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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

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Volume 30, Issue 5

# Upper bounds for geodesic periods over rank one locally symmetric spaces

Jan Frahm
/ Feng Su
Published Online: 2018-01-13 | DOI: https://doi.org/10.1515/forum-2017-0185

## Abstract

We prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic cycles of the ambient manifold and twisted with automorphic forms on the cycles. The upper bounds are in terms of the Laplace eigenvalues of the two automorphic forms, and they generalize previous results for real hyperbolic manifolds to the context of all rank one locally symmetric spaces.

MSC 2010: 11F70; 22E46; 53C35

## References

• [1]

G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia Math. Appl. 71, Cambridge University Press, Cambridge, 1999. Google Scholar

• [2]

J. Bernstein and A. Reznikov, Estimates of automorphic functions, Mosc. Math. J. 4 (2004), no. 1, 19–37, 310. Google Scholar

• [3]

J. Bernstein and A. Reznikov, Periods, subconvexity of L-functions and representation theory, J. Differential Geom. 70 (2005), no. 1, 129–141.

• [4]

J. Bernstein and A. Reznikov, Subconvexity bounds for triple L-functions and representation theory, Ann. of Math. (2) 172 (2010), no. 3, 1679–1718.

• [5]

N. Burq, P. Gérard and N. Tzvetkov, Restrictions of the Laplace–Beltrami eigenfunctions to submanifolds, Duke Math. J. 138 (2007), no. 3, 445–486.

• [6]

M. Cowling, Unitary and uniformly bounded representations of some simple Lie groups, Harmonic Analysis and Group Representations, Liguori, Naples (1982), 49–128. Google Scholar

• [7]

M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), no. 3, 507–549.

• [8]

A. Deitmar, Invariant triple products, Int. J. Math. Math. Sci. 2006 (2006), Article ID 48274. Google Scholar

• [9]

A. Deitmar, Fourier expansion along geodesics on Riemann surfaces, Cent. Eur. J. Math. 12 (2014), no. 4, 559–573.

• [10]

I. M. Gel’fand, M. I. Graev and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions, W. B. Saunders, Philadelphia, 1969. Google Scholar

• [11]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Elsevier, Amsterdam, 2007. Google Scholar

• [12]

F. Knop, B. Krötz, T. Pecher and H. Schlichtkrull, Classification of reductive real spherical pairs II. The semisimple case, preprint (2017), https://arxiv.org/abs/1703.08048.

• [13]

T. Kobayashi and B. Speh, Symmetry breaking for representations of rank one orthogonal groups, Mem. Amer. Math. Soc. 238 (2015), no. 1126, 1–110. Google Scholar

• [14]

J. Möllers and B. Ørsted, Estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles, Int. Math. Res. Not. IMRN 2017 (2017), no. 11, 3209–3236. Google Scholar

• [15]

J. Möllers, B. Ørsted and Y. Oshima, Knapp–Stein type intertwining operators for symmetric pairs, Adv. Math. 294 (2016), 256–306.

• [16]

J. Möllers and F. Su, The second moment of period integrals and Rankin–Selberg L-functions for $\mathrm{GL}\left(3\right)×\mathrm{GL}\left(2\right)$, preprint (2017), https://arxiv.org/abs/1706.05167.

• [17]

A. Reznikov, Rankin–Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms, J. Amer. Math. Soc. 21 (2008), no. 2, 439–477. Google Scholar

• [18]

A. Reznikov, A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces, Forum Math. 27 (2015), no. 3, 1569–1590.

• [19]

F. Su, Upper bounds for geodesic periods over hyperbolic manifolds, preprint (2016), https://arxiv.org/abs/1605.02999.

• [20]

S. Zelditch, Kuznecov sum formulae and Szegö limit formulae on manifolds, Comm. Partial Differential Equations 17 (1992), no. 1–2, 221–260.

Revised: 2017-11-29

Published Online: 2018-01-13

Published in Print: 2018-09-01

Citation Information: Forum Mathematicum, Volume 30, Issue 5, Pages 1065–1077, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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