×

Functoriality, Smith theory, and the Brauer homomorphism. (English) Zbl 1358.11063

This paper describes a functorial lift of mod \(p\) automorphic forms.
Let \(\mathbf{G}\) be a semisimple group over a number field \(F\), and let \(k\) be an algebraically closed field of characteristic \(p\). The authors consider “mod \(p\) automorphic forms”, which are Hecke eigenclasses in the cohomology of congruence subgroups with \(k\)-coefficients.
Let \(\sigma\) be an automorphism of \(\mathbf{G}\) of order \(p\), defined over \(F\). Assume that the group \(\mathbf{G}^\sigma\) is connected. For almost all places \(v\), the authors construct a homomorphism \(\psi_v\) from a Hecke algebra for \(G_v=\mathbf{G}(F_v)\) to a Hecke algebra for \(G^\sigma_v\).
Main theorem. If a \(\text{mod}\, p\) automorphic form for \(\mathbf{G}^\sigma\) has Satake parameters \(\{a_v \}\), then there exists a \(\text{mod}\, p\) automorphic form for \(\mathbf{G}\) with Satake parameters \(\{\psi_v^*(a_v) \}\).
The authors also describe the correspondence at ramified places. Finally, they compute \(\psi_v\) in terms of dual groups.
As explained in the paper, if a mod \(p\) automorphic form lifts to characteristic zero, then functoriality can also be obtained by classical methods (by comparison of trace formulae), and the classical methods sometimes give stronger results. Indeed, the functoriality results for mod \(p\) classes that do not lift to characteristic zero are the main novelty of this paper.

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F33 Congruences for modular and \(p\)-adic modular forms

References:

