arXiv subjects

Itay Glazer

Publications and source records attributed to Itay Glazer.

17 recordsLinked to original sources

arXiv2026

A lower bound on the analytic log-canonical threshold over local fields of positive characteristic

Given a local field $F$ of positive characteristic, an $F$-analytic manifold $X$ and an analytic function $f:X\rightarrow F$, the $F$-analytic log-canonical threshold $\mathrm{lct}_{F}(f;x_{0})$ is the supremum over the values $s\geq0$ such that $\left|f\right|_{F}^{-s}$ is integrable near $x_{0}\in X$. We show that $\mathrm{lct}_{F}(f;x_{0})>0$. Moreover, if $f$ is a regular function on a smooth algebraic $F$-variety, we obtain an effective lower bound $\mathrm{lct}_{F}(f;x_{0})>C$, where $C>0$ is explicit and depends only on the complexity class of $X$ and $f$.

math.AG↗

arXiv2026

Anti-concentration of polynomials: $L^{p}$ balls and symmetric measures

We begin with the observation, based on previous results, that dimension-free lower bounds on the variance of a polynomial under a log-concave measure yield dimension-free small-ball and Fourier decay estimates. Motivated by this, we establish variance bounds for polynomials on log-concave random vectors beyond the classical setting of product measures. First, we consider the family of uniform measures on the $n$-dimensional isotropic $L^{p}$ balls. We show that for a degree-$d$ homogeneous polynomial $f=\sum_{I}a_{I}x^{I}$, with $\sum_{I}a_{I}^{2}=1$, the only obstruction to a dimension-free lower bound on its variance occurs when $p=d$ is an even integer and the coefficients of $f$ are close to those of $\frac{1}{\sqrt{n}}\left\Vert x\right\Vert _{p}^{p}$. Second, we consider general isotropic log-concave measures that are invariant under coordinate permutations and reflections, and determine the minimal variance for quadratic and cubic polynomials. These variance bounds lead to new dimension-free anti-concentration results in both settings, addressing a natural extension of a question posed by Carbery and Wright.

math.PR↗

arXiv2026

On Harish-Chandra's integrability theorem in positive characteristic

The celebrated Harish-Chandra's integrability theorem states that the distributional character of an irreducible smooth representation of a p-adic group $G(F)$ is integrable, that is represented by an $L^1_{loc}(G(F))$ function. Here $F$ is a non-Archimedean local field of characteristic $0$ and $G$ is a reductive algebraic group defined over $F$. In this paper we focus on cuspidal representations of $GL_n(F)$ for a field $F$ of positive characteristic. We show that in this case the integrability holds under the hypothesis of existence of desingularization of (certain) algebraic varieties in positive characteristics. Furthermore, in the case $char(F)>n/2$ we establish the regularity of such characters unconditionally.

math.RT↗

arXiv2025

Word maps and random words

We discuss some recent results by a number of authors regarding word maps on algebraic groups and finite simple groups, their mixing properties and the geometry of their fibers, emphasizing the role played by equidistribution results in finite fields via recent advances on character bounds and non-abelian arithmetic combinatorics. In particular, we discuss character varieties of random groups. In the last section, we give a new proof of a recent theorem of Hrushovski about the geometric irreducibility of the generic fibers of convolutions of dominant morphisms to simply connected algebraic groups. These notes stem out of lectures given by the authors in Oxford, and by the first author in ICTS Bangalore, in spring 2024.

