# Étale homological stability and arithmetic statistics

Étale homological stability and arithmetic statistics Abstract We relate asymptotic point counts and arithmetic statistics over finite fields to homological stability and representation stability over ℂ in the example of configuration spaces of n points on smooth varieties. In order to accomplish this, we establish subexponential bounds on the growth of the unstable cohomology of such spaces. We apply this and étale homological stability to compute the large n limits of various arithmetic statistics of configuration spaces of varieties over Fq. 1. Introduction Let X be a scheme defined over Z. The Weil conjectures provide a fundamental link between the topology of X(C) and the arithmetic of X(Fq). As first indicated by work of Ellenberg–Venkatesh–Westerland [13], followed by Vakil–Wood [26], Church–Ellenberg–Farb [4] and others, this correspondence should convert homological stability phenomena in topology to asymptotic point counts on the arithmetic side. We summarize this in the following table, with the rows going from least to most general. Topology Arithmetic H*(X(C)) ∣X(Fq)∣ Homological stability of Xn Asymptotics of ∣Xn(Fq)∣ as n→∞ Representation stability Asymptotics of arithmetic Statistics on Xn(Fq) Topology Arithmetic H*(X(C)) ∣X(Fq)∣ Homological stability of Xn Asymptotics of ∣Xn(Fq)∣ as n→∞ Representation stability Asymptotics of arithmetic Statistics on Xn(Fq) Topology Arithmetic H*(X(C)) ∣X(Fq)∣ Homological stability of Xn Asymptotics of ∣Xn(Fq)∣ as n→∞ Representation stability Asymptotics of arithmetic Statistics on Xn(Fq) Topology Arithmetic H*(X(C)) ∣X(Fq)∣ Homological stability of Xn Asymptotics of ∣Xn(Fq)∣ as n→∞ Representation stability Asymptotics of arithmetic Statistics on Xn(Fq) One of the main goals of the present paper is to realize the bottom rows of this table for varieties PConfn(X) (resp. UConfn(X)) of configurations of ordered (resp. unordered) n-tuples of points on a smooth variety X. We first state our main theorem, after which we will define all of the terms appearing in its statement. Theorem A (Arithmetic statistics of configuration spaces). Let Xbe a scheme, smooth over Z[1/N]for some N, with geometrically connected fibers of finite type. Let pbe a prime not dividing N, and let qbe a power of p. Let Frobqdenote the qthpower Frobenius. Let Pbe any character polynomial, and denote by Heti(PConf(X))Pthe stable P-isotypic part of the étale cohomology of the co-FI-scheme PConf•(X)F¯q (see Section2below for a precise definition). Denote by Heti(UConf(X))the stable étale cohomology of UConfn(X)F¯q. Then limn→∞q−ndimX∑y∈UConfn(X)(Fq)P(y)=∑i=0∞(−1)iTr(Frobq⥀Heti(PConf(X))P*),in particular, both sides of the above converge. Specializing to P=1, we obtain limn→∞q−ndimX∣UConfn(X)(Fq)∣=∑i=0∞(−1)iTr(Frobq⥀Heti(UConf(X))*). The proof of Theorem A has two main steps: We prove what we call étale homological (and representation) stability for Het*(Xn/F¯q;Qℓ). This allows us to break up Het*(Xn/F¯q;Qℓ) into two parts: stable and unstable. We obtain subexponential bounds on the growth of the unstable part of Het*(Xn/F¯q;Qℓ). This allows us to prove that this unstable part does not contribute, via the Grothendieck–Lefschetz trace formula, to the limiting density of ∣Xn(Fq)∣. As we explain below, the absence of such bounds is a significant obstruction to understanding the asymptotic point counts of many families of interest. In the étale context, stability of each Heti(Xn/F¯q;Qℓ) as a Galois representation, not just as a vector space, is crucial. It is the difference between proving that limits such as limn→∞q−dimXn∣Xn(Fq)∣ exist, and actually computing the limiting answer. We now discuss these two steps in more detail. Homological stability: A sequence {Xn} of spaces or groups is said to satisfy homological stability over a ring R if Hi(Xn;R) or Hi(Xn;R) is independent of n for n≥D(i); the number D(i) is called the stable range. Typically, but not always, there are maps ψn:Xn→Xn+1 or ϕn:Xn+1→Xn inducing isomorphisms (ψn)*:Hi(Xn;R)→Hi(Xn+1;R)orϕn*:Hi(Xn;R)→Hi(Xn+1;R). Examples of Xn satisfying homological stability include classifying spaces of symmetric groups Sn (Nakaoka), arithmetic groups like SLnZ (Borel), the moduli spaces Ag (Borel) and Mg (Harer) and also configuration spaces UConfn(M) of unordered n-tuples of distinct points on a manifold M (Arnol’d, McDuff, Segal, Church). Homological stability has been a powerful tool in topology. It converts an a priori infinite computation to a finite one. Further, the stable answer, Hi(Xn;R) for n≥D(i), can often be computed explicitly. Many natural sequences Xn come equipped with actions of groups Gn by automorphisms. A basic example is the space PConfn(M) of ordered n-tuples of distinct points on a manifold M, on which the symmetric group Sn acts by permuting the ordering. Such spaces almost never satisfy homological stability, but they instead often satisfy representation stability: the decomposition of Hi(Xn;Q) into a sum of irreducible Sn-representations stabilizes in a precise sense (see [3, 6], Section 2.1 below, and [14] for a survey). When R=Q, plugging the trivial representation into this theory gives classical homological stability for the sequence Xn/Sn. So for example representation stability for PConfn(M) gives classical homological stability for the space UConfn(M)=PConfn(M)/Sn of unordered n-tuples of distinct points on M; see [8]. The theory of representation stability, initiated by Church, Ellenberg and Farb, is currently undergoing a rapid development. Étale homological stability: Consider a scheme Y, smooth over Z[1/N] for some N. We can extend scalars to C and consider the complex points Y(C), and we can also reduce modulo p for any prime p∤N. This gives a variety defined over Fp, and for any positive power q=pd we can consider both the Fq-points and the F¯q-points of Y, where F¯q is the algebraic closure of Fq. One of the most fundamental arithmetic invariants attached to Y is its ℓ-adic étale cohomology Het*(Y/K¯;Qℓ), where K is a number field or finite field of characteristic prime p∤N, and where ℓ≠p is prime. The Galois action on Y/K¯ induces a Galois action on each Qℓ-vector space Heti(Y/K¯;Qℓ), and this action is a crucial part of the data. Now let Xn be a sequence of schemes that are smooth and quasi-projective over Z[1/N] for some N, for example Xn=PConfn(Y) or Xn=UConfn(Y) for Y smooth over Z[1/N]. Given the usefulness of homological stability in topology, one wants to prove such stability for Heti(Xn/K¯;Qℓ) for K a number field or a finite field with characteristic p∤N. There are a number of different notions of what ‘stability’ means in this context (see Section 2 below). We adopt the strongest of these possibilities. Definition 1.1 (Étale homological stability). We say that a sequence Xn of schemes satisfies étale homological stability over a field K if for each i≥0, there exists D=D(i) so that the isomorphism type of Heti(Xn/K¯;Qℓ) as a Gal(K¯/K)-representation does not depend on n for n≥D. The function D(i) is called the stable range. When each Xn in addition admits an Sn-action, such as Xn=PConfn(Y) or Yn, we have a corresponding notion of étale representation stability over K. This definition is a bit more involved; see Section 2.2. It implies étale homological stability for the sequence of varieties Xn/Sn. To state our results in this direction, we need two different descriptions of representations of the symmetric groups Sn. Let Xi be the class function on all symmetric groups Sn,n≥1 given by setting Xi(σ) to be the number of i-cycles in the cycle decomposition of σ. A character polynomial is any polynomial P∈Q[X1,X2,…]; it is a class function on each Sn,n≥1. The degree of a character polynomial is defined by setting degXi≔i. See Section 2.1 below for more details. As shown by Church–Ellenberg–Farb [3], character polynomials give a compact and uniform way of describing the characters of certain infinite sequences of Sn-representations for n=1,2,…. A partition of n is a sequence λ=(λ1≥⋯≥λr≥0) with ∑iλi=n. The irreducible representations V(λ) of Sn are classified by partitions λ⊢n. A partition λ⊢k gives a sequence V(λ)n of irreducible Sn-representations for n≥k+λ1 by defining V(λ)n to be the irreducible representation of Sn corresponding to the partition (n−k,λ1,…,λr). Every irreducible representation of Sn is of the form V(λ)n for a unique partition λ; for example the trivial and standard representations of Sn are V(0)n and V(1)n, respectively. Similarly, every character polynomial is a linear combination of a finite number of class functions of the form χV(λ). Theorem B (Étale representation stability). Let Ybe a scheme, smooth over Z[1/N]for some N, with geometrically connected fibers of finite type. Let Kbe either a number field or a finite field of characteristic p∤N. For each i≥0, the sequence Heti(PConfn(Y)/K¯;Qℓ)of Gal(K¯/K)-modules satisfies étale representation stability (see Section2.2for the precise definition of étale representation stability) over Kwith stable range D(i)=2ifor dimY≥2and D(i)=4ifor dimY=1. In particular: 1. Inductive description: For all n≥0, there is an isomorphism of Gal(K¯/K)-representations: Heti(PConfn(Y)/K¯;Qℓ)≅colimSHeti(PConf∣S∣(Y)/K¯;Qℓ), (1.1)where the colimit is taken over the poset of all subsets S⊂{1,…,n}such that ∣S∣≤D(i). This gives, for each n≥D(i), a recipe for building the Gal(K¯/K)-representation Heti(PConfn(Y)/K¯;Qℓ)from a fixed finite collection of Gal(K¯/K)-representations. 2. Stability of isotypics: For each character polynomial P, there exists a unique virtual Gal(K¯/K)-representation Heti(PConf(Y))Pover Qℓ, linear in P, so that when P=χV(λ)for some λ⊢k, there exists Dsuch that for all n≥D: Heti(PConf(Y))χV(λ)=Heti(PConfn(Y)/K¯;Qℓ)⊗Qℓ[Sn]V(λ)nand the right-hand side is independent of nas a Gal(K¯/K)-representation. 3. Polynomial characters: There exists a character polynomial Q(X1,…,Xr)so that for all n≥D(i): χHeti(PConfn(Y)/K¯;Qℓ)(σ)=Q(X1(σ),…,Xr(σ))forallσ∈Sn.,where deg(Q)≤iif dimY>1and deg(Q)≤2iif dimY=1. Remarks Theorems A and B in the special case Y=A1 was proved by Church–Ellenberg–Farb [4]. This special case is much simpler since the eigenvalues of Frobq on Heti(PConfn(A1)/F¯q;Qℓ) are known explicitly; they equal qi. Our proof of Theorem B uses the theory of FI-modules, developed in [3]. What we actually prove is that for each i≥0, the FI- Gal(K¯/K)-module (see Section 2.2 below for the precise definition) Heti(PConf•(Y);Qℓ) is finitely generated. Items (1–3) of Theorem B then follow from the general theory of FI-modules, in particular theorems from [3, 5]. As one consequence, the proof shows that Item (1) of Theorem B holds with Qℓ replaced by Zℓ or Z/ℓnZ. Nir Gadish [17] has recently isolated a concept of finitely generated I-poset, for a wide class of categories I, and has used this to prove étale representation stability for a rich class of sequences of complements of linear subspace arrangements. Plugging in P=1 into Item (2) of Theorem B gives the following. Corollary B′ (Étale homological stability). With terminology as in TheoremB, the sequence Heti(UConfn(Y)/K¯;Qℓ)satisfies étale homological stability over K: these Gal(K¯/K)-representations do not depend on nfor n≥D(i). Remarks Quoc Ho [18] has recently given an independent proof of Corollary B′ for Y smooth over any ground field. His method is based on factorization homology, and is quite different from the methods of this paper. Dan Petersen [24] has recently extended Theorem B to a wider class of configuration-like spaces, also dropping the smoothness assumption. Stability of arithmetic statistics: The application of homological stability to arithmetic statistics was pioneered by Ellenberg–Venkatesh–Westerland in [13]. The fundamental link is provided by the Grothendieck–Lefschetz Trace Formula (Here we have assumed that Z is smooth and applied Poincaré Duality to the usual Grothendieck–Lefschetz Formula.): ∣Z(Fq)∣=qdim(Z)∑i≥0(−1)iTr(Frobq:Heti(Z/F¯q;Qℓ)*→Heti(Z/F¯q;Qℓ)*) (1.2) and its twisted version (see (4.1) below). Given Deligne’s theorem [10, Theorem 1.6] that any eigenvalue λ of Frobq on Heti(Z/F¯q;Qℓ)* satisfies ∣λ∣≤q−i/2, one can bound the number ∣Z(Fq)∣ of Fq-points via ∣Z(Fq)∣≤qdimZ∑i=02dimZbiq−i/2, where bi≔dimHeti(Z/F¯q;Qℓ)*. Applying this reasoning to a sequence Zn of smooth varieties gives q−dimZn∣Zn(Fq)∣≤∑i=02dimZnbi(n)q−i/2, (1.3) where we have emphasized via notation that bi is a function of n. It seems that étale homological stability, namely the fact that bi(n) is constant for n≥D(i), should imply that the limit as n→∞ of the left-hand side of (1.3) exists. However, it could be that dim(Zn) goes to ∞ with n and that bi(n) grows more quickly than qi/2, even for