Publications Mathématiques de Besançon

Algèbre et Théorie des Nombres
no. 2
On the sup-norm of SL 3 Hecke–Maass cusp forms
Roman Holowinsky1; Kevin Nowland1; Guillaume Ricotta2; Emmanuel Royer3
1 Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210-1174
2 Université de Bordeaux, IMB, 351 cours de la libération, 33405 Talence, France
3 Université Blaise Pascal, Laboratoire de mathématiques, Les Cézeaux, BP 80026, 63171 Aubière Cedex, France
Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 2 (2019), pp. 53-80.

This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL 3 (ℤ) Hecke–Maass cusp form restricted to a compact set.

Ce travail contient une preuve d’une borne non-triviale explicite quantitative par rapport à la valeur propre pour la norme infinie d’une forme de Hecke–Maass cuspidale de SL 3 (ℤ) restreinte à un ensemble compact.

Received:
Published online:
DOI: 10.5802/pmb.36
Classification: 11F55, 11F60, 11F72, 11H55, 11D75, 43A90, 43A80
Keywords: Automorphic forms, sup-norm, pre-trace formula, amplification method, Paley–Wiener theorem, Helgason transform, spherical function
Roman Holowinsky 1; Kevin Nowland 1; Guillaume Ricotta 2; Emmanuel Royer 3

1 Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210-1174
2 Université de Bordeaux, IMB, 351 cours de la libération, 33405 Talence, France
3 Université Blaise Pascal, Laboratoire de mathématiques, Les Cézeaux, BP 80026, 63171 Aubière Cedex, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{PMB_2019___2_53_0,
     author = {Roman Holowinsky and Kevin Nowland and Guillaume Ricotta and Emmanuel Royer},
     title = {On the sup-norm of $SL_3$ {Hecke{\textendash}Maass} cusp forms},
     journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
     pages = {53--80},
     publisher = {Presses universitaires de Franche-Comt\'e},
     number = {2},
     year = {2019},
     doi = {10.5802/pmb.36},
     language = {en},
     url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.36/}
}
TY  - JOUR
AU  - Roman Holowinsky
AU  - Kevin Nowland
AU  - Guillaume Ricotta
AU  - Emmanuel Royer
TI  - On the sup-norm of $SL_3$ Hecke–Maass cusp forms
JO  - Publications mathématiques de Besançon. Algèbre et théorie des nombres
PY  - 2019
SP  - 53
EP  - 80
IS  - 2
PB  - Presses universitaires de Franche-Comté
UR  - https://pmb.centre-mersenne.org/articles/10.5802/pmb.36/
UR  - https://doi.org/10.5802/pmb.36
DO  - 10.5802/pmb.36
LA  - en
ID  - PMB_2019___2_53_0
ER  - 
%0 Journal Article
%A Roman Holowinsky
%A Kevin Nowland
%A Guillaume Ricotta
%A Emmanuel Royer
%T On the sup-norm of $SL_3$ Hecke–Maass cusp forms
%J Publications mathématiques de Besançon. Algèbre et théorie des nombres
%D 2019
%P 53-80
%N 2
%I Presses universitaires de Franche-Comté
%U https://doi.org/10.5802/pmb.36
%R 10.5802/pmb.36
%G en
%F PMB_2019___2_53_0
Roman Holowinsky; Kevin Nowland; Guillaume Ricotta; Emmanuel Royer. On the sup-norm of $SL_3$ Hecke–Maass cusp forms. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 2 (2019), pp. 53-80. doi : 10.5802/pmb.36. https://pmb.centre-mersenne.org/articles/10.5802/pmb.36/

[1] Valentin Blomer; Gergely Harcos; Péter Maga On the global sup-norm of GL (3) cusp forms, Israel J. Math., Volume 229 (2019) no. 1, pp. 357-379 | DOI | MR | Zbl

[2] Valentin Blomer; Gergely Harcos; Djordje Milićević Bounds for eigenforms on arithmetic hyperbolic 3-manifolds, Duke Math. J., Volume 165 (2016) no. 4, pp. 625-659 | DOI | MR | Zbl

[3] Valentin Blomer; Péter Maga The sup-norm problem for PGL(4), Int. Math. Res. Not. (2015) no. 14, pp. 5311-5332 | DOI | MR | Zbl

