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Zvi Shem-Tov

Publications and source records attributed to Zvi Shem-Tov.

6 recordsLinked to original sources
arXiv2026

Mass equidistribution for lifts on hyperbolic $4$-manifolds

This paper is part of a broader project to establish the quantum unique ergodicity conjecture for Hecke-Maass forms on arithmetic hyperbolic $4$-manifolds. The conjecture is known in dimensions $2$ and $3$, but dimensions $4$ and higher present a new difficulty: ruling out concentration along certain homogeneous submanifolds whose stabilizers are non-tempered. We overcome it here for the Pitale lifts, a family of non-holomorphic analogues of Saito-Kurokawa lifts. Our main new ingredient is a construction of an amplifier with exceptional geometric properties. To the best of our knowledge, this is the first successful use of amplification to rule out concentration along a non-tempered subgroup.

math.NT↗
arXiv2025

Norm rigidity for arithmetic and profinite groups

Let $A$ be a commutative ring, and assume every non-trivial ideal of $A$ has finite-index. We show that if ${\rm{SL}}_n(A)$ has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. If $G$ is any group satisfying this dichotomy we say that $G$ has the \emph{dichotomy property}. We relate the dichotomy property, as well as some natural variants of it, to other rigidity results in the theory of arithmetic and profinite groups such as the celebrated normal subgroup theorem of Margulis and the seminal work of Nikolov and Segal. As a consequence we derive constraints to the possible approximations of certain non residually finite central extensions of arithmetic groups, which we hope might have further applications in the study of sofic groups. In the last section we provide several open problems for further research.

math.GR↗
arXiv2025

Arithmetic quantum unique ergodicity for products of hyperbolic $2$- and $3$-spaces

We prove the arithemtic quantum unique ergodicity (AQUE) conjecture for sequences of Hecke--Maass forms on quotients $Γ\backslash (\mathbb{H}^{(2)})^r \times (\mathbb{H}^{(3)})^s$. An argument by induction on dimension of the orbit allows us to rule out the limit measure concentrating on closed orbits of proper subgroups despite many returns of the Hecke correspondence to neighborhoods of the orbit.

math.DS↗
arXiv2024

Homogeneity of arithmetic quantum limits for hyperbolic $4$-manifolds

We work toward the arithmetic quantum unique ergodicity (AQUE) conjecture for sequences of Hecke--Maass forms on hyperbolic $4$-manifolds. We show that limits of such forms can only scar on totally geodesic $3$-submanifolds, and in fact that all ergodic components of the microlocal lift other than the uniform measure arise from the uniform measures on these submanifolds.

math.NT↗
arXiv2020

Positive entropy using Hecke operators at a single place

We prove the following statement: Let $X=\text{SL}_n(\mathbb{Z})\backslash \text{SL}_n(\mathbb{R})$, and consider the standard action of the diagonal group $A<\text{SL}_n(\mathbb{R})$ on it. Let $μ$ be an $A$-invariant probability measure on $X$, which is a limit $$ μ=λ\lim_i|ϕ_i|^2dx, $$ where $ϕ_i$ are normalized eigenfunctions of the Hecke algebra at some fixed place $p$, and $λ>0$ is some positive constant. Then any regular element $a\in A$ acts on $μ$ with positive entropy on almost every ergodic component. We also prove a similar result for lattices coming from division algebras over $\mathbb{Q}$, and derive a quantum unique ergodicity result for the associated locally symmetric spaces. This generalizes a result of Brooks and Lindenstrauss.

math.RT↗