any q; this would imply the divergence of the right-hand side of (1.3). This super-exponential growth is known to occur in natural examples, for example for Zn the moduli space of genus n smooth algebraic curves, and also for Zn the moduli space of n-dimensional principally polarized abelian varieties. In the latter example, recent work of Lipnowski–Tsimerman [20] shows that this growth actually does change the point count ∣Zn(Fq)∣, as they show that this number grows more quickly than the expected qdimZn. Thus, in order to apply étale homological stability to obtain the existence of asymptotic point counts in a given example, it is necessary to prove subexponential (in i) bounds on bi(n), independent of n. In other words, control of the unstable étale cohomology Heti(Zn/F¯q;Qℓ)* is needed. Proving such bounds is a major obstruction for arithmetic applications; see Section 3 for a discussion. This problem is a very special case (namely the case P≡1) of more general arithmetic statistics, where one needs a twisted version of the Grothendieck–Lefschetz formula, and where the control on the ‘representation unstable cohomology’ is even more difficult to prove; see Section 3 below. A significant part of this paper, Section 3, is devoted to overcoming this problem for the examples Xn and PConfn(X). Theorem C (Bounding the representation unstable cohomology). Let Xbe either a smooth, orientable manifold with dim(H*(X;Q))<∞ (for example, Xcompact), or a scheme, smooth over Z[1/N]for some N, with geometrically connected fibers of finite type. In the first case let Hidenote singular cohomology with Qcoefficients; in the second, let Hidenote étale cohomology with Qℓcoefficients. Then for any character polynomial P, there exists a function FP(i), subexponential in i, such that for all n≥1: ⟨P,(Hi(PConfn(X)))⟩Sn≤FP(i). In Section 4, we apply Theorems B and C to obtain Theorem A. A different description of the left-hand side of Theorem C, established using analytic methods, will appear in forthcoming work of Chen [7]. En route to proving Theorems C and A, we also prove the analogous statements for Symn(X); see Section 4. However, we note that Theorem C requires more than just bounding the betti numbers Symn(X). 2. Étale representation stability In this section, we briefly summarize the theory of representation stability and FI-modules, as it is used in topology, as well as some of its consequences. This theory was developed by Church–Ellenberg–Farb [3, 6], and later with Nagpal [5]; see [14] for a survey. We refer the reader to these references for details. We then give a general setup for proving similar stability theorems in étale cohomology. 2.1. Quick summary of representation stability and FI-modules An FI-module V over a Noetherian ring R is a functor from the category FI of finite sets and injections to the category of R-modules. Thus, to each natural number n, we have associated an R-module Vn with an Sn action, with a map Vm→Vn for each injection {1,…,m}→{1,…,n}. Recall that the opposite category FIop is the same as FI but with arrows reversed. A co-FI module over R is a functor from FIop to R-modules. We also have the associated notions of FI-space, FI-scheme, etc., and the associated co-FI versions. An FI-module V is finitely generated if there is a finite set S of elements in ∐iVi so that no proper sub-FI-module of V contains S. One of the reasons that we care about finitely generated FI-modules is the following theorem. Theorem 2.1 (Structural properties of finitely generated FI-modules). Let Vbe an FI-module over a commutative Noetherian ring R. If Vis finitely generated then: Representation stability [3]: When Ris a field of characteristic 0, finite generation of Vimplies representation stability in the sense of [6] for the sequence {Vn}of Sn-representations. Inductive description [5]: Let V•be a finitely generated FI-module over a Noetherian ring R. Then there exists some N≥0such that for all n∈N, there is a natural isomorphism Vn≅colimS⊂[n],∣S∣≤NVS,that is these isomorphisms commute with homomorphisms of FI-modules. By definition, the stable range of V• N(V)is the minimal such N. Isomorphism of trivial isotypics [8]: Let V•be a finitely generated FI-module over a Noetherian ring Rwith stable range N(V). Then for all n≥N(V), the map Vn→Vn+1, given by averaging the structure maps, induces an isomorphism VnSn→≅Vn+1Sn+1. We remark that the isomorphism of trivial isotypics illustrates one of the key advantages of considering (co-)FI-spaces: while stabilization maps for many natural sequences of spaces or schemes do not naively exist, a (co-)FI-space Z• comes equipped with canonical rational correspondences from Zn+1/Sn+1 to Zn/Sn. We will also need the following. Lemma 2.2 Let V•and W•be finitely generated FI-modules, and let Nbe the sum of their stable ranges. Then for all n≥N, the maps Vn→Vn+1and Wn→Wn+1 (associated to {1,…,n}⊂{1,…,n+1}) induce isomorphisms Vn⊗Q[Sn]Wn→≅Vn+1⊗Q[Sn+1]Wn+1that are natural in both variables with respect to homomorphisms of FI-modules. Proof By [3, Proposition 2.3.6], the tensor product V•⊗QW• is finitely generated since V• and W• are. Applying the co-invariants functor, we obtain the functor n↦Vn⊗Q[Sn]Wn. Because stable ranges add under tensor product [3, Proposition 2.3.6], and because the stability degree (cf. [3, Definition 3.1.3]) is less than or equal to the stable range [3, Proposition 3.3.3], the map Vn⊗Q[Sn]Wn→Vn+1⊗Q[Sn+1]Wn+1 is an isomorphism for n≥N.□ Character polynomials: Character polynomials and their degree were defined in the introduction. Let ⟨P,Q⟩ denote the inner product of Sn-characters. The expectations of character polynomials Eσ∈SnPn(σ)≔1n!∑σ∈SnPn(σ)=⟨Pn,1⟩ compute the averages of natural combinatorial statistics with respect to the uniform distribution on Sn. As shown in [4, Proposition 2.2], the inner product ⟨Pn,Qn⟩ of character polynomials P,Q∈Q[X1,X2,…] is independent of n once n≥degP+degQ. One remarkable property of finitely generated FI-modules V is that the characters of the Sn-representations Vn are, for large enough n, given by a single polynomial. Theorem 2.3 (Polynomiality of characters [5]). Let Vbe an FI-module over a field of characteristic 0. If Vis finitely generated then the characters χVnof the Sn-representations Vnare eventually polynomial: there exists N≥0and a polynomial P(X1,…,Xr), for some r>0, so that χVn=P(X1,…,Xr)foralln≥N. (2.1) In particular, if Qis any character polynomial, then ⟨χVn,Q⟩is independent of n≥degP+degQ. We note that evaluating (2.1) on the identity permutation gives a polynomial P(T)∈Q[T] so that dimkVn=P(n) for all n≥N. Étale Representation Stability: Given a co-FI-scheme Z• defined over Fq, its étale cohomology Heti(Z•/F¯q;Qℓ) has additional structure beyond that of an FI-module over Qℓ. The geometric Frobenius Frobq gives a natural endomorphism of Z•/F¯q, and this gives rise to an action of Gal(F¯q/Fq) on the FI-module Heti(Z•/F¯q;Qℓ). As noted in the introduction, the eigenvalues of Frobq and the action of Gal(F¯q/Fq) are crucial parts (as observed for example by Milne [22], the Tate conjecture implies that the eigenvalues of Frobq determine the Gal(F¯q/Fq)-action. But, this is not known at present) of the data here. Weaker than knowing an eigenvalue λ of Frobq on Hetj(Zn/F¯q;Qℓ) is knowing its weight. Deligne proved that λ is an algebraic number with ∣λ∣=qr/2 for some j≤r≤2j, with r=j if Zn is smooth and proper. The number r is the weight of the eigenvalue λ. Similarly, for Z• defined over a number field K, the action of Gal(K¯/K) on Z•/K¯ induces an action on Heti(Z•/K¯;Qℓ), and this action is a fundamental part of the data. In increasing order of strength, we could ask that for each i there exists D so that for all n≥D: The isomorphism type of Heti(Zn/K¯;Qℓ) as a Qℓ-vector space does not depend on n. In addition, the list of weights of Frobq on Heti(Zn/K¯;Qℓ) does not depend on n. In addition, the list of eigenvalues of Frobq on Heti(Zn/K¯;Qℓ) does not depend on n. The isomorphism type of Heti(Zn/K¯;Qℓ) as a Gal(K¯/K)-representation does not depend on n. We have adopted the strongest of these as our definition of étale homological stability. (For applications to counting problems, for example Theorem A, only the stability of the eigenvalues is needed.) 2.2. Étale representation stability Let Z be a co-FI scheme smooth over Z[1/N] for some fixed N, with geometrically connected fibers. Let p∤N be prime, and let ℓ≠p be a prime. For each i≥0, the étale cohomology Heti(Z/F¯p;Zℓ) is an FI-module. In addition, for each q=pd, the Frobenius Frobq acts on each Heti(Zn/F¯p;Zℓ), endowing it with the structure of a Gal(F¯p/Fq)-module. The Sn-action on Heti(Zn/F¯p;Zℓ) coming from its structure as an FI-module commutes with the action of Gal(F¯p/Fq), as do all automorphisms of Zn. Similarly, for any number field K, the action of Gal(K¯/K) on Heti(Z/K¯;Qℓ) commutes with the FI-structure. This discussion shows that, for K a number field or a finite field of characteristic prime to N, Heti(Z/K¯;Qℓ) is an FI- Gal(K¯/K)-module; that is, an FI-module equipped with an action of Gal(K¯/K) by FI-automorphisms. We have the corresponding notions of finitely generated FI- Gal(K¯/K)-module: there is a finite set S⊂∐nHeti(Zn/K¯;Qℓ) so that no proper sub-FI-module of Heti(Z/K¯;Qℓ) contains S. Definition 2.4 (Étale representation stability). We say that a sequence Zn of Gal(K¯/K)-modules satisfies étale representation stability if {1,…,n}↦Zn is a finitely generated FI- Gal(K¯/K)-module. Theorem C in [5] gives an inductive description of finitely generated FI-modules V over any Noetherian ring R. Namely, there exists D≥0 such that for all n∈N, there is a natural isomorphism Vn≅colimSVS, (2.2) where the colimit is taken over the poset of all subsets S⊂{1,…,n} such that ∣S∣≤D. If V is a finitely generated FI- Gal(K¯/K)-module then (2.2) gives is an isomorphism of Gal(K¯/K)-modules. Thus (2.2) gives, for each n≥D, a recipe for building the Gal(K¯/K)-representation Vn from a fixed finite collection of Gal(K¯/K)-representations. Étale representation stability for products and configuration spaces: Attached to any scheme X there is an associated configuration space PConfn(X) of ordered n-tuples in X, defined by PConfn(X)≔{(x1,…,xn)∈Xn:xi≠xj∀i≠j}=Xn⧹Δ, where Δ is the fat diagonal and where we write x∈X to denote an arbitrary R-point of X. The group Sn acts freely on PConfn by permuting the coordinates. The quotient UConfn(X)≔PConfn(X)/Sn is the configuration space of unordered n-tuples of points in X. For any scheme X, denote by X• the co-FI scheme that sends {1,…,n} to the cartesian product Xn, and associates to any injection between finite sets the natural projection maps between cartesian powers. The open subsets PConfn(X)⊂Xn are preserved under the co-FI structure maps, and so we obtain a co-FI scheme PConf•(X). In particular, Hi(PConf•(X);Qℓ) is an FI-module for any i≥0. With this setup, we can now prove Theorem B from the introduction. The proof also gives the following result. Theorem 2.5 TheoremBwith PConfn(Y) (resp. UConfn(Y)) replaced by Yn (resp. Symn(Y)) holds. Proof of Theorem B and Theorem 2.5 This theorem follows from the proofs of [3, Theorem 6.1.2] and [3, Theorem 6.2.1]. The only difference is that we now work with étale rather than Betti cohomology. To wit, the co- FI-schemes PConf•(Y) and Y• give rise to FI- Gal(K¯/K)-modules by taking ℓ-adic cohomology. The proof of [3, Theorem 6.1.2] carries over verbatim to the étale setting to show that Heti(Y•;Qℓ) is a finitely generated FI- Gal(K¯/K)-module for all i≥0; see [3, Theorem 4.1.7 and Remark 6.1.3] for the stable range and degree of the character polynomial. Note here that this improved stable range comes from the fact that X• is a so-called FI#-module. For PConf•(Y), the proof of [3, Theorem 6.2.1] applies in étale cohomology just as for singular cohomology. Indeed, as Totaro discusses in [25, p. 1064], for any scheme Y smooth over Z[1/N], the Leray spectral sequence in ℓ-adic cohomology for the inclusion PConfn(Y)↪Yn has E2-page isomorphic to E2p,q=⨁{J⊣n∣∣J∣=n−q/(2dim(Y)−1)}Hetp(YJ;QℓcJ(−qdim(Y)2dim(Y)−1)), where YJ⊂Yn denotes the diagonal where points coincide according to the partition J, dim(Y) denotes the dimension of the scheme, and if J consists of pieces of size j1,…,jn−q/(2dim(Y)−1), then cJ≔(j1−1)!⋯(jn−q/(2dim(Y)−1)−1)!. See also [27] for a detailed proof of this description of the Leray spectral sequence in the étale setting. The proof of [3, Theorem 6.2.1] now carries over verbatim to show that for all i, the FI- Gal(K¯/K)-module Heti(PConf•(Y);Qℓ) is finitely generated for all i≥0. For the claimed stable ranges and degree of character polynomial, see [3, Theorem 6.3.1].