Skip to main content Accessibility help

Lower bounds for Maass forms on semisimple groups

  • Farrell Brumley (a1) and Simon Marshall (a2)


Let $G$ be an anisotropic semisimple group over a totally real number field $F$ . Suppose that $G$ is compact at all but one infinite place $v_{0}$ . In addition, suppose that $G_{v_{0}}$ is $\mathbb{R}$ -almost simple, not split, and has a Cartan involution defined over $F$ . If $Y$ is a congruence arithmetic manifold of non-positive curvature associated with $G$ , we prove that there exists a sequence of Laplace eigenfunctions on $Y$ whose sup norms grow like a power of the eigenvalue.



Hide All

FB is supported by ANR grant 14-CE25 and SM is supported by NSF grant DMS-1902173.



Hide All
[Ava56]Avacumović, G. V., Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltig-keiten, Math. Z. 65 (1956), 327344.
[BP16]Blomer, V. and Pohl, A., The sup-norm problem on the Siegel modular space of rank two, Amer. J. Math. 138 (2016), 9991027.
[Bor63]Borel, A., Compact Clifford–Klein forms of symmetric spaces, Topology 2 (1963), 111122.
[Bor91]Borel, A., Linear algebraic groups, Graduate Texts in Mathematics, vol. 126, second edition (Springer, New York, 1991).
[BT84]Bruhat, F. and Tits, J., Groupes réductifs sur un corps local: II. Schémas en groupes. Existence d’une donnée radicielle valué, Publ. Math. Inst. Hautes Études Sci. 60 (1984), 5184.
[Cas80]Casselman, W., The unramified principal series of p-adic groups I, Compos. Math. 40 (1980), 387406.
[Col85]Colin de Verdière, Y., Ergodicité et fonctions propres du Laplacien, Comm. Math. Phys. 102 (1985), 497502.
[Don07]Donnelly, H., Exceptional sequences of eigenfunctions for hyperbolic manifolds, Proc. Amer. Math. Soc. 135 (2007), 15511555.
[DKV79]Duistermaat, J. J., Kolk, J. A. C. and Varadarajan, V. S., Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math. 52 (1979), 2793.
[FLO12]Feigon, B., Lapid, E. and Offen, O., On representations distinguished by unitary groups, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 185323.
[FL18]Finis, T. and Lapid, E., An approximation principle for congruence subgroups, J. Eur. Math. Soc. (JEMS) 20 (2018), 10751138.
[Gan71]Gangolli, R., On the Plancherel formula and the Paley–Wiener theorem for spherical functions on semisimple Lie groups, Ann. of Math. (2) 93 (1971), 105165.
[Gro96]Gross, B., On the Satake isomorphism, in Galois representations in arithmetic algebraic geometry (Durham, 1996), London Mathematical Society Lecture Note Series, vol. 254 (Cambridge University Press, Cambridge), 223238.
[Gro97]Gross, B., On the motive of a reductive group, Invent. Math. 130 (1997), 287313.
[Har57]Harish-Chandra, A formula for semisimple Lie groups, Amer. J. Math. 79 (1957), 733760.
[Har64]Harish-Chandra, Some results on an invariant integral on a semi-simple Lie algebra, Ann. of Math. (2) 80 (1964), 551593.
[Har66]Harish-Chandra, Discrete series for semisimple Lie groups II, Acta Math. 116 (1966), 1111.
[Hel00]Helgason, S., Groups and geometric analysis, Mathematical Surveys and Monographs, vol. 83. (American Mathematical Society, Providence, RI, 2000).
[Hel01]Helgason, S., Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34 (American Mathematical Society, Providence, RI, 2001).
[Hel08]Helgason, S., Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39 (American Mathematical Society, 2008).
[HW93]Helminck, A. G. and Wang, S. P., On rationality properties of involutions of reductive groups, Adv. Math. 99 (1993), 2697.
[IS95]Iwaniec, H. and Sarnak, P., L norms of eigenfunctions of arithmetic surfaces, Ann. of Math. (2) 141 (1995), 301320.
