Purity of exponential sums on An, II

Purity of exponential sums on An, II Antonio Rojas-Le´on ∗ October 17, 2005 Abstract We give a purity result for exponential sums of the type P x∈knψ(f(x)), where kis a finite field of characteristic p,ψ:k→C?is a non-trivial additive character and f∈k[x1, . . . , xn] is a polynomial whose highest degree homogeneous form splits as a product of factors defining a divisor with normal crossings in Pn−1. 1 Introduction Let kbe a finite field of characteristic pand cardinality q, let f∈k[x1, . . . , xn] be a polynomial of degree dand ψ:k→C?a non-trivial additive character. We are interested in the sum the Px∈knψ(f(x)). Under various different regularity conditions on f(cf. [4] Th´eor`eme 8.4, [10] Theorem 0.4, [1] Theorem 1.4, [20] Theorems 2 and 4) the sum is known to be pure of weight nand rank ≤(d−1)n. In particular, we have the estimate ¯¯¯¯¯X x∈kn ψ(f(x))¯¯¯¯¯ ≤(d−1)n·qn/2. In this article we consider the case where the highest degree homogeneous form of fis reducible, and the hypersurface defined in Pn−1by the product of its distinct factors is a divisor with normal crossings. More precisely, let f=fd+fd0+f0, with fihomogeneous of degree iand f0of degree < d0, and suppose that fd=gα1 1· · · gαr rin ¯ k[x1, . . . , xn], where gi1=. . . =gik= 0 and gi1=...=gik=fd0= 0 define smooth subvarieties of codimension kand k+ 1 respectively in Pn−1 ¯ kfor all {i1, . . . , ik}⊆{1, . . . , r}. These hypotheses appear for the first time in [2], where Adolphson and Sperber show that for d,d0,α1, . . . , αrprime to pthe L-function associated to the exponential sum Px∈knψ(f(x)) is a polynomial or the reciprocal of a polynomial, depending on the parity of n. This suggest that the cohomology of the sum is ∗Partially supported by MTM2004-07203-C02-01 and FEDER 12000 AMS Mathematics Subject Classification: 11L07, 11T23. 1 concentrated in degree n. Our main result (Theorem 1) will show that, under these hypotheses, the sum is indeed cohomologically pure of weight n. The proof consists of two parts. First we will show that, for a fixed k, the property of being pure and the rank of the sum do not depend on the particular fchosen, but only on the combinatorial data: d,d0, the αi’s and the degrees of the gi’s. For this we use an argument similar to the one in ([5], 3.7.3). In order to finish the proof it would then be enough to find a particular ffor which the result is true. In loc.cit., Deligne uses the diagonal polynomial Pn i=1 xd i. Unfortunately, in general we will not have such an explicit example for all possible choices of combinatorial data. Instead, we use the theory of perverse sheaves to show that the sum is pure for a certain subfamily of the family of all polynomials of a given type. This will prove the purity of the sum, but still will not give the value of the rank. In Section 3 we will give some bounds for this rank that will allow us to effectively estimate the absolute value of the sum (cf. Corollary 9). Then in Section 4 we will give en explicit formula for it in terms of the well known formulas for the Euler characteristic of complete intersections in projective space. In particular, this will show that the rank is also independent of the base field k. In the last section we will give some examples of sums that can be bounded using these results. We will find that several previously known estimates, like the ones for non-singular polynomials and for Kloosterman sums, arise as particular cases of this more general purity theorem. Notice that, in this case, the general results in [15] about exponential sums where the highest degree form of fis singular do not give good estimates for the sum, due to the high dimension of the singular locus of the hypersurface fd= 0 (n−3 if αi= 1 for all i= 1, . . . , r and n−2 otherwise). 2 Purity of the sums Throughout this section we will fix a prime p, a finite field kof characteristic p and cardinality qand an integer n≥1. An admissible type will consist of the following data: - a positive integer dprime to p - a positive integer d0< d prime to p - a positive integer r - two r-tuples of positive integers e= (e1, . . . , en) and α= (α1, . . . , αn) such that αiis prime to pfor all iand d=Pieiαi. Given an admissible type (d, d0, r, e, α), we will say that a polynomial f∈ k[x1, . . . , xn] is admissible of type (d, d0, r, e, α) if we can write f=fd+fd0+f0, where fd(respectively fd0) is homogeneous of degree d(resp. d0), f0has degree less than d0and: -fdfactors in ¯ k[x1, . . . , xn] as gα1 1···gαr r, where giis homogeneous of degree ei. 2 - For every subset {i1, . . . , ik} ⊆ {1, . . . , r}, the subscheme of Pn−1 ¯ kdefined by the homogeneous ideal (gi1, . . . , gik) is smooth of codimension k(empty if k≥n). - For every subset {i1, . . . , ik} ⊆ {1, . . . , r}, the subscheme of Pn−1 ¯ kdefined by the homogeneous ideal (fd0, gi1, . . . , gik) is smooth of codimension k+ 1 (empty if k≥n−1). Our main result is the following: Theorem 1 Let (d, d0, r, e, α)be an admissible type. There is a constant M= M(p, n, d, d0, r, e, α)such that, for every admissible polynomial f∈k[x1, . . . , xn] of type (d, d0, r, e, α)and every non-trivial additive character ψ:k→C?, the exponential sum X x∈kn ψ(f(x)) is pure of weight nand rank M. In particular we have the estimate ¯¯¯¯¯X x∈kn ψ(f(x))¯¯¯¯¯ ≤M·qn/2. We will translate the result to cohomological language. Fix a prime `6=p and an isomorphism ι:¯ Q`→C. Consider the smooth Artin-Schreier ¯ Q`-sheaf Lψon A1 kassociated to the non-trivial additive character ψ:k→C?∼ →¯ Q? `(cf. [6], 1.7) and, for every f∈k[x1, . . . , xn], let Lψ(f)be its pull-back to An k. The cohomology groups with compact support Hi c(An ¯ k,Lψ(f)) are endowed with an action of the absolute Galois group Gal(¯ k/k) and, in particular, of the geometric Frobenius element F∈Gal(¯ k/k). By the Grothendieck trace formula we have X x∈kn ψ(f(x)) = 2n X i=0 (−1)iTrace(F|Hi c(An ¯ k,Lψ(f))). Then Theorem 1 is a consequence of the following cohomological result: Theorem 2 Let (d, d0, r, e, α)be an admissible type. There is a constant M= M(p, n, d, d0, r, e, α)such that, for every admissible polynomial f∈k[x1, . . . , xn] of type (d, d0, r, e, α)and every non-trivial additive character ψ:k→C?, the cohomology groups Hi c(An ¯ k,Lψ(f))vanish for i6=n, and Hn c(An ¯ k,Lψ(f))has dimension Mand is pure of weight n. The remainder of this section will be devoted to the proof of Theorem 2. For every positive integer m, let Sm(respectively Sh m) be the affine k-space parameterizing all polynomials (resp. all homogeneous polynomials) of degree ≤m(resp. of degree m) in nvariables. Given an admissible type (d, d0, r, e, α), let U⊂Sh e1× · · · × Sh er×Sd0be the dense open set parameterizing all (r+ 1)- tuples (g1, . . . , gr, h) such that gα1 1· · · gαr r+his admissible of type (d, d0, r, e, α) (Notice that this is not a parametrization of the set of admissible polynomials 3 of type (d, d0, r, e, α) if there are i6=jsuch that ei=ejand αi=αj, since in that case the same polynomial can appear more than once). On the affine n-space X=An Uover Uwe have the universal polynomial F∈Γ(X, OX) such that, for every u= (g1, . . . , gr, h)∈U(k), the restriction of Fto the fiber Xu∼ =An kis fu:= gα1 1···gαr r+h, and the corresponding Artin- Schreier sheaf Lψ(F)whose restriction to Xuis Lψ(fu). Write F=Fd+Fd0+F0 where Fd(respectively Fd0) is the homogeneous component of degree d(resp. of degree d0) and F0=F−Fd−Fd0. Let ¯ X=Pn Ube the projective n-space over U,j:X ,→¯ Xthe inclusion, H=¯ X−X,π:X→Uand ¯π:¯ X→Uthe projections and F=j!Lψ(F). By the proper base change theorem, the fiber of Rπ!Lψ(F)= R¯π?Fat a geometric point ¯uover u∈U(k) is RΓc(An ¯ k,Lψ(fu)). Since Uis connected, Theorem 2 will then be a consequence of the following Proposition 3 The sheaves Riπ!Lψ(F)vanish on Ufor i6=n, and Rnπ!Lψ(F) is smooth and punctually pure of weight non U. We will first show that the Riπ!Lψ(F)are all smooth on U. Recall (cf. [7], 2.12) that a morphism g:Z→Sis said to be locally acyclic for a constructible sheaf Gon Zif, for every geometric point zof Zand every geometric point sof Sg(z)(the henselization of Sat g(z)), the induced map RΓ(Zz,G)→RΓ((Zz)s,G) is an isomorphism, where Zzis the henselization of Zat zand (Zz)sthe fiber at sof the canonical map Zz→Sg(z). Lemma 4 The projection ¯π:¯ X→Uis locally acyclic for F. Proof. Following ([5], 3.7.3) we will show that, locally for the ´etale topology on ¯ X, the pair ( ¯ X, F) is U-isomorphic to the product of Uand a scheme endowed with a ¯ Q`-sheaf. Let x∈¯ Xbe a closed point, we have to distinguish several cases: a) If x∈X, then Fis smooth in a neighborhood of x, and in fact it is trivialized by the ´etale neighborhood (Y, y) of x, where Yis defined by the equation td−t−F= 0 and y∈Yis any point mapping to x, so the assertion is clear in this case. b) Let x∈Hsuch that Fd(x)6= 0. Let Fhbe the homogenization of Fwith respect to the variable x0, where x0= 0 defines the hyperplane at infinity H. Then Fis given by Fh/xd 0. Pick j∈ {1, . . . , n}such that the j-th coordinate of xis non-zero, then a=Fh/xd jand z=x0/xjare well defined regular functions around x, and a(x)6= 0. Take the ´etale neighborhood (Y, y) of xdefined by the equation vd=a,y∈Ymapping to x. Pick a system of parameters t1, . . . , tnat ysuch that t1=z/v, they define an ´etale U-map φfrom a Zariski neighborhood of yin Yto An×Uwith F|Y=ˆ F◦φ, where ˆ F(t1, . . . , tn, u) = t−d 1is independent of u. c) Let x∈Hsuch that Fd(x) = 0 but Fd0(x)6= 0. For simplicity, we can assume without loss of generality that there is some k∈ {1, . . . , r}such that gi(x) = 0 for i≤kand gi(x)6= 0 for i > k. Let jand zbe as in the previous case, Fh0= (Fh−Fd)/xd−d0 0the homogenization of Fd0+F0with respect to 4 x0,a=Fh0/xd0 jand bi=gi/xei jfor i= 1, . . . , r. Then aand the bi’s are regular in a neighborhood of x,a(x)6= 0, bi(x)6= 0 for i > k and b1, . . . , bk, z are part of a system of parameters at x. Let (Y, y) be the ´etale neighborhood of xdefined by the equations vd0=aand wαk=v−dbαk+1 k+1 · · · bαr r, and y∈Y mapping to x. Pick a system of parameters t1, . . . , tnat ysuch that tn=z/v, ti=bifor i= 1, . . . , k −1 and tk=bkw. They define an ´etale U-map φ from a Zariski neighborhood of yin Yto An×Uwith F|Y=ˆ F◦φ, where ˆ F(t1, . . . , tn, u) = t−d ntα1 1tα2 2. . . tαk k+t−d0 nis independent of u. d) Finally, let x∈Hsuch that Fd(x) = Fd0(x) = 0 and assume, as in case (c), that there is some k∈ {1, . . . , r}such that gi(x) = 0 for i≤kand gi(x)6= 0 for i > k. Let j,z,Fh0,aand bibe as above. Then aand the bi are regular in a neighborhood of x,bi(x)6= 0 for i>kand b1, . . . , bk, a, z are part of a system of parameters at x. Let (Y, y) be the ´etale neighborhood