□ 3. Convergent cohomology In this section, we provide the necessary bounds for the ‘representation unstable cohomology’ of Xn and of PConfn(X) that will be necessary for the arithmetic applications in Section 4. 3.1. Definition of convergent cohomology A function F:N→Nhas exponential growth rate λ if limn→∞logf(n)n=λ. (3.1) If (3.1) holds with λ=0, we say that F has subexponential growth. Let Z be a co-FI-scheme over Z[1/N]. For each i≥0, let Hi(Zn) denote either the singular cohomology Hi(Zn(C);Q) or the étale cohomology Heti(Zn/K¯;Qℓ) for K a number field or finite field of characteristic prime to N. In each case, Hi(Z•) is an FI-module (over Q and Qℓ, respectively). For any class function P on Sn, denote by ⟨P,Hi(Zn)⟩, the inner product of (the character of) Hi(Zn) with P. In order to compute arithmetic statistics for a co-FI scheme Z, one needs to control the ‘representation unstable’ cohomology of Z; see Section 4. More precisely, one needs to prove one of the following two properties, which were shown to be equivalent in [4, Section 3]: For each 0≤a≤n there is a function Fa(i), subexponential in i and not depending on n, so that dimHi(Zn)Sn−a≤Fa(i)forallnandi. (3.2) For each character polynomial P∈Q[X1,X2,…], there exists a function FP(i), subexponential in i and not depending on n, such that ∣⟨P,Hi(Zn)⟩∣≤FP(i)forallnandi. (3.3) It is crucial that these bounds hold independently of n. While the second condition is the one that applies to arithmetic statistics (see Section 4 below), it is quite difficult to check. Thus the equivalence with the first condition is quite useful. Definition 3.1 (Convergent cohomology). We say that the co-FI scheme (or space) Z has convergent (singular or étale) cohomology if either of the two equivalent properties 1 or 2 in equations (3.2) or (3.3) holds. If these properties hold with FP(i) having exponential growth 0<λ<∞, we say that Z has weakly convergent cohomology with convergence rate λ. These kinds of bounds are typically not easy to prove. In [13], this is accomplished ([13] only needs to deal with the classical, not representation stable, case; that is, the a=0 case) for the cohomology of certain Hurwitz spaces by obtaining an exponential upper bound for the number of i-cells, via an explicit cell decomposition. In [4], such bounds for the example Hi(PConfn(C);Q) are obtained by a detailed knowledge of these Sn-representations. The rest of this section is devoted to giving such bounds for two natural classes of co-FI schemes. We then apply this in Section 4 to arithmetic statistics for Fq-points on these schemes. 3.2. Polynomial bounds on Betti numbers of symmetric products Let X be a topological space. The n-fold cartesian product Xn is endowed with a natural action of the symmetric group Sn, given by permuting the factors. The quotient SymnX≔Xn/Sn is called the the nthsymmetric product of X. Proposition 3.2 (Growth of Betti numbers of symmetric products). Let Xbe either a space with dim(H*(X;Q))<∞ (for example, Xcompact), or a scheme over Z[1/N]for some N, with geometrically connected fibers of finite type. In the first case let Hidenote singular cohomology with Qcoefficients; in the second, let Hidenote étale cohomology with Qℓcoefficients. In either case, let bibe the associated ithBetti number. Then bi(Symn(X))is bounded above by a polynomial in i, independent of n. Proof We claim that if m<n then bi(Symm(X))≤bi(Symn(X)); further, bi(Symn(X))=bi(Symi(X)) for all i≥n. To see the statement for m<n, observe that for any graded vector space V (over a field of characteristic 0), a choice of ‘unit’ 1∈V0, determines an injection Symm(V)i→Symn(V)iv⃗↦v⃗⊗1⊗⋯⊗1. In particular, dim(Symm(V)i)≤dim(Symn(V)i) for all m<n and i. Considering V=H*(X;Q), Künneth and transfer imply that Symn(H*(X;Q))i≅Hi(Symn(X);Q) and the first part of the claim follows. For the second, we note that for a graded vector space V=V0⊕⋯⊕Vm, with V0=Q, we have Symn(V)≅⨁a0+⋯am=n⊗j=1mSymaj(Vj), and thus Symn(V)i≅⨁(a1,…,am)⊗j=1mSymaj(Vj), where the direct sum is over partitions a1+2a2+⋯mam=i such that a1+⋯+am=n. In particular, the number of pieces in the partition is at most n, and since for any i, the largest number of pieces in any partition is i, we see that for n≥i, the direct sum is independent of n. We conclude the claim by taking V=H*(X;Q) and invoking Künneth and transfer as above. We have just shown that for each fixed i≥0 bi(Symn(X))≤bi(Symi(X))=bi(Sym∞(X))foralln≥1. To prove the proposition, it is therefore enough to bound bi(Sym∞(X)) by a polynomial in i. Well, note that the function f(z)≔∑i=0∞bi(Sym∞(X))zi is just the Poincaré series for the symmetric algebra on the vector space H*(X;Q). By an elementary argument, this Poincaré series is, writing Bi≔bi(X), the following rational function: f(z)=(1+z)B1(1+z)B3…(1−z2)B2(1−z4)B4…. Since each pole of f(z) lies on the unit circle, it follows (see, for example [16], Theorem IV.9) that the ith coefficient bi(Sym∞(X)) of f(z) is bounded above by a polynomial in i.□ Consequence: bounding the representation unstable cohomology of products. The following corollary is also a key ingredient in bounding the representation unstable cohomology of configuration spaces. Corollary 3.3 Let Xbe as in Proposition3.2, and use the notation of that theorem. For each 0≤a≤n, we have that dim(Hi(Xn)Sn−a)is bounded above by a polynomial in i, independent of n. Proof Since the action Sn−a leaves invariant the first n−a factors of Xn and acts as the identity on the last a factors, there is, for each i≥0, an isomorphism: Hi(Xn)Sn−a=⨁p+q=iHp(Xn−a)Sn−a⊗Hq(Xa). (3.4) Since this sum has i+1 terms, it suffices to bound the dimension of each summand by a polynomial in i. Since dimH*(X;Q)<∞ and since a is fixed, there is a constant C, not depending on q, so that dim(Hq(Xa))≤C. It thus suffices to bound each Hp(Xn−a)Sn−a by a polynomial in i. But this follows from transfer together with Proposition 3.2, noting that p≤i.□ 3.3. Bounding the representation unstable cohomology of configuration spaces We build on the subexponential upper bounds for products in the last section to prove the corresponding result for configuration spaces. Theorem 3.4 ( PConf•(X) has convergent cohomology). Let Xbe either a smooth, orientable manifold with dim(H*(X;Q))<∞ (for example, Xcompact), or a scheme, smooth over Z[1/N]for some N, with geometrically connected fibers of finite type. Then the co-FI manifold (resp. scheme) PConf•(X)has convergent singular (resp. étale) cohomology. Proof For the case, when X is a manifold, we let Hi denote singular cohomology with Q coefficients; in the case, when X is a scheme, we let Hi denote étale cohomology with Qℓ coefficients. Fix a≥0. Denote by Sn−a the subgroup Sn−a×1⊂Sn. We will prove that there is a function Fa(i), subexponential in i, so that dim(Hi(PConfn(X);Q))Sn−a≤Fa(i), for all i≥0. Let m be the real dimension of X, and denote by A(n,m) the graded commutative algebra A(n,m)≔Q[{Gab}1≤a≠b≤n]/I, where ∣Gab∣=2m−1 and I is the ideal generated by the elements Gab−GbaGabGac+GbcGba+GcaGcb, for a<b<c distinct. The group Sn acts on A(n,m) via σ·Gab≔Gσ(a)σ(b). Totaro [25, Theorem 4] has shown that H*(PConfn(X);Q) is isomorphic, as a graded Sn-representation, to a sub-quotient of H*(Xn;Q)⊗A(n,m), with the natural action on each factor. As Totaro indicates, this result holds for both the singular and étale cohomology. Note that, for any short exact sequence of Sn-representations 0→V0→V1→V2→0 over a field of characteristic 0, there exists an Sn-equivariant splitting V1≅V0⊕V2. In particular, dimV1Sn=dimV0Sn+dimV2Sn, and, more generally, if V is any sub-quotient of an Sn-representation W, we have dimVSn≤dimWSn. Let V and W be any two Sn-representations. The identity dim(VSn)=⟨χV,1⟩ and the Cauchy–Schwarz inequality give dim(V⊗W)Sn=⟨χV⊗W,1⟩=1n!∑σ∈SnχV(σ)χW(σ)≤1n!(∑σ∈SnχV(σ)2)(∑σ∈SnχW(σ)2)=⟨χV⊗2,1⟩⟨χW⊗2,1⟩=dim((V⊗2)Sn)dim((W⊗2)Sn). Specializing to our setting, we conclude that it suffices to show that (dim(Hi(X(C)n;Q)⊗2)Sn−a)·(dim(A(n,m)⊗2)Sn−a)≤Fa(i) for some Fa(i) subexponential in i. For the first factor, by Künneth and the definition of the action, we have (Hi(Xn;Q)⊗2)Sn−a⊂H2i(Xn×Xn;Q)Sn−a≅H2i((X×X)n;Q)Sn−a≅⨁j=02iHj((X×X)n−a;Q)Sn−a⊗H2i−j((X×X)a;Q). By transfer, this is isomorphic to ⨁j=02iHj(Symn−a(X×X);Q)⊗H2i−j((X×X)a;Q). Let C=maxibi(X×X), and let D=C·2m. By Künneth, for all j<2i, dimH2i−j(X2a;Q)≤(2i−j)Ca. Combining this with Proposition 3.2, we see that dim(Hi(Xn;Q)⊗2)Sn−a≤∑j=02idim(Hj(Symn−a(X×X);Q))(2i−j)Ca≤∑j=02i2DD!P(j)(2i−j)Ca≤2DD!CaQ(i) for some polynomials P,Q. It remains to bound dim(A(n,m)⊗A(n,m))iSn−a. Well, (A(n,m)⊗A(n,m))i=⨁p+q=i(A(n,m)p⊗A(n,m)q). (3.5) Since the right-hand side of (3.5) has at most 2i terms, it suffices to bound each [A(n,m)p⊗A(n,m)q]Sn−a. By the Cauchy–Schwartz inequality, as above, it suffices to bound [A(n,m)p⊗A(n,m)p]Sn−a for each 1≤p≤i. To obtain this bound, first note that the algebra A(n,m) is isomorphic to A(n,2) via an isomorphism that takes the pth graded piece of A(n,2) to the (2m−1)pth graded piece of A(n,m). Since m is fixed and so 2m−1 is fixed, it suffices to bound [A(n,2)p⊗A(n,2)p]Sn−a in terms of i, for each 1≤p≤i. Lehrer–Solomon [19] give an explicit description of A(n,2) as a sum of induced representations A(n,2)p=⨁μIndZ(cμ)Sn(ξμ), where μ runs over the set of conjugacy classes in Sn of permutations having n−p cycles, cμ is any element of the conjugacy class μ, and ξμ is a one-dimensional character of the centralizer Z(cμ) of cμ in Sn (we will not need an explicit description of ξμ). It follows that (A(n,2)p⊗A(n,2)p)Sn−a=⨁μ,ν[IndZ(cμ)Sn(ξμ)⊗IndZ(cν)Sn(ξν)]Sn−a, (3.6) where ν is defined similarly to μ. The summands contributing to the first (resp. second) A(n,2)p factor in (3.6) correspond to conjugacy classes cμ (resp. cν) in Sn decomposing into n−p cycles. The number of such conjugacy classes is in bijection with the set of partitions of p, which is less than the number of partitions of i since p≤i. The Hardy–Ramanujan asymptotic for the number ∣{J⊢i}∣ of partitions of i gives C1,C2>0 so that ∣{J⊢i}∣≤C1eC2i. (3.7) Thus the number of terms in the sum on the right-hand side of (3.6) is, by (3.7), at most [C1eC2i]2=C12e2C2i. As this is subexponential in i, it suffices to bound the dimension of [IndZ(cμ)Sn(ξμ)⊗IndZ(cν)Sn(ξν)]Sn−a. Now, a permutation cμ decomposing into n−p cycles must have at least n−2p fixed points. This implies that the centralizer Z(cμ) contains the subgroup Sn−2p, and thus Sn−2i since p≤i. It follows that IndZ(cμ)Sn(ξμ) is a subrepresentation of IndSn−2iSn(ξμ). Thus, [IndZ(cμ)Sn(ξμ)⊗IndZ(cν)Sn(ξν)]Sn−a⊂[IndSn−2iSn(ξμ)⊗IndSn−2iSn(ξν)]Sn−a. (3.8) Let χμ and χν denote the characters of ξμ and ξν, respectively. The right-hand side of (3.8) consists of the set of bilinear functions f:Sn×Sn→C, satisfying f(σ·g,τ·h)=χμ(σ)χν(τ)f(g,h)∀σ,τ∈Sn−2iand∀g,h∈Sn and f(g·β,h·β)=f(g,h)∀β∈Sn−aand∀g,h∈Sn. It follows that the dimension of this vector space is at most the number of double cosets Sn−a⧹[Sn/Sn−2i×Sn/Sn−2i]. We claim that this number is polynomial in i. Indeed, it is equal to the number of maps f:{1,…,a}→{1,…,2i,⋆}×{1,…,2i,⋆} such that ∣f−1(j,k)∣≤1 and ∣f−1(j,⋆)∣,∣f−1(⋆,k)∣≤(n−2i)2. Since a is fixed, this number is bounded by a constant times the number of subsets of {1,…,2i,⋆}×{1,…,2i,⋆} of size ≤a, which is O(i2a). This completes the proof of Theorem 3.4.