The sup-norm problem for automorphic cusp forms of 𝑃𝐺𝐿(𝑛,ℤ[𝕚])

Maga, Péter; Zábrádi, Gergely

doi:10.1090/proc/16576

The sup-norm problem for automorphic cusp forms of $\mathrm {PGL}(n,\mathbb {Z}[i])$
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by Péter Maga and Gergely Zábrádi

Proc. Amer. Math. Soc. 152 (2024), 559-572

DOI: https://doi.org/10.1090/proc/16576

Published electronically: November 21, 2023

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Abstract:

Let $\phi$ be an $L^2$-normalized Hecke–Maaß cusp form on the locally symmetric space $X≔\mathrm {PGL}_n(\mathbb {Z}[i])\backslash \mathrm {PGL}_n(\mathbb {C}) / \mathrm {PU}_n$. If $\Omega$ is a compact subset of $X$, then we prove the bound $\|\phi |_{\Omega }\|_{\infty }\ll _{\Omega } \lambda _{\phi }^{n(n-1)/4-\delta }$ for some $\delta >0$ depending only on $n$, where $\lambda _{\phi }$ is the Laplace eigenvalue of $\phi$.

References

A. S. Besicovitch, On the linear independence of fractional powers of integers, J. London Math. Soc. 15 (1940), 3–6. MR 2327, DOI 10.1112/jlms/s1-15.1.3
Valentin Blomer and Gergely Harcos, Twisted $L$-functions over number fields and Hilbert’s eleventh problem, Geom. Funct. Anal. 20 (2010), no. 1, 1–52. MR 2647133, DOI 10.1007/s00039-010-0063-x
Valentin Blomer, Gergely Harcos, and Djordje Milićević, Bounds for eigenforms on arithmetic hyperbolic 3-manifolds, Duke Math. J. 165 (2016), no. 4, 625–659. MR 3474814, DOI 10.1215/00127094-3166952
Valentin Blomer, Gergely Harcos, Péter Maga, and Djordje Milićević, The sup-norm problem for $\rm GL(2)$ over number fields, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 1, 1–53. MR 4046009, DOI 10.4171/jems/916
Valentin Blomer and Péter Maga, The sup-norm problem for PGL(4), Int. Math. Res. Not. IMRN 14 (2015), 5311–5332. MR 3384442, DOI 10.1093/imrn/rnu100
Valentin Blomer and Péter Maga, Subconvexity for sup-norms of cusp forms on $\rm {PGL}(n)$, Selecta Math. (N.S.) 22 (2016), no. 3, 1269–1287. MR 3518551, DOI 10.1007/s00029-015-0219-5
Valentin Blomer and Anke Pohl, The sup-norm problem on the Siegel modular space of rank two, Amer. J. Math. 138 (2016), no. 4, 999–1027. MR 3538149, DOI 10.1353/ajm.2016.0032
Farrell Brumley and Nicolas Templier, Large values of cusp forms on $\textrm {GL}_n$, Selecta Math. (N.S.) 26 (2020), no. 4, Paper No. 63, 71. MR 4150475, DOI 10.1007/s00029-020-00589-z
Nate Gillman, Explicit subconvexity savings for sup-norms of cusp forms on $\textrm {PGL}_n(\Bbb {R})$, J. Number Theory 206 (2020), 46–61. MR 4013163, DOI 10.1016/j.jnt.2019.06.002
Amit Ghosh, Andre Reznikov, and Peter Sarnak, Nodal domains of Maass forms I, Geom. Funct. Anal. 23 (2013), no. 5, 1515–1568. MR 3102912, DOI 10.1007/s00039-013-0237-4
H. Iwaniec and P. Sarnak, $L^\infty$ norms of eigenfunctions of arithmetic surfaces, Ann. of Math. (2) 141 (1995), no. 2, 301–320. MR 1324136, DOI 10.2307/2118522
S. Marshall, Upper bounds for Maass forms on semisimple groups, Available at arXiv:1405.7033, 2014.
Zeév Rudnick and Peter Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), no. 1, 195–213. MR 1266075, DOI 10.1007/BF02099418
P. Sarnak, Letter to Morawetz, Available at http://publications.ias.edu/sites/default/files/Sarnak_Letter_to_Morawetz.pdf.