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The orbit method in number theory through the sup-norm problem for $\operatorname{GL}(2)$

Edgar Assing, Radu Toma

TL;DR

The paper applies the quantitative orbit method of Nelson–Venkatesh to the GL(2) sup-norm problem, deriving a new hybrid bound for Hecke–Maaß newforms with large prime-power level N = p^{4n} and large eigenvalue λ: ||φ||∞ ≪ (Nλ)^{5/24+ε} ⋅ ||φ||2. It develops an archimedean microlocal calculus and its p-adic analogue, using microlocalised vectors tied to coadjoint orbits, an Op-calculus, and a refined test-function framework to drive a relative trace formula argument. Amplification and transversality are crucial to control geometric sums and off-diagonal contributions, enabling a bound that matches the Iwaniec–Sarnak exponent spectrally while improving depth aspect bounds in the non-compact, high-level setting. The work also provides a thorough exposition of the microlocal orbit-method machinery in the classical PGL(2) case, with detailed treatments of local periods, model theories (Kirillov/Whittaker), and both archimedean and non-archimedean localization, offering a comprehensive guide for applying the quantitative orbit method to automorphic questions.

Abstract

The orbit method in its quantitative form due to Nelson and Venkatesh has played a central role in recent advances in the analytic theory of higher rank $L$-functions. The goal of this note is to explain how the method can be applied to the sup-norm problem for automorphic forms on $\operatorname{PGL}(2)$. Doing so, we prove a new hybrid bound for newforms $\varphi$ of large prime-power level $N = p^{4n}$ and large eigenvalue $λ$. It states that $\| \varphi \|_\infty \ll_p (λN)^{5/24 + \varepsilon}$, recovering the result of Iwaniec and Sarnak spectrally and improving the local bound in the depth aspect for the first time in this non-compact setting. We also provide an exposition of the microlocal tools used, illustrating and motivating the theory through the classical case of $\operatorname{PGL}(2)$, following notes and lectures of Nelson and Venkatesh.

The orbit method in number theory through the sup-norm problem for $\operatorname{GL}(2)$

TL;DR

The paper applies the quantitative orbit method of Nelson–Venkatesh to the GL(2) sup-norm problem, deriving a new hybrid bound for Hecke–Maaß newforms with large prime-power level N = p^{4n} and large eigenvalue λ: ||φ||∞ ≪ (Nλ)^{5/24+ε} ⋅ ||φ||2. It develops an archimedean microlocal calculus and its p-adic analogue, using microlocalised vectors tied to coadjoint orbits, an Op-calculus, and a refined test-function framework to drive a relative trace formula argument. Amplification and transversality are crucial to control geometric sums and off-diagonal contributions, enabling a bound that matches the Iwaniec–Sarnak exponent spectrally while improving depth aspect bounds in the non-compact, high-level setting. The work also provides a thorough exposition of the microlocal orbit-method machinery in the classical PGL(2) case, with detailed treatments of local periods, model theories (Kirillov/Whittaker), and both archimedean and non-archimedean localization, offering a comprehensive guide for applying the quantitative orbit method to automorphic questions.

Abstract

The orbit method in its quantitative form due to Nelson and Venkatesh has played a central role in recent advances in the analytic theory of higher rank -functions. The goal of this note is to explain how the method can be applied to the sup-norm problem for automorphic forms on . Doing so, we prove a new hybrid bound for newforms of large prime-power level and large eigenvalue . It states that , recovering the result of Iwaniec and Sarnak spectrally and improving the local bound in the depth aspect for the first time in this non-compact setting. We also provide an exposition of the microlocal tools used, illustrating and motivating the theory through the classical case of , following notes and lectures of Nelson and Venkatesh.

Paper Structure

This paper contains 79 sections, 45 theorems, 610 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The main theorem
  3. A short history of the sup-norm problem for Hecke congruence subgroups
  4. Sketch of the proof
  5. Uniqueness, periods, and matrix coefficients
  6. The relative trace formula
  7. Localised vectors
  8. Matrix coefficients of localised vectors
  9. Test functions
  10. A simple test function
  11. A refined test function
  12. The baseline bound
  13. Amplification
  14. Transversality
  15. Archimedean microlocalisation
  16. ...and 64 more sections

Key Result

Theorem 1.1

Let $p$ be an odd prime and $n$ a positive integer. Let $N=p^{4n}$ and let $f\colon \Gamma_H(N)\backslash \mathbb{H}\to \mathbb{C}$ be cuspidal Hecke--Maaß newform with trivial nebentypus and Laplace-Beltrami eigenvalue $\lambda$. If $\lambda$ is sufficiently large, then we have

Figures (7)

  • Figure 1: The region $U_\tau$ on the orbit $\mathcal{O}_{\pi_\lambda}$
  • Figure 2: The circular band $K \mathcal{U} \tau$ and the tilted circle $\gamma K \tau$
  • Figure 3: The orbit $\mathcal{O}_A$
  • Figure 4: The orbit $\mathcal{O}_C$
  • Figure 5: The union of orbits $\mathcal{O}_{B+C} \cup \mathcal{O}_0$, called the nilcone
  • ...and 2 more figures

Theorems & Definitions (117)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Proposition 3.2: Classification of representations
  • Definition 3.1
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • ...and 107 more