EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS OVER FINITE FIELDS

EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS OVER FINITE FIELDS Abstract. 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). Any motivic zeta function ζ µ of a measure µ factoring through the Grothendieck ring of Chow motives is itself exponentiable; we show that exponentiability also holds in the finite field case. We apply our formula to compute Z(Sym n X, t) in a number of explicit cases.. Introduction Consider the category Var k of varieties over a field k (reduced schemes of finite type over Spec k) and for simplicity let k be perfect. The Grothendieck ring of varieties K 0 (Var k ) is 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. This 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 964; it behaves as a shadow (decategorification) of a (still conjectural) category of motives and is sometimes known as the ring of baby motives. A motivic measure µ defines a ring homomorphism on the Grothendieck ring of varieties µ : K 0 (Var k ) R. For a quasiprojective variety X, the Kapranov motivic zeta function for X associated to µ is a power series ζ µ (X, t) = µ([sym n X])t n + tr[[t]] n=0 where [Sym n X] is the class of the nth symmetric power. By the construction of the Grothendieck ring of varieties, this suffices to define a map on K 0 (Var k ), as quasi-projective varieties additively generate K 0 (Var k ). 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 [2] that in the case of varieties over finite fields, the Weil zeta function takes products in W (Z) the big ring of Witt vectors. This motivates the following definition: a measure µ : K 0 (Var k ) R is said to be exponentiable if the motivic zeta function ζ µ associated to that measure is a ring homomorphism taking values in W (R). That is, the motivic zeta function of a product X Y is described by the product ζ µ (X) W ζ µ (Y ) in the Witt ring. The motivic zeta function of an exponentiable measure defines a ring homomorphism on the Grothendieck ring of varieties. Since motivic zeta functions of exponentiable measures are themselves motivic measures, a question raised in [2] is to determine when ζ µ is itself exponentiable. This amounts to providing a generating series for ζ µ (Sym n X, t), the zeta function of the nth symmetric power. In this note, we explore results along these lines.

2 EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS OVER FINITE FIELDS After reviewing some background information on Witt rings and motivic zeta functions, we consider the case of measures which factor through the Grothendieck group of Chow motives for smooth projective varieties over a field of characteristic zero. It is shown by Ramachandran and Tabuada [3] that any motivic measure factoring through the Gillet-Soule motivic measure is exponentiable. We compare this with the case of varieties over finite fields, where we prove our main result. Some explicit computations of zeta functions of Sym n X are provided. 2. Preliminaries 2.. Witt rings and λ-rings. We collect some facts about Witt rings and λ-rings. For a more comprehensive exposition, see Hazewinkel ([7]). Let R be a commutative ring with identity. Consider Λ(R) = ( + tr[[t]], ) the subgroup of invertible power series with coefficients in R. The big ring of Witt vectors W (R) (Witt ring, for short) has Λ(R) as its underlying group, with multiplication of power series as its group operation, sometime denoted + W. For a R, the Teichmuller element [a] W (R) is defined as [a] = ( at) = + at + a2 t 2 + and the ring structure on W (R) is uniquely determined by its product, sometimes denoted W, on Teichmuller elements with [a] W [b] = [ab]. Another way to represent the Witt ring is to use ghost coordinates. The ghost map on W (R) is an injective ring homomorphism gh : W (R) R N with P (t) (b, b 2,...) where t d P (t) dt P (t) = + b t + b 2 t 2 + The coordinates b n are called ghost coordinates gh n (P (t)) = b n. For example. the ghost coordinates of the Teichmuller element are gh n ([a]) = a n. The benefit of working with ghost coordinates is that Witt sum and Witt product become actual sum and product in R. 2.2. Motivic measures and exponentiation. Recall the category Var k of reduced schemes of finite type over k. 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 satisfying the scissor relation µ(x) = µ(y ) + µ(x \ Y ) for closed subschemes Y of X and µ(x Y ) = µ(x)µ(y ). That is, a motivic measure determines a ring homomorphism on the Grothendieck ring of varieties. For example, consider the case of k = C. Then the Euler characteristic χ c defined as the the alternating sum of ranks of cohomology with compact support b i (X) = rank Hc(X(C)) i is a motivic measure taking values in R = Z. Another motivic measure is the Poincaré polynomial µ P defined as µ P (X) = i ( ) i b i (X)z i, taking values in the polynomial ring R = Z[z]. This is essentially the only motivic measure for k = C (see []). In the case of k = F q, we have the counting measure µ # which counts the number of points over F q : µ # (X) = #(X)(F q ). Sometimes also known as an Euler-Poincaré characteristic on Vark

EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS OVER FINITE FIELDS 3 Given a motivic measure µ, Kapranov ([0]) 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 +tr[[t]], and defines a homomorphism ζ µ : K 0 (Var k ) ( + tr[[t]], ) Recall this subgroup of invertible power series is the underlying additive group for the Witt ring W (R). As pointed out in Ramachandran [2], one may ask that the product structure of K 0 (Var k ) be reflected by the Witt product structure on + tr[[t]]. Definition. A motivic measure µ : K 0 (Var k ) R is exponentiable if its associated Kapranov motivic zeta function ζ µ defines a ring homomorphism ζ µ : K 0 (Var k ) W (R) Here are some examples of exponentiable measures: Example. Let k = C. Consider µ(x) = P (X, z) the Poincaré polynomial. Then a nice formula of MacDonald ([6]) provides a closed form for the generating series of symmetric powers P (Sym n X, z): P (Sym n X, z)t n = ( z t) b ( z 3 t) b3 ( z 2n t) b2n ( z t) b ( z 2 t) b2 ( z n t) b2n n=0 where n is the dimension of X. This shows that gh n (ζ P (X, t)) = P (X, z n ), the Poincaré polynomial in z n, which is multiplicative in X, and thus P (X, t) is exponentiable. This specializes to the case of µ = χ c where the formula due to MacDonald ([6]) takes the following form ( ) χc(x) χ c (Sym n X)t n =. t n= This shows that the motivic zeta function ζ χc may be written in the Witt ring W (Z) as χ c (X)[], and exponentiability follows from the fact that the measure χ c is a ring homomorphism. Example 2. Let k = F q. Consider µ # (X) = #(X)(F q ) the counting measure. Then the Weil zeta function of X is defined as [ ] Z(X, t) = exp N m (X) tm m m= where N m = #(X)(F q m). This shows that gh n (Z(X, t)) = N m (X), which is multiplicative in X. That is, the Weil zeta function naturally takes values in the Witt ring W (Z). Lemma. The Kapranov motivic zeta function ζ µ# associated to the counting measure µ # is nothing other than the Weil zeta function Z(X, t).

4 EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS OVER FINITE FIELDS 2.3. Exponentiation of zeta functions. The motivic zeta function ζ µ of an exponentiable motivic measure µ defines a motivic measure itself. Setting Z(X, t) = ζ µ (X, t), we may form the motivic zeta function ζ Z associated to µ = Z: ζ Z (X, u) = Z(Sym n X, t)u n + ua[[u]]. n=0 Here the variable u is used to distinguish it from the variable t used in A = W (R). This may be viewed as a generating series for zeta functions of symmetric powers. One may ask that the product structure of K 0 (Var k ) be reflected by the Witt product structure on Λ(A) = + ua[[u]], in the Witt ring W (A) = W (W (R)). Note that any motivic measure µ which factors through an exponentiable motivic measure µ is itself exponentiable. Indeed, if K 0 (Var k ) µ µ then ζ µ (X, t) = W (f)(ζ µ ) where W (f) is the induced ring homomorphism on Witt rings. R S 3. The Gillet-Soulé motivic measure In a seminal paper [4], Gillet and Soulé extended the theory of Chow motives to a motivic measure on the Grothendieck ring of varieties in the case of char(k) = 0. Associated to the functor h : SmP roj(k) Chow(k) Q from the category of smooth projective varieties over k to the category of rational pure effective Chow motives, they construct a motivic measure µ GS : K 0 (Var k ) K 0 (Chow(k) Q ) such that in the case X is smooth and projective, µ GS ([X]) = [h(x)] the class of the motive. Proposition (Ramachandran, Tabuada). The motivic zeta function ζ µgs is itself a motivic measure which factors through µ GS. That is, µ GS is an exponentiable measure. Proof. It is shown in [3], that any motivic measure µ taking values in a λ-ring R that factors through its defining measure K 0 (Var k ) µ f R ζ µ σ t W (R) is exponentiable, provided that the opposite λ-structure map σ t is a ring homomorphism. The ring K 0 (C) is a special λ-ring for any Q-linear additive idempotent complete symmetric monoidal category (Heinloth [9]). Moreover, results of del Baño Rollin and Aznar ([2]) show that h(sym n X) = Sym n h(x) in Chow(k) Q for

EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS OVER FINITE FIELDS 5 smooth projective X and results of Heinloth ([8]) show that this is sufficient to prove the same holds for µ GS. Therefore, ζ µgs = σ t µ GS. Corollary 2. For char(k) = 0, the motivic measures χ c and µ P factor through µ GS and are therefore exponentiable. Thus, for a broad variety of motivic measures, the Witt product describes the multiplicative structure on K 0 (Var k ). Remark. Another way to show χ c and µ P are exponentiable is to make use of the MacDonald formula. Del Baño Rollin ([]) has shown a motivic version of the MacDonald formula holds, which can be used to proved exponentiability of measures taking value in a suitable category of motives. Corollary 3. The motivic zeta function ζ µgs, itself a motivic measure, is exponentiable. Proof. If a measure µ is exponentiable, then any measure factoring through µ is also exponentiable. 4. Varieties over Finite Fields The Weil zeta function of a variety X over F q can be written in the Witt ring W (Q l ) as Z(X, t) = i,j ( ) i+ [α ij ] where the α ij are the (inverse) eigenvalues of the Frobenius action on Het(X; i Q l ) (Deligne [3]). Again, we restrict ourselves to quasi-projective varieties X in order to make sense of Sym n X. Recall that the zeta function Z(X, t) is the motivic zeta function ζ µ# associated to counting measure µ # (see [2]). There is no MacDonald formula to prove exponentiability; however, recall that the Weil zeta function can be represented as follows: [ ] Z(X, t) = #(Sym n X)(F q )t n = exp #(X)(F q n) tn. n n=0 That is, the ghost coordinates gh n (Z(X, t)) = #(X)(F q n) which is multiplicative in X. The multiplicativity of ghost coordinates suffices to show that a measure is exponentiable. Remark. In addition to the lack of a MacDonald formula for µ #, it is still conjectural whether there exists a suitable motivic category and whether the Gillet-Soulé measure exists for the Grothendieck ring of varieties over finite fields. More recent results of Gillet-Soulé ([5]) extend their construction to the case char(k) > 0; however, it is still unknown whether this exhibits a motivic measure. To wit, the previous arguments for ζ µ factoring through µ GS cannot be used. Theorem 4. The zeta function of a variety X over F q is itself a motivic measure taking values in W (W (Z)) satisfying n= ζ Z (X, u) = n Z(Sym n X, t)u n = i,j ( ) i+ [[ [α ij ] ]]

