Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Hostname: page-component-559fc8cf4f-6pznq Total loading time: 0.345 Render date: 2021-03-08T13:30:45.270Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }
EnglishFrançais
Journal of the Institute of Mathematics of Jussieu Journal of the Institute of Mathematics of Jussieu

Article contents

Hybrid bounds for automorphic forms on ellipsoids over number fields

Part of: Partial differential equations on manifolds; differential operators Algebraic number theory: global fields Forms and linear algebraic groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  20 December 2012

Valentin Blomer  and
Philippe Michel
Valentin Blomer
Affiliation:
Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany (blomer@uni-math.gwdg.de)
Philippe Michel
Affiliation:
EPFL/SB/IMB/TAN, Station 8, CH-1015 Lausanne, Switzerland (philippe.michel@epfl.ch)
Corresponding
E-mail address:
blomer@uni-math.gwdg.de
philippe.michel@epfl.ch
Get access
Rights & Permissions[Opens in a new window]

Abstract

We prove upper bounds for Hecke–Laplace eigenfunctions on certain Riemannian manifolds $X$ of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are $d$ -fold products of $2$ -spheres or $3$ -spheres, realized as adelic quotients of quaternion algebras over totally real number fields. In the volume aspect we prove a (‘Weyl-type’) saving of $\mathrm{vol} \hspace{0.167em} (X)^{- 1/ 6+ \varepsilon } $ .

Keywords

definite quaternion algebras trace formula sup-norms hybrid bounds norm forms spherical harmonics

MSC classification

Secondary: 11R52: Quaternion and other division algebras 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 58J50: Spectral problems; spectral geometry; scattering theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 4 , October 2013 , pp. 727 - 758
Copyright
©Cambridge University Press 2012 

Access options

Get access to the full version of this content by using one of the access options below.
If you should have access and can't see this content please contact technical support.

References

Blomer, V. and Holowinsky, R., Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (3) (2010), 645–681.CrossRefGoogle Scholar
Blomer, V. and Michel, Ph., Sup-norm bounds for eigenfunctions on ellipsoids, Int. Math. Res. Not. 2011 (21) (2011), 4934–4966.Google Scholar
Cassels, J. W. S., Rational quadratic forms, London Mathematical Society Monographs, Volume 13 (Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1978).Google Scholar
Duke, W., Friedlander, J. and Iwaniec, H., Class group $L$-functions, Duke Math. J. 79 (1) (1995), 1–56.CrossRefGoogle Scholar
Erdélyi, T., Magnus, A. P. and Nevai, P., Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 2 (1994), 602–614.Google Scholar
Faraut, J., Analysis on Lie groups. An introduction, Cambridge Studies in Advanced Mathematics, Volume 110 (Cambridge University Press, Cambridge, 2008).CrossRefGoogle Scholar
Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series, and products, 7th ed. (Elsevier/Academic Press, Amsterdam, 2007).Google Scholar
Harcos, G. and Templier, N., On the sup-norm of Maass cusp forms of large level. III, Math. Ann., in press.Google Scholar
Hiraga, K. and Saito, H., On $L$-packets for inner forms of ${\mathrm{SL} }_{n} $, Mem. Amer. Math. Soc. 215 (2012).Google Scholar
Humbert, P., Théorie de la réduction des formes quadratiques définies positives dans un corps algébrique $K$ fini, Comment. Math. Helv. 12 (1940), 263–306.CrossRefGoogle Scholar
Iwaniec, H. and Sarnak, P., ${L}^{\infty } $ norms of eigenfunctions of arithmetic surfaces, Ann. of Math. (2) 141 (2) (1995), 301–320.CrossRefGoogle Scholar
Kitaoka, Y., Arithmetic of quadratic forms, Cambridge Tracts in Mathematics, Volume 106 (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
Ponomarev, P., Arithmetic of quaternary quadratic forms, Acta Arith. 29 (1) (1976), 1–48.CrossRefGoogle Scholar
Sarnak, P., Arithmetic quantum chaos, in The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., Volume 8, pp. 183–236 (Bar-Ilan Univ., Ramat Gan, 1995).Google Scholar
Sarnak, P., Letter to Morawetz, available at http://www.math.princeton.edu/sarnak.Google Scholar
Templier, N., On the sup-norm of Maass cusp forms of large level, Selecta Math. 16 (2010), 501–531.CrossRefGoogle Scholar
Templier, N., Large values of modular forms , preprint arXiv:1207.6134.Google Scholar
VanderKam, J. M., ${L}^{\infty } $ norms and quantum unique ergodicity on the sphere, Int. Math. Res. Not. 1997 (1997), 329–347.CrossRefGoogle Scholar
Venkatesh, A., Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2) 172 (2) (2010), 989–1094.CrossRefGoogle Scholar
Vignéras, M.-F., Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, Volume 800 (Springer, Berlin, 1980) (in French).CrossRefGoogle Scholar
Vilenkin, N. Ja. and Klimyk, A. U., Representation of Lie groups and special functions. Vol. 1, Mathematics and its Applications (Soviet Series), Volume 72 (Kluwer Academic Publishers Group, Dordrecht, 1991).CrossRefGoogle Scholar

Altmetric attention score

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 56 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 8th March 2021. This data will be updated every 24 hours.

4 Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Hybrid bounds for automorphic forms on ellipsoids over number fields
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Hybrid bounds for automorphic forms on ellipsoids over number fields
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Hybrid bounds for automorphic forms on ellipsoids over number fields
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *