L-invariants for cohomological representations of PGL(2) over arbitrary number fields. (English) Zbl 07862238
Summary: Let \(\pi\) be a cuspidal, cohomological automorphic representation of an inner form \(G\) of \(\operatorname{PGL}_2\) over a number field \(F\) of arbitrary signature. Further, let \(\mathfrak{p}\) be a prime of \(F\) such that \(G\) is split at \(\mathfrak{p}\) and the local component \(\pi_{\mathfrak{p}}\) of \(\pi\) at \(\mathfrak{p}\) is the Steinberg representation. Assuming that the representation is noncritical at \(\mathfrak{p}\), we construct automorphic \(\mathcal{L}\)-invariants for the representation \(\pi\). If the number field \(F\) is totally real, we show that these automorphic \(\mathcal{L}\)-invariants agree with the Fontaine-Mazur \(\mathcal{L}\)-invariant of the associated \(p\)-adic Galois representation. This generalizes a recent result of Spieß respectively Rosso and the first named author from the case of parallel weight \(2\) to arbitrary cohomological weights.
MSC:
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 11F33 | Congruences for modular and \(p\)-adic modular forms |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F75 | Cohomology of arithmetic groups |
| 11F80 | Galois representations |
References:
| [1] | Ash, A. and Stevens, G., ‘ \(p\) -adic deformations of arithmetic cohomology’, Preprint, 2008. |
| [2] | Barrera Salazar, D., Dimitrov, M. and Jorza, A., ‘ \(p\) -adic \(L\) -functions of Hilbert cusp forms and the trivial zero conjecture’, J. Eur. Math. Soc. (JEMS)14(10) (2022), 3439-3503. · Zbl 1523.11092 |
| [3] | Barrera Salazar, D. and Williams, C., ‘Exceptional zeros and L-invariants of Bianchi modular forms’, Trans. Amer. Math. Soc.371 (2019), 1-34. · Zbl 1443.11060 |
| [4] | Salazar, D. Barrera and Williams, C., ‘ \(p\) -adic \(L\) -functions for \({GL}_2\) ’, Canad. J. Math.71(5) (2019), 1019-1059. · Zbl 1470.11103 |
| [5] | Bergdall, J. and Hansen, D., ‘On p-adic L-functions for Hilbert modular forms’, Mem. Amer. Math. Soc. To appear. · Zbl 07913525 |
| [6] | Bertolini, M., Darmon, H. and Iovita, A., ‘Families of automorphic forms on definite quaternion algebras and Teitelbaum’s conjecture’, Astérisque331 (2010), 29-64. · Zbl 1251.11033 |
| [7] | Borel, A. and Serre, J.-P., ‘Corners and arithmetic groups’, Comment. Math. Helv.48(1) (1973), 436-491. · Zbl 0274.22011 |
| [8] | Borel, A. and Wallach, N., Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Mathematical Surveys and Monographs (American Mathematical Society, 2000). · Zbl 0980.22015 |
| [9] | Breuil, C., ‘Invariant et série spéciale p-adique’, Ann. Sci. Ec. Norm. Super.37(4) (2004), 559-610. · Zbl 1166.11331 |
| [10] | Breuil, C., ‘Série spéciale p-adique et cohomologie étale complétée’, Astérisque331 (2010), 65-115. · Zbl 1246.11106 |
| [11] | Breuil, C., ‘Remarks on some locally \({\mathbb{Q}}_p\) -analytic representations of \({GL}_2(F)\) in the crystalline case’, in Non-Abelian Fundamental Groups and Iwasawa Theory, London Mathematical Society Lecture Note Series (Cambridge University Press, 2011), 212-238. · Zbl 1312.11045 |
| [12] | Bump, D., Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics (Cambridge University Press, 1997). · Zbl 0911.11022 |
| [13] | Casselman, W., ‘On some results of Atkin and Lehner’, Math. Ann.201(4) (1973), 301-314. · Zbl 0239.10015 |
| [14] | Chida, M., Mok, C. and Park, J., ‘On Teitelbaum type L-invariants of Hilbert modular forms attached to definite quaternions’, J. Number Theory147 (2015), 633-665. · Zbl 1380.11048 |
| [15] | Dat, J., ‘Espaces symétriques de Drinfeld et correspondance de Langlands locale’, Ann. Sci. Ec. Norm. Super. 4e série, 39(1) (2006), 1-74. · Zbl 1141.22004 |
| [16] | Ding, Y., ‘ℒ-invariants and local-global compatibility for the group \({\rm GL}_2/F\) ’, Forum Math. Sigma4 (2016), Paper No. e13, 49. · Zbl 1376.11082 |
| [17] | Ding, Y., ‘ℒ-invariants, partially de Rham families, and local-global compatibility’, Ann. Inst. Fourier (Grenoble)67(4) (2017), 1457-1519. · Zbl 1433.11131 |
