On singularity properties of convolutions of algebraic morphisms

Glazer, Itay; Hendel, Yotam I.

doi:10.1007/s00029-019-0457-z

Cited by 6 publications

(49 citation statements)

References 22 publications

(24 reference statements)

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“…In this paper we verify a conjecture of Aizenbud and Avni (see [20, Conjecture 1.6]), showing that every strongly dominant morphism into an algebraic group becomes (FRS) after finitely many self‐convolutions: Theorem Let

X

be a smooth

K

‐variety,

G

be a connected algebraic

K

‐group and let

φ : X \to G

be a strongly dominant morphism. Then there exists

N \in N

such that for any

n > N

, the

n

‐th convolution power

φ^{* n}

is (FRS).…”

Section: Introductionsupporting

confidence: 60%

“…It is a consequence of [17] and [1, Corollary 2.2], that the (FRS) property is preserved under small deformations. This allows us to extend our main result, Theorem A, to families of morphisms (thus generalizing [20, Theorem 7.1]): Theorem Let

K

and

G

be as in Theorem A, let

Y

be a

K

‐variety, let

\overset{X}{true}

be a family of varieties over

Y

, and let

\tilde{φ} : \tilde{X} \to G \times Y

be a

Y

‐morphism. Denote by

{\overset{φ}{true}}_{y} : {\overset{X}{true}}_{y} \to G

the fiber of

\overset{φ}{true}

y \in Y

.…”

Section: Introductionmentioning

confidence: 87%

“…The rest of this paper is devoted to the study of the second part of Question 1.2, and generalizes the results from [20] in which the case where

G = V

is a vector space was dealt with.…”

Section: Introductionmentioning

confidence: 92%

“…In this paper we continue the investigation of the algebraic convolution operation, which was initiated in [20]. Namely, we study the following operation: Definition Let

X_{1}

and

X_{2}

be algebraic varieties,

G

an algebraic group and let

φ_{1} : X_{1} \to G

and

φ_{2} : X_{2} \to G

be algebraic morphisms.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the results and methods of this paper are used in [19] to investigate families of random walks on compact

p

‐adic groups which are induced by words. For a precise discussion of this probabilistic interpretation see [19, Section 9], and for further discussion of the (FRS) property and its implications see [20, Section 1.3] or [1, 2].…”

Section: Introductionmentioning

confidence: 99%

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On singularity properties of convolutions of algebraic morphisms ‐ the general case

Glazer

Hendel

2020

J. London Math. Soc.

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Let K be a field of characteristic zero, X and Y be smooth K‐varieties, and let G be an algebraic K‐group. Given two algebraic morphisms φ:X→G and ψ:Y→G, we define their convolution φ∗ψ:X×Y→G by φ∗ψ(x,y)=φ(x)·ψ(y). We then show that this operation yields morphisms with improved smoothness properties. More precisely, we show that for any morphism φ:X→G which is dominant when restricted to each geometrically irreducible component of X, by convolving it with itself finitely many times, one obtains a flat morphism with reduced fibers of rational singularities. 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 false{fdouble-struckQp:Qpn→double-struckCfalse}p∈normalprimes is a collection of motivic functions, and fdouble-struckQp is L1 for any p large enough, then in fact there exists ε>0 such that fdouble-struckQp is L1+ε for any p large enough.

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X

be a smooth

K

‐variety,

G

be a connected algebraic

K

‐group and let

φ : X \to G

be a strongly dominant morphism. Then there exists

N \in N

such that for any

n > N

, the

n

‐th convolution power

φ^{* n}

is (FRS).…”

Section: Introductionsupporting

confidence: 60%

K

and

G

be as in Theorem A, let

Y

be a

K

‐variety, let

\overset{X}{true}

be a family of varieties over

Y

, and let

\tilde{φ} : \tilde{X} \to G \times Y

be a

Y

‐morphism. Denote by

{\overset{φ}{true}}_{y} : {\overset{X}{true}}_{y} \to G

the fiber of

\overset{φ}{true}

y \in Y

.…”

