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The sup-norm problem beyond the newform

Edgar Assing

Abstract

In this note we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $π$ we consider a special small $GL_2(\mathbb{Z}_p)$-type $V$ in $π$ and proof global sup-norm bounds for an average over an orthonormal basis of $V$. We achieve a non-trivial saving when the dimension of $V$ grows.

The sup-norm problem beyond the newform

Abstract

In this note we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation we consider a special small -type in and proof global sup-norm bounds for an average over an orthonormal basis of . We achieve a non-trivial saving when the dimension of grows.

Paper Structure

This paper contains 13 sections, 20 theorems, 142 equations, 1 table.

Table of Contents

  1. Introduction
  2. Set-up and main result
  3. Preliminary considerations
  4. Local considerations
  5. A generating domain
  6. The Whittaker bound
  7. Reduction to a local problem
  8. Computing the local averages
  9. The Steinberg representation
  10. Twist minimal Principal Series
  11. Supercuspidal representations
  12. Conclusion
  13. A bound via the pre-trace formula

Key Result

Theorem \oldthetheorem

Let $p>3$ be prime and suppose $\pi$ is twist minimal. In the notation above we have If the (arithmetic)-conductor of $\pi$ is a perfect square (i.e. the exponent-conductor of the $p$-component $\pi_p$ of $\pi$ is even) or the $p$-component $\pi_p$ of $\pi$ is not supercuspidal, then we have the better bound

Theorems & Definitions (37)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 27 more