math.GR↗

arXiv2024

Integrability and singularities of Harish-Chandra characters

Let $G$ be a reductive group over a local field $F$ of characteristic $0$. By Harish-Chandra's regularity theorem, the character $Θ_π$ of an irreducible, admissible representation $π$ of $G$ is given by a locally integrable function $θ_π$ on $G$. It is a natural question whether $θ_π$ has better integrability properties, namely, whether it is locally $L^{1+ε}$-integrable for some $ε>0$. It turns out that the answer is positive, and this gives rise to a new singularity invariant of representations $ε_{\star}(π):=\sup\left\{ ε:θ_π\in L_{Loc}^{1+ε}(G)\right\} $, which we explore in this paper. We provide a lower bound on $ε_{\star}(π)$ which depends only on the absolute root system of $G$, and explicitly determine $ε_{\star}(π)$ in the case of a $p$-adic $\mathrm{GL}_{n}$. This is done by studying integrability properties of the Fourier transforms $\widehatξ_{\mathcal{O}}$ of stable Richardson nilpotent orbital integrals $ξ_{\mathcal{O}}$. We express $ε_{\star}(\widehatξ_{\mathcal{O}})$ as the log-canonical threshold of a suitable relative Weyl discriminant, and use a resolution of singularities algorithm coming from the theory of hyperplane arrangements, to compute it in terms of the partition associated with the orbit. We obtain several applications; firstly, we provide bounds on the multiplicities of $K$-types in irreducible representations of $G$ in the $p$-adic case, where $K$ is an open compact subgroup. We further obtain bounds on the multiplicities of the irreducible representations appearing in the space $L^{2}(K/L)$, where $K$ is a compact simple Lie group, and $L\leq K$ is a Levi subgroup. Finally, we discover surprising applications in random matrix theory, namely to the study of the eigenvalue distribution of powers of random unitary matrices.

math.RT↗

arXiv2024

On the Fourier coefficients of word maps on unitary groups

Given a word $w(x_{1},\ldots,x_{r})$, i.e., an element in the free group on $r$ elements, and an integer $d\geq1$, we study the characteristic polynomial of the random matrix $w(X_{1},\ldots,X_{r})$, where $X_{i}$ are Haar-random independent $d\times d$ unitary matrices. If $c_{m}(X)$ denotes the $m$-th coefficient of the characteristic polynomial of $X$, our main theorem implies that there is a positive constant $ε(w)$, depending only on $w$, such that \[ \left|\mathbb{E}\left(c_{m}\left(w(X_{1},\ldots,X_{r})\right)\right)\right|\leq\left(\begin{array}{c} d\\ m \end{array}\right)^{1-ε(w)}, \] for every $d$ and every $1\leq m\leq d$. Our main computational tool is the Weingarten Calculus, which allows us to express integrals on unitary groups such as the expectation above, as certain sums on symmetric groups. We exploit a hidden symmetry to find cancellations in the sum expressing $\mathbb{E}\left(c_{m}(w)\right)$. These cancellations, coming from averaging a Weingarten function over cosets, follow from Schur's orthogonality relations.

math.PR↗

arXiv2024

Integrability of pushforward measures by analytic maps

Given a map $ϕ:X\rightarrow Y$ between $F$-analytic manifolds over a local field $F$ of characteristic $0$, we introduce an invariant $ε_{\star}(ϕ)$ which quantifies the integrability of pushforwards of smooth compactly supported measures by $ϕ$. We further define a local version $ε_{\star}(ϕ,x)$ near $x\in X$. These invariants have a strong connection to the singularities of $ϕ$. When $Y$ is one-dimensional, we give an explicit formula for $ε_{\star}(ϕ,x)$, and show it is asymptotically equivalent to other known singularity invariants such as the $F$-log-canonical threshold $\operatorname{lct}_{F}(ϕ-ϕ(x);x)$ at $x$. In the general case, we show that $ε_{\star}(ϕ,x)$ is bounded from below by the $F$-log-canonical threshold $λ=\operatorname{lct}_{F}(\mathcal{J}_ϕ;x)$ of the Jacobian ideal $\mathcal{J}_ϕ$ near $x$. If $\dim Y=\dim X$, equality is attained. If $\dim Y<\dim X$, the inequality can be strict; however, for $F=\mathbb{C}$, we establish the upper bound $ε_{\star}(ϕ,x)\leqλ/(1-λ)$, whenever $λ<1$. Finally, we specialize to polynomial maps $φ:X\rightarrow Y$ between smooth algebraic $\mathbb{Q}$-varieties $X$ and $Y$. We geometrically characterize the condition that $ε_{\star}(φ_{F})=\infty$ over a large family of local fields, by showing it is equivalent to $φ$ being flat with fibers of semi-log-canonical singularities.

math.AG↗

arXiv2024

Eventual tightness of projective dimension growth bounds: quadratic in the degree

