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Optimal sup norm bounds for newforms on GL2 with maximally ramified central character

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From the journal Forum Mathematicum

Abstract

Recently, the problem of bounding the sup norms of L2-normalized cuspidal automorphic newforms ϕ on GL2 in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to

ϕN14+ϵ,

at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound ϕN12+ϵ in this setup (due to [A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 2017, 1009–1045]) and matches a lower bound due to [N. Templier, Large values of modular forms, Camb. J. Math. 2 2014, 1, 91–116], thus our result is essentially optimal in this case.

MSC 2010: 11F03; 11F70

Communicated by Valentin Blomer


Acknowledgements

I wish to thank Abhishek Saha for suggesting me this problem as well as helpful comments and discussions, and the anonymous referee whose suggestions helped improving this paper.

References

[1] E. Assing, On sup-norm bounds part I: Ramified Maaß newforms over number fields, preprint (2017), https://arxiv.org/abs/1710.00362. Search in Google Scholar

[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. Search in Google Scholar

[3] E. Assing, On the size of p-adic Whittaker functions, Trans. Amer. Math. Soc. 372 (2019), no. 8, 5287–5340. 10.1090/tran/7685Search in Google Scholar

[4] V. Blomer and R. Holowinsky, Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), no. 3, 645–681. 10.1007/s00222-009-0228-0Search in Google Scholar

[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. 10.1007/s00222-005-0468-6Search in Google Scholar

[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. 10.1007/s00208-012-0844-7Search in Google Scholar

[7] J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero, Ann. of Math. (2) 140 (1994), no. 1, 161–181. 10.2307/2118543Search in Google Scholar

[8] Y. Hu, P. D. Nelson and A. Saha, Some analytic aspects of automorphic forms on GL(2) of minimal type, Comment. Math. Helv. 94 (2019), no. 4, 767–801. 10.4171/CMH/473Search in Google Scholar

[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. 10.1112/S0010437X20007460Search in Google Scholar

[10] H. H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2. With an appendix by Dinakar Ramakrishnan, with an appendix co-authored by Peter Sarnak, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. 10.1090/S0894-0347-02-00410-1Search in Google Scholar

[11] P. Michel and A. Venkatesh, The subconvexity problem for GL2, Publ. Math. Inst. Hautes Études Sci. (2010), no. 111, 171–271. 10.1007/s10240-010-0025-8Search in Google Scholar

[12] P. Nelson, Microlocal lifts and quantum unique ergodicity on GL(p), Algebra Number Theory 12 (2018), 2033–2064. 10.2140/ant.2018.12.2033Search in Google Scholar

[13] A. Saha, Large values of newforms on GL(2) with highly ramified central character, Int. Math. Res. Not. IMRN 2016 (2016), no. 13, 4103–4131. 10.1093/imrn/rnv259Search in Google Scholar

[14] A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 (2017), 1009–1045. 10.2140/ant.2017.11.1009Search in Google Scholar

[15] A. Saha, Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, Math. Ann. 376 (2020), no. 1–2, 609–644. 10.1007/s00208-019-01923-3Search in Google Scholar

[16] R. Schmidt, Some remarks on local newforms for GL(2), J. Ramanujan Math. Soc. 17 (2002), no. 2, 115–147. Search in Google Scholar

[17] N. Templier, Large values of modular forms, Camb. J. Math. 2 (2014), no. 1, 91–116. 10.4310/CJM.2014.v2.n1.a3Search in Google Scholar

Received: 2020-03-28
Revised: 2020-08-05
Published Online: 2020-09-01
Published in Print: 2021-01-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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