An explicit computation of p-stabilized vectors
Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 531-558.

Nous donnons une méthode concrète pour calculer les vecteurs p-stables dans l’espace des éléments fixés par un sous-groupe parahorique d’un groupe réductif p-adique. Nous discutons d’une application globale et, en particulier, nous donnons un exemple explicite d’un relèvement de Saito-Kurokawa p-stable.

In this paper, we give a concrete method to compute p-stabilized vectors in the space of parahori-fixed vectors for connected reductive groups over p-adic fields. An application to the global setting is also discussed. In particular, we give an explicit p-stabilized form of a Saito-Kurokawa lift.

Reçu le : 2012-11-29
Accepté le : 2013-09-02
Publié le : 2015-03-09
DOI : https://doi.org/10.5802/jtnb.878
Classification : 11F85,  22E50
@article{JTNB_2014__26_2_531_0,
     author = {Michitaka MIYAUCHI and Takuya YAMAUCHI},
     title = {An explicit computation of $p$-stabilized vectors},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {531--558},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {2},
     year = {2014},
     doi = {10.5802/jtnb.878},
     mrnumber = {3320491},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2014__26_2_531_0/}
}
Michitaka MIYAUCHI; Takuya YAMAUCHI. An explicit computation of $p$-stabilized vectors. Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 531-558. doi : 10.5802/jtnb.878. https://jtnb.centre-mersenne.org/item/JTNB_2014__26_2_531_0/

[1] A. N. Andrianov, Quadratic forms and Hecke operators, Springer Berlin (1987). | MR 884891 | Zbl 0613.10023

[2] S. Böcherer, Siegfried On the Hecke operator U(p). With an appendix by Ralf Schmidt. J. Math. Kyoto Univ. 45, 4 (2005), 807–829. | MR 2226631 | Zbl 1114.11044

[3] A. Borel, Automorphic L -functions. Automorphic forms, representations and L -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I.,(1979) 27–61. | MR 546608 | Zbl 0412.10017

[4] A. Borel and H. Jacquet, Automorphic forms and automorphic representations. With a supplement “On the notion of an automorphic representation” by R. P. Langlands. Proc. Sympos. Pure Math., XXXIII, Automorphic forms, representations and L -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, 189–207. | MR 546598 | Zbl 0414.22020

[5] C. J. Bushnell and P. C. Kutzko, Smooth representations of reductive p -adic groups: structure theory via types. Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634. | MR 1643417 | Zbl 0911.22014

[6] W. Casselman, Introduction to admissible representations of p-adic groups, available at his homepage.

[7] Robert F. Coleman, Classical and overconvergent modular forms. Invent. Math. 124, 1-3 (1996), 215–241. | MR 1369416 | Zbl 0851.11030

[8] P. Garrett, Representations with Iwahori-fixed vectors. Note available at his homepage .

[9] M. Goresky, Compactifications and cohomology of modular varieties. Harmonic analysis, the trace formula, and Shimura varieties, , Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, (2005), 551–582. | MR 2192016 | Zbl 1158.14306

[10] R. Howe Harish-Chandra homomorphisms for p-adic groups, 59 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, (1985). With the collaboration of A. Moy. | MR 821216 | Zbl 0593.22014

[11] T. Ibukiyama, Saito-Kurokawa liftings of level N and practical construction of Jacobi forms, Kyoto J. Math. 52, 1 (2012), 141–178. | MR 2892771 | Zbl 1284.11085

[12] H. Jacquet, Sur les représentations des groupes réductifs p-adiques, C. R. Acad. Sci. Paris Ser. A-B 280 (1975), Aii, A1271–A1272. | MR 369624 | Zbl 0309.22012

[13] D. Keys, Principal series representations of special unitary groups over local fields, Compositio Math. 51, 1 (1984), 115–130. | Numdam | MR 734788 | Zbl 0547.22009

[14] B. Mazur, An “infinite fern” in the universal deformation space of Galois representations, Collect. Math., 48, 1-2 (1997), 155–193. Journées Arithmétiques (Barcelona, 1995). | MR 1464022 | Zbl 0865.11046

[15] J. S. Milne, Introduction to Shimura varieties. In Harmonic analysis, the trace formula, and Shimura varieties, 4 of Clay Math. Proc., (2005) 265–378. Amer. Math. Soc., Providence, RI. | MR 2192012 | Zbl 1148.14011

[16] B. Roberts and R. Schmidt, Local newforms for GSp(4). Lecture Notes in Mathematics, 1918, Springer, Berlin, (2007), viii+307 pp. | MR 2344630 | Zbl 1126.11027

[17] R. Salvati Manni and J. Top, Cusp forms of weight 2 for the group Γ(4,8), Amer. J. Math. 115, (1993), 455–486. | MR 1216438 | Zbl 0780.11024

[18] I. Satake, Algebraic structures of symmetric domains Kano Memorial Lectures, 4. Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J., (1980). xvi+321 pp. | MR 591460 | Zbl 0483.32017

[19] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Reprint of the 1971 original. Publications of the Mathematical Society of Japan, 11, Kano Memorial Lectures, 1. Princeton University Press, Princeton, NJ, (1994). | MR 1291394 | Zbl 0872.11023

[20] C. Skinner and E. Urban, Sur les déformations p-adiques de certaines représentations automorphes. J. Inst. Math. Jussieu 5, 4 (2006), 629–698. | MR 2261226 | Zbl 1169.11314

[21] J. Tilouine, Nearly ordinary rank four Galois representations and p-adic Siegel modular forms. With an appendix by Don Blasius. Compos. Math. 142, 5 (2006), 1122–1156. | MR 2264659 | Zbl 1159.11018