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Mass equidistribution for lifts on hyperbolic $4$-manifolds

Alexandre de Faveri, Zvi Shem-Tov

Abstract

We prove the quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak for the sequence of Pitale lifts, which are Hecke-Maass forms on a congruence quotient of $\mathbb{H}^4$ constructed as lifts from half-integral weight forms (i.e. non-holomorphic analogues of the Saito-Kurokawa lifts). The result is unconditional, unlike other mass equidistribution results for similar lifts. Our main innovation is the delicate construction of an amplifier with favorable geometric properties (while we do use the non-temperedness of the lifts, it alone is not enough). To the best of our knowledge, this is the first successful use of the amplification method for escaping a non-tempered subgroup.

Mass equidistribution for lifts on hyperbolic $4$-manifolds

Abstract

We prove the quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak for the sequence of Pitale lifts, which are Hecke-Maass forms on a congruence quotient of constructed as lifts from half-integral weight forms (i.e. non-holomorphic analogues of the Saito-Kurokawa lifts). The result is unconditional, unlike other mass equidistribution results for similar lifts. Our main innovation is the delicate construction of an amplifier with favorable geometric properties (while we do use the non-temperedness of the lifts, it alone is not enough). To the best of our knowledge, this is the first successful use of the amplification method for escaping a non-tempered subgroup.
Paper Structure (44 sections, 25 theorems, 118 equations)

This paper contains 44 sections, 25 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Non-concentration for lifts
  4. Statement of results
  5. Sketch of the proof
  6. The parallel/transverse argument
  7. Hecke operators avoiding the large subgroup $\mathrm{SL}_2(\mathbb{C})$
  8. Reduction to non-concentration
  9. Preliminaries
  10. Hamilton quaternions
  11. Isometries of
  12. Subgroups of
  13. The lattice
  14. Maass forms on
  15. Hecke operators on
  16. ...and 29 more sections

Key Result

Theorem 1

The probability measures $\mu_n$ converge in the weak-$*$ topology to the uniform probability measure on $\Gamma\backslash \mathbb{H}^4$.

Theorems & Definitions (57)

  • Theorem 1: QUE for Pitale lifts
  • Theorem 2: Non-concentration
  • Lemma 1
  • proof
  • proof : Proof of \ref{['mainthm']}
  • Definition 1
  • Remark 1
  • Lemma 2: Bound is multiplicative for different primes
  • proof
  • Lemma 3
  • ...and 47 more