Hostname: page-component-7dd5485656-g8tfn Total loading time: 0 Render date: 2025-10-31T17:07:03.376Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces

Part of: Discontinuous groups and automorphic forms Partial differential equations on manifolds; differential operators Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  26 September 2025

Louis Soares Open the ORCID record for Louis Soares [Opens in a new window]
Louis Soares*
Affiliation:
Independent Researcher, Arbon, Switzerland
*
Email: louis.soares@gmx.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and let $X=\Gamma \backslash {\mathbb{H}}^2$ be the associated hyperbolic surface. We consider the family of Hecke congruence coverings of $X$, which we denote as usual by $ X_0(q) = \Gamma _0(q)\backslash {\mathbb{H}}^2$. Conditional on the Lindelöf Hypothesis for quadratic L-functions, we establish a uniform and explicit spectral gap for the Laplacian on $ X_0(q)$ for “almost” all prime levels $q$. Assuming the generalized Riemann hypothesis for quadratic $L$-functions, we obtain an even larger spectral gap.

MSC classification

Primary: 58J50: Spectral problems; spectral geometry; scattering theory
Secondary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas 11F06: Structure of modular groups and generalizations; arithmetic groups

Information

Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 41
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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.)

Article purchase

Temporarily unavailable

References

Borthwick, D., Spectral theory of infinite-area hyperbolic surfaces, 2nd edition, (Birkhäuser/Springer, Basel, 2016).Google Scholar
Borthwick, D., Judge, C. and Perry, P. A., Determinants of Laplacians and isopolar metrics on surfaces of infinite area, Duke Math. J. 118 (2003), 61–102.10.1215/S0012-7094-03-11814-1CrossRefGoogle Scholar
Bourgain, J. and Dyatlov, S., Fourier dimension and spectral gaps for hyperbolic surfaces, Geom. Funct. Anal. 27(4) (2017), 744–771.10.1007/s00039-017-0412-0CrossRefGoogle Scholar
Bourgain, J., Gamburd, A. and Sarnak, P., Generalization of Selberg’s $\frac {3}{16}$ theorem and affine sieve, Acta Math. 207(2) (2011), 255–290, MR2892611.10.1007/s11511-012-0070-xCrossRefGoogle Scholar
Bourgain, J. and Kontorovich, A., On representations of integers in thin subgroups of $\mathrm{SL}_2(\mathbb{Z})$ , Geom. Funct. Anal. 20(5) (2010), 1144–1174.10.1007/s00039-010-0093-4CrossRefGoogle Scholar
Button, J., All Fuchsian Schottky groups are classical Schottky groups, in The Epstein birthday schrift, Geom. Topol. Monogr., Vol. 1 (Geom. Topol. Publ, Coventry, 1998), 117–125, MR1668339Google Scholar
Calderón, I. and Magee, M., Explicit spectral gap for Schottky subgroups of $\mathrm{SL}(2,\mathbb{Z})$ , J. Eur. Math. Soc. (2025), published online first.CrossRefGoogle Scholar
Duren, P. and Schuster, A., Bergman spaces, (Providence, RI: American Mathematical Society, 2004).10.1090/surv/100CrossRefGoogle Scholar
Dyatlov, S. and Zworski, M., Fractal uncertainty for transfer operators, Int. Math. Res. Not. 2020(3) (2018), 781–812.10.1093/imrn/rny026CrossRefGoogle Scholar
Ehrman, M., Almost primes in thin orbits of pythagorean triangles, Int. Math. Res. Not. 2019(11) (2017), 3498–3526.CrossRefGoogle Scholar
Fedosova, K. and Pohl, A., Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy, Selecta Mathematica 26(1) (2020), 649–670, Paper No. 9.10.1007/s00029-019-0534-3CrossRefGoogle Scholar
Gamburd, A., On the spectral gap for infinite index “congruence” subgroups of $\textrm {SL}_2(\mathbb{Z})$ , Israel J. Math. 127 (2002), 157–200, MR1900698.10.1007/BF02784530CrossRefGoogle Scholar
Gohberg, I. C., Goldberg, S. and Krupnik, N., Traces and determinants of linear operators, Vol. 116 (Springer Basel AG, Basel, 2000).CrossRefGoogle Scholar
Guillopé, L. and Zworski, M., Scattering asymptotics for Riemann surfaces, Ann. Math. 145(3) (1997), 597–660, MR1454705, (2).10.2307/2951846CrossRefGoogle Scholar
Hong, J. and Kontorovich, A., Almost prime coordinates for anisotropic and thin pythagorean orbits, Israel J. Math. 209(1) (2015), 397–420.10.1007/s11856-015-1223-3CrossRefGoogle Scholar
Iwaniec, H., Small eigenvalues of Laplacian for $\Gamma _0(N)$ , Acta Arith. 56(1) (1990), 65–82.10.4064/aa-56-1-65-82CrossRefGoogle Scholar
Iwaniec, H., The lowest eigenvalue for congruence groups, in Topics in geometry. Progress in nonlinear differential equations and their applications (Gindikin, S., editor), vol. 20, (Basel: Birkhäuser, 1996), 203–212.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic number theory, Vol. 53, (Providence, RI: Colloquium Publications, American Mathematical Society, 2004).Google Scholar
Kontorovich, A., Bourgain, J. and Magee, M., Thermodynamic expansion to arbitrary moduli, J. Reine Angew. Math. 753 (2019), 89–135, Appendix.Google Scholar
Jakobson, D. and Naud, F., Resonances and density bounds for convex co-compact congruence subgroups of $ \mathrm{SL}_{2}(\mathbb{Z})$ , Israel J. Math. 213(1) (2000), 443–473.10.1007/s11856-016-1332-7CrossRefGoogle Scholar
Jakobson, D., Naud, F. and Soares, L., Large degree covers and sharp resonances of hyperbolic surfaces, Ann. Institut Fourier 70(2) (2020), 523–596.10.5802/aif.3319CrossRefGoogle Scholar
Kim, H., Ramakrishnan, D. and Sarnak, P., Functoriality for the exterior square of $\mathrm{GL}_4$ and the symmetric fourth of $\mathrm{GL}_2$ , J. Am. Math. Soc. 16(1) (2003), 139–183.10.1090/S0894-0347-02-00410-1CrossRefGoogle Scholar
Kim, H. and Shahidi, F., Functorial products for $\mathrm{GL}_2 \times \mathrm{GL}_3$ and the symmetric cube for $\mathrm{GL}_2$ , Ann. Math. 155(3) (2002), 837–893, (2)10.2307/3062134CrossRefGoogle Scholar
Kontorovich, A., The hyperbolic lattice point count in infinite volume with applications to sieves, Duke Math. J. 149(1) (2009), 1–36.10.1215/00127094-2009-035CrossRefGoogle Scholar
Kontorovich, A. and Oh, H., Almost prime Pythagorean triples in thin orbits, J. Reine Angew. Math. 2012 (2011), 128.Google Scholar
Lax, P. and Phillips, R. S., Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. I, Commun. Pure Appl. Math. 37(3) (1984), 303–328.10.1002/cpa.3160370304CrossRefGoogle Scholar
Luo, W., Rudnick, Z. and Sarnak, P., On Selberg’s eigenvalue conjecture, GAFA 5(2) (1995), 387–401.Google Scholar
Montgomery, H. L. and Vaughan, R. C., Multiplicative number theory. I Classical theory, in Cambridge Studies in advanced mathematics, Vol. 97, reprint of the 2007 hardback edition, (Cambridge University Press, 2012).Google Scholar
Naud, F., Density and location of resonances for convex co-compact hyperbolic surfaces, Invent. Math. 195(3) (2014), 723–750, MR3166217.CrossRefGoogle Scholar
Naud, F. and Magee, M., Explicit spectral gaps for random covers of Riemann surfaces, Publ. Math. IHES 132(1) (2020), 137–179.Google Scholar
Oh, H. and Winter, D., Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of ${\rm SL}_2(\Bbb {Z})$ , J. Amer. Math. Soc. 29(4) (2016), 1069–1115, MR3522610.10.1090/jams/849CrossRefGoogle Scholar
Patterson, S., The limit set of a Fuchsian group, Acta Math. 136 (1976), 241–273.10.1007/BF02392046CrossRefGoogle Scholar
Patterson, S. J. and Perry, P. A., The divisor of Selberg’s Zeta function for Kleinian Groups, Duke Math. J. 106 (2001), 321–390, Appendix A by Charles Epstein.10.1215/S0012-7094-01-10624-8CrossRefGoogle Scholar
Pohl, A. and Soares, L., Density of resonances for covers of Schottky surfaces, J. Spectr. Theory 10(3) (2020), 1053–1101.10.4171/jst/321CrossRefGoogle Scholar
Sarnak, P., Selberg’s eigenvalue conjecture, Notices Amer. Math. Soc. 42(11) (1995), 1272–1277.Google Scholar
Sarnak, P., Harmonic analysis, the trace formula, and Shimura varieties, in Clay Math. Proc, Vol. 4 (Amer. Math. Soc, Providence, R.I, 2005), 659–685.Google Scholar
Sarnak, P. and Xue, X., Bounds for multiplicities of automorphic representations, Duke Math. J. 64(1) (1991), 207–227.10.1215/S0012-7094-91-06410-0CrossRefGoogle Scholar
Selberg, A., On the estimation of fourier coefficients of modular forms, in Proc. Sympos. Pure Math., Vol. VIII (Amer. Math. Soc, Providence, R.I, 1965), 1–15.10.1090/pspum/008/0182610CrossRefGoogle Scholar
Soares, L., Improved fractal weyl bounds for convex cocompact hyperbolic surfaces and large resonance-free regions, (2023), arXiv preprint: 2301.03023.Google Scholar
Soares, L., Uniform resonance free regions for convex cocompact hyperbolic surfaces and expanders, (2023), arXiv: 2101.05757.Google Scholar
Venkov, A., Spectral theory of automorphic functions, Proc. Steklov Inst. Math. 153(4) (1982), ix+163, pp., A translation of Trudy Mat. Inst. Steklov. 153 (1981).Google Scholar
Venkov, A. and Zograf, P., Analogues of Artin’s factorization formulas in the spectral theory of automorphic functions associated with induced representations of Fuchsian groups, Izv. Akad. Nauk SSSR Ser. Mat. 46(6) (1982), 1150–1158, 1343. MR682487.Google Scholar