Optimal sup norm bounds for newforms on \(\mathrm{GL}_2\) with maximally ramified central character. (English) Zbl 1469.11065
Summary: Recently, the problem of bounding the sup norms of \(L^2\)-normalized cuspidal automorphic newforms \(\phi\) on \(\mathrm{GL}_2\) in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character \(\chi\) of \(\phi\) is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general \(\chi \). If the level \(N\) is a square, our result reduces to
\[
\|\phi\|_{\infty}\ll N^{\frac{1}{4}+\epsilon},
\]
at least under the Ramanujan Conjecture. In particular, when \(\chi\) has conductor \(N\), this improves upon the previous best known bound \(\|\phi\|_{\infty}\ll N^{\frac{1}{2}+\epsilon}\) in this setup (due to A. Saha [Algebra Number Theory 11, No. 5, 1009–1045 (2017; Zbl 1432.11044)]) and matches a lower bound due to N. Templier [Camb. J. Math. 2, No. 1, 91–116 (2014; Zbl 1307.11062)], thus our result is essentially optimal in this case.
MSC:
| 11F03 | Modular and automorphic functions |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
References:
| [1] | E. Assing, On sup-norm bounds part I: Ramified Maaß newforms over number fields, preprint (2017), https://arxiv.org/abs/1710.00362. |
| [2] | E. Assing, Local analysis of Whittaker new vectors and global applications, Ph.D. thesis, The University of Bristol, 2019, https://research-information.bris.ac.uk/en/theses/local-analysis-of-whittaker-new-vectors-and-global-applications(fd1d8115-513c-48db-94de-79abb60c5c89).html. |
| [3] | E. Assing, On the size of p-adic Whittaker functions, Trans. Amer. Math. Soc. 372 (2019), no. 8, 5287-5340. · Zbl 1471.11168 |
| [4] | V. Blomer and R. Holowinsky, Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), no. 3, 645-681. · Zbl 1243.11059 |
| [5] | G. Harcos and P. Michel, The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points. II, Invent. Math. 163 (2006), no. 3, 581-655. · Zbl 1111.11027 |
| [6] | G. Harcos and N. Templier, On the sup-norm of Maass cusp forms of large level. III, Math. Ann. 356 (2013), no. 1, 209-216. · Zbl 1332.11049 |
| [7] | J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero, Ann. of Math. (2) 140 (1994), no. 1, 161-181. · Zbl 0814.11032 |
| [8] | Y. Hu, P. D. Nelson and A. Saha, Some analytic aspects of automorphic forms on \rm GL(2) of minimal type, Comment. Math. Helv. 94 (2019), no. 4, 767-801. · Zbl 1443.11062 |
| [9] | Y. Hu and A. Saha, Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces. II: Newforms and subconvexity, preprint (2019), https://arxiv.org/abs/1905.06295. |
| [10] | H. H. Kim, Functoriality for the exterior square of \(\rm GL}_{4\) and the symmetric fourth of \(\rm GL}_{2\). With an appendix by Dinakar Ramakrishnan, with an appendix co-authored by Peter Sarnak, J. Amer. Math. Soc. 16 (2003), no. 1, 139-183. · Zbl 1018.11024 |
| [11] | P. Michel and A. Venkatesh, The subconvexity problem for \(\rm GL}_{2\), Publ. Math. Inst. Hautes Études Sci. (2010), no. 111, 171-271. · Zbl 1376.11040 |
| [12] | P. Nelson, Microlocal lifts and quantum unique ergodicity on \(\text{GL}(\mathbb{Q}_p)\), Algebra Number Theory 12 (2018), 2033-2064. · Zbl 1406.58019 |
| [13] | A. Saha, Large values of newforms on {\rm GL}(2) with highly ramified central character, Int. Math. Res. Not. IMRN 2016 (2016), no. 13, 4103-4131. · Zbl 1404.11055 |
| [14] | A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 (2017), 1009-1045. · Zbl 1432.11044 |
| [15] | A. Saha, Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, Math. Ann. 376 (2020), no. 1-2, 609-644. · Zbl 1471.11157 |
| [16] | R. Schmidt, Some remarks on local newforms for \rm GL(2), J. Ramanujan Math. Soc. 17 (2002), no. 2, 115-147. · Zbl 0997.11040 |
| [17] | N. Templier, Large values of modular forms, Camb. J. Math. 2 (2014), no. 1, 91-116. · Zbl 1307.11062 |
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