Overconvergent cohomology of Hilbert modular varieties and p-adic L-functions
Annales de l'Institut Fourier, Volume 68 (2018) no. 5, p. 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.
Received : 2015-08-08
Revised : 2017-06-17
Accepted : 2017-09-14
Published online : 2018-11-23
DOI : https://doi.org/10.5802/aif.3206
Classification:  11F41,  11F67,  11S80
Keywords: p-adic L-functions, Hilbert modular forms
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {5},
     year = {2018},
     pages = {2177-2213},
     doi = {10.5802/aif.3206},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_5_2177_0}
}
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/item/AIF_2018__68_5_2177_0/

[1] Amice, Yvette; Vélu, Jacques Distributions p-adiques associées aux séries de Hecke, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974), Société Mathématique de France (Astérisque) Tome 24-25 (1975), pp. 119-131 | MR 0376534 | Zbl 0332.14010

[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., Tome 48 (1974), pp. 436-491 | Zbl 0274.22011

[7] Bredon, Glen E. Sheaf theory, Springer, Graduate Texts in Mathematics, Tome 170 (1997), xii+502 pages | Article | MR 1481706 | Zbl 0874.55001

[8] Dabrowski, Andrzej p-adic L-functions of Hilbert modular forms, Ann. Inst. Fourier, Tome 44 (1994) no. 4, pp. 1025-1041 | Zbl 0808.11035

[9] Dimitrov, Mladen Automorphic symbols, p-adic L-functions and ordinary cohomology of Hilbert modular varieties, Am. J. Math., Tome 135 (2013) no. 4, pp. 1117-1155 | Article | MR 3086071 | Zbl 1325.11044

[10] Hida, Haruzo On the critical values of L-functions of GL 2 and GL 2 ×GL 2 , Duke Math. J., Tome 74 (1994) no. 2, pp. 431-529 | Zbl 0838.11036

[11] Januszewski, Fabian On p-adic L-functions for GL(n)×GL(n-1) over totally real fields, Int. Math. Res. Not., Tome 2015 (2015) no. 17, pp. 7884-7949 | Zbl 1385.11029

[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, Springer (Contributions in Mathematical and Computational Sciences) Tome 7 (2014), pp. 357-378 | Zbl 1384.11100

[13] Manin Non-archimedean integration and p-adic Jacquet-Langlands L-functions, Uspehi Mat. Nauk, Tome 31 (1976), pp. 5-54 | Zbl 0348.12016

[14] Mazur, Barry; Tate, John; Teitelbaum, Jeremy On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math., Tome 84 (1986) no. 1, pp. 1-48 | Article | MR 830037 | Zbl 0699.14028

[15] Mok, Chung Pang The exceptional zero conjecture for Hilbert modular forms, Compos. Math., Tome 145 (2009) no. 1, pp. 1-55 | Article | MR 2480494 | Zbl 1247.11071

[16] Munkres, James R. Elementary differential topology, Princeton University Press, Annals of Mathematics Studies, Tome 54 (1966), xii+112 pages | Zbl 0161.20201

[17] Panchishkin, Alexei A. Motives over totally real fields and p-adic L-functions, Ann. Inst. Fourier, Tome 44 (1994) no. 4, pp. 989-1023 | Zbl 0808.11034

[18] Pollack, Robert; Stevens, Glenn Overconvergent modular symbols and p-adic L-functions, Ann. Sci. Éc. Norm. Supér., Tome 44 (2011) no. 4, pp. 1-42 | Zbl 1268.11075

[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 2667884 | Zbl 1233.11075

[22] Urban, Eric Eigenvarieties for reductive groups, Ann. Math., Tome 174 (2011) no. 3, pp. 1685-1784 | Article | MR 2846490 | Zbl 1285.11081

[23] Vishik, M. Non-archimedean measures connected with Dirichlet series, Math. USSR, Sb., Tome 28 (1978), pp. 216-228 | Zbl 0369.14010

[24] Williams, Chris P-adic L-functions of Bianchi modular forms, Proc. Lond. Math. Soc., Tome 114 (2017) no. 4, pp. 614-656 | Zbl 06775303