×

Bounds for the multiplicities of cohomological automorphic forms on \(\mathrm{GL}_2\). (English) Zbl 1319.11026

Let \(F\) be a number field that is not totally real, the paper considers the bound of the dimension of the space of cusp forms on \(\mathrm{GL}_2\) with fixed level and growing weight. There is a “trivial” bound \(\Delta({\mathbf d})\) (determined by the weight \({\mathbf d}\)) which is given by the trace formula method. The paper breaks this trivial bound and gives a bound of the form \(\min({\mathbf d})^{-\frac13+\varepsilon}\Delta({\mathbf d})\). When \(F\) is a totally real number field, previously Shimizu established an asymptotic formula again in terms of \(\Delta({\mathbf d})\).
The automorphic forms considered are tempered but not in the discrete series. Nontrivial bounds for the dimension of the space of such forms are quite rare. The proof uses the theory of \(p\)-adically completed cohomology developed by F. Calegari and M. Emerton [Ann. Math. (2) 170, No. 3, 1437–1446 (2009; Zbl 1195.22015)].

MSC:

11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F75 Cohomology of arithmetic groups

Citations:

Zbl 1195.22015
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] A. Borel, ”Stable real cohomology of arithmetic groups,” Ann. Sci. École Norm. Sup., vol. 7, pp. 235-272 (1975), 1974. · Zbl 0316.57026
[2] A. Borel, ”Stable real cohomology of arithmetic groups. II,” in Manifolds and Lie Groups, Mass.: Birkhäuser, 1981, vol. 14, pp. 21-55. · Zbl 0483.57026
[3] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Second ed., Providence, RI: Amer. Math. Soc., 2000, vol. 67. · Zbl 0980.22015
[4] F. Calegari and M. Emerton, ”Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms,” Ann. of Math., vol. 170, iss. 3, pp. 1437-1446, 2009. · Zbl 1195.22015 · doi:10.4007/annals.2009.170.1437
[5] F. Calegari and M. Emerton, ”Mod-\(p\) cohomology growth in \(p\)-adic analytic towers of 3-manifolds,” Groups Geom. Dyn., vol. 5, iss. 2, pp. 355-366, 2011. · Zbl 1242.57014 · doi:10.4171/GGD/131
[6] W. Duke, ”The dimension of the space of cusp forms of weight one,” Internat. Math. Res. Notices, vol. 1995, iss. 2, p. no. 2, 99-109. · Zbl 0843.11024 · doi:10.1155/S1073792895000080
[7] J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic Pro-\(p\) Groups, Second ed., Cambridge: Cambridge Univ. Press, 1999, vol. 61. · Zbl 0934.20001 · doi:10.1017/CBO9780511470882
[8] M. Emerton, ”On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms,” Invent. Math., vol. 164, iss. 1, pp. 1-84, 2006. · Zbl 1090.22008 · doi:10.1007/s00222-005-0448-x
[9] T. Finis, F. Grunewald, and P. Tirao, ”The cohomology of lattices in \({ SL}(2,\mathbb C)\),” Experiment. Math., vol. 19, iss. 1, pp. 29-63, 2010. · Zbl 1225.11072 · doi:10.1080/10586458.2010.10129067
[10] G. Harder, ”Eisenstein cohomology of arithmetic groups. The case \({ GL}_2\),” Invent. Math., vol. 89, iss. 1, pp. 37-118, 1987. · Zbl 0629.10023 · doi:10.1007/BF01404673
[11] M. Harris, ”\(p\)-adic representations arising from descent on abelian varieties,” Compositio Math., vol. 39, iss. 2, pp. 177-245, 1979. · Zbl 0417.14034
[12] E. Lucas, ”Sur les congruences des nombres eulériens et les coefficients différentiels des functions trigonométriques suivant un module premier,” Bull. Soc. Math. France, vol. 6, pp. 49-54, 1878.
[13] P. Michel and A. Venkatesh, ”On the dimension of the space of cusp forms associated to 2-dimensional complex Galois representations,” Int. Math. Res. Not., vol. 2002, p. no. 38, 2021-2027. · Zbl 1008.11019 · doi:10.1155/S1073792802206121
[14] C. S. Rajan, ”On the non-vanishing of the first Betti number of hyperbolic three manifolds,” Math. Ann., vol. 330, iss. 2, pp. 323-329, 2004. · Zbl 1077.11040 · doi:10.1007/s00208-004-0552-z
[15] H. Shimizu, ”On discontinuous groups operating on the product of the upper half planes,” Ann. of Math., vol. 77, pp. 33-71, 1963. · Zbl 0218.10045 · doi:10.2307/1970201
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.