In projective dimension growth results, one bounds the number of rational points of height at most $H$ on an irreducible hypersurface in $\mathbb P^n$ of degree $d>3$ by $C(n)d^2 H^{n-1}(\log H)^{M(n)}$, where the quadratic dependence in $d$ has been recently obtained by Binyamini, Cluckers and Kato in 2024 [1]. For these bounds, it was already shown by Castryck, Cluckers, Dittmann and Nguyen in 2020 [3] that one cannot do better than a linear dependence in $d$. In this paper we show that, for the mentioned projective dimension growth bounds, the quadratic dependence in $d$ is eventually tight when $n$ grows. More precisely the upper bounds cannot be better than $c(n)d^{2-2/n} H^{n-1}$ in general. Note that for affine dimension growth (for affine hypersurfaces of degree $d$, satisfying some extra conditions), the dependence on $d$ is also quadratic by [1], which is already known to be optimal by [3]. Our projective case thus complements the picture of tightness for dimension growth bounds for hypersurfaces.

math.NT↗

arXiv2024

Fourier and small ball estimates for word maps on unitary groups

To a non-trivial word $w(x_{1},...,x_{r})$ in a free group $F_{r}$ on $r$ elements and a group $G$, one can associate the word map $w_{G}:G^{r}\rightarrow G$ that takes an $r$-tuple $(g_{1},...,g_{r})$ in $G^{r}$ to $w(g_{1},...,g_{r})$. If $G$ is compact, we further associate the word measure $τ_{w,G}$, defined as the distribution of $w_{G}(\mathsf{X}_{1},...,\mathsf{X}_{r})$, where $\mathsf{X}_{1},...,\mathsf{X}_{r}$ are independent and Haar-random elements in $G$. In this paper we study word maps and word measures on the family of special unitary groups $\left\{ \mathrm{SU}_{n}\right\} _{n\geq2}$. Our first result is a small ball estimate for $w_{\mathrm{SU}_{n}}$. We show that for every $w\in F_{r}\smallsetminus\left\{ 1\right\} $ there are $ε(w),δ(w)>0$ such that if $B\subseteq\mathrm{SU}_{n}$ is a ball of radius at most $δ(w)\mathrm{diam}(\mathrm{SU}_{n})$ in the Hilbert-Schmidt metric, then $τ_{w,\mathrm{SU}_{n}}(B)\leq(μ_{\mathrm{SU}_{n}}(B))^{ε(w)}$, where $μ_{\mathrm{SU}_{n}}$ is the Haar probability measure. Our second main result is about the random walks generated by $τ_{w,\mathrm{SU}_{n}}$. We provide exponential upper bounds on the large Fourier coefficients of $τ_{w,\mathrm{SU}_{n}}$, and as a consequence we show there exists $t(w)\in\mathbb{N}$, such that $τ_{w,\mathrm{SU}_{n}}^{*t}$ has bounded density for every $t\geq t(w)$ and every $n\geq2$, answering a conjecture by the first two authors. As a key step in the proof, we establish, for every large irreducible character $ρ$ of $\mathrm{SU}_{n}$, an exponential upper bound of the form $\left|ρ(g)\right|<ρ(1)^{1-ε}$, for elements $g$ in $\mathrm{SU}_{n}$ whose eigenvalues are sufficiently spread out on the unit circle in $\mathbb{C^{\times}}$.

math.GR↗

arXiv2022

Pro-isomorphic zeta functions of nilpotent groups and Lie rings under base extension

We consider pro-isomorphic zeta functions of the groups $Γ(\mathcal{O}_K)$, where $Γ$ is a unipotent group scheme defined over $\mathbb{Z}$ and $K$ varies over all number fields. Under certain conditions, we show that these functions have a fine Euler decomposition with factors indexed by primes $\mathfrak{p}$ of $K$ and depending only on the structure of $Γ$, the degree $[K : \mathbb{Q}]$, and the cardinality of the residue field $\mathcal{O}_K / \mathfrak{p}$. We show that the factors satisfy a certain uniform rationality and study their dependence on $[K : \mathbb{Q}]$. Explicit computations are given for several families of unipotent groups. These include an apparently novel identity involving permutation statistics on the hyperoctahedral group.

math.GR↗

arXiv2022

Anti-concentration of polynomials: dimension-free covariance bounds and decay of Fourier coefficients

