×

zbMATH — the first resource for mathematics

Triple product \(p\)-adic \(L\)-functions for balanced weights. (English) Zbl 07159950
Summary: We construct \(p\)-adic triple product \(L\)-functions that interpolate (square roots of) central critical \(L\)-values in the balanced region. Thus, our construction complements that of Harris and Tilouine. There are four central critical regions for the triple product \(L\)-functions and two opposite settings, according to the sign of the functional equation. In the first case, three of these regions are of interpolation, having positive sign; they are called the unbalanced regions and one gets three \(p\)-adic \(L\)-functions, one for each region of interpolation (this is the Harris-Tilouine setting). In the other setting there is only one region of interpolation, called the balanced region: we produce the corresponding \(p\)-adic \(L\)-function. Our triple product \(p\)-adic \(L\)-function arises as \(p\)-adic period integrals interpolating normalizations of the local archimedean period integrals. The latter encode information about classical representation theoretic branching laws. The main step in our construction of \(p\)-adic period integrals is showing that these branching laws vary in a \(p\)-adic analytic fashion. This relies crucially on the Ash-Stevens theory of highest weight representations over affinoid algebras.
MSC:
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Andreatta, F., Iovita, A.: Triple product \(p\)-adic \(L\)-functions associated to finite slope \(p\)-adic modular forms. Preprint (2019). http://www.mat.unimi.it/users/andreat/research.html
[2] Andreatta, F.; Iovita, A.; Pilloni, V., On overconvergent Hilbert modular cusp forms, Astérisque, 382, 163-193 (2016) · Zbl 1408.11037
[3] Andreatta, F.; Iovita, A.; Stevens, G., Overconvergent Eichler-Shimura isomorphisms, J. Inst. Math. Jussieu, 14, 221-274 (2015) · Zbl 1379.11062
[4] Ash, A., Stevens, G.: \(p\)-adic deformations of arithmetic cohomology. Submitted Preprint (2008). https://www2.bc.edu/avner-ash/Papers/Ash-Stevens-Oct-07-DRAFT-copy.pdf · Zbl 0866.20038
[5] Bellaïche, J.; Chenevier, G., Families of Galois representations and Selmer groups, Astérisque, 324, 1-314 (2009) · Zbl 1192.11035
[6] Bertolini, M., Seveso, M.A., Venerucci, R.: On exceptional zeros of triple product \(p\)-adic \(L\)-functions. In progress. https://sites.google.com/site/sevesomarco/publications
[7] Böcherer, S., Schulze-Pillot, R.: On central critical values of triple product L-functions. In: Sinnou D (ed) Number theory (Paris, 1994-1995), Cambridge University Press, Lond. Math. Soc. Lect. Note Ser. 235, 1-46 (1996) · Zbl 0924.11034
[8] Carayol, H., Sur les représentations \(l\) -adiques associées aux formes modulaires de Hilbert, Ann. Sci. Ecole Norm. Sup., 19, 3, 409-468 (1986) · Zbl 0616.10025
[9] Chenevier, G., Familles \(p\)-adiques de formes automorphes pour \(\mathbf{GL}_n \), J. Reine Angew. Math., 570, 143-217 (2004) · Zbl 1093.11036
[10] Chenevier, G., Une correspondance de Jacquet-Langlands \(p\)-adique, Duke Math. J., 126, 161-194 (2005) · Zbl 1070.11016
[11] Coleman, Rf, \(p\)-adic Banach spaces and families of modular forms, Invent. Math., 127, 417-479 (1997) · Zbl 0918.11026
[12] Coleman, Rf; Edixhoven, B., On the semi-simplicity of the \(U_p\)-operator on modular forms, Math. Ann., 310, 119-127 (1998) · Zbl 0902.11020
[13] Coleman, R.; Mazur, B., The Eigencurve, Galois Representations in Arithmetic Algebraic Geometry, 1-114 (1998) · Zbl 0932.11030
[14] Collins, D.J.: Anticyclotomic \(p\)-adic \(L\) functions and Ichino’s formula, PhD thesis
[15] Darmon, H.; Rotger, V., Diagonal cycles and Euler systems I: a \(p\)-adic Gross-Zagier formula, Ann. Scient. Ec. Norm. Sup., 4e ser, 47, 4, 779-832 (2014) · Zbl 1356.11039
[16] Darmon, H.; Rotger, V., Diagonal cycles and Euler systems II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-series, J. Am. Math. Soc., 30, 601-672 (2017) · Zbl 1397.11090
[17] Dimitrov, M.; Nyssen, L., Test vectors for trilinear forms when at least one representation is not supersingular, Manuscr. Math., 133, 479-504 (2010) · Zbl 1219.22015
