Overconvergent cohomology of Hilbert modular varieties and p-adic L-functions
Annales de l'Institut Fourier, Volume 68 (2018) no. 5, pp. 2177-2213.

For each Hilbert modular form of non-critical slope we construct a p-adic distribution on the Galois group of the maximal abelian extension unramified outside p and of the totally real field. We prove that the distribution is admissible and interpolates the critical values of the complex L-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 p-adique sur le groupe de Galois de l’extension abélienne maximale du corps totalement réel, non-ramifiée en dehors de p et . On démontre que la distribution obtenue est admissible et interpole les valeurs critiques de la fonction L 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.

Published online:
DOI: 10.5802/aif.3206
Classification: 11F41, 11F67, 11S80
Keywords: $p$-adic $L$-functions, Hilbert modular forms
Barrera Salazar, Daniel 1

1 Universitat Politecnica de Catalunya Campus Nord Calle Jordi Girona, 1-3 Barcelona, 08034 (Spain)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Barrera Salazar, Daniel},
     title = {Overconvergent cohomology of {Hilbert} modular varieties and $p$-adic $L$-functions},
     journal = {Annales de l'Institut Fourier},
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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] Amice, Yvette; Vélu, Jacques Distributions p-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] Ash, A.; Stevens, Glenn p-adic deformations of arithmetic cohomology (2008) (preprint)

[3] Barrera Salazar, Daniel; Dimitrov, Mladen; Jorza, A. p-adic L-functions for nearly finite slope Hilbert modular forms and the exceptional zero conjecture (2017) (https://arxiv.org/abs/1709.08105v2)

[4] Barrera Salazar, Daniel; Williams, Chris P-adic L-functions for GL 2 (2017) (to appear in Can. J. Math.)

[5] Bergdall, John; Hansen, David On p-adic L-functions for Hilbert modular forms (2017) (preprint)

[6] Borel, Armand; Serre, Jean-Pierre Corners and arithmetic groups, Comment. Math. Helv., Volume 48 (1974), pp. 436-491 | Zbl

[7] Bredon, Glen E. Sheaf theory, Graduate Texts in Mathematics, 170, Springer, 1997, xii+502 pages | DOI | MR | Zbl

[8] Dabrowski, Andrzej p-adic L-functions of Hilbert modular forms, Ann. Inst. Fourier, Volume 44 (1994) no. 4, pp. 1025-1041 | Zbl

[9] Dimitrov, Mladen Automorphic symbols, p-adic L-functions and ordinary cohomology of Hilbert modular varieties, Am. J. Math., Volume 135 (2013) no. 4, pp. 1117-1155 | DOI | MR | Zbl

[10] Hida, Haruzo On the critical values of L-functions of GL 2 and GL 2 ×GL 2 , Duke Math. J., Volume 74 (1994) no. 2, pp. 431-529 | Zbl

[11] Januszewski, Fabian On p-adic L-functions for GL(n)×GL(n-1) over totally real fields, Int. Math. Res. Not., Volume 2015 (2015) no. 17, pp. 7884-7949 | Zbl

[12] Loeffler, David P-adic integration on ray class groups and non-ordinary p-adic L-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] Manin Non-archimedean integration and p-adic Jacquet-Langlands L-functions, Uspehi Mat. Nauk, Volume 31 (1976), pp. 5-54 | Zbl

[14] Mazur, Barry; Tate, John; Teitelbaum, Jeremy On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math., Volume 84 (1986) no. 1, pp. 1-48 | DOI | MR | Zbl

[15] Mok, Chung Pang The exceptional zero conjecture for Hilbert modular forms, Compos. Math., Volume 145 (2009) no. 1, pp. 1-55 | DOI | MR | Zbl

[16] Munkres, James R. Elementary differential topology, Annals of Mathematics Studies, 54, Princeton University Press, 1966, xii+112 pages | Zbl

[17] Panchishkin, Alexei A. Motives over totally real fields and p-adic L-functions, Ann. Inst. Fourier, Volume 44 (1994) no. 4, pp. 989-1023 | Zbl

[18] Pollack, Robert; Stevens, Glenn Overconvergent modular symbols and p-adic L-functions, Ann. Sci. Éc. Norm. Supér., Volume 44 (2011) no. 4, pp. 1-42 | Zbl

[19] Stevens, Glenn Families of overconvergent modular symbols (unpublished)

[20] Stevens, Glenn Rigid analytic symbols (1994) (preprint)

[21] Stevens, Glenn Coleman’s -invariant and families of modular forms, Astérisque (2010) no. 331, pp. 1-12 | MR | Zbl

[22] Urban, Eric Eigenvarieties for reductive groups, Ann. Math., Volume 174 (2011) no. 3, pp. 1685-1784 | DOI | MR | Zbl

[23] Vishik, M. Non-archimedean measures connected with Dirichlet series, Math. USSR, Sb., Volume 28 (1978), pp. 216-228 | Zbl

[24] Williams, Chris P-adic L-functions of Bianchi modular forms, Proc. Lond. Math. Soc., Volume 114 (2017) no. 4, pp. 614-656 | Zbl

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