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Large values of Maass forms on hyperbolic Grassmannians in the volume aspect

Thibaut Menes

TL;DR

This work establishes the existence of exceptional Maass forms on compact quotients of the hyperbolic Grassmannians in the volume aspect, extending Rudnick–Sarnak–Donnelly–Brumley–Marshall to higher rank with dual reductive pairs $(U(n,m),U(m,m))$ or $(O(n,m),Sp(2m))$. The authors deploy a hybrid strategy: a counting argument based on a global period relation to detect theta lifts from an auxiliary group, together with a careful choice of $\tau$-spherical test functions to control spectral contribution and level-independence. A central technical achievement is a mean-square analysis of discrete $H$-periods, yielding lower bounds on sup-norms in terms of a volume ratio of congruence manifolds, up to a logarithmic loss from partial trace methods. They also establish a Weyl-type law for cusp forms in this setting, with a quantitative upper bound that, combined with the period analysis, underpins the main volume-aspect exceptional-form results. Overall, the paper advances higher-rank quantum chaos in arithmetic manifolds by linking theta lifts, global periods, and spectral counting to produce sharp volume-dependent lower bounds on Maass form amplitudes.

Abstract

Let $n > m \geq 1$ be integers such that $n+ m \geq 4$ is even. We prove the existence, in the volume aspect, of exceptional Maass forms on compact quotients of the hyperbolic Grassmannian of signature $(n,m)$. The method builds upon the work of Rudnick and Sarnak, extended by Donnelly and then generalized by Brumley and Marshall to higher rank manifolds. It combines a counting argument with a period relation, showing that a certain period distinguishes theta lifts from an auxiliary group. The congruence structure is defined with respect to this period and the auxiliary group is either $U(m,m)$ or $Sp_{2m}(\mathbb{R})$, making $(U(n,m),U(m,m))$ or $(O(n,m),Sp_{2m}(\mathbb{R}))$ a type 1 dual reductive pair. The lower bound is naturally expressed, up to a logarithmic factor, as the ratio of the volumes, with the principal congruence structure on the auxiliary group.

Large values of Maass forms on hyperbolic Grassmannians in the volume aspect

TL;DR

This work establishes the existence of exceptional Maass forms on compact quotients of the hyperbolic Grassmannians in the volume aspect, extending Rudnick–Sarnak–Donnelly–Brumley–Marshall to higher rank with dual reductive pairs or . The authors deploy a hybrid strategy: a counting argument based on a global period relation to detect theta lifts from an auxiliary group, together with a careful choice of -spherical test functions to control spectral contribution and level-independence. A central technical achievement is a mean-square analysis of discrete -periods, yielding lower bounds on sup-norms in terms of a volume ratio of congruence manifolds, up to a logarithmic loss from partial trace methods. They also establish a Weyl-type law for cusp forms in this setting, with a quantitative upper bound that, combined with the period analysis, underpins the main volume-aspect exceptional-form results. Overall, the paper advances higher-rank quantum chaos in arithmetic manifolds by linking theta lifts, global periods, and spectral counting to produce sharp volume-dependent lower bounds on Maass form amplitudes.

Abstract

Let be integers such that is even. We prove the existence, in the volume aspect, of exceptional Maass forms on compact quotients of the hyperbolic Grassmannian of signature . The method builds upon the work of Rudnick and Sarnak, extended by Donnelly and then generalized by Brumley and Marshall to higher rank manifolds. It combines a counting argument with a period relation, showing that a certain period distinguishes theta lifts from an auxiliary group. The congruence structure is defined with respect to this period and the auxiliary group is either or , making or a type 1 dual reductive pair. The lower bound is naturally expressed, up to a logarithmic factor, as the ratio of the volumes, with the principal congruence structure on the auxiliary group.

Paper Structure

This paper contains 58 sections, 36 theorems, 389 equations.

Table of Contents

  1. Introduction
  2. The sup norm problem
  3. Spectral aspect
  4. Volume aspect
  5. Main result
  6. The choice of $K_H(\mathfrak{n})$
  7. A hybrid result in the spectral-volume aspect
  8. Global period relation and test function
  9. Outline
  10. $\tau$-spherical harmonic analysis
  11. Notations
  12. $\tau$-spherical function
  13. $\tau$-spherical representation
  14. Additional notations
  15. $\tau$-spherical inversion formula
  16. ...and 43 more sections

Key Result

Theorem 1.1

Fix integers $n > m \geq 1$ with $n + m \geq 4$ even and assume $W$ is split. Fix an archimedean place $v_0$ of $F$ and assume $V$ has signature $(n,m)$ at $v_0$, $V$ is positive definite at every other real places and $X$ is negative definite at $v_0$. Let $\nu \in i \mathfrak{a}^*$ be sufficiently where $\mathcal{N}(\mathfrak{n}) = \vert \mathcal{O} / \mathfrak{n} \vert$ is the norm of $\mathfra

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • Corollary 2.2
  • proof : Proof of Lemma \ref{['lem3.3.1']}
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • ...and 60 more