ABSTRACT EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS

Transcription

1 ABSTRACT Title of dissertation: EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS Jonathan A Huang, Doctor of Philosophy, 2017 Dissertation directed by: Professor Niranjan Ramachandran Department of Mathematics We provide a formula for the generating series of the Weil zeta function Z(X, t) of symmetric powers Sym n X of varieties X over finite fields. This realizes the zeta function Z(X, t) as an exponentiable measure whose associated Kapranov motivic zeta function takes values in W (R) the big Witt ring of R = W (Z). We apply our formula to compute Z(Sym n X, t) in a number of explicit cases. Any motivic zeta function ζ µ of a measure µ factoring through the Grothendieck ring of Chow motives is itself exponentiable; in fact, this applies to ζ µ as a motivic measure itself. We prove a condition for which any motivic measure taking values in a λ-ring has an associated motivic zeta function Z = ζ µ that is itself an exponentiable measure µ Z = Z, and this process is shown to iterate indefinitely. This involves a study of the case of λ-ring-valued motivic measures. Finally, we provide an understanding of MacDonald s formula in this context.

2 EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS by Jonathan Andrew Huang Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2017 Advisory Committee: Professor Niranjan Ramachandran, Chair/Advisor Professor Amin Gholampour Professor Jonathan Rosenberg Professor Lawrence Washington Professor William Gasarch

3 c Copyright by Jonathan Andrew Huang 2017

4 Dedication To my parents. ii

5 Acknowledgments I am deeply indebted to my thesis adviser Niranjan Ramachandran for suggesting this project, for eagerly and openly sharing his ideas and insights, and for being an extremely patient and understanding academic guide throughout my graduate studies. I want to express my appreciation to the Department of Mathematics at the University of Maryland, College Park in particular to helpful discussions with Professors Larry Washington, Jonathan Rosenberg, and Patrick Fitzpatrick, as well as fellow students E. H. Brooks, D. Karpuk, S. Rostami, S. Schmieding, and A. Sehanobish. I am also grateful to my friends and family whose support was essential in allowing me to complete this project. In particular, I am very lucky have had the constant and enduring love and support of my parents and my siblings Christine, Jennifer, and Justin. Finally, I would like to thank my dear friend Sean Kelly for the invaluable help, encouragement, advice, and companionship he has provided throughout these many years. iii

6 Table of Contents Introduction 1 1 Preliminaries Lambda rings and Witt rings The Grothendieck Ring of Varieties Motivic Measures and Motivic Zeta Functions Exponentiable Motivic Measures The Case of Varieties over Finite Fields Exponentiation of the Weil zeta function Examples Proof of the Main Theorem Motivic Measures with Values in Lambda-Rings Lambda-ring valued motivic measures Exponentiation of motivic zeta functions MacDonald formulae Appendices 55 A Symmetric Polynomials 56 A.1 The universal polynomials P n and P n,m A.2 The complete homogeneous and power symmetric polynomials A.3 Relation to vector spaces B Symmetric Powers 61 B.1 Symmetric powers of vector spaces B.2 Symmetric powers of graded vector spaces B.3 Symmetric powers of varieties Bibliography 65 iv

7 Introduction Zeta functions play a central role in number theory; the theme of many conjectures is that the special values of certain zeta functions, which are analytically defined objects, can be expressed in terms of algebraic invariants. These algebraic invariants (l-adic cohomology, Weil-étale cohomology,...) satisfy usual properties, for example they are typically additive and multiplicative (via a Künneth formula, for instance). These are so-called Euler-Poincaré characteristics on the category of certain spaces (varieties over Spec k, schemes of finite type over Spec Z,...). If we view the zeta function as a map on the category of such spaces, then it is natural to ask if zeta functions themselves can be considered Euler-Poincaré characteristics. This would shed light on why the special values of zeta functions take the form of Euler characteristic formulas: the zeta function is itself an Euler characteristic. To better understand the relation of zeta functions to these algebraic invariants, an increasingly important object of study in recent years has been the Grothendieck ring of varieties K 0 (Var k ). This is the value group of the universal Euler- Poincaré characteristics on the category Var k of varieties over Spec k, as Heinloth describes in [12]. Ring homomorphisms on K 0 (Var k ) are called motivic measures. Given a motivic measure µ, Kapranov in [15] constructs a motivic zeta function 1

8 ζ µ (X, t) for X Var k using these algebraic invariants µ(x). It is the generating series for (the measure of) symmetric powers Sym n X: for µ : K 0 (Var k ) R, ζ µ (X, t) = µ(sym n X)t n R[[t]]. n=0 This is a group homomorphism on K 0 (Var k ) taking values in the group of invertible power series (1 + tr[[t]], ). To understand ζ µ as an Euler-Poincaré characteristic, we must study its (possible) structure as a ring homomorphism. Indeed, the Grothendieck ring of varieties is a ring where multiplication is induced by X k Y. We can endeavor to describe the product ζ µ (X k Y ) using a product structure on (1 + tr[[t]], ). As it turns out, the correct product structure is often given by the big Witt ring W (R). Take for instance the case of varieties over finite fields, Var Fq. The counting measure µ # is the motivic measure that counts the number of points over F q : given a variety X Var Fq, set µ # (X) = #X(F q ), taking values in the ring Z. The zeta function of a variety over a finite field is the Kapranov motivic zeta function associated to the counting measure, Z(X, t) = ζ µ#. In [22], Ramachandran points out that for varieties over finite fields, the zeta function Z(X Y, t) is the product Z(X, t) W Z(Y, t) in the Witt ring W (Z). This leads to the following definition: a measure µ : K 0 (Var k ) R is exponentiable (see [22] and [23]) if the product structure of K 0 (Var k ) is reflected by the product in W (R). That is, motivic zeta functions (of exponentiable measures) are Euler-Poincaré characteristics ζ µ : K 0 (Var k ) W (R) taking values in the big Witt ring. In this thesis, we answer several questions posed in [22]. Specifically, our main 2

9 results are as follows: We prove a formula for the generating series of the Weil zeta function Z(X, t) of symmetric powers Sym n X of varieties X over finite fields (Theorem 2.3). This shows that the induced motivic zeta function measure µ Z is exponentiable for Var Fq. We prove various conditions for lambda-ring-valued exponentiable motivic measures µ to have an induced motivic zeta function measure µ Z that is itself exponentiable, and show that this process iterates (Theorem 3.5, Corollary 3.6, and Theorem 3.8). We relate these results to the classical MacDonald formula in the setting of λ-ring valued measures and argue that the formula in Theorem 2.3 provides a MacDonald formula for the Weil zeta function Z(X, t) (Section 3.3). As the motivic zeta function ζ µ can be thought of as a motivic measure, we can ask whether ζ µ is itself an exponentiable measure. We show that this is the case for varieties over finite fields Var Fq and the Weil zeta function Z(X, t). We prove a formula reminiscent of the Euler-Poincaré characteristic formula for the Weil zeta function as an alternating sum of Teichmüller elements [α]: Z(X, t) = i,j ( 1) i+1 [α ij ] where this sum is taking place in the Witt ring W (Z). As Z(X, t) = ζ µ# (X, t) takes values in the ring A = W (Z), we show that the zeta function measure µ Z (X) = Z(X, t) is exponentiable and its associated motivic zeta function ζ µz takes values in 3

