Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T01:40:57.803Z Has data issue: false hasContentIssue false
  • English
  • Français

EXTREME VALUES OF GEODESIC PERIODS ON ARITHMETIC HYPERBOLIC SURFACES

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  22 December 2020

Bart Michels Open the ORCID record for Bart Michels [Opens in a new window]
Bart Michels*
Affiliation:
Université Sorbonne Paris Nord, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse, France ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Given a closed geodesic on a compact arithmetic hyperbolic surface, we show the existence of a sequence of Laplacian eigenfunctions whose integrals along the geodesic exhibit nontrivial growth. Via Waldspurger’s formula we deduce a lower bound for central values of Rankin-Selberg L-functions of Maass forms times theta series associated to real quadratic fields.

Keywords

L-functionsMaass formsclosed geodesicsamplification method

MSC classification

Primary: 11F03: Modular and automorphic functions
Secondary: 11F72: Spectral theory; Selberg trace formula
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 5 , September 2022 , pp. 1507 - 1542
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abert, M., Bergeron, N. and Le Masson, E., Eigenfunctions and random waves in the Benjamini-Schramm limit. Preprint 2018, https://arxiv.org/abs/1810.05601.Google Scholar
Aurich, R. and Steiner, F., Exact theory for the quantum eigenstates of the Hadamard-Gutzwiller model, Phys. D, 48(2–3) (1991), 445–470.CrossRefGoogle Scholar
Aurich, R. and Steiner, F., Statistical properties of highly excited quantum eigenstates of a strongly chaotic system, Phys. D 64(1–3) (1993), 185–214.Google Scholar
Avakumović, V. G., Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten [On the eigenfunctions on closed Riemannian manifolds], Math. Z. 65 (1956) 327–344.CrossRefGoogle Scholar
Berry, M. V., Regular and irregular semiclassical wavefunctions, J. Phys. A 10(12) (1977), 2083–2091.Google Scholar
Blomer, V., Fouvry, É., Kowalski, E., Ph. Michel, Milićević, D. and Sawin, W., The second moment theory of families of $L$ -functions, Mem. Amer. Math. Soc., 93–119.Google Scholar
Bondarenko, A. and Seip, K., Large greatest common divisor sums and extreme values of the Riemann zeta function, Duke Math. J., 166(9) (2017), 1685–1701.Google Scholar
Brumley, F. and Marshall, S., Lower bounds for Maass forms on semisimple groups, Compos. Math. 156(5) (2020), 959–1003.Google Scholar
Chen, X. and Sogge, C. D., On integrals of eigenfunctions over geodesics, Proc. Amer. Math. Soc. 143(1) (2015), 151–161.Google Scholar
de la Bretèche, R. and Tenenbaum, G., Sommes de Gál et applications [Gál sums and applications], Proc. Lond. Math. Soc. (3) 119(3) (2019), 104–134.CrossRefGoogle Scholar
Dyatlov, S. and Zworski, M., Quantum ergodicity for restrictions to hypersurfaces, Nonlinearity 26(1) (2013), 35–52.Google Scholar
Eichler, M., Lectures on Modular Correspondences, Vol. 9 of Lectures on Mathematics and Physics: Mathematics (Tata Institute of Fundamental Research, Mumbai, India, 1965).Google Scholar
Farmer, D. W., Gonek, S. M. and Hughes, C. P., The maximum size of $L$ -functions, J. Reine Angew. Math. 609 (2007), 215–236.Google Scholar
Friedberg, S. and Hoffstein, J., Nonvanishing theorems for automorphic $L$ -functions on $GL(2)$ , Ann. Math. (2) 142 (1995), 385–423.Google Scholar
Goldfeld, D., Hoffstein, J. and Lieman, D., Appendix: an effective zero-free region, Ann. Math. (2) 140(1) (1994), 177–181.CrossRefGoogle Scholar
Guo, J., On the positivity of the central critical values of automorphic $L$ -functions for $GL(2)$ , Duke Math. J. 83(1) (1996), 157–190.CrossRefGoogle Scholar
Hejhal, D. A. and Rackner, B. N., On the topography of Maass waveforms for PSL(2,Z), Exp. Math. 1(4) (1992), 275–305.Google Scholar
Hilberdink, T., An arithmetical mapping and applications to $\omega$ -results for the Riemann zeta function, Acta Arith. 139(4) (2009), 341–367.Google Scholar
Hörmander, L., The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218.CrossRefGoogle Scholar
Iwaniec, H., Spectral Methods of Automorphic Forms, Vol. 53 of Graduate Studies in Mathematics, 2nd ed. (American Mathematical Society, Providence, Rhode Island, 2002).Google Scholar
Iwaniec, H. and Sarnak, P., ${L}^{\infty }$ norms of eigenfunctions on arithmetic surfaces, Ann. Math. (2) 141 (1995), 301–320.CrossRefGoogle Scholar
Kahane, J.-P., Some Random Series of Functions , Vol. 5 of Cambridge Studies in Advanced Mathematics, 2nd ed. (Cambridge University Press, Cambridge, United Kingdom, 1985).Google Scholar
Li, X., Upper bounds on $L$ -functions at the edge of the critical strip, Int. Math. Res. Not. 2010(4) (2010), 727–755.Google Scholar
Marshall, S., Geodesic restrictions of arithmetic eigenfunctions, Duke Math. J. 165(3) (2016), 463–508.Google Scholar
Milićević, D., Large values of eigenfunctions on arithmetic hyperbolic surfaces, Duke Math. J. 155(2) (2010), 365–401.Google Scholar
Milićević, D., Large values of eigenfunctions on arithmetic hyperbolic 3-manifolds, Geom. Funct. Anal. 21(6) (2011), 1375–1418.CrossRefGoogle Scholar
Popa, A. A., Central values of Rankin $L$ -series over real quadratic fields, Compos. Math. 142(4) (2006), 811–866.Google Scholar
Ratner, M., The central limit theorem for geodesic flows on $n$ -dimensional manifolds of negative curvature, Israel J. Math. 16 (1973), 181–197.Google Scholar
Reznikov, A., A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces, Forum Math. 27(3) (2015), 1569–1590.Google Scholar
Rudnick, Z. and Sarnak, P., The behaviour of eigenstates of arithmetic hyperbolic manifolds, Commun. Math. Phys. 161 (1994), 195–213.CrossRefGoogle Scholar
Salem, R. and Zygmund, A., Some properties of trigonometric series whose terms have random signs, Acta Math. 91 (1954), 245–301.CrossRefGoogle Scholar
Sands, J. W., Generalization of a theorem of Siegel, Acta Arith. 58(1) (1991), 47–56.CrossRefGoogle Scholar
Sarnak, P., Arithmetic quantum chaos, in The Schur Lectures, Vol. 8 of Israel Mathematical Conference Proceedings (American Mathematical Society, Providence, Rhode Island, 1995).Google Scholar
Sarnak, P., Reciprocal geodesics, Clay Math. Proc. 7 (2007), 217–237.Google Scholar
Sinai, Ya. G., The central limit theorem for geodesic flows on manifolds of constant negative curvature, Proc. USSR Acad. Sci. 133(6) (1960), 1303–1306 (in Russian).Google Scholar
Soundararajan, K., Extreme values of zeta and $L$ -functions, Math. Ann. 342 (2008), 467–486.CrossRefGoogle Scholar
Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals Princeton Mathematical Series (Princeton University Press, Princeton, NJ, 1993).Google Scholar
Toth, J. A. and Zelditch, S., Quantum ergodic restriction theorems. I: interior hypersurfaces in domains with ergodic billiards, Ann. Henri Poincaré 13(4) (2012), 599–670.Google Scholar
Toth, J. A. and Zelditch, S., Quantum ergodic restriction theorems: manifolds without boundary, Geom. Funct. Anal. 23(2) (2013), 715–775.Google Scholar
Vignèras, M., Arithmétique des algèbres de quaternions [Arithmetic of quaternion algebras], in Lecture Notes in Mathematics, Vol. 800 (Springer-Verlag, Heidelberg, Germany, 1980), pp 104.Google Scholar
Waldspurger, J.-L., Sur les valeurs de certaines fonctions $l$ automorphes et leur centre de symétrie [On the values of certain automorphic L-functions and their center of symmetry], Compos. Math. 54 (1985), 173–242.Google Scholar
Young, M., The quantum unique ergodicity conjecture for thin sets, Adv. Math. 286 (2016), 958–1016.Google Scholar
Young, M., Equidistribution of Eisenstein series on geodesic segments, Adv. Math. 340 (2018), 1166–1218.Google Scholar
Zelditch, S., Kuznecov sum formulae and Szegő limit formulae on manifolds, Commun. Partial Differ. Equ. 17(1–2) (1992), 221–260.Google Scholar
Zelditch, S., Quantum ergodicity and mixing of eigenfunctions, in Encyclopedia of Mathematical Physics, Academic Press, Cambridge, Massachusetts, pages 183–196 (2006).Google Scholar
Zhang, S., Gross-Zagier formula for ${GL}_2$ , Asian J. Math. 5(2) (2001), 183–290.Google Scholar