| [18] | Ding, Y., ‘Simple-invariants for \({GL}_n\) ’, Trans. Amer. Math. Soc.372(11) (2019), 7993-8042. · Zbl 1446.11106 |
| [19] | Emerton, M., ‘ \(p\) -adic \(L\) -functions and unitary completions of representations of \(p\) -adic reductive groups’, Duke Math. J.130(2) (2005), 353-392. · Zbl 1092.11024 |
| [20] | Fornea, M. and Gehrmann, L., ‘Plectic Stark-Heegner points’, Adv. Math.414 (2023), Article 108861. · Zbl 1525.11049 |
| [21] | Gehrmann, L., ‘On Shalika models and p-adic L-functions’, Isr. J. Math.226(1) (2018), 237-294. · Zbl 1414.11062 |
| [22] | Gehrmann, L., ‘Derived Hecke algebra and automorphic L-invariants’, Trans. Amer. Math. Soc.372(11) (2019), 7767-7784. · Zbl 1472.11141 |
| [23] | Gehrmann, L., ‘Functoriality of automorphic \(L\) -invariants and applications’, Doc. Math.24 (2019), 1225-1243. · Zbl 1469.11106 |
| [24] | Gehrmann, L., ‘Automorphic L-invariants for reductive groups’, J. Reine Angew. Math.2021(779) (2021), 57-103. · Zbl 1482.11076 |
| [25] | Gehrmann, L. and Rosso, G.. ‘Big principal series, \(p\) -adic families and -invariants’, Compos. Math.158(2) (2022), 409-436. · Zbl 1500.11041 |
| [26] | Grothendieck, A., ‘Sur quelques points d’algèbre homologique, I’, Tohoku Math. J.9(2) (1957), 119 - 221. · Zbl 0118.26104 |
| [27] | Hansen, D., ‘Universal eigenvarieties, trianguline Galois representations, and \(p\) -adic Langlands functoriality’, J. Reine Angew. Math.730 (2017), 1-64. With an appendix by James Newton. · Zbl 1425.11117 |
| [28] | Harder, G., ‘Eisenstein cohomology of arithmetic groups. the case GL2’, Invent. Math.89(1) (1987), 37-118. · Zbl 0629.10023 |
| [29] | Januszewski, F., ‘Rational structures on automorphic representations’, Math. Ann.370(3-4) (2018), 1805-1881. · Zbl 1403.11042 |
| [30] | Kohlhaase, J., ‘The cohomology of locally analytic representations’, J. Reine Angew. Math.651 (2011), 187-240. · Zbl 1226.22020 |
| [31] | Kohlhaase, J. and Schraen, B., ‘Homological vanishing theorems for locally analytic representations’, Math. Ann.353(1) (2012), 219-258. · Zbl 1259.22012 |
| [32] | Orlik, S., ‘On extensions of generalized Steinberg representations’, J. Algebra293(2) (2005), 611-630. · Zbl 1080.22008 |
| [33] | Orton, L., ‘An elementary proof of a weak exceptional zero conjecture’, Canad. J. Math.56 (2004), 373-405. · Zbl 1060.11030 |
| [34] | Rotger, V. and Seveso, M., ‘L-invariants and Darmon cycles attached to modular forms. J. Eur. Math. Soc. (JEMS)14(6) (2012), 1955-1999. · Zbl 1292.11069 |
| [35] | Saito, T.. ‘Hilbert modular forms and \(p\) -adic Hodge theory’, Compos. Math.145(5) (2009), 1081-1113. · Zbl 1259.11060 |
| [36] | Schneider, P. and Stuhler, U., ‘Resolutions for smooth representations of the general linear group over a local field’, J. Reine Angew. Math.436 (1993), 19-32. · Zbl 0780.20026 |
| [37] | Schraen, B., ‘Représentations \(p\) -adiques de \({GL}_2(L)\) et Catégories Dérivées’, Isr. J. Math. 176 (2010), 307-361. · Zbl 1210.11066 |
| [38] | Seveso, M., ‘The Teitelbaum conjecture in the indefinite setting’, Amer. J. Math.135(6) (2013), 1525-1557. · Zbl 1288.11051 |
| [39] | Spieß, M., ‘On special zeros of p-adic L-functions of Hilbert modular forms’, Invent. Math.196(1) (2014), 69-138. · Zbl 1392.11027 |
| [40] | Spieß, M., ‘On L-invariants associated to Hilbert modular forms’, Preprint, 2020. |
| [41] | Teitelbaum, J., ‘Values of p-adic L-functions and a p-adic Poisson kernel’, Invent. Math.101(2) (1990), 395-410. · Zbl 0731.11065 |
| [42] | Urban, E., ‘Eigenvarieties for reductive groups’, Ann. of Math.174(3) (2011), 1685-1784. · Zbl 1285.11081 |
| [43] | Xie, B., ‘On noncritical Galois representations’, J. Inst. Math. Jussieu22(1) (2021), 1-38. |
| [44] | Zhang, Y., ‘L-invariants and logarithm derivatives of eigenvalues of Frobenius’, Sci. China Math.57(8) (2014), 1587-1604. · Zbl 1323.11096 |
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