6 EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS OVER FINITE FIELDS Here, the element [[ [α ij ] ]] is the Teichmuller element in W (R) where [α ij ] R = W (Q l ). This formula can be viewed as a generating series for zeta functions of symmetric powers. Moreover, the Weil zeta function is exponentiable. We will need the following result on Newton s identities. Let h n be the complete homogeneous symmetric functions and let p n be the power symmetric functions in a variable set (X, X 2,...). Newton s identities state that nh n = p n + h p n + + h n p Recursively, this gives a way to recover h n from the power functions p, p 2,..., p n. Let us call this relation φ so that φ(h n, p, p 2,..., p n ) = 0 Lemma 5. For P (t) W (R), if P (t) = n α nt n and β n = gh n (P (t)), then φ(α n, β, β 2,..., β n ) = 0 This uniquely determines P (t) in the following sense: if Q(t) W (R) with Q(t) = n α nt n and φ(α n, β, β 2,..., β n ) = 0, then P (t) = Q(t). Lemma 6. For X a variety over a finite field F q, let N r (X) be the number of points over F q r, #(X)(F q r). Then φ(n r (Sym n X), N r (X), N 2r (X),..., N nr (X)) = 0 Proof. Recall that for the zeta function of X, Z(X, t) = [ ] N (Sym n X)t n = exp N r (X) tr r n r By the previous Lemma, this shows that φ(n (Sym n X), N (X), N 2 (X),..., N n (X)) = 0 Then use the Frobenius operator. The rth Frobenius operator on zeta functions is the zeta function of the base change to F q r. But, on ghost coordinates, the Frobenius multiplies by r: Fr r gh n = gh nr. Proof of Theorem. Let the ghost map on W (Z) be denoted by gh t and the ghost map on W (W (Z)) be denoted by gh u. Let the ghost coordinates of ζ Z (X, u) be denoted by gh u n(ζ Z ) = β n, where β n W (Z). It suffices to show that β n = gh u n ( ) i+ [[ [α ij ] ]] = ( ) i+ [α ij ] n W (Z) i,j i,j and for equality in W (Z), it suffices to show that gh t rβ n = gh t r ( ) i+ [αij] n = i,j i,j Recall from Lemma 6 that ( ) i+ α nr ij φ(z(sym n X, t), β, β 2,..., β n ) = 0 W (Z) = N nr (X) Z

EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS OVER FINITE FIELDS 7 Now take the ghost map gh t r in W (Z): φ(gh t r(z(sym n X, t)), gh t rβ, gh t rβ 2,..., gh t rβ n ) = 0 But, by Lemma 7 φ(n r (Sym n X), N r (X), N 2r (X),..., N nr (X)) = 0 and thus gh t rβ n = N nr (X), and the theorem follows. 5. Examples We apply the generating function formula to compute Z(Sym n X, t) for various cases of varieties X over finite fields. 5.. 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) = ( q n t) = [qn ] W (Z) It is also easy to show that N r (P n ) = q nr + q (n )r + + q r + so that Z(P n, t) = ( t)( qt) ( q m t) = [qm ] + [q m ] + + [q] + [] W (Z). Recall that Sym n A = A n and Sym n P = P n. Our formula predicts that Z(Sym n A, t) is the coefficient of u n in [[ [q] ]] W (W (Z)). Similarly, that Z(Sym n P, t) is the coefficient of u n in [[ [] ]]+[[ [q] ]] W (W (Z)). This agrees with the zeta functions Z(A n, t) and Z(P n, t) described above. 5.2. Elliptic curves. Let E be an elliptic curve over F q. In this case, we have Z(E, t) = ( αt)( βt) ( t)( qt) where α + β = a Z and αβ = q. Written in the Witt ring W (Z), this appears as Z(E, t) = [] [α] [β] + [q], where [α] = ( αt) the Teichmuller element. The formula provided by Theorem 5 is [ ] ( [α]u)( [β]u)) Z(Sym n E, t) = coefficient of u n in ( []u)( [q]u)) 5.2.. Small cases n = 2, 3. In the case of Sym 2 E, the formula states that Z(Sym 2 E, t) = ( αt)( βt)( qαt)( qβt) ( t)( qt)( qt)( 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 Teichmuller element, [α], the rth ghost coordinate is α r. The formula exhibits Z(Sym 2 E, t) as a combination of Teichmuller 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) = r + q r + q r + (q 2 ) r α r β r (qα) r (qβ) r

8 EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS OVER FINITE FIELDS This is what we will verify. Recall that N r (E) = 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. N r (Sym 2 E) = 2 N r(n r + ) + 2 (N 2r N r ) Proof. We will need the following combinatorial lemma: Lemma 7. The number ( of unordered ) k-tuples of a set of n elements is given n + k by the binomial coefficient. 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). The first term in Proposition 8 works out as follows ( ) Nr + = 2 2 N r(n r + ) = 2 ( αr β r + q r )(2 α r β r + q r ) The second term works out as follows: 2 (N 2r N r ) = [ ( α 2r β 2r + q 2r ) ( α r β r + q r ) ] 2 Adding the terms yields = 2 (αr α 2r + β r β 2r + q 2r q r ) N r (Sym 2 E) = 2 N r(n r + ) + 2 (N 2r N r ) = 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 (). We can repeat this procedure for Sym 3 E. In this case, the formula states that Z(Sym 3 E, t) = ( αt)( βt)( αqt)( βqt)( αq2 t)( βq 2 t ( t)( qt)( q 2 t)( αβt)( αβqt)( q 3 t) which implies that N r (Sym 3 E) = 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 3. N r (Sym 3 E) = ( Nr + 2 3 ) + 2 (N 2r N r )N r + 3 (N 3r N r )

EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS OVER FINITE FIELDS 9 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 ( ) Nr + 2 = 3 6 (N r + 2)(N r + )(N r ) = 6 (3 αr β r + q r )(2 α r β r + q r )( α r β r + q r ) = 6 (6 αr β r + q r + 6α 2r + 2α r β r 2α r q r 2β 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: 2 (N 2r N r )N r = 2 (αr α 2r + β r β 2r + q 2r q r )( α r β r + q r ) = 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 3 (N 3r N r ) = 3 (αr α 3r + β r β 3r + q 3r q r ) Adding these terms yields the count predicted by the formula. 5.2.2. 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 provided by the Newton identities: nh n = p n + h p n + + h n p These relations allow us to recover h n from p i ; for example h = p, h 2 = 2 p 2 + p 2, h 3 = 6 p3 + 2 p p 2 + 3 p 3 Setting h n = N r (Sym n X) and p n = N nr (X), here are some small cases n = 2 and n = 3: and N r (Sym 2 X) = 2 N 2r(X) + 2 N r(x) 2 = 2 (N r)(n r + ) + 2 (N 2r N r ) N r (Sym 3 X) = 6 N r(x) 3 + 2 N r(x)n 2r (X) + 3 N 3r(X) = 6 (N r + 2)(N r + )N r + 2 (N 2r N r )N r + 3 (N 3r N r ). These are exactly the counts provided by the combinatorial propositions above.

0 EXPONENTIATION OF MOTIVIC ZETA FUNCTIONS OVER FINITE FIELDS 5.3. Grassmannians. Let Gr(n, d) be the Grassmannian of d-dimensional subspaces of a n-dimensional vector space over F q. In this case, we have ( ) n N r (Gr(n, d)) = d(n d)b d i (q r ) i q r = the Gaussian binomial coefficient. Then, Z(Gr(n, d)) = ( t) b0 ( qt) b ( q d(n d) t) b d(n d) Using the formula from Theorem 5, we can compute the zeta function for the symmetric powers Sym m Gr(n, d). i=0 References [] S. del Bano Rollin, On the motive of moduli spaces of rank two vector bundles over a curve, Compos. Math. 3 (2002), no., 30. [2] S. del Bano Rollin and V. Navarro Aznar, On the motive of a quotient variety, Collect. Math. 49 (998), no. 2-3, 203 226. [3] P. Deligne, La Conjecture de Weil. II., Inst. Hautes. Études Sci. Publ. Math. 52 (980), 37 252. [4] H. Gillet and C. Soulé, Descent, motives, and K-theory, J. Reine Angew. Math. 478 (996), 27 76. [5], Motivic weight complexes for arithmetic varieties, J. Algebra 322 (2009), no. 9, 3088 34. [6] I. G. Hazewinkel, The Poincaré polynomial of a symmetric product, Proc. Cambridge Philos. Soc. 58 (962), 563 568. [7] M. Hazewinkel, Witt vectors. i., Handb. Algebr. 6 (2009), 39 472. [8] F. Heinloth, The universal Euler characteristic for varieties of characteristic zero, Compos. Math. 40 (2004), no. 4, 0 032. [9], A note on functional equations for zeta functions with values in chow motives, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 6, 927 945. [0] M. Kapranov, The elliptic curve in s-duality theory and Eisenstein series for Kac-Moody groups. Available at arxiv:math/000005v2. [] M. Mustata, Zeta functions in algebraic geometry. Available at Link. [2] N. Ramachandran, Zeta functions, Grothendieck groups, and the Witt ring, Bull. Sci. Math. 39 (205), no. 6, 599 627. [3] N. Ramachandran and G. Tabuada, Exponentiable motivic measures, J. Ramanujan Math. Soc. 30 (205), no. 4, 349 360.