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Bounds for the periods of eigenfunctions on arithmetic hyperbolic 3-manifolds over surfaces

Jiaqi Hou

TL;DR

The paper studies period bounds for Hecke-Maass forms on compact arithmetic hyperbolic 3-manifolds restricted to totally geodesic surfaces. It implements the arithmetic amplification method of Iwaniec–Sarnak, translating the period into a pretrace kernel and then bounding the geometric side via Hecke returns and oscillatory integrals. A detailed stationary-phase analysis, including Morse–Bott degeneracies, yields a power-saving λ^{-1/74+ε} over the local bound, with a careful division of nondegenerate and degenerate regimes. The results advance restriction-type estimates for automorphic forms on higher-rank arithmetic manifolds and relate to automorphic distinction principles and period formulas in special cases.

Abstract

Let $ψ$ be a Hecke-Maass form on a compact congruence arithmetic hyperbolic 3-manifold $X$, and let $Y$ be a hyperbolic surface in $X$ that is not necessarily closed. We obtain a power saving result over the local bound for the period of $ψ$ along $Y$, by applying the method of arithmetic amplification developed by Iwaniec and Sarnak.

Bounds for the periods of eigenfunctions on arithmetic hyperbolic 3-manifolds over surfaces

TL;DR

The paper studies period bounds for Hecke-Maass forms on compact arithmetic hyperbolic 3-manifolds restricted to totally geodesic surfaces. It implements the arithmetic amplification method of Iwaniec–Sarnak, translating the period into a pretrace kernel and then bounding the geometric side via Hecke returns and oscillatory integrals. A detailed stationary-phase analysis, including Morse–Bott degeneracies, yields a power-saving λ^{-1/74+ε} over the local bound, with a careful division of nondegenerate and degenerate regimes. The results advance restriction-type estimates for automorphic forms on higher-rank arithmetic manifolds and relate to automorphic distinction principles and period formulas in special cases.

Abstract

Let be a Hecke-Maass form on a compact congruence arithmetic hyperbolic 3-manifold , and let be a hyperbolic surface in that is not necessarily closed. We obtain a power saving result over the local bound for the period of along , by applying the method of arithmetic amplification developed by Iwaniec and Sarnak.
Paper Structure (23 sections, 30 theorems, 243 equations)

This paper contains 23 sections, 30 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Acknowledgements
  4. Notation
  5. Quaternion algebras and adelic groups
  6. Lie groups and algebras
  7. Hecke algebras
  8. Arithmetic manifolds and Hecke-Maass forms
  9. Harish-Chandra transforms
  10. Amplification
  11. An amplification inequality and amplifiers
  12. Bounds for periods
  13. Estimates of Hecke returns
  14. Vector spaces over a number field
  15. Bounding $M(g,\delta, \mathfrak{n})$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\psi$ be a Hecke-Maass form on $X$ with spectral parameter $\lambda$. For any totally geodesic surface $Y$ in $X$ and $b\in C_c^\infty(Y)$, we have where the implied constant depends on $X$, the support of $b$, and the $L^\infty$-norms of $b$ and finitely many of its derivatives.

Theorems & Definitions (58)

  • Theorem 1.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • proof : Proof of Theorem \ref{['main']}
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 48 more