□ 4. Stability of arithmetic statistics Throughout this section, we will fix a prime power q=pd and a prime ℓ not divisible by p. 4.1. Point counting and étale cohomology Let Y be a scheme of finite type (not necessarily smooth) over Z[1/N]. We can base change to Spec(Fp) for any prime p∤N, and for any positive power q=pd we can consider both the Fq-points as well as the F¯q-points of Y, where F¯q is the algebraic closure of Fq. The arithmetic Frobenius morphism Frobq:Y→Y acts on Y(F¯q) by acting on the coordinates (y1,…,yd) of any affine chart of y via Frobq(y1,…,yd)≔(y1q,…,ydq). A point y∈Y(F¯q) will be fixed by Frobq precisely when y∈Y(Fq). Thus Y(Fq)=Fix(Frobq:Y(F¯q)→Y(F¯q)). Fix a prime ℓ not dividing q, and let Qℓ denote the ℓ-adic rationals. Let Het*(Y/F¯q;Qℓ) (resp. Het,c*(Y/F¯q;Qℓ)) denote the étale cohomology groups (resp. compactly supported étale cohomology groups) of the base change Y/F¯q of Y to F¯q (see, for example, [11, 21]). Denote by Qℓ(−i) the rank 1 Gal(F¯q/Fq)-representation on which Frobenius acts by qi. Let V be a constructible, rational ℓ-adic sheaf on Y (see, for example, [15]). If y∈Y(F¯q) is a fixed point for the action of Frobq, then Frobq acts on the stalk Vy over y. Attached to this action is its trace Tr(Frobq:Vy→Vy). The twisted Grothendieck–Lefschetz Trace Formula [15, Theorem II.3.14] and [11, 6.1.1.1] gives ∑y∈Y(Fq)Tr(Frobq:Vy→Vy)=∑i=02dim(Y)(−1)iTr(Frobq:Het,ci(Y;V)→Het,ci(Y;V)). (4.1) When Y is smooth, Poincaré duality for étale cohomology [21, Theorem 24.1] gives Het,ci(Y/F¯q;V)≅Het2dim(Y)−i(Y/F¯q;V(−dim(Y)))*. (4.2) Plugging this into Equation (4.1) gives, for smooth Y ∑y∈Y(Fq)Tr(Frobq:Vy→Vy)=qdim(Y)∑i=02dim(Y)(−1)iTr(Frobq:Heti(Y;V)*→Heti(Y;V)*). (4.3) Sn-schemes: Now let Z be smooth and quasi-projective over Z[1/N]. Suppose that the symmetric group Sn acts generically freely on Z by automorphisms, and let p:Z→Y denote the quotient map. By [23, Theorem p. 63 and Remark p. 65 (Chapter 2.7)], Y is a scheme. It is typically not smooth even when Z is smooth: for example if Z=(A2)n with the standard Sn action, then Y=Symn(A2) is singular at the point {0,…,0}. Recall that any finite-dimensional representation of Sn over a field of characteristic 0 is defined over Q. There is a bijective correspondence between isomorphism classes of finite-dimensional Sn-representations and finite-dimensional constructible sheaves on Y that become isomorphic to Qℓ⊕r when pulled back to Z: given an Sn-representation V over Qℓ, one can form an Sn-equivariant, locally constant sheaf V over Z with fiber V. Pushing forward to Y and taking Sn invariants, that is (p*V)Sn, we obtain a constructible sheaf of Qℓ vector spaces over Y which is a sheaf-theoretic analogue of the usual topological diagonal quotient ‘ Z×SnV’. Suppose that y∈Y(F¯q) is fixed by Frobq. Then Frobq acts on the fiber p−1(y). Now Sn acts transitively on p−1(y) with some stabilizer H (not depending, up to conjugacy, on y˜∈p−1(y)), and so we can identify p−1(y) with Sn/H after picking a basepoint y˜∈p−1(y). The Frobq action on p−1(y) commutes with this Sn action, and so it is determined by its action on a single basepoint y˜, with stabilizer H. Now Frobq(H)=σyH for σy∈Sn. Following Gadish [17], for any Sn-representation V and any coset σH of Sn, we set χV(σH)≔1∣H∣∑h∈HχV(σh). With this notation we have Tr(Frobq:Vy→Vy)=χV(σyH), (4.4) which we denote simply by χV(Frobq;Vy); note that this definition is independent of our choice of basepoint y˜. More generally: Definition 4.1 For any class function P, and any y∈Y fixed by Frobq, define P(y) by P(y)≔1∣H∣∑h∈HP(σyh). (4.5) An elementary check shows that the definitions above are independent of the choice of coset H, since the action of Sn is transitive on fibers. Plugging Equation (4.4) into Equation (4.1) now gives ∑y∈Y(Fq)χV(Frobq;Vy)=∑i=02dim(Y)(−1)iTr(Frobq:Het,ci(Y;V)→Het,ci(Y;V)). (4.6) The right-hand side of (4.6) could be computed from the eigenvalues λij of Frobq on each Het,ci(Y;Qℓ). Typically one only has estimates on ∣λij∣. For example, for Y smooth and proper, the Riemann Hypothesis for finite fields (proved by Deligne) gives that ∣λij∣=qi/2. Many natural examples Y, including many of those we study in this paper, are not proper, and finding the λij is more difficult. Given that we only have general bounds on the eigenvalues of Frobq, to bound the traces of Frobq we must determine the dimensions of each Het,ci(Y;V). To do this, we follow the argument in [4, Section 3.3]. First note that the pullback V˜ of V to Z is trivial. We then compute Het,ci(Y;V)≅Het,ci(Z;V˜)Snbytransfer≅(Het,ci(Z;Qℓ)⊗V)SnbytrivialityofV˜∣Z≅(Het2dim(Z)−i(Z;Qℓ(dim(Z)))*⊗V)SnbyPoincarèduality≅Het2dim(Z)−i(Z;Qℓ(dim(Z)))*⊗Qℓ[Sn]V. (4.7) Because every Sn-representation is self-dual, it follows that dimQℓHet,ci(Y;V)=⟨V,Het2dim(Z)−i(Z;Qℓ)⟩Sn, (4.8) where ⟨V,W⟩Sn is the usual inner product of Sn-representations V and W ⟨V,W⟩=dimQℓHomQℓ[Sn](V,W). 4.2. Co-FI schemes with convergent étale cohomology Now that we have discussed schemes, and Sn-schemes, we are ready to discuss sequences of Sn-schemes. Let Z be a co-FI scheme, smooth and quasi-projective over Z[1/N]. For each i≥0, the étale cohomology Heti(Z•/F¯q;Qℓ) is an FI-module over Qℓ. We want to consider the implications of finite generation of this FI-module for point-counting problems over Fq for the sequence of schemes Zn/Sn (cf. [23, Theorem p. 63 and Remark p. 65]). As discussed in [3], any partition λ of any k≥1 determines a finitely generated FI-module V(λ) with V(λ)n being the irreducible representation of Sn corresponding to the partition (n−∣λ∣)+λ. Definition 4.2 Let K be a field, and let M• be a finitely generated FI- Gal(K¯/K)-module over Qℓ with stable range N. Let λ be a partition of n, let V(λ) the associated FI-module, and let D=max{N,λ1}. Define the stable λ-isotypic part Mλof M to be the Gal(F¯q/Fq)-module Mλ≔MD⊗Qℓ[SD]V(λ)D. More generally, for a character polynomial P, we define the stable P-isotypic part of M to be the Qℓ-virtual Galois module MP obtained as a linear combination of the Mλ, with the sum taken in the representation ring of Gal(K¯/K) with Qℓ-coefficients. Lemma 2.2 shows that for n≥D, there are canonical Galois-equivariant isomorphisms Mn⊗Qℓ[Sn]V(λ)n→≅Mn+1⊗Qℓ[Sn+1]V(λ)n+1 and similarly for the stable P-isotypic parts for n≥D. We can now give the following theorem, which generalizes earlier special cases by Ellenberg–Venkatesh–Westerland [13], Ellenberg [12], and Church–Ellenberg–Farb [4]. Its proof is along the exact same lines of the previous proofs. We hope that the generality of the statement here will be useful in future work. Theorem 4.3 (Convergent Grothendieck–Lefschetz) Let Zbe a smooth, quasi-projective co-FI over Fq, and set Yn≔Zn/Sn (we do not assume Ynsmooth over Z[1/N]). Assume that for each i≥0the FI-module Heti(Zn/F¯q;Qℓ)is finitely generated, and, for a character polynomial P, denote by Heti(Z)P*the dual of the stable P-isotypic part. If Zhas convergent étale cohomology over F¯q, then for any character polynomial P: limn→∞q−dimYn∑y∈Yn(Fq)P(y)=∑i=0∞(−1)iTr(Frobq⥀Heti(Z)P*), (4.9)and, taking the absolute value limn→∞q−dimYn∣∑y∈Yn(Fq)P(y)∣≤∑i=0∞⟨P,Heti(Z)⟩qi/2<∞. (4.10) If Zonly has weakly convergent cohomology with convergence rate λ, then (4.9) and (4.10) hold for all q>λ. Remark 4.4 Specializing Theorem 4.3 to the case P=1 gives limn→∞q−dimYn∣Yn(Fq)∣=∑i=0∞(−1)iTr(Frobq⥀Heti(Y)*), (4.11) where Heti(Y)* denotes the stable rational étale cohomology of the sequence Y1,Y2,…. The bound (4.10) is sharp, as is seen by taking Zn=(P1)n, Yn=Pn, and P=1. Proof of Theorem 4.3 Because all of the equations in the statement of the theorem are Qℓ-linear in P, it suffices to prove the theorem for P=Pλ, the character polynomial of the finitely generated FI-module V(λ)(cf. Theorem 2.3 above). Let Vn correspond to the twisted sheaf on Yn corresponding to the representation V(λ)n. We show that the left side of (4.9) converges by showing that the sequences n↦q−dimYn∑y∈Yn(Fq)P(y) (4.12) is Cauchy. To start, note that ∑y∈Yn(Fq)P(y)=∑i=02dim(Yn)(−1)iTr(Frobq:Het,ci(Yn;Vn)→Het,ci(Yn;Vn))=∑i=02dim(Zn)(−1)iTr(Frobq⥀Het2dim(Zn)−i(Zn;Qℓ(dim(Zn)))*⊗Qℓ[Sn]V(λ)n)byEquation(4.7)=∑i=02dim(Zn)(−1)iTr(Frobq⥀Heti(Zn;Qℓ(dim(Zn)))*⊗Qℓ[Sn]V(λ)n*), (4.13) where the last equation uses the self-duality of Sn-representations. Denote by N(n,P) the slope of stability of Het*(Z•;Qℓ) for V(λ), that is the number such that for all i≤N(n,λ), Heti(Zn;Qℓ)⊗Qℓ[Sn]V(λ)n≅Heti(Z)P. Let FP(i) denote the subexponential function in Definition 3.1 guaranteed by the assumption that Z has convergent étale cohomology. Then, for n>m ∣q−dimYn(∑y∈Yn(Fq)P(y))−q−dimYm(∑y∈Ym(Fq)P(y))∣=∣∑i=02dim(Zn)(−1)iq−dim(Zn)Tr(Frobq⥀Heti(Zn;Qℓ(dim(Zn)))*⊗Qℓ[Sn]V(λ)n*)−∑i=02dim(Zm)(−1)iq−dim(Zm)Tr(Frobq⥀Heti(Zm;Qℓ(dim(Zm)))*⊗Qℓ[Sm]V(λ)m)∣byEquations(4.1)and(4.13)≤∑i=0∞q−i/2∣⟨P,Heti(Zn;Qℓ)⟩−⟨P,Heti(Zm;Qℓ)⟩∣byDeligne=∑i=N(m,P)∞q−i/2∣⟨P,Heti(Zn;Qℓ)⟩−⟨P,Heti(Zm;Qℓ)⟩∣byétalerepresentationstability≤∑i=N(m,P)∞2q−i/2FP(i)byconvergentcohomology. Because N(m,P) tends to ∞ with m, and because FP(i) is subexponential in i, we see that the sequence (4.12) is Cauchy. Similarly, we see that the right-hand side of (4.9) ∑i=0∞(−1)iTr(Frobq⥀Heti(Z)P*) converges as a consequence of the existence of the stable P-isotypic part, Deligne’s bounds on the eigenvalues of Frobq and the existence of the subexponential bounds FP(i). It remains to show that the two limits agree. For this, we have ∣q−dimYn∑y∈Yn(Fq)P(y)−∑i=02dim(Zn)(−1)iTr(Frobq⥀Heti(Z)P*)∣=∣∑i=02dim(Zn)(−1)i(Tr(Frobq⥀Heti(Zn;Qℓ(dim(Zn)))*⊗Qℓ[Sn]V(λ)n*)−Tr(Frobq⥀Heti(Z)P*))∣byEquation(4.13)=∣∑i=N(n,P)+12dim(Zn)(−1)i(Tr(Frobq⥀Heti(Zn;Qℓ(dim(Zn)))*⊗Qℓ[Sn]V(λ)n*)−Tr(Frobq⥀Heti(Z)P*))∣byétalerepresentationstability≤∑i=N(n,P)+12dim(Zn)2q−i/2FP(i)byDeligneandconvergentcohomology. (4.14) Because FP(i) is subexponential in i, we conclude that (4.14) becomes arbitrarily small as n approaches ∞, which proves the theorem.□ We can now prove Theorem A from the introduction, as well as the following. Theorem 4.5 (Statistics for SymnX). TheoremAwith PConfn(Y) (resp. UConfn(Y)) replaced by Yn (resp. Symn(Y)) holds. Proof of Theorems A and 4.5 Theorem B gives that, for each i≥0, the FI-modules Het*(X•;Qℓ) and Het*(PConf•(X);Qℓ) are finitely generated. Corollary 3.3 (resp. Theorem 3.4) gives that the singular cohomology H*(X•;Q) (resp. H*(PConf•(X);Q)) of the co-FI scheme X• (resp. PConf•(X)) is convergent. Now apply Theorem 4.3.□ In special cases, it is possible to compute the right-hand side of Equation (4.9) explicitly. Example 4.6 When X=Ar, we can explicitly compute polynomial statistics on UConfn(Ar), extending the main theorem of [4]. Indeed, the computations of Arnol’d [1] and F. Cohen [9, Section 2] combine with results of Björner–Ekedahl [2, Theorem 4.9] to show that Het*(PConfn(Ar)F¯q;Qℓ) is a graded algebra generated by classes in degree 2r−1 with eigenvalues of Frobq equal to qr. As a result, for any character polynomial Pλ: Tr(Frobq:Heti(UConfn(Ar);V)*→Heti(UConfn(Ar);V)*)={0ifi≠k(2r−1)q−kr⟨Pλ,Heti(PConfn(Ar);Qℓ)⟩ifi=k(2r−1). Here we have, as above, applied Poincarè duality to replace the compactly supported cohomology of the smooth schemes UConfn(Ar) with (the Tate twist of) the dual of ordinary étale cohomology. We thus have, for all P, limn→∞q−nr∑y∈UConfn(Ar)(Fq)P(y)=∑i=0∞(−1)i(2r−1)q−ir⟨P,Heti(2r−1)(PConf(Ar))⟩. Funding B.F. is supported in part by NSF Grant nos. DMS-1105643 and DMS-1406209. J.W. was supported in part by NSF Grant no. DMS-1400349. Acknowledgements It is a pleasure to thank George Andrews, Kathrin Bringmann, Mark Kisin, and Ken Ono for useful discussions. We thank Nir Gadish and Brian Conrad for very helpful conversations about twisted-adic sheaves and transfer. We thank Weiyan Chen, Jordan Ellenberg and Burt Totaro for numerous helpful comments on an earlier draft. It is a pleasure to thank Matt Emerton for his careful reading and many detailed comments on an earlier draft of this paper. Finally, we thank an anonymous referee for making several suggestions that helped to simplify the proofs in this paper. References 1 V. I. Arnol’d , The cohomology ring of the colored braid group , Math. Notes 5 ( 1969 ), 138 – 140 . Google Scholar Crossref Search ADS 2 A. Björner and T. Ekedahl , Subspace arrangements over finite fields: cohomological and enumerative aspects , Adv. Math. 129 ( 1997 ), 159 – 187 . Google Scholar Crossref Search ADS 3 T. Church , J. S. Ellenberg and B. Farb , FI-modules: a new approach to stability for Sn-representations , Duke Math. J. 164 ( 2015 ), 1833 – 1910 . Google Scholar Crossref Search ADS 4 T. Church , J. S. Ellenberg and B. Farb , Representation stability in cohomology and asymptotics for families of varieties over finite fields, to appear in Algebraic Topology: Applications and New Directions , AMS Contemp. Math. Ser. 620 ( 2014 ), 1 – 54 . Google Scholar Crossref Search ADS 5 T. Church , J. S. Ellenberg , B. Farb and R. Nagpal , FI-modules over Noetherian rings , Geom. Topol. 18 ( 2014 ), 2951 – 2984 . Google Scholar Crossref Search ADS 6 T. Church and B. Farb , Representation theory and homological stability , Adv. Math. 245 ( 2013 ), 250 – 314 . Google Scholar Crossref Search ADS 7 W. Chen , Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting, preprint, March 2016 , arXiv:1603.03931v1. 8 T. Church , Homological stability for configuration spaces of manifolds , Invent. Math. 188 ( 2012 ), 465 – 504 . Google Scholar Crossref Search ADS 9 F. Cohen , On configuration spaces, their homology, and Lie algebras , J. Pure Appl. Alg. 100 ( 1995 ), 19 – 42 . Google Scholar Crossref Search ADS 10 P. Deligne , La conjecture de Weil, II , Inst. Hautes Études Sci. Publ. Math. 52 ( 1980 ), 137 – 252 . Google Scholar Crossref Search ADS 11 P. Deligne , Cohomologie a supports propre, Theorie des Topos et Cohomologie Étale des Schemas (SGA 4), Lecture Notes in Mathematics, Vol. 269 ( 1972 ). 12 J. Ellenberg , Arizona Winter School 2014 notes, http://swc.math.arizona.edu/aws/2014/2014EllenbergNotes.pdf. 13 J. Ellenberg , A. Venkatesh and C. Westerland , Homological stability and Cohen–Lenstra over function fields , Ann. Math. 183 ( 2016 ), 729 – 786 . Google Scholar Crossref Search ADS 14 B. Farb , Representation Stability, Proc. ICM, Seoul, 2014 . 15 E. Freitag and R. Kiehl , Étale Cohomology and the Weil Conjectures , Springer-Verlag , Berlin-Heidelberg GMb, 1980 . 16 P. Flajolet and R. Sedgewick , Analytic Combinatorics , Cambridge University Press , Cambridge, 2009 . 17 N. Gadish , representation stability for families of linear subspace arrangements, preprint, March 2016 , arXiv:1603.08547v3. 18 Q. Ho , Free factorization algebras and homology of configuration spaces in algebraic geometry, preprint, December 2015 , arXiv:1512.04490v1. 19 G. Lehrer and L. Solomon , On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes , J. Algebra 104 ( 1986 ), 410 – 424 . Google Scholar Crossref Search ADS 20 M. Lipnowski and J. Tsimerman , How large is Ag(Fq)?, arXiv:1511.02212v1, November, 2015 . 21 J. Milne , Lectures on étale cohomology, Version 2.21, March 22 2013 . 22 J. Milne , Values of Zeta Functions of Varieties over Finite Fields , Amer. J. Math. 108 ( 1986 ), 297 – 360 . Google Scholar Crossref Search ADS 23 D. Mumford , Abelian Varieties , 2nd ed , Hindustan Book Agency, New Delhi, 2008. 24 D. Petersen , A spectral sequence for stratified spaces and configuration spaces of points, preprint, arXiv:1603.01137, March 2016 . 25 B. Totaro , Configuration spaces of algebraic varieties , Topology 35 ( 1996 ), 1057 – 1067 . Google Scholar Crossref Search ADS 26 R. Vakil and M. M. Wood , Discriminants in the Grothendieck ring , Duke Math. J. 164 ( 2015 ), 1139 – 1185 . Google Scholar Crossref Search ADS 27 A. Weber , The Leray spectral sequence for complements of certain arrangements of smooth submanifolds, arXiv:1505.01365v1, May 6, 2015 . © The Author(s) 2018. Published by Oxford University Press. All rights reserved. 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# Étale homological stability and arithmetic statistics

, Volume 69 (3) – Sep 1, 2018
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### Abstract

Abstract We relate asymptotic point counts and arithmetic statistics over finite fields to homological stability and representation stability over ℂ in the example of configuration spaces of n points on smooth varieties. In order to accomplish this, we establish subexponential bounds on the growth of the unstable cohomology of such spaces. We apply this and étale homological stability to compute the large n limits of various arithmetic statistics of configuration spaces of varieties over Fq. 1. Introduction Let X be a scheme defined over Z. The Weil conjectures provide a fundamental link between the topology of X(C) and the arithmetic of X(Fq). As first indicated by work of Ellenberg–Venkatesh–Westerland [13], followed by Vakil–Wood [26], Church–Ellenberg–Farb [4] and others, this correspondence should convert homological stability phenomena in topology to asymptotic point counts on the arithmetic side. We summarize this in the following table, with the rows going from least to most general. Topology Arithmetic H*(X(C)) ∣X(Fq)∣ Homological stability of Xn Asymptotics of ∣Xn(Fq)∣ as n→∞ Representation stability Asymptotics of arithmetic Statistics on Xn(Fq) Topology Arithmetic H*(X(C)) ∣X(Fq)∣ Homological stability of Xn Asymptotics of ∣Xn(Fq)∣ as n→∞ Representation stability Asymptotics of arithmetic Statistics on Xn(Fq) Topology Arithmetic H*(X(C)) ∣X(Fq)∣ Homological stability of Xn Asymptotics of ∣Xn(Fq)∣ as n→∞ Representation stability Asymptotics of arithmetic Statistics on Xn(Fq) Topology Arithmetic H*(X(C)) ∣X(Fq)∣ Homological stability of Xn Asymptotics of ∣Xn(Fq)∣ as n→∞ Representation stability Asymptotics of arithmetic Statistics on Xn(Fq) One of the main goals of the present paper is to realize the bottom rows of this table for varieties PConfn(X) (resp. UConfn(X)) of configurations of ordered (resp. unordered) n-tuples of points on a smooth variety X. We first state our main theorem, after which we will define all of the terms appearing in its statement. Theorem A (Arithmetic statistics of configuration spaces). Let Xbe a scheme, smooth over Z[1/N]for some N, with geometrically connected fibers of finite type. Let pbe a prime not dividing N, and let qbe a power of p. Let Frobqdenote the qthpower Frobenius. Let Pbe any character polynomial, and denote by Heti(PConf(X))Pthe stable P-isotypic part of the étale cohomology of the co-FI-scheme PConf•(X)F¯q (see Section2below for a precise definition). Denote by Heti(UConf(X))the stable étale cohomology of UConfn(X)F¯q. Then limn→∞q−ndimX∑y∈UConfn(X)(Fq)P(y)=∑i=0∞(−1)iTr(Frobq⥀Heti(PConf(X))P*),in particular, both sides of the above converge. Specializing to P=1, we obtain limn→∞q−ndimX∣UConfn(X)(Fq)∣=∑i=0∞(−1)iTr(Frobq⥀Heti(UConf(X))*). The proof of Theorem A has two main steps: We prove what we call étale homological (and representation) stability for Het*(Xn/F¯q;Qℓ). This allows us to break up Het*(Xn/F¯q;Qℓ) into two parts: stable and unstable. We obtain subexponential bounds on the growth of the unstable part of Het*(Xn/F¯q;Qℓ). This allows us to prove that this unstable part does not contribute, via the Grothendieck–Lefschetz trace formula, to the limiting density of ∣Xn(Fq)∣. As we explain below, the absence of such bounds is a significant obstruction to understanding the asymptotic point counts of many families of interest. In the étale context, stability of each Heti(Xn/F¯q;Qℓ) as a Galois representation, not just as a vector space, is crucial. It is the difference between proving that limits such as limn→∞q−dimXn∣Xn(Fq)∣ exist, and actually computing the limiting answer. We now discuss these two steps in more detail. Homological stability: A sequence {Xn} of spaces or groups is said to satisfy homological stability over a ring R if Hi(Xn;R) or Hi(Xn;R) is independent of n for n≥D(i); the number D(i) is called the stable range. Typically, but not always, there are maps ψn:Xn→Xn+1 or ϕn:Xn+1→Xn inducing isomorphisms (ψn)*:Hi(Xn;R)→Hi(Xn+1;R)orϕn*:Hi(Xn;R)→Hi(Xn+1;R). Examples of Xn satisfying homological stability include classifying spaces of symmetric groups Sn (Nakaoka), arithmetic groups like SLnZ (Borel), the moduli spaces Ag (Borel) and Mg (Harer) and also configuration spaces UConfn(M) of unordered n-tuples of distinct points on a manifold M (Arnol’d, McDuff, Segal, Church). Homological stability has been a powerful tool in topology. It converts an a priori infinite computation to a finite one. Further, the stable answer, Hi(Xn;R) for n≥D(i), can often be computed explicitly. Many natural sequences Xn come equipped with actions of groups Gn by automorphisms. A basic example is the space PConfn(M) of ordered n-tuples of distinct points on a manifold M, on which the symmetric group Sn acts by permuting the ordering. Such spaces almost never satisfy homological stability, but they instead often satisfy representation stability: the decomposition of Hi(Xn;Q) into a sum of irreducible Sn-representations stabilizes in a precise sense (see [3, 6], Section 2.1 below, and [14] for a survey). When R=Q, plugging the trivial representation into this theory gives classical homological stability for the sequence Xn/Sn. So for example representation stability for PConfn(M) gives classical homological stability for the space UConfn(M)=PConfn(M)/Sn of unordered n-tuples of distinct points on M; see [8]. The theory of representation stability, initiated by Church, Ellenberg and Farb, is currently undergoing a rapid development. Étale homological stability: Consider a scheme Y, smooth over Z[1/N] for some N. We can extend scalars to C and consider the complex points Y(C), and we can also reduce modulo p for any prime p∤N. This gives a variety defined over Fp, and for any positive power q=pd we can consider both the Fq-points and the F¯q-points of Y, where F¯q is the algebraic closure of Fq. One of the most fundamental arithmetic invariants attached to Y is its ℓ-adic étale cohomology Het*(Y/K¯;Qℓ), where K is a number field or finite field of characteristic prime p∤N, and where ℓ≠p is prime. The Galois action on Y/K¯ induces a Galois action on each Qℓ-vector space Heti(Y/K¯;Qℓ), and this action is a crucial part of the data. Now let Xn be a sequence of schemes that are smooth and quasi-projective over Z[1/N] for some N, for example Xn=PConfn(Y) or Xn=UConfn(Y) for Y smooth over Z[1/N]. Given the usefulness of homological stability in topology, one wants to prove such stability for Heti(Xn/K¯;Qℓ) for K a number field or a finite field with characteristic p∤N. There are a number of different notions of what ‘stability’ means in this context (see Section 2 below). We adopt the strongest of these possibilities. Definition 1.1 (Étale homological stability). We say that a sequence Xn of schemes satisfies étale homological stability over a field K if for each i≥0, there exists D=D(i) so that the isomorphism type of Heti(Xn/K¯;Qℓ) as a Gal(K¯/K)-representation does not depend on n for n≥D. The function D(i) is called the stable range. When each Xn in addition admits an Sn-action, such as Xn=PConfn(Y) or Yn, we have a corresponding notion of étale representation stability over K. This definition is a bit more involved; see Section 2.2. It implies étale homological stability for the sequence of varieties Xn/Sn. To state our results in this direction, we need two different descriptions of representations of the symmetric groups Sn. Let Xi be the class function on all symmetric groups Sn,n≥1 given by setting Xi(σ) to be the number of i-cycles in the cycle decomposition of σ. A character polynomial is any polynomial P∈Q[X1,X2,…]; it is a class function on each Sn,n≥1. The degree of a character polynomial is defined by setting degXi≔i. See Section 2.1 below for more details. As shown by Church–Ellenberg–Farb [3], character polynomials give a compact and uniform way of describing the characters of certain infinite sequences of Sn-representations for n=1,2,…. A partition of n is a sequence λ=(λ1≥⋯≥λr≥0) with ∑iλi=n. The irreducible representations V(λ) of Sn are classified by partitions λ⊢n. A partition λ⊢k gives a sequence V(λ)n of irreducible Sn-representations for n≥k+λ1 by defining V(λ)n to be the irreducible representation of Sn corresponding to the partition (n−k,λ1,…,λr). Every irreducible representation of Sn is of the form V(λ)n for a unique partition λ; for example the trivial and standard representations of Sn are V(0)n and V(1)n, respectively. Similarly, every character polynomial is a linear combination of a finite number of class functions of the form χV(λ). Theorem B (Étale representation stability). Let Ybe a scheme, smooth over Z[1/N]for some N, with geometrically connected fibers of finite type. Let Kbe either a number field or a finite field of characteristic p∤N. For each i≥0, the sequence Heti(PConfn(Y)/K¯;Qℓ)of Gal(K¯/K)-modules satisfies étale representation stability (see Section2.2for the precise definition of étale representation stability) over Kwith stable range D(i)=2ifor dimY≥2and D(i)=4ifor dimY=1. In particular: 1. Inductive description: For all n≥0, there is an isomorphism of Gal(K¯/K)-representations: Heti(PConfn(Y)/K¯;Qℓ)≅colimSHeti(PConf∣S∣(Y)/K¯;Qℓ), (1.1)where the colimit is taken over the poset of all subsets S⊂{1,…,n}such that ∣S∣≤D(i). This gives, for each n≥D(i), a recipe for building the Gal(K¯/K)-representation Heti(PConfn(Y)/K¯;Qℓ)from a fixed finite collection of Gal(K¯/K)-representations. 2. Stability of isotypics: For each character polynomial P, there exists a unique virtual Gal(K¯/K)-representation Heti(PConf(Y))Pover Qℓ, linear in P, so that when P=χV(λ)for some λ⊢k, there exists Dsuch that for all n≥D: Heti(PConf(Y))χV(λ)=Heti(PConfn(Y)/K¯;Qℓ)⊗Qℓ[Sn]V(λ)nand the right-hand side is independent of nas a Gal(K¯/K)-representation. 3. Polynomial characters: There exists a character polynomial Q(X1,…,Xr)so that for all n≥D(i): χHeti(PConfn(Y)/K¯;Qℓ)(σ)=Q(X1(σ),…,Xr(σ))forallσ∈Sn.,where deg(Q)≤iif dimY>1and deg(Q)≤2iif dimY=1. Remarks Theorems A and B in the special case Y=A1 was proved by Church–Ellenberg–Farb [4]. This special case is much simpler since the eigenvalues of Frobq on Heti(PConfn(A1)/F¯q;Qℓ) are known explicitly; they equal qi. Our proof of Theorem B uses the theory of FI-modules, developed in [3]. What we actually prove is that for each i≥0, the FI- Gal(K¯/K)-module (see Section 2.2 below for the precise definition) Heti(PConf•(Y);Qℓ) is finitely generated. Items (1–3) of Theorem B then follow from the general theory of FI-modules, in particular theorems from [3, 5]. As one consequence, the proof shows that Item (1) of Theorem B holds with Qℓ replaced by Zℓ or Z/ℓnZ. Nir Gadish [17] has recently isolated a concept of finitely generated I-poset, for a wide class of categories I, and has used this to prove étale representation stability for a rich class of sequences of complements of linear subspace arrangements. Plugging in P=1 into Item (2) of Theorem B gives the following. Corollary B′ (Étale homological stability). With terminology as in TheoremB, the sequence Heti(UConfn(Y)/K¯;Qℓ)satisfies étale homological stability over K: these Gal(K¯/K)-representations do not depend on nfor n≥D(i). Remarks Quoc Ho [18] has recently given an independent proof of Corollary B′ for Y smooth over any ground field. His method is based on factorization homology, and is quite different from the methods of this paper. Dan Petersen [24] has recently extended Theorem B to a wider class of configuration-like spaces, also dropping the smoothness assumption. Stability of arithmetic statistics: The application of homological stability to arithmetic statistics was pioneered by Ellenberg–Venkatesh–Westerland in [13]. The fundamental link is provided by the Grothendieck–Lefschetz Trace Formula (Here we have assumed that Z is smooth and applied Poincaré Duality to the usual Grothendieck–Lefschetz Formula.): ∣Z(Fq)∣=qdim(Z)∑i≥0(−1)iTr(Frobq:Heti(Z/F¯q;Qℓ)*→Heti(Z/F¯q;Qℓ)*) (1.2) and its twisted version (see (4.1) below). Given Deligne’s theorem [10, Theorem 1.6] that any eigenvalue λ of Frobq on Heti(Z/F¯q;Qℓ)* satisfies ∣λ∣≤q−i/2, one can bound the number ∣Z(Fq)∣ of Fq-points via ∣Z(Fq)∣≤qdimZ∑i=02dimZbiq−i/2, where bi≔dimHeti(Z/F¯q;Qℓ)*. Applying this reasoning to a sequence Zn of smooth varieties gives q−dimZn∣Zn(Fq)∣≤∑i=02dimZnbi(n)q−i/2, (1.3) where we have emphasized via notation that bi is a function of n. It seems that étale homological stability, namely the fact that bi(n) is constant for n≥D(i), should imply that the limit as n→∞ of the left-hand side of (1.3) exists. However, it could be that dim(Zn) goes to ∞ with n and that bi(n) grows more quickly than qi/2, even for any q; this would imply the divergence of the right-hand side of (1.3). This super-exponential growth is known to occur in natural examples, for example for Zn the moduli space of genus n smooth algebraic curves, and also for Zn the moduli space of n-dimensional principally polarized abelian varieties. In the latter example, recent work of Lipnowski–Tsimerman [20] shows that this growth actually does change the point count ∣Zn(Fq)∣, as they show that this number grows more quickly than the expected qdimZn. Thus, in order to apply étale homological stability to obtain the existence of asymptotic point counts in a given example, it is necessary to prove subexponential (in i) bounds on bi(n), independent of n. In other words, control of the unstable étale cohomology Heti(Zn/F¯q;Qℓ)* is needed. Proving such bounds is a major obstruction for arithmetic applications; see Section 3 for a discussion. This problem is a very special case (namely the case P≡1) of more general arithmetic statistics, where one needs a twisted version of the Grothendieck–Lefschetz formula, and where the control on the ‘representation unstable cohomology’ is even more difficult to prove; see Section 3 below. A significant part of this paper, Section 3, is devoted to overcoming this problem for the examples Xn and PConfn(X). Theorem C (Bounding the representation unstable cohomology). Let Xbe either a smooth, orientable manifold with dim(H*(X;Q))<∞ (for example, Xcompact), or a scheme, smooth over Z[1/N]for some N, with geometrically connected fibers of finite type. In the first case let Hidenote singular cohomology with Qcoefficients; in the second, let Hidenote étale cohomology with Qℓcoefficients. Then for any character polynomial P, there exists a function FP(i), subexponential in i, such that for all n≥1: ⟨P,(Hi(PConfn(X)))⟩Sn≤FP(i). In Section 4, we apply Theorems B and C to obtain Theorem A. A different description of the left-hand side of Theorem C, established using analytic methods, will appear in forthcoming work of Chen [7]. En route to proving Theorems C and A, we also prove the analogous statements for Symn(X); see Section 4. However, we note that Theorem C requires more than just bounding the betti numbers Symn(X). 2. Étale representation stability In this section, we briefly summarize the theory of representation stability and FI-modules, as it is used in topology, as well as some of its consequences. This theory was developed by Church–Ellenberg–Farb [3, 6], and later with Nagpal [5]; see [14] for a survey. We refer the reader to these references for details. We then give a general setup for proving similar stability theorems in étale cohomology. 2.1. Quick summary of representation stability and FI-modules An FI-module V over a Noetherian ring R is a functor from the category FI of finite sets and injections to the category of R-modules. Thus, to each natural number n, we have associated an R-module Vn with an Sn action, with a map Vm→Vn for each injection {1,…,m}→{1,…,n}. Recall that the opposite category FIop is the same as FI but with arrows reversed. A co-FI module over R is a functor from FIop to R-modules. We also have the associated notions of FI-space, FI-scheme, etc., and the associated co-FI versions. An FI-module V is finitely generated if there is a finite set S of elements in ∐iVi so that no proper sub-FI-module of V contains S. One of the reasons that we care about finitely generated FI-modules is the following theorem. Theorem 2.1 (Structural properties of finitely generated FI-modules). Let Vbe an FI-module over a commutative Noetherian ring R. If Vis finitely generated then: Representation stability [3]: When Ris a field of characteristic 0, finite generation of Vimplies representation stability in the sense of [6] for the sequence {Vn}of Sn-representations. Inductive description [5]: Let V•be a finitely generated FI-module over a Noetherian ring R. Then there exists some N≥0such that for all n∈N, there is a natural isomorphism Vn≅colimS⊂[n],∣S∣≤NVS,that is these isomorphisms commute with homomorphisms of FI-modules. By definition, the stable range of V• N(V)is the minimal such N. Isomorphism of trivial isotypics [8]: Let V•be a finitely generated FI-module over a Noetherian ring Rwith stable range N(V). Then for all n≥N(V), the map Vn→Vn+1, given by averaging the structure maps, induces an isomorphism VnSn→≅Vn+1Sn+1. We remark that the isomorphism of trivial isotypics illustrates one of the key advantages of considering (co-)FI-spaces: while stabilization maps for many natural sequences of spaces or schemes do not naively exist, a (co-)FI-space Z• comes equipped with canonical rational correspondences from Zn+1/Sn+1 to Zn/Sn. We will also need the following. Lemma 2.2 Let V•and W•be finitely generated FI-modules, and let Nbe the sum of their stable ranges. Then for all n≥N, the maps Vn→Vn+1and Wn→Wn+1 (associated to {1,…,n}⊂{1,…,n+1}) induce isomorphisms Vn⊗Q[Sn]Wn→≅Vn+1⊗Q[Sn+1]Wn+1that are natural in both variables with respect to homomorphisms of FI-modules. Proof By [3, Proposition 2.3.6], the tensor product V•⊗QW• is finitely generated since V• and W• are. Applying the co-invariants functor, we obtain the functor n↦Vn⊗Q[Sn]Wn. Because stable ranges add under tensor product [3, Proposition 2.3.6], and because the stability degree (cf. [3, Definition 3.1.3]) is less than or equal to the stable range [3, Proposition 3.3.3], the map Vn⊗Q[Sn]Wn→Vn+1⊗Q[Sn+1]Wn+1 is an isomorphism for n≥N.□ Character polynomials: Character polynomials and their degree were defined in the introduction. Let ⟨P,Q⟩ denote the inner product of Sn-characters. The expectations of character polynomials Eσ∈SnPn(σ)≔1n!∑σ∈SnPn(σ)=⟨Pn,1⟩ compute the averages of natural combinatorial statistics with respect to the uniform distribution on Sn. As shown in [4, Proposition 2.2], the inner product ⟨Pn,Qn⟩ of character polynomials P,Q∈Q[X1,X2,…] is independent of n once n≥degP+degQ. One remarkable property of finitely generated FI-modules V is that the characters of the Sn-representations Vn are, for large enough n, given by a single polynomial. Theorem 2.3 (Polynomiality of characters [5]). Let Vbe an FI-module over a field of characteristic 0. If Vis finitely generated then the characters χVnof the Sn-representations Vnare eventually polynomial: there exists N≥0and a polynomial P(X1,…,Xr), for some r>0, so that χVn=P(X1,…,Xr)foralln≥N. (2.1) In particular, if Qis any character polynomial, then ⟨χVn,Q⟩is independent of n≥degP+degQ. We note that evaluating (2.1) on the identity permutation gives a polynomial P(T)∈Q[T] so that dimkVn=P(n) for all n≥N. Étale Representation Stability: Given a co-FI-scheme Z• defined over Fq, its étale cohomology Heti(Z•/F¯q;Qℓ) has additional structure beyond that of an FI-module over Qℓ. The geometric Frobenius Frobq gives a natural endomorphism of Z•/F¯q, and this gives rise to an action of Gal(F¯q/Fq) on the FI-module Heti(Z•/F¯q;Qℓ). As noted in the introduction, the eigenvalues of Frobq and the action of Gal(F¯q/Fq) are crucial parts (as observed for example by Milne [22], the Tate conjecture implies that the eigenvalues of Frobq determine the Gal(F¯q/Fq)-action. But, this is not known at present) of the data here. Weaker than knowing an eigenvalue λ of Frobq on Hetj(Zn/F¯q;Qℓ) is knowing its weight. Deligne proved that λ is an algebraic number with ∣λ∣=qr/2 for some j≤r≤2j, with r=j if Zn is smooth and proper. The number r is the weight of the eigenvalue λ. Similarly, for Z• defined over a number field K, the action of Gal(K¯/K) on Z•/K¯ induces an action on Heti(Z•/K¯;Qℓ), and this action is a fundamental part of the data. In increasing order of strength, we could ask that for each i there exists D so that for all n≥D: The isomorphism type of Heti(Zn/K¯;Qℓ) as a Qℓ-vector space does not depend on n. In addition, the list of weights of Frobq on Heti(Zn/K¯;Qℓ) does not depend on n. In addition, the list of eigenvalues of Frobq on Heti(Zn/K¯;Qℓ) does not depend on n. The isomorphism type of Heti(Zn/K¯;Qℓ) as a Gal(K¯/K)-representation does not depend on n. We have adopted the strongest of these as our definition of étale homological stability. (For applications to counting problems, for example Theorem A, only the stability of the eigenvalues is needed.) 2.2. Étale representation stability Let Z be a co-FI scheme smooth over Z[1/N] for some fixed N, with geometrically connected fibers. Let p∤N be prime, and let ℓ≠p be a prime. For each i≥0, the étale cohomology Heti(Z/F¯p;Zℓ) is an FI-module. In addition, for each q=pd, the Frobenius Frobq acts on each Heti(Zn/F¯p;Zℓ), endowing it with the structure of a Gal(F¯p/Fq)-module. The Sn-action on Heti(Zn/F¯p;Zℓ) coming from its structure as an FI-module commutes with the action of Gal(F¯p/Fq), as do all automorphisms of Zn. Similarly, for any number field K, the action of Gal(K¯/K) on Heti(Z/K¯;Qℓ) commutes with the FI-structure. This discussion shows that, for K a number field or a finite field of characteristic prime to N, Heti(Z/K¯;Qℓ) is an FI- Gal(K¯/K)-module; that is, an FI-module equipped with an action of Gal(K¯/K) by FI-automorphisms. We have the corresponding notions of finitely generated FI- Gal(K¯/K)-module: there is a finite set S⊂∐nHeti(Zn/K¯;Qℓ) so that no proper sub-FI-module of Heti(Z/K¯;Qℓ) contains S. Definition 2.4 (Étale representation stability). We say that a sequence Zn of Gal(K¯/K)-modules satisfies étale representation stability if {1,…,n}↦Zn is a finitely generated FI- Gal(K¯/K)-module. Theorem C in [5] gives an inductive description of finitely generated FI-modules V over any Noetherian ring R. Namely, there exists D≥0 such that for all n∈N, there is a natural isomorphism Vn≅colimSVS, (2.2) where the colimit is taken over the poset of all subsets S⊂{1,…,n} such that ∣S∣≤D. If V is a finitely generated FI- Gal(K¯/K)-module then (2.2) gives is an isomorphism of Gal(K¯/K)-modules. Thus (2.2) gives, for each n≥D, a recipe for building the Gal(K¯/K)-representation Vn from a fixed finite collection of Gal(K¯/K)-representations. Étale representation stability for products and configuration spaces: Attached to any scheme X there is an associated configuration space PConfn(X) of ordered n-tuples in X, defined by PConfn(X)≔{(x1,…,xn)∈Xn:xi≠xj∀i≠j}=Xn⧹Δ, where Δ is the fat diagonal and where we write x∈X to denote an arbitrary R-point of X. The group Sn acts freely on PConfn by permuting the coordinates. The quotient UConfn(X)≔PConfn(X)/Sn is the configuration space of unordered n-tuples of points in X. For any scheme X, denote by X• the co-FI scheme that sends {1,…,n} to the cartesian product Xn, and associates to any injection between finite sets the natural projection maps between cartesian powers. The open subsets PConfn(X)⊂Xn are preserved under the co-FI structure maps, and so we obtain a co-FI scheme PConf•(X). In particular, Hi(PConf•(X);Qℓ) is an FI-module for any i≥0. With this setup, we can now prove Theorem B from the introduction. The proof also gives the following result. Theorem 2.5 TheoremBwith PConfn(Y) (resp. UConfn(Y)) replaced by Yn (resp. Symn(Y)) holds. Proof of Theorem B and Theorem 2.5 This theorem follows from the proofs of [3, Theorem 6.1.2] and [3, Theorem 6.2.1]. The only difference is that we now work with étale rather than Betti cohomology. To wit, the co- FI-schemes PConf•(Y) and Y• give rise to FI- Gal(K¯/K)-modules by taking ℓ-adic cohomology. The proof of [3, Theorem 6.1.2] carries over verbatim to the étale setting to show that Heti(Y•;Qℓ) is a finitely generated FI- Gal(K¯/K)-module for all i≥0; see [3, Theorem 4.1.7 and Remark 6.1.3] for the stable range and degree of the character polynomial. Note here that this improved stable range comes from the fact that X• is a so-called FI#-module. For PConf•(Y), the proof of [3, Theorem 6.2.1] applies in étale cohomology just as for singular cohomology. Indeed, as Totaro discusses in [25, p. 1064], for any scheme Y smooth over Z[1/N], the Leray spectral sequence in ℓ-adic cohomology for the inclusion PConfn(Y)↪Yn has E2-page isomorphic to E2p,q=⨁{J⊣n∣∣J∣=n−q/(2dim(Y)−1)}Hetp(YJ;QℓcJ(−qdim(Y)2dim(Y)−1)), where YJ⊂Yn denotes the diagonal where points coincide according to the partition J, dim(Y) denotes the dimension of the scheme, and if J consists of pieces of size j1,…,jn−q/(2dim(Y)−1), then cJ≔(j1−1)!⋯(jn−q/(2dim(Y)−1)−1)!. See also [27] for a detailed proof of this description of the Leray spectral sequence in the étale setting. The proof of [3, Theorem 6.2.1] now carries over verbatim to show that for all i, the FI- Gal(K¯/K)-module Heti(PConf•(Y);Qℓ) is finitely generated for all i≥0. For the claimed stable ranges and degree of character polynomial, see [3, Theorem 6.3.1].□ 3. Convergent cohomology In this section, we provide the necessary bounds for the ‘representation unstable cohomology’ of Xn and of PConfn(X) that will be necessary for the arithmetic applications in Section 4. 3.1. Definition of convergent cohomology A function F:N→Nhas exponential growth rate λ if limn→∞logf(n)n=λ. (3.1) If (3.1) holds with λ=0, we say that F has subexponential growth. Let Z be a co-FI-scheme over Z[1/N]. For each i≥0, let Hi(Zn) denote either the singular cohomology Hi(Zn(C);Q) or the étale cohomology Heti(Zn/K¯;Qℓ) for K a number field or finite field of characteristic prime to N. In each case, Hi(Z•) is an FI-module (over Q and Qℓ, respectively). For any class function P on Sn, denote by ⟨P,Hi(Zn)⟩, the inner product of (the character of) Hi(Zn) with P. In order to compute arithmetic statistics for a co-FI scheme Z, one needs to control the ‘representation unstable’ cohomology of Z; see Section 4. More precisely, one needs to prove one of the following two properties, which were shown to be equivalent in [4, Section 3]: For each 0≤a≤n there is a function Fa(i), subexponential in i and not depending on n, so that dimHi(Zn)Sn−a≤Fa(i)forallnandi. (3.2) For each character polynomial P∈Q[X1,X2,…], there exists a function FP(i), subexponential in i and not depending on n, such that ∣⟨P,Hi(Zn)⟩∣≤FP(i)forallnandi. (3.3) It is crucial that these bounds hold independently of n. While the second condition is the one that applies to arithmetic statistics (see Section 4 below), it is quite difficult to check. Thus the equivalence with the first condition is quite useful. Definition 3.1 (Convergent cohomology). We say that the co-FI scheme (or space) Z has convergent (singular or étale) cohomology if either of the two equivalent properties 1 or 2 in equations (3.2) or (3.3) holds. If these properties hold with FP(i) having exponential growth 0<λ<∞, we say that Z has weakly convergent cohomology with convergence rate λ. These kinds of bounds are typically not easy to prove. In [13], this is accomplished ([13] only needs to deal with the classical, not representation stable, case; that is, the a=0 case) for the cohomology of certain Hurwitz spaces by obtaining an exponential upper bound for the number of i-cells, via an explicit cell decomposition. In [4], such bounds for the example Hi(PConfn(C);Q) are obtained by a detailed knowledge of these Sn-representations. The rest of this section is devoted to giving such bounds for two natural classes of co-FI schemes. We then apply this in Section 4 to arithmetic statistics for Fq-points on these schemes. 3.2. Polynomial bounds on Betti numbers of symmetric products Let X be a topological space. The n-fold cartesian product Xn is endowed with a natural action of the symmetric group Sn, given by permuting the factors. The quotient SymnX≔Xn/Sn is called the the nthsymmetric product of X. Proposition 3.2 (Growth of Betti numbers of symmetric products). Let Xbe either a space with dim(H*(X;Q))<∞ (for example, Xcompact), or a scheme over Z[1/N]for some N, with geometrically connected fibers of finite type. In the first case let Hidenote singular cohomology with Qcoefficients; in the second, let Hidenote étale cohomology with Qℓcoefficients. In either case, let bibe the associated ithBetti number. Then bi(Symn(X))is bounded above by a polynomial in i, independent of n. Proof We claim that if m<n then bi(Symm(X))≤bi(Symn(X)); further, bi(Symn(X))=bi(Symi(X)) for all i≥n. To see the statement for m<n, observe that for any graded vector space V (over a field of characteristic 0), a choice of ‘unit’ 1∈V0, determines an injection Symm(V)i→Symn(V)iv⃗↦v⃗⊗1⊗⋯⊗1. In particular, dim(Symm(V)i)≤dim(Symn(V)i) for all m<n and i. Considering V=H*(X;Q), Künneth and transfer imply that Symn(H*(X;Q))i≅Hi(Symn(X);Q) and the first part of the claim follows. For the second, we note that for a graded vector space V=V0⊕⋯⊕Vm, with V0=Q, we have Symn(V)≅⨁a0+⋯am=n⊗j=1mSymaj(Vj), and thus Symn(V)i≅⨁(a1,…,am)⊗j=1mSymaj(Vj), where the direct sum is over partitions a1+2a2+⋯mam=i such that a1+⋯+am=n. In particular, the number of pieces in the partition is at most n, and since for any i, the largest number of pieces in any partition is i, we see that for n≥i, the direct sum is independent of n. We conclude the claim by taking V=H*(X;Q) and invoking Künneth and transfer as above. We have just shown that for each fixed i≥0 bi(Symn(X))≤bi(Symi(X))=bi(Sym∞(X))foralln≥1. To prove the proposition, it is therefore enough to bound bi(Sym∞(X)) by a polynomial in i. Well, note that the function f(z)≔∑i=0∞bi(Sym∞(X))zi is just the Poincaré series for the symmetric algebra on the vector space H*(X;Q). By an elementary argument, this Poincaré series is, writing Bi≔bi(X), the following rational function: f(z)=(1+z)B1(1+z)B3…(1−z2)B2(1−z4)B4…. Since each pole of f(z) lies on the unit circle, it follows (see, for example [16], Theorem IV.9) that the ith coefficient bi(Sym∞(X)) of f(z) is bounded above by a polynomial in i.□ Consequence: bounding the representation unstable cohomology of products. The following corollary is also a key ingredient in bounding the representation unstable cohomology of configuration spaces. Corollary 3.3 Let Xbe as in Proposition3.2, and use the notation of that theorem. For each 0≤a≤n, we have that dim(Hi(Xn)Sn−a)is bounded above by a polynomial in i, independent of n. Proof Since the action Sn−a leaves invariant the first n−a factors of Xn and acts as the identity on the last a factors, there is, for each i≥0, an isomorphism: Hi(Xn)Sn−a=⨁p+q=iHp(Xn−a)Sn−a⊗Hq(Xa). (3.4) Since this sum has i+1 terms, it suffices to bound the dimension of each summand by a polynomial in i. Since dimH*(X;Q)<∞ and since a is fixed, there is a constant C, not depending on q, so that dim(Hq(Xa))≤C. It thus suffices to bound each Hp(Xn−a)Sn−a by a polynomial in i. But this follows from transfer together with Proposition 3.2, noting that p≤i.□ 3.3. Bounding the representation unstable cohomology of configuration spaces We build on the subexponential upper bounds for products in the last section to prove the corresponding result for configuration spaces. Theorem 3.4 ( PConf•(X) has convergent cohomology). Let Xbe either a smooth, orientable manifold with dim(H*(X;Q))<∞ (for example, Xcompact), or a scheme, smooth over Z[1/N]for some N, with geometrically connected fibers of finite type. Then the co-FI manifold (resp. scheme) PConf•(X)has convergent singular (resp. étale) cohomology. Proof For the case, when X is a manifold, we let Hi denote singular cohomology with Q coefficients; in the case, when X is a scheme, we let Hi denote étale cohomology with Qℓ coefficients. Fix a≥0. Denote by Sn−a the subgroup Sn−a×1⊂Sn. We will prove that there is a function Fa(i), subexponential in i, so that dim(Hi(PConfn(X);Q))Sn−a≤Fa(i), for all i≥0. Let m be the real dimension of X, and denote by A(n,m) the graded commutative algebra A(n,m)≔Q[{Gab}1≤a≠b≤n]/I, where ∣Gab∣=2m−1 and I is the ideal generated by the elements Gab−GbaGabGac+GbcGba+GcaGcb, for a<b<c distinct. The group Sn acts on A(n,m) via σ·Gab≔Gσ(a)σ(b). Totaro [25, Theorem 4] has shown that H*(PConfn(X);Q) is isomorphic, as a graded Sn-representation, to a sub-quotient of H*(Xn;Q)⊗A(n,m), with the natural action on each factor. As Totaro indicates, this result holds for both the singular and étale cohomology. Note that, for any short exact sequence of Sn-representations 0→V0→V1→V2→0 over a field of characteristic 0, there exists an Sn-equivariant splitting V1≅V0⊕V2. In particular, dimV1Sn=dimV0Sn+dimV2Sn, and, more generally, if V is any sub-quotient of an Sn-representation W, we have dimVSn≤dimWSn. Let V and W be any two Sn-representations. The identity dim(VSn)=⟨χV,1⟩ and the Cauchy–Schwarz inequality give dim(V⊗W)Sn=⟨χV⊗W,1⟩=1n!∑σ∈SnχV(σ)χW(σ)≤1n!(∑σ∈SnχV(σ)2)(∑σ∈SnχW(σ)2)=⟨χV⊗2,1⟩⟨χW⊗2,1⟩=dim((V⊗2)Sn)dim((W⊗2)Sn). Specializing to our setting, we conclude that it suffices to show that (dim(Hi(X(C)n;Q)⊗2)Sn−a)·(dim(A(n,m)⊗2)Sn−a)≤Fa(i) for some Fa(i) subexponential in i. For the first factor, by Künneth and the definition of the action, we have (Hi(Xn;Q)⊗2)Sn−a⊂H2i(Xn×Xn;Q)Sn−a≅H2i((X×X)n;Q)Sn−a≅⨁j=02iHj((X×X)n−a;Q)Sn−a⊗H2i−j((X×X)a;Q). By transfer, this is isomorphic to ⨁j=02iHj(Symn−a(X×X);Q)⊗H2i−j((X×X)a;Q). Let C=maxibi(X×X), and let D=C·2m. By Künneth, for all j<2i, dimH2i−j(X2a;Q)≤(2i−j)Ca. Combining this with Proposition 3.2, we see that dim(Hi(Xn;Q)⊗2)Sn−a≤∑j=02idim(Hj(Symn−a(X×X);Q))(2i−j)Ca≤∑j=02i2DD!P(j)(2i−j)Ca≤2DD!CaQ(i) for some polynomials P,Q. It remains to bound dim(A(n,m)⊗A(n,m))iSn−a. Well, (A(n,m)⊗A(n,m))i=⨁p+q=i(A(n,m)p⊗A(n,m)q). (3.5) Since the right-hand side of (3.5) has at most 2i terms, it suffices to bound each [A(n,m)p⊗A(n,m)q]Sn−a. By the Cauchy–Schwartz inequality, as above, it suffices to bound [A(n,m)p⊗A(n,m)p]Sn−a for each 1≤p≤i. To obtain this bound, first note that the algebra A(n,m) is isomorphic to A(n,2) via an isomorphism that takes the pth graded piece of A(n,2) to the (2m−1)pth graded piece of A(n,m). Since m is fixed and so 2m−1 is fixed, it suffices to bound [A(n,2)p⊗A(n,2)p]Sn−a in terms of i, for each 1≤p≤i. Lehrer–Solomon [19] give an explicit description of A(n,2) as a sum of induced representations A(n,2)p=⨁μIndZ(cμ)Sn(ξμ), where μ runs over the set of conjugacy classes in Sn of permutations having n−p cycles, cμ is any element of the conjugacy class μ, and ξμ is a one-dimensional character of the centralizer Z(cμ) of cμ in Sn (we will not need an explicit description of ξμ). It follows that (A(n,2)p⊗A(n,2)p)Sn−a=⨁μ,ν[IndZ(cμ)Sn(ξμ)⊗IndZ(cν)Sn(ξν)]Sn−a, (3.6) where ν is defined similarly to μ. The summands contributing to the first (resp. second) A(n,2)p factor in (3.6) correspond to conjugacy classes cμ (resp. cν) in Sn decomposing into n−p cycles. The number of such conjugacy classes is in bijection with the set of partitions of p, which is less than the number of partitions of i since p≤i. The Hardy–Ramanujan asymptotic for the number ∣{J⊢i}∣ of partitions of i gives C1,C2>0 so that ∣{J⊢i}∣≤C1eC2i. (3.7) Thus the number of terms in the sum on the right-hand side of (3.6) is, by (3.7), at most [C1eC2i]2=C12e2C2i. As this is subexponential in i, it suffices to bound the dimension of [IndZ(cμ)Sn(ξμ)⊗IndZ(cν)Sn(ξν)]Sn−a. Now, a permutation cμ decomposing into n−p cycles must have at least n−2p fixed points. This implies that the centralizer Z(cμ) contains the subgroup Sn−2p, and thus Sn−2i since p≤i. It follows that IndZ(cμ)Sn(ξμ) is a subrepresentation of IndSn−2iSn(ξμ). Thus, [IndZ(cμ)Sn(ξμ)⊗IndZ(cν)Sn(ξν)]Sn−a⊂[IndSn−2iSn(ξμ)⊗IndSn−2iSn(ξν)]Sn−a. (3.8) Let χμ and χν denote the characters of ξμ and ξν, respectively. The right-hand side of (3.8) consists of the set of bilinear functions f:Sn×Sn→C, satisfying f(σ·g,τ·h)=χμ(σ)χν(τ)f(g,h)∀σ,τ∈Sn−2iand∀g,h∈Sn and f(g·β,h·β)=f(g,h)∀β∈Sn−aand∀g,h∈Sn. It follows that the dimension of this vector space is at most the number of double cosets Sn−a⧹[Sn/Sn−2i×Sn/Sn−2i]. We claim that this number is polynomial in i. Indeed, it is equal to the number of maps f:{1,…,a}→{1,…,2i,⋆}×{1,…,2i,⋆} such that ∣f−1(j,k)∣≤1 and ∣f−1(j,⋆)∣,∣f−1(⋆,k)∣≤(n−2i)2. Since a is fixed, this number is bounded by a constant times the number of subsets of {1,…,2i,⋆}×{1,…,2i,⋆} of size ≤a, which is O(i2a). This completes the proof of Theorem 3.4.□ 4. Stability of arithmetic statistics Throughout this section, we will fix a prime power q=pd and a prime ℓ not divisible by p. 4.1. Point counting and étale cohomology Let Y be a scheme of finite type (not necessarily smooth) over Z[1/N]. We can base change to Spec(Fp) for any prime p∤N, and for any positive power q=pd we can consider both the Fq-points as well as the F¯q-points of Y, where F¯q is the algebraic closure of Fq. The arithmetic Frobenius morphism Frobq:Y→Y acts on Y(F¯q) by acting on the coordinates (y1,…,yd) of any affine chart of y via Frobq(y1,…,yd)≔(y1q,…,ydq). A point y∈Y(F¯q) will be fixed by Frobq precisely when y∈Y(Fq). Thus Y(Fq)=Fix(Frobq:Y(F¯q)→Y(F¯q)). Fix a prime ℓ not dividing q, and let Qℓ denote the ℓ-adic rationals. Let Het*(Y/F¯q;Qℓ) (resp. Het,c*(Y/F¯q;Qℓ)) denote the étale cohomology groups (resp. compactly supported étale cohomology groups) of the base change Y/F¯q of Y to F¯q (see, for example, [11, 21]). Denote by Qℓ(−i) the rank 1 Gal(F¯q/Fq)-representation on which Frobenius acts by qi. Let V be a constructible, rational ℓ-adic sheaf on Y (see, for example, [15]). If y∈Y(F¯q) is a fixed point for the action of Frobq, then Frobq acts on the stalk Vy over y. Attached to this action is its trace Tr(Frobq:Vy→Vy). The twisted Grothendieck–Lefschetz Trace Formula [15, Theorem II.3.14] and [11, 6.1.1.1] gives ∑y∈Y(Fq)Tr(Frobq:Vy→Vy)=∑i=02dim(Y)(−1)iTr(Frobq:Het,ci(Y;V)→Het,ci(Y;V)). (4.1) When Y is smooth, Poincaré duality for étale cohomology [21, Theorem 24.1] gives Het,ci(Y/F¯q;V)≅Het2dim(Y)−i(Y/F¯q;V(−dim(Y)))*. (4.2) Plugging this into Equation (4.1) gives, for smooth Y ∑y∈Y(Fq)Tr(Frobq:Vy→Vy)=qdim(Y)∑i=02dim(Y)(−1)iTr(Frobq:Heti(Y;V)*→Heti(Y;V)*). (4.3) Sn-schemes: Now let Z be smooth and quasi-projective over Z[1/N]. Suppose that the symmetric group Sn acts generically freely on Z by automorphisms, and let p:Z→Y denote the quotient map. By [23, Theorem p. 63 and Remark p. 65 (Chapter 2.7)], Y is a scheme. It is typically not smooth even when Z is smooth: for example if Z=(A2)n with the standard Sn action, then Y=Symn(A2) is singular at the point {0,…,0}. Recall that any finite-dimensional representation of Sn over a field of characteristic 0 is defined over Q. There is a bijective correspondence between isomorphism classes of finite-dimensional Sn-representations and finite-dimensional constructible sheaves on Y that become isomorphic to Qℓ⊕r when pulled back to Z: given an Sn-representation V over Qℓ, one can form an Sn-equivariant, locally constant sheaf V over Z with fiber V. Pushing forward to Y and taking Sn invariants, that is (p*V)Sn, we obtain a constructible sheaf of Qℓ vector spaces over Y which is a sheaf-theoretic analogue of the usual topological diagonal quotient ‘ Z×SnV’. Suppose that y∈Y(F¯q) is fixed by Frobq. Then Frobq acts on the fiber p−1(y). Now Sn acts transitively on p−1(y) with some stabilizer H (not depending, up to conjugacy, on y˜∈p−1(y)), and so we can identify p−1(y) with Sn/H after picking a basepoint y˜∈p−1(y). The Frobq action on p−1(y) commutes with this Sn action, and so it is determined by its action on a single basepoint y˜, with stabilizer H. Now Frobq(H)=σyH for σy∈Sn. Following Gadish [17], for any Sn-representation V and any coset σH of Sn, we set χV(σH)≔1∣H∣∑h∈HχV(σh). With this notation we have Tr(Frobq:Vy→Vy)=χV(σyH), (4.4) which we denote simply by χV(Frobq;Vy); note that this definition is independent of our choice of basepoint y˜. More generally: Definition 4.1 For any class function P, and any y∈Y fixed by Frobq, define P(y) by P(y)≔1∣H∣∑h∈HP(σyh). (4.5) An elementary check shows that the definitions above are independent of the choice of coset H, since the action of Sn is transitive on fibers. Plugging Equation (4.4) into Equation (4.1) now gives ∑y∈Y(Fq)χV(Frobq;Vy)=∑i=02dim(Y)(−1)iTr(Frobq:Het,ci(Y;V)→Het,ci(Y;V)). (4.6) The right-hand side of (4.6) could be computed from the eigenvalues λij of Frobq on each Het,ci(Y;Qℓ). Typically one only has estimates on ∣λij∣. For example, for Y smooth and proper, the Riemann Hypothesis for finite fields (proved by Deligne) gives that ∣λij∣=qi/2. Many natural examples Y, including many of those we study in this paper, are not proper, and finding the λij is more difficult. Given that we only have general bounds on the eigenvalues of Frobq, to bound the traces of Frobq we must determine the dimensions of each Het,ci(Y;V). To do this, we follow the argument in [4, Section 3.3]. First note that the pullback V˜ of V to Z is trivial. We then compute Het,ci(Y;V)≅Het,ci(Z;V˜)Snbytransfer≅(Het,ci(Z;Qℓ)⊗V)SnbytrivialityofV˜∣Z≅(Het2dim(Z)−i(Z;Qℓ(dim(Z)))*⊗V)SnbyPoincarèduality≅Het2dim(Z)−i(Z;Qℓ(dim(Z)))*⊗Qℓ[Sn]V. (4.7) Because every Sn-representation is self-dual, it follows that dimQℓHet,ci(Y;V)=⟨V,Het2dim(Z)−i(Z;Qℓ)⟩Sn, (4.8) where ⟨V,W⟩Sn is the usual inner product of Sn-representations V and W ⟨V,W⟩=dimQℓHomQℓ[Sn](V,W). 4.2. Co-FI schemes with convergent étale cohomology Now that we have discussed schemes, and Sn-schemes, we are ready to discuss sequences of Sn-schemes. Let Z be a co-FI scheme, smooth and quasi-projective over Z[1/N]. For each i≥0, the étale cohomology Heti(Z•/F¯q;Qℓ) is an FI-module over Qℓ. We want to consider the implications of finite generation of this FI-module for point-counting problems over Fq for the sequence of schemes Zn/Sn (cf. [23, Theorem p. 63 and Remark p. 65]). As discussed in [3], any partition λ of any k≥1 determines a finitely generated FI-module V(λ) with V(λ)n being the irreducible representation of Sn corresponding to the partition (n−∣λ∣)+λ. Definition 4.2 Let K be a field, and let M• be a finitely generated FI- Gal(K¯/K)-module over Qℓ with stable range N. Let λ be a partition of n, let V(λ) the associated FI-module, and let D=max{N,λ1}. Define the stable λ-isotypic part Mλof M to be the Gal(F¯q/Fq)-module Mλ≔MD⊗Qℓ[SD]V(λ)D. More generally, for a character polynomial P, we define the stable P-isotypic part of M to be the Qℓ-virtual Galois module MP obtained as a linear combination of the Mλ, with the sum taken in the representation ring of Gal(K¯/K) with Qℓ-coefficients. Lemma 2.2 shows that for n≥D, there are canonical Galois-equivariant isomorphisms Mn⊗Qℓ[Sn]V(λ)n→≅Mn+1⊗Qℓ[Sn+1]V(λ)n+1 and similarly for the stable P-isotypic parts for n≥D. We can now give the following theorem, which generalizes earlier special cases by Ellenberg–Venkatesh–Westerland [13], Ellenberg [12], and Church–Ellenberg–Farb [4]. Its proof is along the exact same lines of the previous proofs. We hope that the generality of the statement here will be useful in future work. Theorem 4.3 (Convergent Grothendieck–Lefschetz) Let Zbe a smooth, quasi-projective co-FI over Fq, and set Yn≔Zn/Sn (we do not assume Ynsmooth over Z[1/N]). Assume that for each i≥0the FI-module Heti(Zn/F¯q;Qℓ)is finitely generated, and, for a character polynomial P, denote by Heti(Z)P*the dual of the stable P-isotypic part. If Zhas convergent étale cohomology over F¯q, then for any character polynomial P: limn→∞q−dimYn∑y∈Yn(Fq)P(y)=∑i=0∞(−1)iTr(Frobq⥀Heti(Z)P*), (4.9)and, taking the absolute value limn→∞q−dimYn∣∑y∈Yn(Fq)P(y)∣≤∑i=0∞⟨P,Heti(Z)⟩qi/2<∞. (4.10) If Zonly has weakly convergent cohomology with convergence rate λ, then (4.9) and (4.10) hold for all q>λ. Remark 4.4 Specializing Theorem 4.3 to the case P=1 gives limn→∞q−dimYn∣Yn(Fq)∣=∑i=0∞(−1)iTr(Frobq⥀Heti(Y)*), (4.11) where Heti(Y)* denotes the stable rational étale cohomology of the sequence Y1,Y2,…. The bound (4.10) is sharp, as is seen by taking Zn=(P1)n, Yn=Pn, and P=1. Proof of Theorem 4.3 Because all of the equations in the statement of the theorem are Qℓ-linear in P, it suffices to prove the theorem for P=Pλ, the character polynomial of the finitely generated FI-module V(λ)(cf. Theorem 2.3 above). Let Vn correspond to the twisted sheaf on Yn corresponding to the representation V(λ)n. We show that the left side of (4.9) converges by showing that the sequences n↦q−dimYn∑y∈Yn(Fq)P(y) (4.12) is Cauchy. To start, note that ∑y∈Yn(Fq)P(y)=∑i=02dim(Yn)(−1)iTr(Frobq:Het,ci(Yn;Vn)→Het,ci(Yn;Vn))=∑i=02dim(Zn)(−1)iTr(Frobq⥀Het2dim(Zn)−i(Zn;Qℓ(dim(Zn)))*⊗Qℓ[Sn]V(λ)n)byEquation(4.7)=∑i=02dim(Zn)(−1)iTr(Frobq⥀Heti(Zn;Qℓ(dim(Zn)))*⊗Qℓ[Sn]V(λ)n*), (4.13) where the last equation uses the self-duality of Sn-representations. Denote by N(n,P) the slope of stability of Het*(Z•;Qℓ) for V(λ), that is the number such that for all i≤N(n,λ), Heti(Zn;Qℓ)⊗Qℓ[Sn]V(λ)n≅Heti(Z)P. Let FP(i) denote the subexponential function in Definition 3.1 guaranteed by the assumption that Z has convergent étale cohomology. Then, for n>m ∣q−dimYn(∑y∈Yn(Fq)P(y))−q−dimYm(∑y∈Ym(Fq)P(y))∣=∣∑i=02dim(Zn)(−1)iq−dim(Zn)Tr(Frobq⥀Heti(Zn;Qℓ(dim(Zn)))*⊗Qℓ[Sn]V(λ)n*)−∑i=02dim(Zm)(−1)iq−dim(Zm)Tr(Frobq⥀Heti(Zm;Qℓ(dim(Zm)))*⊗Qℓ[Sm]V(λ)m)∣byEquations(4.1)and(4.13)≤∑i=0∞q−i/2∣⟨P,Heti(Zn;Qℓ)⟩−⟨P,Heti(Zm;Qℓ)⟩∣byDeligne=∑i=N(m,P)∞q−i/2∣⟨P,Heti(Zn;Qℓ)⟩−⟨P,Heti(Zm;Qℓ)⟩∣byétalerepresentationstability≤∑i=N(m,P)∞2q−i/2FP(i)byconvergentcohomology. Because N(m,P) tends to ∞ with m, and because FP(i) is subexponential in i, we see that the sequence (4.12) is Cauchy. Similarly, we see that the right-hand side of (4.9) ∑i=0∞(−1)iTr(Frobq⥀Heti(Z)P*) converges as a consequence of the existence of the stable P-isotypic part, Deligne’s bounds on the eigenvalues of Frobq and the existence of the subexponential bounds FP(i). It remains to show that the two limits agree. For this, we have ∣q−dimYn∑y∈Yn(Fq)P(y)−∑i=02dim(Zn)(−1)iTr(Frobq⥀Heti(Z)P*)∣=∣∑i=02dim(Zn)(−1)i(Tr(Frobq⥀Heti(Zn;Qℓ(dim(Zn)))*⊗Qℓ[Sn]V(λ)n*)−Tr(Frobq⥀Heti(Z)P*))∣byEquation(4.13)=∣∑i=N(n,P)+12dim(Zn)(−1)i(Tr(Frobq⥀Heti(Zn;Qℓ(dim(Zn)))*⊗Qℓ[Sn]V(λ)n*)−Tr(Frobq⥀Heti(Z)P*))∣byétalerepresentationstability≤∑i=N(n,P)+12dim(Zn)2q−i/2FP(i)byDeligneandconvergentcohomology. (4.14) Because FP(i) is subexponential in i, we conclude that (4.14) becomes arbitrarily small as n approaches ∞, which proves the theorem.□ We can now prove Theorem A from the introduction, as well as the following. Theorem 4.5 (Statistics for SymnX). TheoremAwith PConfn(Y) (resp. UConfn(Y)) replaced by Yn (resp. Symn(Y)) holds. Proof of Theorems A and 4.5 Theorem B gives that, for each i≥0, the FI-modules Het*(X•;Qℓ) and Het*(PConf•(X);Qℓ) are finitely generated. Corollary 3.3 (resp. Theorem 3.4) gives that the singular cohomology H*(X•;Q) (resp. H*(PConf•(X);Q)) of the co-FI scheme X• (resp. PConf•(X)) is convergent. Now apply Theorem 4.3.□ In special cases, it is possible to compute the right-hand side of Equation (4.9) explicitly. Example 4.6 When X=Ar, we can explicitly compute polynomial statistics on UConfn(Ar), extending the main theorem of [4]. Indeed, the computations of Arnol’d [1] and F. Cohen [9, Section 2] combine with results of Björner–Ekedahl [2, Theorem 4.9] to show that Het*(PConfn(Ar)F¯q;Qℓ) is a graded algebra generated by classes in degree 2r−1 with eigenvalues of Frobq equal to qr. As a result, for any character polynomial Pλ: Tr(Frobq:Heti(UConfn(Ar);V)*→Heti(UConfn(Ar);V)*)={0ifi≠k(2r−1)q−kr⟨Pλ,Heti(PConfn(Ar);Qℓ)⟩ifi=k(2r−1). Here we have, as above, applied Poincarè duality to replace the compactly supported cohomology of the smooth schemes UConfn(Ar) with (the Tate twist of) the dual of ordinary étale cohomology. We thus have, for all P, limn→∞q−nr∑y∈UConfn(Ar)(Fq)P(y)=∑i=0∞(−1)i(2r−1)q−ir⟨P,Heti(2r−1)(PConf(Ar))⟩. Funding B.F. is supported in part by NSF Grant nos. DMS-1105643 and DMS-1406209. J.W. was supported in part by NSF Grant no. DMS-1400349. Acknowledgements It is a pleasure to thank George Andrews, Kathrin Bringmann, Mark Kisin, and Ken Ono for useful discussions. We thank Nir Gadish and Brian Conrad for very helpful conversations about twisted-adic sheaves and transfer. We thank Weiyan Chen, Jordan Ellenberg and Burt Totaro for numerous helpful comments on an earlier draft. It is a pleasure to thank Matt Emerton for his careful reading and many detailed comments on an earlier draft of this paper. Finally, we thank an anonymous referee for making several suggestions that helped to simplify the proofs in this paper. References 1 V. I. Arnol’d , The cohomology ring of the colored braid group , Math. Notes 5 ( 1969 ), 138 – 140 . Google Scholar Crossref Search ADS 2 A. Björner and T. Ekedahl , Subspace arrangements over finite fields: cohomological and enumerative aspects , Adv. Math. 129 ( 1997 ), 159 – 187 . 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