We study random variables of the form $f(X)$, when $f$ is a degree $d$ polynomial, and $X$ is a random vector on $\mathbb{R}^{n}$, motivated towards a deeper understanding of the covariance structure of $X^{\otimes d}$. For applications, the main interest is to bound $\mathrm{Var}(f(X))$ from below, assuming a suitable normalization on the coefficients of $f$. Our first result applies when $X$ has independent coordinates, and we establish dimension-free bounds. We also show that the assumption of independence can be relaxed and that our bounds carry over to uniform measures on isotropic $L_{p}$ balls. Moreover, in the case of the Euclidean ball, we provide an orthogonal decomposition of $\mathrm{Cov}(X^{\otimes d})$. Finally, we utilize the connection between anti-concentration and decay of Fourier coefficients to prove a high-dimensional analogue of the van der Corput lemma, thus partially answering a question posed by Carbery and Wright.

math.PR↗

arXiv2021

A number theoretic characterization of $E$-smooth and (FRS) morphisms: estimates on the number of $\mathbb{Z}/p^{k}\mathbb{Z}$-points

We provide uniform estimates on the number of $\mathbb{Z}/p^{k}\mathbb{Z}$-points lying on fibers of flat morphisms between smooth varieties whose fibers have rational singularities, termed (FRS) morphisms. For each individual fiber, the estimates were known by work of Avni and Aizenbud, but we render them uniform over all fibers. The proof technique for individual fibers is based on Hironaka's resolution of singularities and Denef's formula, but breaks down in the uniform case. Instead, we use recent results from the theory of motivic integration. Our estimates are moreover equivalent to the (FRS) property, just like in the absolute case by Avni and Aizenbud. In addition, we define new classes of morphisms, called $E$-smooth morphisms ($E\in\mathbb{N}$), which refine the (FRS) property, and use the methods we developed to provide uniform number-theoretic estimates as above for their fibers. Similar estimates are given for fibers of $\varepsilon$-jet flat morphisms, improving previous results by the last two authors.

math.AG↗

arXiv2020

On singularity properties of convolutions of algebraic morphisms -- the general case (with an appendix joint with Gady Kozma)

Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth $K$-varieties, and let $G$ be a algebraic $K$-group. Given two algebraic morphisms $φ:X\rightarrow G$ and $ψ:Y\rightarrow G$, we define their convolution $φ*ψ:X\times Y\to G$ by $φ*ψ(x,y)=φ(x)\cdotψ(y)$. We then show that this operation yields morphisms with improved smoothness properties. More precisely, we show that for any morphism $φ:X\rightarrow G$ which is dominant when restricted to each absolutely irreducible component of $X$, by convolving it with itself finitely many times, one can obtain a flat morphism with reduced fibers of rational singularities, generalizing the main result of our previous paper. Uniform bounds on families of morphisms are given as well. Moreover, as a key analytic step, we also prove the following result in motivic integration; if $\{f_{\mathbb{Q}_{p}}:\mathbb{Q}_{p}^{n}\rightarrow\mathbb{C}\}_{p\in\mathrm{primes}}$ is a collection of functions which is motivic in the sense of Denef-Pas, and $f_{\mathbb{Q}_{p}}$ is $L^{1}$ for any $p$ large enough, then in fact there exists $ε>0$ such that $f_{\mathbb{Q}_{p}}$ is $L^{1+ε}$ for any $p$ large enough.

math.AG↗

arXiv2020

On singularity properties of word maps and applications to probabilistic Waring type problems