10 W (A) = W (W (Z)). To do so, we provide a closed formula for the generating series of the zeta function of symmetric powers, ζ µz (X, u) = Z(Sym n X, t)u n. n=0 Specifically, the formula put forth in Theorem 2.3 presents ζ µz in W (W (Z)) as a sum of double Teichmüller elements [[ [α] ]]: ζ µz (X, u) = i,j ( 1) i+1 [[ [α ij ] ]]. We then generalize these results for µ = µ # and varieties over finite fields to the general case of abstract motivic measures µ. We prove that if µ is any motivic measure taking values in a λ-ring R and its motivic zeta function factors through µ via the opposite λ-ring structure σ t : R W (R), the motivic zeta function is itself an exponentiable measure µ Z (X) := ζ µ (X, t) with its associated motivic zeta function ζ µz taking values in W (W (R)). This involves a study of λ-ring-valued motivic measures. The idea of studying the Grothendieck ring of varieties using motivic measures is well-established in the literature. Moreover, the importance of λ-ring-valued motivic measures is observed in Larsen and Lunts [17], Heinloth [12] [13], Gusein- Zade, Lluengo, and Melle-Hernandez [10] [9], del Baño Rollin [2] [3], and Maxim and Schurmann [20]. Although Witt-type phenomena of zeta functions appears in prior studies, identifying the Kapranov motivic zeta function as an object in the big Witt ring is only earnestly present in Ramachandran [22] and Ramachandran- Tabuada [23]. In this work, we make heavy use of the fact that the big Witt ring W (R) is a λ-ring and study zeta functions as measures taking values in this ring. 4

11 We begin in Chapter 1 with a brief survey of preliminary results on λ-rings, Witt rings, motivic measures and motivic zeta functions. We then analyze the finite field case, proving a generating series formula for the Weil zeta function of symmetric powers of a variety, and we then apply this formula in a number of explicit cases (Chapter 2). This shows that the zeta function for varieties over finite fields, when considered as a motivic measure, is exponentiable. In Chapter 3 we study exponentiation of λ-ring-valued motivic measures along the lines of Ramachandran and Tabuada in [23]. We show for certain motivic measures, the induced motivic zeta function measure µ Z = ζ µ is exponentiable, and this process iterates indefinitely. Finally, we contrast this with the existence of a MacDonald formula by reframing these in the setting of λ-rings. 5

12 Chapter 1 Preliminaries This chapter is a survey of some well-known results on λ-rings and Witt rings, the Grothendieck ring of varieties K 0 (Var k ), and motivic measures and the Kapranov motivic zeta functions. We end with an explanation of what it means for a motivic measure to be exponentiable. 1.1 Lambda rings and Witt rings We begin with a review of the theory of λ-rings and Witt rings. For a more comprehensive introduction, the reader may consult Knutson [16], Yao [25], and Hazewinkel [11]. Grothendieck originally introduced the general concept of a λ-ring in order to study the Riemann-Roch theorem. Let A be a commutative ring with identity. We denote by Λ(A) the following subgroup of invertible power series under the usual multiplication of power series: Λ(A) := (1 + ta[[t]], ). We will seek to endow Λ(A) with a product structure in the sequel; however, unless otherwise stated, Λ(A) will mean the abelian group under multiplication of power series. 6

13 Given a ring map f : R S, the induced map Λ f : Λ(R) Λ(S) acts on each coefficient: Λ f (a 0 + a 1 t + a 2 t 2 + ) = f(a 0 ) + f(a 1 )t + f(a 2 )t 2 + Note that f needs to be a ring homomorphism in order for Λ f to be a group homomorphism. We shall make heavy use of the ghost map gh : Λ(A) A N. Given P (t) Λ(A), we define gh(p (t)) = (b 1, b 2, b 3,...), where t dp P dt = b n t n. n=1 This is a functorial group homomorphism. Essentially, instead of considering the power series P (t) = a 0 + a 1 t + a 2 t 2, we use another sequence of coordinates (b 1, b 2,...) to represent P (t) in A N. These coordinates b n are called the ghost coordinates gh n of P (t) where gh n (P (t)) = b n where gh n : A A. As we will see, it is often more convenient to use the ghost coordinates of a power series. The relation between the power series coefficients and the ghost coordinates can be made explicit: Lemma 1.1. For P (t) = n a nt n Λ(A) and b n = gh n (P (t)), we have na n = b n + a 1 b n 1 + a n 1 b 1. This relation uniquely determines P (t) in terms of its ghost coordinates b n in the case where A has no Z-torsion. 7

14 Proof. By definition, t dp P dt = n=1 b nt n so that ( ) P (t) b n t n = t dp dt n=1 (a 0 + a 1 t + a 2 t 2 + )(b 1 t + b 2 t 2 + ) = t(a 1 + 2a 2 t + 3a 3 t 2 ) = a 1 t + 2a 2 t 2 + 3a 3 t 3 + and we simply identify coefficients. Lemma 1.2. For A = Z, the ghost coordinates of P (t) = exp [ r=1 ] t r b r r are given by gh(p ) = (b 1, b 2, b 3,...). This is true for any ring A where exp makes sense. For example, ( ) [ ] 1 = exp a r tr 1 at r r=1 and ( ) [ a ] 1 = exp a tr. 1 t r r=1 and thus gh r ((1 at) 1 ) = a r and gh r ((1 t) a ) = a Multiplicative structure on Λ(A) The first multiplicative structure we will study on Λ(A) comes from certain universal polynomials. Given two power series f(t) = i a it i and g(t) = i b it i, we define their product as the power series f(t) Λ g(t) = n P n (a 1,..., a n ; b 1, b 2,..., b n )t n 8

15 where the P n are the universal polynomials described in Appendix A. For example, P 0 = 1 P 1 (a 1 ; b 1 ) = a 1 b 1 P 2 (a 1, a 2 ; b 1 b 2 ) = a 2 1b 2 2a 2 b 2 + a 2 b 2 1 This commutative product is inspired by the behavior of the exterior product Λ n V on vector spaces V (see Appendix A). We will refer to this as the universal multiplicative structure on Λ(A) Lambda-rings We wish to define operations λ n on A that behave like exterior powers on vector spaces, and we will use the power series coefficients in Λ(A) to enumerate them. Definition 1.3. A pre-λ-ring structure on A is a group homomorphism λ t : A Λ(A) denoted by λ t (a) = 1 + λ n (a)t n, such that λ 1 (a) = a. n=1 The group homomorphism requirement is equivalent to λ t (a + b) = λ t (a)λ t (b) the product of power series. Thus we obtain a sequence of maps λ n : A A such that λ n (a + b) = λ i (a)λ j (b) for all n. i+j=n The pair (A, λ t ) is called a pre-λ-ring. 9