Section: Introductionmentioning

confidence: 87%

“…The rest of this paper is devoted to the study of the second part of Question 1.2, and generalizes the results from [20] in which the case where

G = V

is a vector space was dealt with.…”

Section: Introductionmentioning

confidence: 92%

“…In this paper we continue the investigation of the algebraic convolution operation, which was initiated in [20]. Namely, we study the following operation: Definition Let

X_{1}

and

X_{2}

be algebraic varieties,

G

an algebraic group and let

φ_{1} : X_{1} \to G

and

φ_{2} : X_{2} \to G

be algebraic morphisms.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the results and methods of this paper are used in [19] to investigate families of random walks on compact

p

Section: Introductionmentioning

confidence: 99%

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On singularity properties of convolutions of algebraic morphisms ‐ the general case

Glazer

Hendel

2020

J. London Math. Soc.

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On Singularity Properties of Word Maps and Applications to Probabilistic Waring Type Problems

Glazer,

Hendel

2024

Memoirs of the AMS

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We study singularity properties of word maps on semisimple Lie algebras, semisimple algebraic groups and matrix algebras and obtain various applications to random walks induced by word measures on compact p p -adic groups. Given a word w w in a free Lie algebra L r \mathcal {L}_{r} , it induces a word map φ w : g r → g \varphi _{w}:\mathfrak {g}^{r}\rightarrow \mathfrak {g} for every semisimple Lie algebra g \mathfrak {g} . Given two words w 1 ∈ L r 1 w_{1}\in \mathcal {L}_{r_{1}} and w 2 ∈ L r 2 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 : g r 1 + r 2 → g \varphi _{w_{1}}*\varphi _{w_{2}}≔\varphi _{w_{1}}+\varphi _{w_{2}}:\mathfrak {g}^{r_{1}+r_{2}}\rightarrow \mathfrak {g} . By introducing new degeneration techniques, we show that for any word w ∈ L r w\in \mathcal {L}_{r} of degree d d , and any simple Lie algebra g \mathfrak {g} with φ w ( g r ) ≠ 0 \varphi _{w}(\mathfrak {g}^{r})\neq 0 , one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking O ( d 4 ) O(d^{4}) self-convolutions of φ w \varphi _{w} . Similar results are obtained for matrix word maps. We deduce that a group word map of length ℓ \ell becomes (FRS), locally around identity, after O ( ℓ 4 ) O(\ell ^{4}) self-convolutions, for every semisimple algebraic group G _ \underline {G} . We furthermore provide uniform lower bounds on the log canonical threshold of the fibers of Lie algebra, matrix and group word maps. For the commutator word w 0 = [ X , Y ] w_{0}=[X,Y] , we show that φ w 0 ∗ 4 \varphi _{w_{0}}^{*4} is (FRS) for any semisimple Lie algebra, improving a result of Aizenbud-Avni, and obtaining applications in representation growth of compact p 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 Z / p k Z \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 p -adic groups. We explore these connections, characterizing some of the singularity properties discussed in probabilistic terms, and provide applications to p p -adic probabilistic Waring type problems.

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A number theoretic characterization of E-smooth and (FRS) morphisms : estimates on the number of ℤ∕pkℤ-points

Cluckers,

Glazer,

Hendel

2023

Alg. Number Th.

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We provide uniform estimates on the number of ‫/ޚ‬ p k ‫-ޚ‬points 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 ∈ ‫,)ގ‬ 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 ε-jet flat morphisms, improving previous results by the last two authors.

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On singularity properties of convolutions of algebraic morphisms

Cited by 6 publications

References 22 publications

On singularity properties of convolutions of algebraic morphisms ‐ the general case

On singularity properties of convolutions of algebraic morphisms ‐ the general case

On Singularity Properties of Word Maps and Applications to Probabilistic Waring Type Problems

A number theoretic characterization of E-smooth and (FRS) morphisms : estimates on the number of ℤ∕pkℤ-points

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