[18] Emerton, E.; Pollack, R.; Weston, T., Variation of Iwasawa invariants in Hida families, Invent. Math., 163, 523-580 (2006) · Zbl 1093.11065
[19] Fouquet, O.; Ochiai, T., Control theorems for Selmer groups of nearly ordinary deformations, J. Reine Angew. Math., 666, 163-187 (2012) · Zbl 1248.11041
[20] Greenberg, M.; Seveso, Ma, \(p\)-adic families of cohomological modular forms for indefinite quaternion algebras and the Jacquet-Langlands correspondence, Can. J. Math., 68, 961-998 (2016) · Zbl 1406.11054
[21] Greenberg, M.; Seveso, Ma, \(p\)-families of modular forms and \(p\)-adic Abel-Jacobi maps, Ann. Math. Qué., 40, 397-434 (2016) · Zbl 1417.11101
[22] Greenberg, M., Seveso, M.A.: On the rationality of period integrals and special value formulas in the compact case. To appear in Rendiconti del Seminario Matematico della Università di Padova · Zbl 1416.11067
[23] Greenberg, R.; Stevens, G., \(p\)-adic \(L\)-functions and \(p\)-adic periods of modular forms, Invent. Math., 111, 401-447 (1993) · Zbl 0778.11034
[24] Harris, M.; Kudla, Ss, The central critical value of a triple product \(L\)-function, Ann. Math. (2), 133, 3, 605-672 (1991) · Zbl 0731.11031
[25] Harris, M.; Tilouine, J., \(p\)-adic measures and square roots of special values of triple product \(L\)-functions, Math. Ann., 320, 127-147 (2001) · Zbl 1034.11034
[26] Hida, H., Congruence of cusp forms and special values of their zeta functions, Invent. Math., 63, 2, 225-261 (1981) · Zbl 0459.10018
[27] Hida, H., Galois representations into \(\mathbf{GL}_2\left( \mathbb{Z}_p\left[\left[X\right] \right] \right)\) attached to ordinary cusp forms, Invent. Math., 85, 3, 545-613 (1986) · Zbl 0612.10021
[28] Hida, H., Modules of congruence of Hecke algebras and \(L\)-functions associated with cusp forms, Am. J. Math., 110, 323-382 (1988) · Zbl 0645.10029
[29] Hsieh, M.L.: Hida families and \(p\)-adic triple product \(L\)-functions. Am. J.Math. (To Appear). https://www.math.sinica.edu.tw/mlhsieh/research.htm
[30] Hu, Y., The subconvexity bound for the triple product \(L\) -function in level aspect, Am. J. Math., 139, 1, 215-259 (2017) · Zbl 1393.11041
[31] Ichino, A., Trilinear forms and the central values of triple product \(L\)-functions, Duke Math. J., 145, 2, 281-307 (2008) · Zbl 1222.11065
[32] Kings, G.; Loeffler, D.; Zerbes, S-L, Rankin-Eisenstein classes and explixit reciprocity laws, Camb. J. Math., 5, 1, 1-122 (2017) · Zbl 1428.11103
[33] Mazur, B.; Tate, J.; Teitelbaum, J., On \(p\)-adic analogs of the conjectures of Birch and Swinnerton-Dyer, Invent. Math., 84, 1-48 (1986) · Zbl 0699.14028
[34] Prasad, D., Trilinear forms for representations of GL (2) and local \(\epsilon \)-factors, Compos. Math., 75, 1, 1-46 (1990) · Zbl 0731.22013
[35] Pollack, R.; Weston, T., On anticyclotomic \({\mu } \)-invariants of modular forms, Compos. Math., 147, 5, 1353-1381 (2011) · Zbl 1259.11101
[36] Saha, Jp, Purity for families of Galois representations, Ann. I. Fourier, 67, 879-910 (2017) · Zbl 1441.11134
[37] Saha, Jp, Conductors in \(p\)-adic families, Ramanujan J., 44, 359-366 (2017) · Zbl 1434.11104
[38] Seveso, Ma, Heegner cycles and derivatives of p-adic L-functions, J. Reine Angew. Math., 686, 111-148 (2014) · Zbl 1356.11040
[39] Venerucci, R.: \(p\)-adic regulators and \(p\)-adic families of modular forms. Ph. D. thesis · Zbl 1406.11061
[40] Venerucci, R., Exceptional zero formulae and a conjecture of Perrin-Riou, Invent. Math., 203, 923-972 (2016) · Zbl 1406.11060
[41] Venerucci, R., On the p-converse of the Kolyvagin-Gross-Zagier theorem, Comment. Math. Helv., 91, 397-444 (2016) · Zbl 1406.11061
[42] Weil, A.: Adeles and algebraic groups. Progress in Mathematics vol. 23, Birkhäuser Boston (1982) · Zbl 0118.15801
[43] Woodbury, M.: Explicit trilinear forms and triple product \(L\)-functions. Preprint. http://www.mi.uni-koeln.de/ woodbury/research/researchindex.html
[44] Yu, C-F, Variations of Mass formulas for definite division algebras, J. Algebra, 422, 166-186 (2015) · Zbl 1395.11131
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.