16 Given two pre-λ-rings (A, λ t ) and (B, λ t), a ring homomorphism f : A B is a map of pre-λ-rings if it preserves the pre-λ-ring structure map. That is, Λ f λ t = λ t f, i.e. commutes. f A B λ t Λ(A) Λ f λ t Λ(B) Examples. Here are a few examples of rings with pre-λ-ring structure: Z with λ t (a) = (1 + t) a for a Z. In this case, the λ maps are given by binomial coefficients λ n (a) = R with λ t (x) = (1 + t) x. a. n R with λ t (x) = e xt. K 0 (F ) the Grothendieck ring of (isomorphism classes of) finite dimensional (virtual) vector spaces over a field F. In this case, λ n [V ] = [Λ n V ]. Definition 1.4. Given a pre-λ-ring A, the nth Adams operation Ψ n : A A is defined as t dλ t (a) λ t (a) dt = ( 1) n+1 Ψ n (a)t n n=1 The nth Adams operation is simply (up to ±) the nth ghost coordinate of λ t (a) Λ(A): Ψ n (a) = ( 1) n+1 gh n (λ t (a)). 10

17 Thus, we have Ψ n (a + b) = Ψ n (a) + Ψ n (b), i.e. the Adams operations are group homomorphisms, since λ t is a group homomorphism. Examples. Here are a few examples of Adams operations: In the case of Z with λ t (a) = (1 + t) a, Ψ n (a) = a. In the case of R with λ t (x) = (1 + t) x, Ψ n (x) = x. In the case of R with λ t (x) = e xt, x for n = 1 Ψ n (x) = 0 for n > 1 If λ t (a) = 1 + at, then Ψ n (a) = a n. In the case of K 0 (F ), the Adams operations are the classical Adams operations on vector spaces. Example 1.5. Recall from Section that the abelian group Λ(A) is a commutative ring, given a universal multiplicative structure. It has a natural pre-λ-ring structure denoted (Λ(A), λ u ). For m 1 the λ-ring maps λ m : Λ(A) Λ(A) are given by λ m (1 + a 1 t + a 2 t 2 + ) = 1 + P n,m (a 1, a 2,..., a nm )t n Λ(A) n=1 with λ 0 = 1 and λ u = m 0 λm u m Λ(Λ(A)), where the P n,m are the universal polynomials described in Appendix A. For example, P 0,m = 1 P 1,m (a 1, a 2,..., a m ) = a m 11

18 These polynomials can quickly get complicated. For example, P 4,2 (a 1, a 2,..., a 8 ) = a 1 a 3 a 4 3a 1 a 2 a 5 + a 3 1a 5 a 2 1a 6 + a 1 a 7 + 2a 2 a 6 a 8. For a table of polynomials P n,m, see Hopkinson ( [14]). Proposition 1.6. This pre-λ-ring structure is functorial in the following sense: given any ring homomorphism f : A A, the induced map Λ f : Λ(A) Λ(A ) is a map of pre-λ-rings. Proof. For Q = 1 + a 1 t + a 2 t 2 + Λ(A), we have λ m (Λ f Q) = λ m (f(1) + f(a 1 t + f(a 2 )t 2 + )) = 1 + P n,m (f(a 1 ), f(a 2 ),..., f(a nm ))t n n=1 = f(1) + f(p n,m (a 1, a 2,..., a nm ))t n = Λ f (λ m (Q)) n=1 Definition 1.7. A pre-λ-ring (A, λ t ) is a λ-ring if λ t is a map of pre-λ-rings. Specifically, this means for (Λ(A), λ u ) Λ λt (λ t (a)) = λ u (λ t (a)). and the diagram A λ t Λ(A) λ t Λ(A) Λ λt λ u Λ(Λ(A)) commutes. A map of λ-rings is simply a map of the underlying pre-λ-rings. 12

19 Note. In the literature, pre-λ-rings are sometimes known as λ-rings and λ-rings sometimes known as special λ-rings. Proposition 1.8. A pre-λ-ring (A, λ t ) is a λ-ring if and only if both conditions on Adams operations Ψ n : A A hold Ψ n (ab) = Ψ n (a)ψ n (b) for all n. Ψ n Ψ m = Ψ nm. Proof. The first condition says that λ t is a ring homomorphism. The second says that it is a map of pre-λ-rings. See Knutson [16, p. 49]. Examples. The pre-λ-ring structures described above are λ-rings: Z with λ t (a) = (1 + t) a for a Z is a λ-ring. This is in fact the unique λ-ring structure on Z. R with λ t (x) = (1 + t) x is a λ-ring. R with λ t (x) = e xt is not a λ-ring (see the Adams operations Ψ n (x) in this case). K 0 (F ) with λ n [V ] = [Λ n V ] is a λ-ring. In fact, K 0 (R) is a λ-ring. See Grothendieck [8]. The following propositions will be useful later: Proposition 1.9. The pre-λ-ring structure defined on Λ(A) is a λ-ring structure. Proof. See Knutson [16]. 13

20 Proposition For A a Z-torsion free ring, any λ-ring structure λ t on A is uniquely determined by the Adams operations Ψ n. Moreover, given f : A A a ring homomorphism between λ-rings (A, λ t ) and (A, λ t), f is a λ-ring map if and only if f respects the Adams operations. That is, for Ψ n and Ψ n on A and A respectively, f Ψ n = Ψ n f for all n. Proof. See Yau [25, Thm 3.15 and Cor 3.16]. Proposition If A is a λ-ring then so is the polynomial ring A[z]. For A = Z, we have λ t (a 0 + a 1 z + a 2 z 2 + a n z n ) = i (1 + z i t) a i For P [z] A[z], the Adams operations are Ψ n (P (z)) = P (z n ). Proof. Define the λ-ring structure on A[z] by λ t (z) = (1+zt). Now use the universal polynomials to define the λ-maps on products az i, using the λ-ring structure on A. In particular, for A = Z, λ t (nz) = λ t (z + z + + z) = (1 + zt) n, and a straightforward computation shows we have Ψ n (z) = z n. Definition Given a λ-ring (A, λ t ), the opposite λ-ring structure map σ t is defined as σ t = λ 1 t, with σ n : A A given by σ t (a) = σ n (a)t n. n=0 Note that this is not a ring homomorphism into Λ(A). Whereas the the λ-ring structure map λ t is inspired by exterior powers of vector spaces, the opposite λ-ring structure map corresponds to symmetric powers. Note that Ψ n (a) = gh n (σ t (a)). 14

21 1.1.3 Witt rings We now describe another multiplicative structure on Λ(A) given by the universal ring of Witt vectors. While the universal multiplicative structure on Λ(A) described above is inspired by the behavior of exterior products on vector spaces, the Witt ring is inspired by the behavior of Chern character Ch(V ) of vector bundles V. Our presentation here follows Bloch [1] and Ramachandran [22]. Definition For a commutative ring A with identity, the big Witt ring W (A) has underlying additive group Λ(A) and has multiplication W defined such that for a, b A, (1 at) 1 W (1 bt) 1 = (1 abt) 1. We require that the association A W (A) is functorial the induced map W (f) : W (A) W (A ) given by Λ f is a ring homomorphism and the resulting diagram commutes. This uniquely determines the structure of a commutative ring on W (A). For a A, the Teichmüller element [a] W (A) is [a] = (1 at) 1. Multiplication on W (A) is defined such that [a] W [b] = [ab]. The Teichmüller element [1] is the multiplicative identity in W (A); the additive identity element in W (A) is 1 = [0] A. It is often convenient to recast the big Witt ring in terms of ghost coordinates. In fact, one reason to consider the Witt product is that the ghost map 15

22 (previously defined) induces a ring homomorphism gh : W (A) A N, where A N has a component-wise ring structure. Thus for P and Q in W (A), gh n (P Q) = gh n (P ) + gh n (Q), and gh n (P W Q) = gh n (P ) gh n (Q). Note that on Teichmüller elements, gh n ([a]) = a n. Note. The big Witt ring is traditionally described in terms of Witt coordinates a i A, and P = (a 1, a 2,...) A N. These are related to the ghost coordinates by the Witt polynomials w n = d n d(a d ) n/d = gh n (P ). Thought of as a power series P Λ(A) is recovered from the Witt coordinates as P (t) = (1 a n t n ) 1. n=1 The Witt polynomials w n are used to describe addition and multiplication in the Witt ring. Note. The Witt ring W (A) has descending filtration Filt n W (A) = (1 + t n+1 A[[t]]) and W n (A) = W (A)/Filt n W (A). We have W (A) = lim W n (A), giving the Witt ring an induced topology where each W n (A) is given the discrete topology. Definition For each positive integer n, the Frobenius Fr n : W (A) W (A) is a ring endomorphism defined as Fr m (P (t)) = P (ζt 1/m ) ζ m =1 On Teichmüller elements, Fr n ([a]) = [a n ]. On ghost coordinates, gh m (Fr n (P )) = gh nm (P ) Clearly, we have Fr n Fr m = Fr nm. 16

23 1.1.4 Witt rings of lambda-rings The Witt ring W (A) has a natural λ-ring structure λ u (see Knutson, [16, p. 18]). On Teichmüller elements, it is given by λ 0 ([a]) = [1], λ 1 ([a]) = [a], λ n ([a]) = 1, n > 1 and so λ u ([a]) = [1] + [a]u Λ(W (A)). The Frobenius morphism Fr n : W (A) W (A) is equal to the nth Adams operation Ψ n (see Hazewinkel [11]). Note that the Adams operation requirement Ψ n Ψ m = Ψ nm in Proposition 1.8 is readily satisfied by the Frobenius on W (A), as noted above. To compare the λ-ring structures on the universal λ-ring Λ(A) and the Witt ring W (A), there is a ring isomorphism ι : Λ(A) W (A), P (t) P ( t) 1 induced by the so-called Artin-Hasse exponential. Proposition Given its natural λ-ring structure λ u, W (A) has opposite λ-ring structure map σ u : W (A) W (W (A)) satisfying σ u ([a]) = ([1] [a]u) 1 This may be denoted by the double Teichmüller [[ [a] ]]. Note that this only holds for Teichmüller elements [a] and not arbitrary P (t) W (A). Proposition Given any ring homomorphism f : A A, the induced map W (f) : W (A) W (A ) is a λ-ring map. Proof. See Example 1.5. We are particularly interested in the case of Witt rings W (A) where the commutative ring A is a λ-ring, (A, λ t ). Just as in the case of the λ-ring structure on 17

24 Λ(A), there are two λ-ring structures: (A, λ t ) and (W (A), λ t ). As A is a λ-ring, λ t : A Λ(A) is a ring homomorphism. Using the isomorphism ι : Λ(A) W (A), we see that σ t : A W (A) is also a ring homomorphism. In fact, we have Proposition If A is a λ-ring, its opposite λ-structure map σ t : A W (A) is a λ-ring map. In particular, Λ σt (σ t (a)) = σ u (σ t (a)) where σ u is the opposite λ-structure map on W (A). Notice that Λ σt was also identified previously as W (σ t ). 1.2 The Grothendieck Ring of Varieties Consider Var k, the category of algebraic varieties X over a field k. For us, an algebraic variety over k will mean a reduced scheme of finite type over Spec k. In particular, these are not necessarily separated nor irreducible. For simplicity let k be perfect; thus, for X and Y reduced, the product X k Y is also reduced. Definition The Grothendieck ring of varieties K 0 (Var k ) is the abelian group generated by symbols [X] of isomorphism classes of X Var k subject to the scissor relation [X] = [Y ] + [X \ Y ] for Y any closed subvariety of X. K 0 (Var k ) is a commutative ring under the product [X] [Y ] = [X k Y ]. The idea of studying such a ring originates in a letter of Grothendieck to Serre in 1964; it behaves as a shadow (decategorification) of a (still conjectural) category of motives and is sometimes known as the ring of baby motives. Any 18

25 additive invariant on varieties (known generally as Euler-Poincaré characteristics, see below), must factor through this universal value group K 0 (Var k ). The scissor relation is a violent operation on varieties making it possible to formally add and subtract varieties. For a comprehensive survey of the Grothendieck ring of varieties, see Mustata [21, Chap 7]. Examples. Here are a few examples of elements in K 0 (Var k ): X = Spec k = pt and [pt] K 0 (Var k ) is the multiplicative identity 1. The empty set [ ] K 0 (Var k ) is the additive identity 0. For X = A 1 the affine line, the class [A 1 ] K 0 (Var k ) is often denoted L, the Lefschetz motive. For P n projective space and A n affine space, [P 1 ] = [A 1 ] + [pt] and [P n ] = [A n ] + [A 1 ] + [pt]. We now identify certain important properties of K 0 (Var k ): Proposition For the Grothendieck ring of varieties: Every map of fields k k induces a base change ring homomorphism on b : K 0 (Var k ) K 0 (Var k ). K 0 (Var k ) is additively generated by the classes [X] for X quasi-projective. Proof. See Mustata [21, Lem 7.6, Prop 7.27]. In the case of char k = 0, Bittner provides a presentation of the Grothendieck ring as generated by smooth projective varieties and smooth closed subvarieties. 19

26 Proposition 1.20 (Bittner [12]). The Grothendieck group K 0 (Var k ) is the abelian group generated by isomorphism classes [X] of smooth projective varieties X Var k subject to the scissor relation [X] = [Y ]+[X \Y ] for Y any closed smooth subvariety of X. 1.3 Motivic Measures and Motivic Zeta Functions A motivic measure µ is a map that associates to every X Var k up to isomorphism an element µ(x) in a ring R such that µ(x) = µ(x red ). µ(x \ Y ) = µ(x) µ(y ) for Y a closed subvariety of X. µ(x k Y ) = µ(x)µ(y ). Thus every motivic measure determines a ring homomorphism on the Grothendieck ring of varieties, also denoted µ : K 0 (Var k ) R. These are sometimes described as Euler-Poincaré characteristics, reflecting the theme that all Euler-Poincaré characteristics should factor through K 0 (Var k ), the universal value group. Examples. Here are some examples of motivic measures Consider the case of k = C. Then the (classical) Euler characteristic χ c (X) defined as the the alternating sum of ranks of cohomology with compact support b i (X) = rank H i c(x(c)), χ c (X) = 2m i=0 ( 1) i b i (X) where m is the dimension of X, is a motivic measure taking values in R = Z. 20

27 For X Var C and m the dimension of X, the Poincaré polynomial P (X, z), P (X, z) = 2m i=0 ( 1) i b i (X)z i, defines a motivic measure µ P (X) = P (X, z) taking values in the polynomial ring R = Z[z]. Consider the case of k = F q. The counting measure µ # : K 0 (Var Fq ) Z returns the number of points over F q : µ # (X) = #(X)(F q ). For any k, there is the universal motivic measure u : K 0 (Var k ) K 0 (Var k ), given by the identity map. Given a motivic measure µ, Kapranov in [15] associates to µ a motivic zeta function ζ µ defined on quasi-projective varieties X as ζ µ (X, t) = µ(sym n X)t n n=0 where Sym n X is the nth symmetric power of X. This is a power series in 1+tR[[t]], and as the Grothendieck ring of varieties is additively generated by quasi-projective varieties, it defines a map ζ µ : K 0 (Var k ) Λ(R) Recall this subgroup of invertible power series Λ(R) = (1 + tr[[t]], ) is the underlying additive group for the Witt ring W (R). For all µ, the Kapranov motivic zeta function is a group homomorphism. Note. In the case of char k > 0, in order to properly define the motivic zeta function, we need to consider a quotient K 0 (Var k ) of K 0 (Var k ) where [X] = [Y ] if there exists 21

28 a so-called radicial morphism f : X Y. This is required in order to define a group homomorphism on K 0 (Var k ) using symmetric products Sym n X (as these fail to behave nicely under the scissor relation). See Appendix B or Mustata [21] for details. For char k = 0, we have K 0 (Var k ) = K 0 (Var k ). Moreover, for char k > 0, our focus will be on ζ µ for the counting measure µ = µ #, which factors through K 0 (Var k ). Examples. Here are a few examples of motivic zeta functions: 1. For any motivic measure µ, ζ µ (Spec k, t) = (1 t) 1 and ζ µ (, t) = For µ = u, ζ u is sometimes denoted Z mot the universal motivic zeta function taking values in Λ(K 0 (Var k )). Here we have Z mot (L, t) = 1 (1 Lt) and Z mot (P 1, t) = 1 (1 t)(1 Lt) 1.4 Exponentiable Motivic Measures Since the Grothendieck ring is in fact a ring, it is natural to ask how products behave under the motivic zeta function map. It is pointed out in Ramachandran [22] that in the case of varieties over finite fields, the Weil zeta function takes products in the Witt ring W (Z). Thus one may ask that for the Kapranov motivic zeta function, the product structure of K 0 (Var k ) be reflected by the Witt product structure on Λ(R). Definition A motivic measure µ : K 0 (Var k ) R is exponentiable or expo- 22

29 nentiates 1 if its associated Kapranov motivic zeta function ζ µ defines a ring homomorphism ζ µ : K 0 (Var k ) W (R). Lemma A motivic measure µ is exponentiable if and only if the ghost coordinates gh n (ζ µ ) are multiplicative for all n. Example Let k = C. Consider µ P (X) = P (X, z) the Poincaré polynomial measure. Then a classical formula of MacDonald [18] provides a closed form for the generating series of symmetric powers P (Sym n X, z): P (Sym n X, z)t n = (1 z1 t) b 1 (1 z 3 t) b3 (1 z 2m 1 t) b 2m 1 (1 z 1 t) b 1 (1 z2 t) b 2 (1 z 2m t) b 2m [ ] = exp P (X, z r ) tr r n=0 r=1 where m is the dimension of X. This shows that gh n (ζ µp (X, t)) = P (X, z n ), the Poincaré polynomial in z n, which is multiplicative in X, and thus P (X, t) is exponentiable. Moreover, the associated motivic zeta function may be written in the Witt ring W (Z[z]) as ζ µp (X, t) = 2m i=1 ( 1) i b i (X)[z i ]. Example The previous example specializes to the case of µ = χ c where the formula due to MacDonald [18] takes the following form χ c (Sym n X)t n = n=1 ( ) [ χc(x) ] 1 = exp χ c (X) tr. 1 t r r=1 1 This terminology comes from the observation that the symmetric power Sym n X corresponds to xn n! since n! is the size of the symmetric group S n and the motivic zeta function thus corresponds to e x = n=0 xn n!. 23

30 Thus the motivic zeta function ζ χc may be written in the Witt ring W (Z) as χ c (X)[1] (and its ghost coordinates are constant χ c (X)), and exponentiability follows from the fact that the measure χ c is a ring homomorphism. There are many benefits of understanding the multiplicative structure expressed by the motivic zeta function. The class of exponentiable motivic measures is also well-behaved. Here are a few of the results detailed in [22] and [23]: Corollary If µ is an exponentiable motivic measure µ : K 0 (Var k ) R, then ζ µ is a motivic measure ζ µ : K 0 (Var k ) W (R). Proof. The zeta function ζ µ is always a group homomorphism; µ being exponentiable makes it a ring homomorphism. Corollary If µ is an exponentiable motivic measure and µ factors through µ, then µ is an exponentiable measure. Proof. Suppose that µ : K 0 (Var k ) R and µ : K 0 (Var k ) S with K 0 (Var k ) µ R f µ S for f : R S some ring homomorphism. Then ζ µ (X, t) = µ (Sym n X)t n = n=0 f µ(sym n X)t n = W (f)(ζ µ ) Thus ζ µ is a composition of ring homomorphisms W (f) and ζ µ. n=0 Proposition For µ an exponentiable measure, if ζ µ (X) and ζ µ (Y ) are rational functions, then so is ζ µ (X Y ). 24

31 Proof. The subset of rational functions in W (R) is an ideal, in particular, closed under multiplication. In the following chapters, we will be exploring certain examples of motivic measures which are exponentiable. We end this chapter with an example of a nonexponentiable measure. Example For k = C and X a smooth projective surface of Kodaira dimension 0, Larsen-Lunts construct a measure µ LL such that the associated motivic zeta function ζ µll (X, t) is not rational (Theorem 7.6 in [17]). However, the universal motivic zeta function for a smooth projective curve C is always rational (see [15]), and so ζ µ (C, t) is rational for any motivic measure µ. Applying this to the smooth projective surface C 1 C 2 implies that ζ µll cannot take values in the Witt ring, since in the Witt ring, the product of rational functions is rational. Note. The Grothendieck ring of varieties has some interesting properties which can be probed using motivic measures and motivic zeta functions. For instance, K 0 (Var k ) is a pre-λ-ring via the universal motivic zeta function ζ u ζ u (X) = [Sym n X]t n Λ(K 0 (Var k )). n=0 This is simply because ζ u (X) is a group homomorphism ζ u : K 0 (Var k ) Λ(K 0 (Var k )). However, this motivic zeta function is not a λ-ring structure map since it is not necessarily a ring homomorphism into W (K 0 (Var k )). For instance, in the case k = C the construction of the Larsen-Lunts measure suffices to show ζ u is not a suitable (opposite) λ-structure map since ζ µ = Λ µ ζ u and Λ µ preserves rationality. See [17] Section 8. 25

32 Chapter 2 The Case of Varieties over Finite Fields In this chapter, we provide a formula for the generating series of the zeta function Z(X, t) of symmetric powers of varieties over finite fields. The zeta function Z(X, t) is the Kapranov motivic zeta function ζ µ# associated to the counting measure µ # and counting measure µ # is exponentiable; that is, Z(X, t) takes values in the big Witt ring W (Z). We use our formula to prove that the zeta function, when thought of as a motivic measure itself µ Z (X) = Z(X, t), is also exponentiable, so that its motivic zeta function ζ µz takes values in W (W (Z)). A number of explicit examples of zeta functions of symmetric powers are computed. Given a variety X over a finite field F q, the Weil zeta function (also known as the Hasse-Weil zeta function or simply the zeta function) is defined as ζ X (s) = (1 N(x) s ) 1 x X where the product ranges over all closed points x X and N(x) is the degree of the residue field κ(x). Setting t = q s, we may rewrite the zeta function as ζ X (s) = Z(X, q s ), Z(X, t) = [ ] (1 t deg x ) 1 = exp N r (X) tr 1 + tz[[t]] (2.1) r x X r=1 where N r (X) = #X(F q r) the number of points over F q r. 26

33 Due to the work of Grothendieck and others on the Weil conjectures, we can understand the zeta function in terms the action of the Frobenius on (compactly supported) étale cohomology. Namely, there is an algebraic Lefschetz fixed point theorem, for l p, whereby the number of points is given by an alternating trace N r (X) = i ( 1) i Tr(Φ r H i et,c(x; Q l )) for Φ the Frobenius on X. This can be used to show that Z(X, t) = i P 2i+1(t) i P 2i(t), P i(t) = j (1 α ij t) 1 (2.2) where α ij are the inverse eigenvalues of the Frobenius action Φ on H i et,c(x; Q l ). These are so-called Weil q-numbers α i,j = q r/2 for some r i (Deligne) and Z(X, t) is a rational function (Dwork). The Weil zeta function Z(X, t) is the Kapranov motivic zeta function associated to the counting measure µ #. This was Kapranov s initial motivation as well as justification for calling the generating series for symmetric powers a zeta function. Proposition 2.1. For X a variety over F q, ζ µ# (X, t) = #(Sym n X)(F q )t n = Z(X, t) n=0 Proof. We may rewrite the zeta function as Z(X, t) = x X 1 (1 t deg x ) = Y X t deg Y where Y ranges over all effective zero cycles Y X. These are classified by #(Sym n X)(F q ). See Mustata [21, Prop 7.31]. 27

34 From the form of Z(X, t) in (2.1), it is clear that the zeta function is additive over closed subvarieties Y X in Λ(Z). The zeta function Z(X, t) takes values in W (Z); that is, the zeta function of a product is the Witt product of the zeta functions. Again, from (2.1) we have that gh r (Z(X, t)) = N r (X) (see Lemma 1.2) and N r (X) is multiplicative which suffices to prove that µ # is exponentiable. In fact, the following is shown in [22, Thm 2.1]: Proposition 2.2. For X and Y Var Fq, the zeta function of the product X Fq Y is the Witt product of the zeta functions Z(X Fq Y, t) = Z(X, t) W Z(Y, t). Moreover, the Frobenius operator Fr n on W (Z) corresponds to base change X Fq n, so that Fr n (Z(X/F q, t)) = Z(X/F q r, t). The base change b m to F q m induces a map on the Grothendieck ring, and the following K 0 (Var Fq) b m K 0 (Var Fq m) Z W (Z) Fr m Z W (Z) is a commutative diagram of ring homomorphisms. 2.1 Exponentiation of the Weil zeta function W (Z) as The Weil zeta function of a variety X over F q can be written in the Witt ring Z(X, t) = i,j ( 1) i+1 [α ij ] 28

35 where the sum is taking place in the Witt ring and [α] is the Teichmüller element (1 αt) 1. This directly follows from the presentation in (2.2) of the zeta function as an alternating product of polynomials P i = (1 α ij t) 1 and Proposition 2.2 above. Although each Teichmüller element [α ij ] is in W (Q l ), the sum nevertheless lies in the subring W (Z). We now state our main result. Theorem 2.3 (Main Theorem). The Weil zeta function of a variety X over F q, when considered as a motivic measure µ Z (X) = Z(X, t), has an associated motivic zeta function ζ µz taking the form ζ µz (X, u) = n Z(Sym n X, t)u n = i,j ( 1) i+1 [[ [α ij ] ]] where the second sum is taking place in the Witt ring W (W (Q l )). This can be viewed as a closed product formula for the generating function for zeta series of symmetric powers. We call the induced measure µ Z the Weil zeta function measure. 2.2 Examples Before proving the main theorem, we exhibit some explicit examples. We apply the generating series formula to compute Z(Sym n X, t) for various cases of varieties X over finite fields. 29

36 2.2.1 Affine and projective space Consider n-dimensional affine space A n and n-dimensional projective space P n over F q. Then N r (A n ) = q nr and we have Z(A n, t) = 1 (1 q n t) It is also easy to show that N r (P n ) = q nr + q (n 1)r + + q r + 1 so that Z(P n, t) = 1 (1 t)(1 qt) (1 q m t) In the Witt ring W (Z), these zeta functions are Z(A n, t) = [q n ] and Z(P n, t) = [q m ] + W [q m 1 ] + W + W [q] + W [1]. Recall that Sym n A 1 = A n and Sym n P 1 = P n, and that Z(A 1, t) = 1 1 qt = [q] and Z(P1, t) = 1 (1 t)(1 qt) = [1] + W [q] The formula provided in Theorem 2.3 predicts that Z(Sym n A 1, t) is the coefficient of u n in [[ [q] ]] W (W (Z)). Similarly, that Z(Sym n P 1, t) is the coefficient of u n in [[ [1] ]] + W [[ [q] ]] W (W (Z)). We have and [[ [q] ]] = 1 [1] [q]u = [1] + [q]u + [q2 ]u 2 + ( ) ( ) 1 1 [[ [1] ]] + W [[ [q] ]] = = [1] + ([1] + [q])u + [1] [1]u [1] [q]u which agrees with the zeta functions Z(A n, t) and Z(P n, t) described above: 30

37 2.2.2 Elliptic curves Let E be an elliptic curve over F q. In this case, we have Z(E, t) = (1 αt)(1 βt) (1 t)(1 qt) where α + β = a Z and αβ = q. Written in the Witt ring W (Z), this appears as Z(E, t) = [1] [α] [β] + [q], where [α] = (1 αt) 1 the Teichmüller element. The formula provided by Theorem 2.3 is Z(Sym n E, t) = coefficient of u n in [ ] (1 [α]u)(1 [β]u)). (1 [1]u)(1 [q]u)) The results in the section in fact work for any curve C over F q. In this general case, the eigenvalues α and β for the action on H 1 (C, Q l ) come in pairs α i and β i for i = 1, 2,..., g where g is the genus. In fact, the case of symmetric powers of smooth projective curves was worked out by MacDonald in [19] in Here we work with the elliptic curve case for simplicity Small cases n = 2, 3 In the case of Sym 2 E, the formula states that Z(Sym 2 E, t) = (1 αt)(1 βt)(1 qαt)(1 qβt) (1 t)(1 qt)(1 qt)(1 q 2 t) The ghost coordinates of Z(X, t) are given by the number of points over F q r, which we denote by N r (X). We verify the formula by counting the number of points over F q r. For a Teichmüller element [α] the rth ghost coordinate is α r. The formula 31

38 exhibits Z(Sym 2 E, t) as a combination of Teichmüller elements, thus whose ghost coordinates are a combination of powers. We will verify this combination with a counting argument for # Sym 2 E(F q r). By the formula for Sym 2 E above, the number of points over F q r should be N r (Sym 2 E) = 1 r + q r + q r + (q 2 ) r α r β r (qα) r (qβ) r (2.3) This is what we will verify. Recall that N r (E) = 1 r α r β r + q r. For notational convenience, we will use N r = N r (E). There is another combinatorial way of counting points on Sym 2 E. Proposition 2.4. N r (Sym 2 E) = 1 2 N r(n r + 1) (N 2r N r ) Proof. We will need the following well-known combinatorial lemma: Lemma 2.5. The number of unordered k-tuples of a set of n elements is given by the binomial coefficient n + k 1. k Thus the first term represents the number of unordered pairs of points in E(F q r). The second term counts the extra points on E(F q 2r) of the form (P, P ) where P is the conjugate in the quadratic extension F q 2r. These points are fixed by Frobenius since (P, P ) = (P, P ) in Sym 2 E and thus they must be counted in Sym 2 E(F q r). 32

39 The first term in Proposition 2.4 works out as follows N r = 1 2 N r(n r + 1) = 1 2 (1 αr β r + q r )(2 α r β r + q r ) The second term works out as follows: 1 2 (N 2r N r ) = 1 [ (1 α 2r β 2r + q 2r ) (1 α r β r + q r ) ] 2 = 1 2 (αr α 2r + β r β 2r + q 2r q r ) Adding the terms yields N r (Sym 2 E) = 1 2 N r(n r + 1) (N 2r N r ) = 1 α r β r + q r + α r β r α r q r + q 2r β r q r which is the number predicted by the formula (2.3). We can repeat this procedure for Sym 3 E. In this case, the formula states that Z(Sym 3 E, t) = (1 αt)(1 βt)(1 αqt)(1 βqt)(1 αq2 t)(1 βq 2 t (1 t)(1 qt)(1 q 2 t)(1 αβt)(1 αβqt)(1 q 3 t) which implies that N r (Sym 3 E) = 1 r +q r α r β r α r q r β r q r +q 2r +α r β r +α r β r q r α r q 2r β r q 2r +q 3r We will verify this against the combinatorial count: Proposition 2.6. N r (Sym 3 E) = N r (N 2r N r )N r (N 3r N r ) 33

40 Proof. The first two terms are similar to the terms in the count for Sym 2 E, where instead we count the number of unordered triples. The third term counts the points of the form (P, σp, σ 2 P ) in a cubic extension. The first term works out as follows N r = 1 6 (N r + 2)(N r + 1)(N r ) = 1 6 (3 αr β r + q r )(2 α r β r + q r )(1 α r β r + q r ) = 1 6 (6 11αr 11β r + 11q r + 6α 2r + 12α r β r 12α r q r 12β r q r +6β 2r + 6q 2r α 3r 3α 2r β r 3β 2r α r + 6q r α r β r + 3α 2r q r +3β 2r q r β 3r + q 3r ) The second term works out as follows: 1 2 (N 2r N r )N r = 1 2 (αr α 2r + β r β 2r + q 2r q r )(1 α r β r + q r ) = 1 2 (αr 2α 2r + β r 2β 2r q r + α 3r 2α r β r + α r β 2r + α 2r β r +β 3r α r q 2r β r q 2r + 2α r q r + 2β r q r q r α 2r q r β 2r + q 3r ) The third term is 1 3 (N 3r N r ) = 1 3 (αr α 3r + β r β 3r + q 3r q r ) Adding these terms yields the count predicted by the formula General case The combinatorial propositions counting the number of points on Sym 2 E and Sym 3 E generalizes to arbitrary varieties X over F q. They reduce to the relation 34

41 provided by the Newton identities: na n = b n + a 1 b n a n 1 b 1 These relations allow us to recover a n from b i ; for example a 1 = b 1, a 2 = 1 2 b 2 + b 2 1, a 3 = 1 6 b b 1b b 3 Setting a n = N r (Sym n X) and b n = N nr (X), here are some small cases n = 2 and n = 3: N r (Sym 2 X) = 1 2 N 2r(X) N r(x) 2 = 1 2 (N r)(n r + 1) (N 2r N r ) and N r (Sym 3 X) = 1 6 N r(x) N r(x)n 2r (X) N 3r(X) = 1 6 (N r + 2)(N r + 1)N r (N 2r N r )N r (N 3r N r ). Notice these are exactly the counts provided by the combinatorial propositions above. Newton s identities provide a relationship between N r (Sym n X) to N ir (X) for i = 1, 2,.... n; this relationship will be formalized using λ-rings in the sequel. While the previous examples served to verify the formula provided in Theorem 2.3, we now directly apply the formula to compute zeta functions of symmetric products. For notational convenience, we will suppress + W, and sums in W (R) will be understood as such. 35

42 2.2.3 Symmetric powers of affine and projective space We use the formula in Theorem 2.3 to compute the zeta functions Z(Sym n A m, t) and Z(Sym n P m, t). Note that these varieties are no longer smooth once m > 1. Recall that Z(A m, t) = 1 (1 q m t) = [qm ] and Z(P m, t) = Then by the formula we have for affine space 1 (1 t)(1 [q m ]) = [1] + [qm ] Z(Sym n A m, t) = coefficient of u n in [[ [q m ] ]] = [q nm ], which agrees with the fact that [Sym n A m ] = [A nm ] (see Appendix B). For projective space, Z(Sym n P m, t) = coefficient of u n in [[ [1] ]] + [[ [q] ]] + + [[ [q m ] ]]. Note that this coefficient is not [1] + [q] + + [q nm ] as Sym n P m is not P nm. For instance, consider n = 2 and m = 2, Z(Sym 2 P 2, t) = [1] + [q] + 2[q 2 ] + [q 3 ] + [q 4 ] whereas Z(P 4, t) = [1] + [q] + [q 2 ] + [q 3 ] + [q 4 ] Product of elliptic curves Consider X = E 1 E 2 a product of elliptic curves over F q. We use the formula in Theorem 2.3 to compute Z(Sym n X, t) in terms of Z(E 1, t) and Z(E 2, t). Let Z(E 1, t) = (1 α 1t)(1 β 1 t) (1 t)(1 qt) and Z(E 2, t) = (1 α 2t)(1 β 2 t) (1 t)(1 qt) 36

43 and recall that in the Witt ring, these can be written as Z(E 1, t) = [1] [α 1 ] [β 1 ] + [q] and Z(E 2, t) = [1] [α 2 ] [β 2 ] + [q]. Then since the zeta function takes products in the Witt ring, we may compute Z(E 1 E 2, t) as Z(E 1 E 2, t) = ([1] [α 1 ] [β 1 ] + [q]) ([1] [α 2 ] [β 2 ] + [q]) = [1] [α 1 ] [α 2 ] [β 1 ] [β 2 ] + [α 1 β 2 ] + [α 2 β 1 ] + [α 1 α 2 ] +[β 1 β 2 ] [α 1 q] [α 2 q] [β 1 q] [β 2 q] + [q 2 ]. Thus the motivic zeta function generating series is ζ Z (E 1 E 2, u) = Z(Sym n (E 1 E 2 ), t)u n n=0 = [[ [1] ]] [[ [α 1 ] ]] [[ [α 2 ] ]] [[ [β 1 ] ]] [[ [β 2 ] ]] +[[ [α 1 β 2 ] ]] + [[ [α 2 β 1 ] ]] + [[ [α 1 α 2 ] ]] + [[ [β 1 β 2 ] ]] [[ [α 1 q] ]] [[ [α 2 q] ]] [[ [β 1 q] ]] [[ [β 2 q] ]] + [[ [q 2 ] ]]. Note that these equalities are being written in W (W (A)). These methods similarly also work for n-fold products of smooth projective curves; however, the formulas quickly become unwieldy Grassmannians Let Gr(n, d) be the Grassmannian of d-dimensional subspaces of a n-dimensional vector space over F q. In order count the number of points # Gr(n, d)(f q r), we will need the Gaussian binomial coefficient. For a prime power q, this is the q-analog of 37

44 the usual binomial coefficient, defined as n d q = (qn 1) (q n d+1 1), (q d 1) (q 1) and it is a polynomial in q of degree d(n d). Let b i (q) be the coefficient of q i in n d q so that n d q = d(n d) i=0 b i (q) q i It turns out that these coefficients b i (q) = b i are independent of the power q. For a detailed description of these computations, see Mustata [21] Section 2.6. For a nice collection of computations of zeta functions of Grassmannians, see Kolhatkar [?]. The number of elements over F q r is in fact the Gaussian binomial coefficient: n N r (Gr(n, d)) = = d(n d)b i (q r ) i. d i=0 q r Thus we have Z(Gr(n, d), t) = exp [ r=1 ] N r (Gr(n, d)) tr r However, since the coefficients b i remain the same for all powers q r, we can realize the zeta function as Z(Gr(n, d)) = d(n d) 1 (1 t) b 0 (1 qt) b 1 (1 q d(n d) t) = b b i [q i ]. d(n d) where the sum in the second equality is taking place in the Witt ring. i=0 38

45 For example, Gr(n, 1) = P n 1 and the Gaussian binomial coefficient is so we recover n 1 Z(Gr(n, 1), t) = q = (qn 1) (q 1) = 1 + q + q2 + + q n 1, (n 1) 1 (1 t)(1 qt) (1 q (n 1) t) = [q i ]. Another small example is Gr(4, 2) where the Gaussian binomial coefficient is so we have 4 2 Z(Gr(4, 2), t) = q i=0 = (q4 1)(q 3 1) (q 2 1)(q 1) = 1 + q + 2q2 + q 3 + q 4, 1 (1 t)(1 qt)(1 q 2 t) 2 (1 q 3 t)(1 q 4 t) = [1] + [q] + 2[q 2 ] + [q 3 ] + [q 4 ]. Using the formula from Theorem 2.3, we can compute the zeta function for the symmetric powers Sym m Gr(n, d). Here we only work out the zeta function Z(Sym 2 Gr(4, 2), t) as the coefficient of u 2 in 1 ([1] [1]u)([1] [q]u)([1] [q 2 ]u) 2 ([1] [q 3 ]u)([1] [q 4 ]u). 2.3 Proof of the Main Theorem We now proceed to the proof of Theorem 2.3. First, we will need to recall the result on Newton s identities from Lemma 1.1: in Λ(A), for P (t) = n a nt n and b n = gh n (P (t)), we have na n = b n + a 1 b n 1 + a n 1 b 1 39

46 Recursively, this gives a way to recover a n from the ghost coordinates b 1, b 2,..., b n. That is, a n can be written purely in terms of b i for i = 1, 2,... n by replacing each a i in the above relation. Let us call this relation φ so that a n = φ(b 1, b 2,..., b n ). (2.4) Lemma 2.7. For a variety X over a finite field F q, let N r (X) be the number of points over F q r, #X(F q r). Then N r (Sym n X) = φ(n r (X), N 2r (X),..., N nr (X)) ( r ) Proof. Recall that for the zeta function, Z(X, t) = [ ] N 1 (Sym n X)t n = exp N r (X) tr r n r (2.5) Recall that this implies gh r (Z(X, t)) = N r (X). Using the relation (2.4), this shows that N 1 (Sym n X) = φ(n 1 (X), N 2 (X),..., N n (X)), which is ( 1 ). To obtain ( r ) from ( 1 ), we now use the Frobenius operator Fr r for W (Z) on the equation (2.5). The proof follows now follows from the observation that on ghost coordinates, gh n (Fr r (P )) = gh nr (P ) and for the Weil zeta function, Fr r Z(X/F q, t) = Z(X/F q r). Proof of Theorem 2.3. Recall the statement in Theorem 2.3: ζ Z (X, t) = n Z(Sym n X, t)u n = i,j ( 1) i+1 [[ [α ij ] ]] (2.6) 40