[4] Valentin Blomer; Péter Maga Subconvexity for sup-norms of cusp forms on PGL (n), Sel. Math., New Ser., Volume 22 (2016) no. 3, pp. 1269-1287 | DOI | MR | Zbl

[5] Valentin Blomer; Philippe Michel Sup-norms of eigenfunctions on arithmetic ellipsoids, Int. Math. Res. Not. (2011) no. 21, pp. 4934-4966 | DOI | MR | Zbl

[6] Valentin Blomer; Anke Pohl The sup-norm problem on the Siegel modular space of rank two, Am. J. Math., Volume 138 (2016) no. 4, pp. 999-1027 | DOI | MR | Zbl

[7] Johannes J. Duistermaat; Johan A. C. Kolk; Veeravalli S. Varadarajan Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math., Volume 52 (1979) no. 1, pp. 27-93 | DOI | MR | Zbl

[8] Ramesh Gangolli On the Plancherel formula and the Paley–Wiener theorem for spherical functions on semisimple Lie groups, Ann. Math., Volume 93 (1971), pp. 150-165 | DOI | MR | Zbl

[9] Dorian Goldfeld Automorphic forms and L-functions for the group GL(n,ℝ), Cambridge Studies in Advanced Mathematics, 99, Cambridge University Press, 2006, xiv+493 pages (With an appendix by Kevin A. Broughan) | MR | Zbl

[10] Roman Holowinsky; Guillaume Ricotta; Emmanuel Royer The amplification method in the GL(3) Hecke algebra, Publ. Math. Besançon, Algèbre Théorie Nombres, Volume 2015 (2015), pp. 13-40 | DOI | MR | Zbl

[11] Henryk Iwaniec; Peter Sarnak L ∞ norms of eigenfunctions of arithmetic surfaces, Ann. Math., Volume 141 (1995) no. 2, pp. 301-320 | DOI | MR | Zbl

[12] Jay Jorgenson; Serge Lang Pos n (R) and Eisenstein series, Lecture Notes in Mathematics, 1868, Springer, 2005, viii+168 pages | MR | Zbl

[13] Shin-Ya Koyama L ∞ -norms on eigenfunctions for arithmetic hyperbolic 3-manifolds, Duke Math. J., Volume 77 (1995) no. 3, pp. 799-817 | DOI | MR | Zbl

[14] Simon Marshall Restrictions of SL 3 Maass forms to maximal flat subspaces, Int. Math. Res. Not. (2015) no. 16, pp. 6988-7015 | DOI | MR | Zbl

[15] Simon Marshall Geodesic restrictions of arithmetic eigenfunctions, Duke Math. J., Volume 165 (2016) no. 3, pp. 463-508 | DOI | MR | Zbl

[16] Simon Marshall L p norms of higher rank eigenfunctions and bounds for spherical functions, J. Eur. Math. Soc., Volume 18 (2016) no. 7, pp. 1437-1493 | DOI | MR | Zbl

[17] Nikolai Nadirashvili; Dzh. Tot; Dmitry Yakobson Geometric properties of eigenfunctions, Usp. Mat. Nauk, Volume 56 (2001) no. 6(342), pp. 67-88 | DOI | MR

[18] Morris Newman Integral matrices, Pure and Applied Mathematics, 45, Academic Press Inc., 1972, xvii+224 pages | MR | Zbl

[19] Peter Sarnak Letter to C. Morawetz on bounds for eigenfunctions on symmetric spaces available at http://publications.ias.edu/sarnak/paper/480 (2004)

[20] Peter Sarnak Arithmetic quantum chaos, The Schur lectures (Tel Aviv, 1992) (Israel Mathematical Conference Proceedings), Volume 8, Bar-Ilan University, 1995, pp. 183-236 | MR | Zbl

[21] Andreas Seeger; Christopher D. Sogge Bounds for eigenfunctions of differential operators, Indiana Univ. Math. J., Volume 38 (1989) no. 3, pp. 669-682 | DOI | MR | Zbl

[22] Goro Shimura Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, 11, Princeton University Press, 1994, xiv+271 pages | MR | Zbl

[23] Jeffrey M. VanderKam L ∞ norms and quantum ergodicity on the sphere, Int. Math. Res. Not. (1997) no. 7, pp. 329-347 | DOI | MR | Zbl

Cited by Sources: