scholar | DOI

[FP23] Fakhruddin, N. and Pilloni, V., ‘Hecke operators and the coherent cohomology of Shimura varieties’, J. Inst. Math. Jussieu 22(1) (2023), 1–69. Google Scholar | DOI

[GK19] Goldring, W. and Koskivirta, J.-S., ‘Strata Hasse invariants, Hecke algebras and Galois representations’, Invent. Math. 217(3) (2019), 887–984. Google Scholar | DOI

[Gro74] Grothendieck, A., Groupes de Barsotti-Tate et Cristaux de Dieudonné (Séminaire de Mathématiques Supérieures) No. 45, Été, 1970 (Les Presses de l’Université de Montréal, Montreal, Que., 1974). Google Scholar

[KN82] Kimura, T. and Niitsuma, H., ‘On Kunz’s conjecture’, J. Math. Soc. Japan 34(2) (1982), 371–378. Google Scholar | DOI

[KW17] Kottwitz, R. and Wake, P., ‘Primitive elements for -divisible groups, Res. Number Theory 3 (2017), Paper No. 20, 11 pp. Google Scholar | DOI

[Pap95] Pappas, G., ‘Arithmetic models for Hilbert modular varieties’, Compos. Math. 98(1) (1995), 43–76. Google Scholar

[PR05] Pappas, G. and Rapoport, M., ‘Local models in the ramified case. II. Splitting models’, Duke Math. J. 127(2) (2005), 193–250. Google Scholar | DOI

[Ray74] Raynaud, M., ‘Schémas en groupes de type ’, Bull. Soc. Math. France 102 (1974), 241–280. Google Scholar | DOI

[RX17] Reduzzi, D. A. and Xiao, L., ‘Partial Hasse invariants on splitting models of Hilbert modular varieties’, Ann. Sci. Éc. Norm. Supér. (4) 50(3) (2017), 579–607. Google Scholar | DOI

[StaX] The Stacks Project Contributors, The Stacks Project, open source textbook, available at http://stacks.math.columbia.edu. Google Scholar

[Vol05] Vollaard, I., ‘On the Hilbert-Blumenthal moduli problem’, J. Inst. Math. Jussieu 4(4) (2005), 653–683. Google Scholar | DOI