We study singularity properties of word maps on semisimple algebraic groups and Lie algebras, generalizing the work of Aizenbud-Avni in the case of the commutator map. Given a word $w$ in a free Lie algebra $\mathcal{L}_{r}$, it induces a word map $φ_{w}:\mathfrak{g}^{r}\rightarrow\mathfrak{g}$ for every semisimple Lie algebra $\mathfrak{g}$. Given two words $w_{1}\in\mathcal{L}_{r_{1}}$ and $w_{2}\in\mathcal{L}_{r_{2}}$, we define and study the convolution of the corresponding word maps $φ_{w_{1}}*φ_{w_{2}}:=φ_{w_{1}}+φ_{w_{2}}:\mathfrak{g}^{r_{1}+r_{2}}\rightarrow\mathfrak{g}$. We show that for any word $w\in\mathcal{L}_{r}$ of degree $d$, and any simple Lie algebra $\mathfrak{g}$ with $φ_{w}(\mathfrak{g}^{r})\neq0$, one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking $O(d^{4})$ self-convolutions of $φ_{w}$. We deduce that a group word map of length $\ell$ becomes (FRS) at $(e,\ldots,e)\in G^{r}$ after $O(\ell^{4})$ self-convolutions, for any semisimple algebraic group $G$. We furthermore bound the dimensions of the jet schemes of the fibers of Lie algebra word maps, and the fibers of group word maps in the case where $G=\mathrm{SL}_{n}$. For the commutator word $ν=[X,Y]$, we show that $φ_ν^{*4}$ is (FRS) for any semisimple Lie algebra, obtaining applications in representation growth of compact $p$-adic and arithmetic groups. The singularity properties we consider, such as the (FRS) property, are intimately connected to the point count of fibers over finite rings of the form $\mathbb{Z}/p^{k}\mathbb{Z}$. This allows us to relate them to properties of some natural families of random walks on finite and compact $p$-adic groups. We explore these connections, and provide applications to $p$-adic probabilistic Waring type problems.

math.AG↗

arXiv2020

On singularity properties of convolutions of algebraic morphisms

Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth $K$-varieties, and let $V$ be a finite dimensional $K$-vector space. For two algebraic morphisms $φ:X\rightarrow V$ and $ψ:Y\rightarrow V$ we define a convolution operation, $φ*ψ:X\times Y\to V$, by $φ*ψ(x,y)=φ(x)+ψ(y)$. We then study the singularity properties of the resulting morphism, and show that as in the case of convolution in analysis, it has improved smoothness properties. Explicitly, we show that for any morphism $φ:X\rightarrow V$ which is dominant when restricted to each irreducible component of $X$, there exists $N\in\mathbb{N}$ such that for any $n>N$ the $n$-th convolution power $φ^{n}:=φ*\dots*φ$ is a flat morphism with reduced geometric fibers of rational singularities (this property is abbreviated (FRS)). By a theorem of Aizenbud and Avni, for $K=\mathbb{Q}$, this is equivalent to good asymptotic behavior of the size of the $\mathbb{Z}/p^{k}\mathbb{Z}$-fibers of $φ^{n}$ when ranging over both $p$ and $k$. More generally, we show that given a family of morphisms $\{φ_{i}:X_{i}\rightarrow V\}$ of complexity $D\in\mathbb{N}$ (i.e. that the number of variables and the degrees of the polynomials defining $X_{i}$ and $φ_{i}$ are bounded by $D$), there exists $N(D)\in\mathbb{N}$ such that for any $n>N(D)$, the morphism $φ_{1}*\dots*φ_{n}$ is (FRS).

math.AG↗

arXiv2017

On rational singularities and counting points of schemes over finite rings

We study the connection between the singularities of a finite type $\mathbb{Z}$-scheme X and the asymptotic point count of X over various finite rings. In particular, if the generic fiber $X_{\mathbb{Q}}=X\times_{\mathrm{Spec}\mathbb{Z}}\mathrm{Spec}\mathbb{Q}$ is a local complete intersection, we show that the boundedness of $\frac{\left|X(\mathbb{Z}/p^{n}\mathbb{Z})\right|}{p^{n\mathrm{dim}X_{\mathbb{Q}}}}$ in p and n is in fact equivalent to the condition that $X_{\mathbb{Q}}$ is reduced and has rational singularities. This paper completes a result of Aizenbud and Avni.

math.AG↗

arXiv2016

Representations of reductive groups distinguished by symmetric subgroups

Let $G$ be a complex reductive group and $H=G^θ$ be its fixed point subgroup under a Galois involution $θ$. We show that any $H$-distinguished representation $π$ (i.e $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\neq0$) satisfies: 1) $π^θ\simeq\tildeπ$, where $\tildeπ$ is the contragredient representation and $π^θ$ is the twist of $π$ under $θ$. 2) $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\leq\left|B\backslash G/H\right|$, where $B$ is a Borel subgroup of $G$. By proving Statement 1), we give a partial answer to a conjecture by Lapid.

math.RT↗