For each Hilbert modular form of non-critical slope we construct a -adic distribution on the Galois group of the maximal abelian extension unramified outside and of the totally real field. We prove that the distribution is admissible and interpolates the critical values of the complex -function of the form. This construction is based on the study of the overconvergent cohomology of Hilbert modular varieties and certain cycles on these varieties.
Pour une forme de Hilbert de pente non critique, l’on construit une distribution -adique sur le groupe de Galois de l’extension abélienne maximale du corps totalement réel, non-ramifiée en dehors de et . On démontre que la distribution obtenue est admissible et interpole les valeurs critiques de la fonction complexe de la forme de Hilbert. Cette construction est basée sur l’étude de la cohomologie surconvergente des variétés modulaires de Hilbert et de certains cycles sur ces variétés.
Revised:
Accepted:
Published online:
Keywords: $p$-adic $L$-functions, Hilbert modular forms
Mot clés : Fonctions $L$ $p$-adiques, Formes modulaires de Hilbert
Barrera Salazar, Daniel 1
@article{AIF_2018__68_5_2177_0, author = {Barrera Salazar, Daniel}, title = {Overconvergent cohomology of {Hilbert} modular varieties and $p$-adic $L$-functions}, journal = {Annales de l'Institut Fourier}, pages = {2177--2213}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {5}, year = {2018}, doi = {10.5802/aif.3206}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3206/} }
TY - JOUR AU - Barrera Salazar, Daniel TI - Overconvergent cohomology of Hilbert modular varieties and $p$-adic $L$-functions JO - Annales de l'Institut Fourier PY - 2018 SP - 2177 EP - 2213 VL - 68 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3206/ DO - 10.5802/aif.3206 LA - en ID - AIF_2018__68_5_2177_0 ER -
%0 Journal Article %A Barrera Salazar, Daniel %T Overconvergent cohomology of Hilbert modular varieties and $p$-adic $L$-functions %J Annales de l'Institut Fourier %D 2018 %P 2177-2213 %V 68 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3206/ %R 10.5802/aif.3206 %G en %F AIF_2018__68_5_2177_0
Barrera Salazar, Daniel. Overconvergent cohomology of Hilbert modular varieties and $p$-adic $L$-functions. Annales de l'Institut Fourier, Volume 68 (2018) no. 5, pp. 2177-2213. doi : 10.5802/aif.3206. https://aif.centre-mersenne.org/articles/10.5802/aif.3206/
[1] Distributions -adiques associées aux séries de Hecke, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (Astérisque), Volume 24-25, Société Mathématique de France, 1975, pp. 119-131 | MR | Zbl
[2] -adic deformations of arithmetic cohomology (2008) (preprint)
[3] -adic -functions for nearly finite slope Hilbert modular forms and the exceptional zero conjecture (2017) (https://arxiv.org/abs/1709.08105v2)
[4] -adic -functions for (2017) (to appear in Can. J. Math.)
[5] On -adic -functions for Hilbert modular forms (2017) (preprint)
[6] Corners and arithmetic groups, Comment. Math. Helv., Volume 48 (1974), pp. 436-491 | Zbl
[7] Sheaf theory, Graduate Texts in Mathematics, 170, Springer, 1997, xii+502 pages | DOI | MR | Zbl
[8] -adic -functions of Hilbert modular forms, Ann. Inst. Fourier, Volume 44 (1994) no. 4, pp. 1025-1041 | Zbl
[9] Automorphic symbols, -adic -functions and ordinary cohomology of Hilbert modular varieties, Am. J. Math., Volume 135 (2013) no. 4, pp. 1117-1155 | DOI | MR | Zbl
[10] On the critical values of -functions of and , Duke Math. J., Volume 74 (1994) no. 2, pp. 431-529 | Zbl
[11] On -adic -functions for over totally real fields, Int. Math. Res. Not., Volume 2015 (2015) no. 17, pp. 7884-7949 | Zbl
[12] -adic integration on ray class groups and non-ordinary -adic -function, Iwasawa Theory 2012: State of the Art and Recent Advances (Contributions in Mathematical and Computational Sciences), Volume 7, Springer, 2014, pp. 357-378 | Zbl
[13] Non-archimedean integration and -adic Jacquet-Langlands -functions, Uspehi Mat. Nauk, Volume 31 (1976), pp. 5-54 | Zbl
[14] On -adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math., Volume 84 (1986) no. 1, pp. 1-48 | DOI | MR | Zbl
[15] The exceptional zero conjecture for Hilbert modular forms, Compos. Math., Volume 145 (2009) no. 1, pp. 1-55 | DOI | MR | Zbl
[16] Elementary differential topology, Annals of Mathematics Studies, 54, Princeton University Press, 1966, xii+112 pages | Zbl
[17] Motives over totally real fields and -adic -functions, Ann. Inst. Fourier, Volume 44 (1994) no. 4, pp. 989-1023 | Zbl
[18] Overconvergent modular symbols and -adic -functions, Ann. Sci. Éc. Norm. Supér., Volume 44 (2011) no. 4, pp. 1-42 | Zbl
[19] Families of overconvergent modular symbols (unpublished)
[20] Rigid analytic symbols (1994) (preprint)
[21] Coleman’s -invariant and families of modular forms, Astérisque (2010) no. 331, pp. 1-12 | MR | Zbl
[22] Eigenvarieties for reductive groups, Ann. Math., Volume 174 (2011) no. 3, pp. 1685-1784 | DOI | MR | Zbl
[23] Non-archimedean measures connected with Dirichlet series, Math. USSR, Sb., Volume 28 (1978), pp. 216-228 | Zbl
[24] -adic -functions of Bianchi modular forms, Proc. Lond. Math. Soc., Volume 114 (2017) no. 4, pp. 614-656 | Zbl
Cited by Sources: