Triple product $p$-adic $L$-functions for Shimura curves over totally real number fields
[Fonctions $L$ $p$-adiques triple produit pour des courbes de Shimura sur des corps de nombres totalement réels]
Barrera Salazar, Daniel1  ; Molina, Santiago2
1Universidad de Santiago de Chile, Alameda 3363, Estación Central, Santiago (Chile)
2Universitat de Lleida, Campus Universitari Igualada-UdL, Av. Pla de la Massa 8, 08700 Igualada (Spain)
Annales de l'Institut Fourier, Online first, 76 p.

Let $F$ be a totally real number field. Using a recent geometric approach developed by Andreatta and Iovita we construct several variables $p$-adic families of finite slope quaternionic automorphic forms over $F$. It is achieved by interpolating the modular sheaves defined over some auxiliary unitary Shimura curves.

Secondly, we attach $p$-adic $L$-functions to triples of ordinary $p$-adic families of quaternionic automorphic eigenforms. This is done by relating trilinear periods to some trilinear products over unitary Shimura curves which can be interpolated adapting the work of Liu–Zhang–Zhang to our families.

Soit $F$ un corps de nombres totalement réel. En utilisant une approche géométrique récemment développé par Andreatta et Iovita, nous construisons des familles $p$-adiques à plusieurs variables de formes automorphes quaternioniques de pente finie sur $F$. Cela est réalisé en interpolant les faisceaux modulaires définis sur certaines courbes de Shimura unitaires auxiliaires.

Deuxièmement, nous attachons des fonctions $L$ $p$-adiques à des triplets de familles ordinaires $p$-adiques de formes automorphes quaternioniques propres. Cela est effectué en reliant des périodes trilinéaires à certains produits trilinéaires sur des courbes de Shimura unitaires, lesquels peuvent être interpolés en adaptant les travaux de Liu–Zhang–Zhang à nos familles.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3791
Classification : 11S40, 11F67, 11R42, 11F85, 14G35
Keywords: Triple product $p$-adic L-functions, Eigenvarieties, families of automorphic forms, unitary Shimura curves.
Mots-clés : Fonctions $L$ $p$-adiques triple produit, variétés de Hecke, familles de formes automorphes, courbes de Shimura unitaires

Barrera Salazar, Daniel  1   ; Molina, Santiago  2

1 Universidad de Santiago de Chile, Alameda 3363, Estación Central, Santiago (Chile)
2 Universitat de Lleida, Campus Universitari Igualada-UdL, Av. Pla de la Massa 8, 08700 Igualada (Spain) 
Barrera Salazar, Daniel; Molina, Santiago. Triple product $p$-adic $L$-functions for Shimura curves over totally real number fields. Annales de l'Institut Fourier, Online first, 76 p.
@unpublished{AIF_0__0_0_A80_0,
     author = {Barrera Salazar, Daniel and Molina, Santiago},
     title = {Triple product $p$-adic $L$-functions for {Shimura} curves over totally real number fields},
     journal = {Annales de l'Institut Fourier},
     year = {2026},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     doi = {10.5802/aif.3791},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Barrera Salazar, Daniel
AU  - Molina, Santiago
TI  - Triple product $p$-adic $L$-functions for Shimura curves over totally real number fields
JO  - Annales de l'Institut Fourier
PY  - 2026
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3791
LA  - en
ID  - AIF_0__0_0_A80_0
ER  - 
%0 Unpublished Work
%A Barrera Salazar, Daniel
%A Molina, Santiago
%T Triple product $p$-adic $L$-functions for Shimura curves over totally real number fields
%J Annales de l'Institut Fourier
%D 2026
%V 0
%N 0
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3791
%G en
%F AIF_0__0_0_A80_0

[1] Andreatta, Fabrizio; Iovita, Adrian Triple product p-adic L-functions associated to finite slope p-adic families of modular forms, Duke Math. J., Volume 170 (2021) no. 9, pp. 1989-2083 | Zbl | MR

[2] Andreatta, Fabrizio; Iovita, Adrian; Pilloni, Vincent p-adic families of Siegel modular cuspforms, Ann. Math. (2), Volume 181 (2015) no. 2, pp. 623-697 | DOI | MR | Zbl

[3] Andreatta, Fabrizio; Iovita, Adrian; Pilloni, Vincent Le halo spectral, Ann. Sci. Éc. Norm. Supér. (4), Volume 51 (2018) no. 3, pp. 603-655 | DOI | MR | Zbl | Numdam

[4] Barrera, Daniel; Cauchy, Antonio; Molina, Santiago; Rotger, Victor Kato’s theorem for elliptic curves over totally real number fields (In progress)

[5] Bertolini, Massimo; Darmon, Henri; Rotger, Victor Beilinson-Flach elements and Euler systems II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil L-series, J. Algebr. Geom., Volume 24 (2015), pp. 569-604 | DOI | MR | Zbl

[6] Bertolini, Massimo; Seveso, Marco Adamo; Venerucci, Rodolfo Diagonal classes and the Bloch–Kato conjecture, Münster J. Math., Volume 13 (2020) no. 2, pp. 317-352 | DOI | MR | Zbl

[7] Bijakowski, Stéphane; Pilloni, Vincent; Stroh, Benoît Classicité de formes modulaires surconvergentes, Ann. Math. (2), Volume 183 (2016) no. 3, pp. 975-1014 | DOI | MR | Zbl

[8] Brasca, Riccardo p-adic modular forms of non-integral weight over Shimura curves, Compos. Math., Volume 149 (2013) no. 1, pp. 32-62 | DOI | MR | Zbl

[9] Bump, Daniel Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, 1997, xiv+574 pages | DOI | MR | Zbl

[10] Buzzard, Kevin Eigenvarieties, L-functions and Galois representations (London Mathematical Society Lecture Note Series), Volume 320, Cambridge University Press, 2007, pp. 59-120 | DOI | MR | Zbl

[11] Carayol, Henri Sur la mauvaise réduction des courbes de Shimura, Compos. Math., Volume 59 (1986) no. 2, pp. 151-230 | MR | Zbl | Numdam

[12] Coleman, Robert F. p-adic Banach spaces and families of modular forms, Invent. Math., Volume 127 (1997) no. 3, pp. 417-479 | DOI | MR | Zbl

[13] Darmon, Henri; Rotger, Victor Diagonal cycles and Euler systems I: A p-adic Gross-Zagier formula, Ann. Sci. Éc. Norm. Supér. (4), Volume 47 (2014) no. 4, pp. 779-832 | DOI | MR | Zbl | Numdam

[14] Darmon, Henri; Rotger, Victor Diagonal cycles and Euler systems II: The Birch and Swinnerton–Dyer conjecture for Hasse–Weil–Artin L-functions, J. Am. Math. Soc., Volume 30 (2017) no. 3, pp. 601-672 | DOI | MR | Zbl

[15] Ding, Yiwen Formes modulaires p-adiques sur les courbes de Shimura unitaires et compatibilité local-global, Mémoires de la Société Mathématique de France. Nouvelle Série, Société Mathématique de France, 2017 no. 155, viii+245 pages | DOI | MR | Zbl

[16] Fan, Yangyu Local expansion in Serre-Tate coordinates and p-adic iteration of Gauss–Manin connections, Ph. D. Thesis, Università Degli Studi di Milano (2018)

[17] Greenberg, Matthew; Seveso, Marco Adamo Triple product p-adic L-functions for balanced weights, Math. Ann., Volume 376 (2020) no. 1-2, pp. 103-176 | DOI | MR | Zbl

[18] Harris, Michael; Kudla, Stephen S. The central critical value of a triple product L-function, Ann. Math. (2), Volume 133 (1991) no. 3, pp. 605-672 | Zbl | DOI | MR

[19] Harris, Michael; Kudla, Stephen S. On a conjecture of Jacquet, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, 2004, pp. 355-371 | MR | Zbl

[20] Harris, Michael; Tilouine, Jacques p-adic measures and square roots of special values of triple product L-functions, Math. Ann., Volume 320 (2001) no. 1, pp. 127-147 | DOI | MR | Zbl

[21] Hsieh, Ming-Lun Hida families and p-adic triple product L-functions, Am. J. Math., Volume 143 (2021) no. 2, pp. 411-532 | DOI | MR | Zbl

[22] Ichino, Atsushi Trilinear forms and the central values of triple product L-functions, Duke Math. J., Volume 145 (2008) no. 2, pp. 281-307 | DOI | MR | Zbl

[23] Jetchev, Dimitar; Skinner, Christopher; Wan, Xin The Birch and Swinnerton-Dyer formula for elliptic curves of analytic rank one, Camb. J. Math., Volume 5 (2017) no. 3, pp. 369-434 | DOI | MR | Zbl

[24] Jones, Owen T. R. An analogue of the BGG resolution for locally analytic principal series, J. Number Theory, Volume 131 (2011) no. 9, pp. 1616-1640 | DOI | MR | Zbl

[25] Kassaei, Payman L. 𝔭-adic modular forms over Shimura curves over totally real fields, Compos. Math., Volume 140 (2004) no. 2, pp. 359-395 | DOI | MR | Zbl

[26] Kassaei, Payman L. A gluing lemma and overconvergent modular forms, Duke Math. J., Volume 132 (2006) no. 3, pp. 509-529 | DOI | MR | Zbl

[27] Kassaei, Payman L. Overconvergence and classicality: the case of curves, J. Reine Angew. Math., Volume 631 (2009), pp. 109-139 | DOI | MR | Zbl

[28] Kazi, Ananyo Twisted Triple Product p-adic L-function for Finite Slope Families of Hilbert Modular Forms (2025) (https://arxiv.org/abs/2401.13230)

[29] Kings, Guido; Loeffler, David; Zerbes, Sarah Livia Rankin–Eisenstein classes and explicit reciprocity laws, Camb. J. Math., Volume 5 (2017) no. 1, pp. 1-122 | DOI | MR | Zbl

[30] Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia Euler systems for Hilbert modular surfaces, Forum Math. Sigma, Volume 6 (2018), e23, 67 pages | DOI | MR | Zbl

[31] Liu, Yifeng; Zhang, Shouwu; Zhang, Wei A p-adic Waldspurger formula, Duke Math. J., Volume 167 (2018) no. 4, pp. 743-833 | DOI | MR | Zbl

[32] Loeffler, David; Zerbes, Sarah Livia Rankin–Eisenstein classes in Coleman families, Res. Math. Sci., Volume 3 (2016), 29, 53 pages | DOI | MR | Zbl

[33] Loeffler, David; Zerbes, Sarah Livia On the Bloch–Kato conjecture for GSp(4) (2024) (https://arxiv.org/abs/2003.05960)

[34] Loke, Hung Yean Trilinear forms of 𝔤𝔩 2 , Pac. J. Math., Volume 197 (2001) no. 1, pp. 119-144 | MR | Zbl | DOI

[35] Molina, Santiago Finite slope triple product p-adic L-functions over totally real number fields, J. Number Theory, Volume 223 (2021), pp. 267-306 | DOI | MR | Zbl

[36] Prasad, Dipendra Trilinear forms for representations of GL(2) and local ϵ-factors, Compos. Math., Volume 75 (1990) no. 1, pp. 1-46 | MR | Zbl

[37] Skinner, Christopher; Urban, Eric The Iwasawa main conjectures for GL 2 , Invent. Math., Volume 195 (2014) no. 1, pp. 1-277 | DOI | MR | Zbl

[38] Urban, Eric Eigenvarieties for reductive groups, Ann. Math. (2), Volume 174 (2011) no. 3, pp. 1685-1784 | DOI | MR | Zbl

[39] Urban, Eric Nearly overconvergent modular forms, Iwasawa theory 2012 (Contributions in Mathematical and Computational Sciences), Volume 7, Springer, 2014, pp. 401-441 | MR | Zbl | DOI

[40] Vignéras, M.-F. Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, 800, Springer, 1980, vii+169 pages | DOI | MR | Zbl

Cité par Sources :