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Mosc. Math. J., 2004, Volume 4, Number 1, Pages 19–37 (Mi mmj141)  

This article is cited in 27 scientific papers (total in 27 papers)

Estimates of automorphic functions

J. H. Bernsteina, A. Reznikovb

a Tel Aviv University
b Bar-Ilan University

Abstract: We present a new method to estimate trilinear period for automorphic representations of $SL_2(\mathbb R)$. The method is based on the uniqueness principle in representation theory. We show how to separate the exponentially decaying factor in the triple period from the essential automorphic factor which behaves polynomially. We also describe a general method which gives an estimate for the average of the automorphic factor and thus prove a convexity bound for the triple period.

Key words and phrases: Automorphic representations, periods, uniqueness.


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MSC: 11F67, 11F70, 22E45
Received: May 6, 2003

Citation: J. H. Bernstein, A. Reznikov, “Estimates of automorphic functions”, Mosc. Math. J., 4:1 (2004), 19–37

Citation in format AMSBIB
\by J.~H.~Bernstein, A.~Reznikov
\paper Estimates of automorphic functions
\jour Mosc. Math.~J.
\yr 2004
\vol 4
\issue 1
\pages 19--37

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    This publication is cited in the following articles:
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    2. Blomer V., Harcos G., “The spectral decomposition of shifted convolution sums”, Duke Math. J., 144:2 (2008), 321–339  crossref  mathscinet  zmath  isi  scopus
    3. Reznikov A., “Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of maass forms”, J. Amer. Math. Soc., 21:2 (2008), 439–477  crossref  mathscinet  zmath  isi  scopus
    4. Dimitrov M., Nyssen L., “Test vectors for trilinear forms when at least one representation is not supercuspidal”, Manuscripta Math, 133:3–4 (2010), 479–504  crossref  mathscinet  zmath  isi  scopus
    5. Bernstein J., Reznikov A., “Subconvexity bounds for triple L-functions and representation theory”, Ann of Math (2), 172:3 (2010), 1679–1718  crossref  mathscinet  zmath  isi  scopus
    6. Clerc J.-L., Kobayashi T., Orsted B., Pevzner M., “Generalized Bernstein-Reznikov integrals”, Math Ann, 349:2 (2011), 395–431  crossref  mathscinet  zmath  isi  scopus
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    8. Clerc J.-L., Orsted B., “Conformally Invariant Trilinear Forms on the Sphere”, Ann. Inst. Fourier, 61:5 (2011), 1807–1838  crossref  mathscinet  zmath  isi  scopus
    9. Anantharaman N., Zelditch S., “Intertwining the Geodesic Flow and the Schrodinger Group on Hyperbolic Surfaces”, Math. Ann., 353:4 (2012), 1103–1156  crossref  mathscinet  zmath  isi  elib  scopus
    10. Kobayashi T., Oshima T., “Finite Multiplicity Theorems for Induction and Restriction”, Adv. Math., 248 (2013), 921–944  crossref  mathscinet  zmath  isi  scopus
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    13. Ben Said S., Koufany Kh., Zhang G., “Invariant Trilinear Forms on Spherical Principal Series of Real Rank One Semisimple Lie Groups”, Int. J. Math., 25:3 (2014), 1450017  crossref  mathscinet  zmath  isi  scopus
    14. Bui Van Binh, V. V. Schechtman, “Invariant Functionals and Zamolodchikovs' Integral”, Funct. Anal. Appl., 49:1 (2015), 57–59  mathnet  crossref  crossref  zmath  isi  elib
    15. Clare P., “Invariant Trilinear Forms For Spherical Degenerate Principal Series of Complex Symplectic Groups”, Int. J. Math., 26:13 (2015), 1550107  crossref  mathscinet  zmath  isi  scopus
    16. Reznikov A., “a Uniform Bound For Geodesic Periods of Eigenfunctions on Hyperbolic Surfaces”, Forum Math., 27:3 (2015), 1569–1590  crossref  mathscinet  zmath  isi  scopus
    17. Moellers J., Orsted B., Oshima Y., “Knapp-Stein Type Intertwining Operators For Symmetric Pairs”, Adv. Math., 294 (2016), 256–306  crossref  mathscinet  zmath  isi  scopus
    18. Hoffstein J., Hulse T.A., Reznikov A., “Multiple Dirichlet Series and Shifted Convolutions”, J. Number Theory, 161:SI (2016), 457–533  crossref  mathscinet  zmath  isi  scopus
    19. Clerc J.-L., “Singular Conformally Invariant Trilinear Forms, i the Multiplicity One Theorem”, Transform. Groups, 21:3 (2016), 619–652  crossref  mathscinet  zmath  isi  scopus
    20. Kroetz B., Sayag E., Schlichtkrull H., “the Harmonic Analysis of Lattice Counting on Real Spherical Spaces”, Doc. Math., 21 (2016), 627–660  mathscinet  zmath  isi
    21. Kobayashi T., Leontiev A., “Symmetry Breaking Operators For the Restriction of Representations of Indefinite Orthogonal Groups O(P, Q)”, Proc. Jpn. Acad. Ser. A-Math. Sci., 93:8 (2017), 86–91  crossref  zmath  isi  scopus
    22. Ghosh A., Reznikov A., Sarnak P., “Nodal Domains of Maass Forms, II”, Am. J. Math., 139:5 (2017), 1395–1447  crossref  zmath  isi  scopus
    23. Clerc J.-L., “Singular Conformally Invariant Trilinear Forms, II the Higher Multiplicity Case”, Transform. Groups, 22:3 (2017), 651–706  crossref  zmath  isi  scopus
    24. Moellers J., Orsted B., “Estimates For the Restriction of Automorphic Forms on Hyperbolic Manifolds to Compact Geodesic Cycles”, Int. Math. Res. Notices, 2017, no. 11, 3209–3236  crossref  isi
    25. Knop F., Kroetz B., Schlichtkrull H., “The Tempered Spectrum of a Real Spherical Space”, Acta Math., 218:2 (2017), 319–383  crossref  zmath  isi  scopus
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    27. Petridis Y.N., Risager M.S., “Averaging Over Heegner Points in the Hyperbolic Circle Problem”, Int. Math. Res. Notices, 2018, no. 16, 4942–4968  crossref  mathscinet  zmath  isi  scopus
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