[Jac05]Jacquet, H., Kloosterman identities over a quadratic extension II, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), 609669.
[JLR93]Jacquet, H., Lai, K. and Rallis, S., A trace formula for symmetric spaces, Duke Math. J. 70 (1993), 305372.
[Kna02]Knapp, A., Lie groups beyond an introduction, Progress in Mathematics, vol. 140 (Birkhäuser, Boston, 2002).
[Kot86]Kottwitz, R., Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), 365400.
[Kot88]Kottwitz, R., Tamagawa numbers, Ann. of Math. (2) 127 (1988), 629646.
[LO07]Lapid, E. and Offen, O., Compact unitary periods, Compos. Math. 143 (2007), 323338.
[Lev52]Levitan, B. M., On the asymptoptic behavior of the spectral function of a self-adjoint differential equation of second order, Isv. Akad. Nauk SSSR Ser. Mat. 16 (1952), 325352.
[Mac71]Macdonald, I. G., Spherical functions on a group of p-adic type, Publications of the Ramanujan Institute no. 2 (Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras, Madras, 1971).
[MR03]Maclachlan, C. and Reid, A. W., The arithmetic of hyperbolic 3-manifolds (Springer, New York, 2003).
[Mar14]Marshall, S., Upper bounds for Maass forms on semisimple groups, Preprint (2014),arXiv:1405.7033.
[MT15]Matz, J. and Templier, N., Sato–Tate equidistribution for families of Hecke–Maass forms on $\text{SL}(n,\mathbb{R})/\text{SO}(n)$, Preprint (2015), arXiv:1505.07285.
[Mil11]Milicevic, D., Large values of eigenfunctions on arithmetic hyperbolic 3-manifolds, Geom. Funct. Anal. 21 (2011), 13751418.
[Nad05]Nadler, D., Perverse sheaves on real loop Grassmannians, Invent. Math. 159 (2005), 173.
[OV94]Onishchik, A. and Vinberg, E., Lie groups and Lie algebras III, Encyclopaedia of Mathematical Sciences, vol. 41 (Springer, Berlin, 1994).
[RS94]Rudnick, Z. and Sarnak, P., The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), 195213.
[Sah15]Saha, A., Large values of newforms on GL (2) with highly ramified central character, Int. Math. Res. Not. IMRN 2016 (2016), 41034131.
[Sak08]Sakellaridis, Y., On the unramified spectrum of spherical varieties over p-adic fields, Compos. Math. 144 (2008), 9781016.
[Sak13]Sakellaridis, Y., Spherical functions on spherical varieties, Amer. J. Math. 135 (2013), 12911381.
[SV17]Sakellaridis, Y. and Venkatesh, A., Periods and harmonic analysis on spherical varieties, Astérisque 396 (2017).
[Sar95]Sarnak, P., Arithmetic quantum chaos, Israel Mathematical Conference Proceedings, Schur Lectures (Bar-Ilan University, Ramat-Gan, Israel, 1995).
[Sar04]Sarnak, P., Letter to Morawetz, available at
[Sch74]Schnirelman, A. I., Ergodic properties of eigenfunctions, Uspekhi Mat. Nauk 29 (1974), 181182.
[ST16]Shin, S.-W. and Templier, N., Sato–Tate theorem for families and low-lying zeros of automorphic L-functions, Invent. Math. 203 (2016), 1177.
[SZ02]Sogge, C. and Zelditch, S., Riemannian manifolds with maximal eigenfunction growth, Duke Math. J. 114 (2002), 387437.
[Ste68]Steinberg, R., Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80 (1968).
[Tem14]Templier, N., Large values of modular forms, Camb. J. Math. 2 (2014), 91116.
[Tit79]Tits, J, Reductive groups over local fields, in Automorphic forms, representations and L-functions, Proceedings of Symposia in Pure Mathematics, vol. 33 (American Mathematical Society, Providence, RI, 1979), 2970; Part 1.
[Vus74]Vust, T., Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France 102 (1974), 317333.
[Zel87]Zelditch, S., Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919941.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

Lower bounds for Maass forms on semisimple groups

  • Farrell Brumley (a1) and Simon Marshall (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.