Home Optimal sup norm bounds for newforms on GL2 with maximally ramified central character
Article
Licensed
Unlicensed Requires Authentication

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

  • Félicien Comtat EMAIL logo
Published/Copyright: September 1, 2020
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 1

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

∥ϕ∥∞≪N14+ϵ,

at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound ∥ϕ∥∞≪N12+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.

Keywords: Automorphic forms; sup norms; Whittaker expansion
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⁢(ℚp), 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

Articles in the same Issue

  1. Frontmatter
  2. Optimal sup norm bounds for newforms on GL2 with maximally ramified central character
  3. On three-variable expanders over finite valuation rings
  4. Completely positive maps of order zero on pro-𝐶∗-algebras
  5. Hybrid subconvexity for class group 𝐿-functions and uniform sup norm bounds of Eisenstein series
  6. Special value formula for the twisted triple product L-function and an application to the restricted L2-norm problem
  7. Generalised Iwasawa invariants and the growth of class numbers
  8. Laws of the iterated logarithm on covering graphs with groups of polynomial volume growth
  9. The geometric sieve for quadrics
  10. On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness
  11. Commutative algebraic monoid structures on affine surfaces
  12. Representations induced from the Zelevinsky segment and discrete series in the half-integral case
  13. A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on SL(2,ℝ)
  14. From subcategories to the entire module categories
  15. Coherent state transform for Landau levels on quasi-tori
  16. On rational homotopy and minimal models
https://doi.org/10.1515/forum-2020-0080
Keywords for this article
Automorphic forms; sup norms; Whittaker expansion

Articles in the same Issue

  1. Frontmatter
  2. Optimal sup norm bounds for newforms on GL2 with maximally ramified central character
  3. On three-variable expanders over finite valuation rings
  4. Completely positive maps of order zero on pro-𝐶∗-algebras
  5. Hybrid subconvexity for class group 𝐿-functions and uniform sup norm bounds of Eisenstein series
  6. Special value formula for the twisted triple product L-function and an application to the restricted L2-norm problem
  7. Generalised Iwasawa invariants and the growth of class numbers
  8. Laws of the iterated logarithm on covering graphs with groups of polynomial volume growth
  9. The geometric sieve for quadrics
  10. On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness
  11. Commutative algebraic monoid structures on affine surfaces
  12. Representations induced from the Zelevinsky segment and discrete series in the half-integral case
  13. A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on SL(2,ℝ)
  14. From subcategories to the entire module categories
  15. Coherent state transform for Landau levels on quasi-tori
  16. On rational homotopy and minimal models
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 9.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2020-0080/html
Scroll to top button