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

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Veröffentlicht/Copyright: 1. September 2020
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 33 Heft 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

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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

Artikel in diesem Heft

  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
Schlagwörter für diesen Artikel
Automorphic forms; sup norms; Whittaker expansion

Artikel in diesem Heft

  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
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