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 → 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 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 [GH]. 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 Qp : Q n p → C}p∈primes is a collection of functions which is motivic in the sense of Denef-Pas, and f Qp is L 1 for any p large enough, then in fact there exists ǫ > 0 such that f Qp is L 1+ǫ for any p large enough.
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.
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 Lr, it induces a word map ϕw : g r → g for every semisimple Lie algebra g. Given two words w1 ∈ Lr 1 and w2 ∈ Lr 2 , we define and study the convolution of the corresponding word maps ϕw 1 * ϕw 2 := ϕw 1 + ϕw 2 : g r 1 +r 2 → g.We show that for any word w ∈ Lr of degree d, and any simple Lie algebra g with ϕw(g r ) = 0, one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking O(d 6 ) self-convolutions of ϕw. We deduce that a group word map of length ℓ becomes (FRS) at (e, . . . , e) ∈ G r after O(ℓ 6 ) 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 = SLn. For the commutator word w0 = [X, Y ], we show that ϕ * 4 w 0 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 Z/p k 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, characterizing some of the singularity properties discussed in probabilistic terms, and provide applications to p-adic probabilistic Waring type problems. 5. Lie algebra word maps: proof of the main theorems 37 5.1. Discussion of the main methods 37 5.2. Proof for Lie algebras of low rank 39 5.3. Proof for d-hypergraphs with few edges 41 5.4. Proof for sln 43 5.5. Matrix word maps 49 5.6. Proof of the general case 50 6. The commutator map revisited 56 7. Local behavior of word maps and a lower bound on the log canonical threshold of their fibers 59 7.1. The degree of a word and of a word map 59 7.2. Algebro-geometric results on word maps 61 8. Number theoretic interpretation of the flatness, ε-jet flatness and (FRS) properties 63 8.1. The Lang-Weil bounds and a number theoretic characterization of flatness 63 8.2. Number theoretic and analytic interpretation of the (FRS) property 65 8.3. The Denef-Pas language and motivic functions 66 8.4. Number theoretic characterization of ε-jet flatness 67 9. Applications to the p-adic probabilistic Waring type problem 70 9.1. Algebraic random walks: general case 71 9.2. Algebraic random walks induced from word maps 74 References 76 * t(w) w : G rt(w) → G is surjective for any G in a nice family G of groups. The minimal such t(w) is called the G-width of w. Similarly, * Cℓ(w) α w is (FRS)? (2) If such α exists, what is its optimal value? What is the optimal value of such α if we just demand that our word map becomes flat? (3) What is the answer to (2) in the case of simple Lie algebras, where ℓ(w) is replaced by deg...
Given a map φ : X → Y between F -analytic manifolds over a local field F of characteristic 0, we introduce an invariant ǫ ⋆ (φ) which quantifies the integrability of pushforwards of smooth compactly supported measures by φ. We further define a local version ǫ ⋆ (φ, x) near x ∈ X. These invariants have a strong connection to the singularities of φ.When Y is one-dimensional, we give an explicit formula for ǫ ⋆ (φ, x), and show it is asymptotically equivalent to other known singularity invariants such as the F -log-canonical threshold lct F (φ − φ(x); x) at x.In the general case, we show that ǫ ⋆ (φ, x) is bounded from below by the F -log-canonical threshold λ = lct F (J φ ; x) of the Jacobian ideal 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 = C, we establish the upper bound ǫ ⋆ (φ, x) ≤ λ/(1 − λ), whenever λ < 1.Finally, we specialize to polynomial maps ϕ : X → Y between smooth algebraic Q-varieties X and Y . We geometrically characterize the condition that ǫ ⋆ (ϕ F ) = ∞ over a large family of local fields, by showing it is equivalent to ϕ being flat with fibers of semi-log-canonical singularities.
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