[1] A. Ash, ”Smith theory and Hecke operators,” J. Algebra, vol. 259, iss. 1, pp. 43-58, 2003. · Zbl 1045.11034 · doi:10.1016/S0021-8693(02)00547-1
[2] A. Ash, ”Direct sums of \(\bmod p\) characters of \({ GAL}(\overline{\mathbb Q}/{\mathbb Q})\) and the homology of \(GL(n,{\mathbb Z})\),” Comm. Algebra, vol. 41, iss. 5, pp. 1751-1775, 2013. · Zbl 1337.11035 · doi:10.1080/00927872.2011.649508
[3] N. Bergeron and A. Venkatesh, ”The asymptotic growth of torsion homology for arithmetic groups,” J. Inst. Math. Jussieu, vol. 12, iss. 2, pp. 391-447, 2013. · Zbl 1266.22013 · doi:10.1017/S1474748012000667
[4] I. N. Bernstein and A. V. Zelevinsky, ”Induced representations of reductive \(\mathfrakp\)-adic groups. I,” Ann. Sci. École Norm. Sup., vol. 10, iss. 4, pp. 441-472, 1977. · Zbl 0412.22015
[5] J. Bernstein, Course notes of a course on \(p\)-adic groups.
[6] A. Borel, ”Automorphic \(L\)-functions,” in Automorphic Forms, Representations and \(L\)-Functions, Providence, R.I.: Amer. Math. Soc., 1979, vol. 33, pp. 27-61. · Zbl 0412.10017
[7] A. Borel and J. Tits, ”Groupes réductifs,” Publ. Math. Inst. Hautes Études Sci., vol. 27, pp. 55-150, 1965. · Zbl 0145.17402 · doi:10.1007/BF02684375
[8] N. Bourbaki, Algebra II. Chapters 4-7, New York: Springer-Verlag, 2003. · Zbl 1017.12001 · doi:10.1007/978-3-642-61698-3
[9] K. Buzzard and T. Gee, The conjectural connections between automorphic representations and Galois representations, 2010. · Zbl 1377.11067
[10] P. Cartier, ”Representations of \(p\)-adic groups: a survey,” in Automorphic Forms, Representations and \(L\)-Functions, Providence, R.I.: Amer. Math. Soc., 1979, vol. 33, pp. 111-155. · Zbl 0421.22010
[11] W. Casselman, Introduction to the theory of admissible representations. · Zbl 0355.20041
[12] L. Clozel, ”Formes modulaires mod \(p\), changement de base et théorie d’Iwasawa,” Rend. Mat. Appl. (7), vol. 35, iss. 1-2, pp. 35-46, 2014. · Zbl 1375.11046
[13] L. Clozel, ”Formes modulaires sur la \(\mathbb Z_p\)-extension cyclotomique de \(\mathbb Q\),” Pacific J. Math., vol. 268, iss. 2, pp. 259-274, 2014. · Zbl 1302.11016 · doi:10.2140/pjm.2014.268.259
[14] P. Deligne and G. Lusztig, ”Representations of reductive groups over finite fields,” Ann. of Math., vol. 103, iss. 1, pp. 103-161, 1976. · Zbl 0336.20029 · doi:10.2307/1971021
[15] M. Emerton, ”A local-global compatibility conjecture in the \(p\)-adic Langlands programme for \({ GL}_{2/{\mathbb Q}}\),” Pure Appl. Math. Q., vol. 2, iss. 2, Special Issue: In honor of John H. Coates. Part 2, pp. 279-393, 2006. · Zbl 1254.11106 · doi:10.4310/PAMQ.2006.v2.n2.a1
[16] W. T. Gan and G. Savin, ”Real and global lifts from \( PGL_3\) to \(G_2\),” Int. Math. Res. Not., vol. 2003, iss. 50, pp. 2699-2724, 2003. · Zbl 1037.22033 · doi:10.1155/S1073792803130607
[17] G. Glauberman, ”Correspondences of characters for relatively prime operator groups.,” Canad. J. Math., vol. 20, pp. 1465-1488, 1968. · Zbl 0167.02602 · doi:10.4153/CJM-1968-148-x
[18] B. H. Gross, ”On the Satake isomorphism,” in Galois Representations in Arithmetic Algebraic Geometry, Cambridge: Cambridge Univ. Press, 1998, vol. 254, pp. 223-237. · Zbl 0996.11038 · doi:10.1017/CBO9780511662010.006
[19] P. Gunnells and D. Yasaki, On the growth of torsion in the cohomology of arithmetic groups. · Zbl 1277.11053
[20] T. J. Haines and S. Rostami, ”The Satake isomorphism for special maximal parahoric Hecke algebras,” Represent. Theory, vol. 14, pp. 264-284, 2010. · Zbl 1251.22013 · doi:10.1090/S1088-4165-10-00370-5
[21] G. Henniart and M. Vignéras, ”A Satake isomorphism for representations modulo \(p\) of reductive groups over local fields,” J. reine angew. Math., vol. 701, pp. 33-75, 2015. · Zbl 1327.22019 · doi:10.1515/crelle-2013-0021
[22] D. Joyner, ”On finite dimensional representations of non-connected reductive groups,” J. Lie Theory, vol. 10, iss. 2, pp. 269-284, 2000. · Zbl 0986.20043
[23] S. Kionke, ”On lower bounds for cohomology growth in \(p\)-adic analytic towers,” Math. Z., vol. 277, iss. 3-4, pp. 709-723, 2014. · Zbl 1354.20032 · doi:10.1007/s00209-013-1273-3
[24] R. E. Kottwitz and D. Shelstad, Foundations of Twisted Endoscopy, , 1999, vol. 255. · Zbl 0958.22013
[25] D. Kraines and C. Schochet, ”Differentials in the Eilenberg-Moore spectral sequence,” J. Pure Appl. Algebra, vol. 2, iss. 2, pp. 131-148, 1972. · Zbl 0237.55016 · doi:10.1016/0022-4049(72)90018-7
[26] R. P. Langlands, ”Representations of abelian algebraic groups,” Pacific J. Math., vol. 181, iss. Special Issue, pp. 231-250, 1997. · Zbl 0910.11045 · doi:10.2140/pjm.1997.181.231
[27] W. Müller and J. Pfaff, ”On the growth of torsion in the cohomology of arithmetic groups,” Math. Ann., vol. 359, iss. 1-2, pp. 537-555, 2014. · Zbl 1318.11072 · doi:10.1007/s00208-014-1014-x
[28] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Boston: Academic Press, 1994, vol. 139. · Zbl 0841.20046
[29] J. Rohlfs and B. Speh, ”Lefschetz numbers and twisted stabilized orbital integrals,” Math. Ann., vol. 296, iss. 2, pp. 191-214, 1993. · Zbl 0808.11039 · doi:10.1007/BF01445102
[30] I. Satake, ”Theory of spherical functions on reductive algebraic groups over \(\mathfrakp\)-adic fields,” Publ. Math. Inst. Hautes Études Sci., vol. 18, pp. 5-69, 1963. · Zbl 0122.28501 · doi:10.1007/BF02684781
[31] M. H. cSengün, ”On the integral cohomology of Bianchi groups,” Exp. Math., vol. 20, iss. 4, pp. 487-505, 2011. · Zbl 1269.22007 · doi:10.1080/10586458.2011.594671
[32] . J-P. Serre, ”Two letters on quaternions and modular forms (mod \(p\)),” Israel J. Math., vol. 95, pp. 281-299, 1996. · Zbl 0870.11030 · doi:10.1007/BF02761043
[33] . J-P. Serre, Cohomologie Galoisienne, New York: Springer-Verlag, 1973, vol. 5. · Zbl 0812.12002
[34] R. Steinberg, ”Regular elements of semisimple algebraic groups,” Publ. Math. Inst. Hautes Études Sci., vol. 25, pp. 49-80, 1965. · Zbl 0136.30002 · doi:10.1007/BF02684397
[35] R. Steinberg, Endomorphisms of Linear Algebraic Groups, Providence, RI: Amer. Math. Soc., 1968, vol. 80. · Zbl 0164.02902
[36] D. Treumann, Smith theory and geometric Hecke algebras, 2011.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.