of xdefined by the equation vαk=bαk+1 k+1 · · · bαr rand y∈Ymapping to x. Pick a system of parameters t1, . . . , tnat ysuch that tn=z,tn−1=a,ti=bifor i= 1, . . . , k −1 and tk=bkv. They define an ´etale U-map φfrom a Zariski neighborhood of yin Yto An×Uwith F|Y=ˆ F◦φ, where ˆ F(t1, . . . , tn, u) = t−d ntα1 1tα2 2. . . tαk k+t−d0 ntn−1is independent of u. So in either case we find that Fis locally isomorphic, for the ´etale topology, to a sheaf of the form Lψ(g)on An×U, where g∈Γ(U, OU)(t1, . . . , tn) (and hence Lψ(g)) does not depend on U, therefore being the product of Uand a scheme endowed with a sheaf. ¤ Corollary 5 The sheaves Riπ!Lψ(F)are smooth on U. Proof. By ([7], A.2.2), for every geometric point uof Uthe specialization map (R¯π?F)u= RΓ( ¯ Xu,F)→RΓ( ¯ X¯η,F) = (R¯π?F)¯ηis a quasi-isomorphism, where ¯ηis a geometric generic point of U. Therefore the cohomology sheaves Riπ!Lψ(F)= Ri¯π?Fof R¯π?Fare smooth. ¤ In particular, it suffices to prove Theorem 2 for one particular admissible polynomial of type (d, d0, r, e, α). This will show that the stalk of Riπ!Lψ(F)at one geometric point of Uvanishes for i6=nand is pure of weight nfor i=n. Since the sheaves are smooth on U, the same will be true at any other geometric point of U. Replacing kby a finite extension if necessary, assume that U(k)6=∅, and pick any u= (g1, . . . , gr, h)∈U(k). Let V⊂ˆ Anbe the dense open set of all linear forms lsuch that (g1, . . . , gr, h +l)∈U(which is the entire ˆ Anif d0>1). Then Vis a closed subset of Uvia the embedding l7→ (g1, . . . , gr, h +l). Proposition 6 The object K= Rπ!Lψ(F)[2n]∈ Db c(V, ¯ Q`)is perverse and pure of weight 2n. Proof. This is a particular case of the results in [18] and [14]. The object K is just the restriction to Vof the Fourier transform of Lψ(fu)[n] with respect to the character ψ(cf. [18], 2.1). Since Lψ(fu)is smooth and punctually pure 5 of weight 0 and Anis smooth, Lψ(fu)[n] is perverse and pure of weight n(cf. loc.cit., 1.2.2(iii) and 1.3.2(iii)). But the Fourier transform preserves perversity and purity and shifts weights by n(cf. loc.cit., 2.1.5(iii) and 2.2.1), therefore K is perverse and pure of weight 2n.¤ We can now finish the proof of Proposition 3. The object Kis perverse by Proposition 6 and has smooth cohomology sheaves by Corollary 5. Therefore all of them must vanish except for H−n(K) = Rnπ!Lψ(F), which is smooth and punctually pure of weight n, since Kis pure of weight 2n(cf. [18], 1.4.1). This proves the proposition (and consequently Theorems 1 and 2) for a particular ψ, but it is clear that Mdoes not depend on the additive character chosen, since any other non-trivial character would be given by t→ψ(at) for some a∈k?, and multiplying an admissible polynomial by a non-zero constant does not change its type. Remark. The following examples show that it is essential to consider the components of lower degree of fin the purity theorem, and therefore it is not possible to give a similar result depending only on the highest degree homogeneous component of f. First, suppose that nis prime to pand let f(x) = Qn x=1 xi=x1· · · xn, which clearly defines a divisor with normal crossings in Pn−1. Then X x∈kn ψ(f(x)) = X a∈kX x1···xn−1=aX xn∈k ψ(axn). But Pxn∈kψ(axn) = 0 for a6= 0 and qfor a= 0, so X x∈kn ψ(f(x)) = X x1···xn−1=0 q=q·#{(x1, . . . , xn−1)∈kn−1:x1···xn−1= 0}= =q(qn−1−(q−1)n−1) = O(qn−1). As a second example, take f(x) = xd 1, with dprime to p. Then, on An= A1×An−1we have Lψ(f)=Lψ(xd)£¯ Q` and therefore, since Hi c(An−1 ¯ k,¯ Q`) = 0 for i6= 2n−2, we deduce that H2n−1 c(An ¯ k,Lψ(f)) = H1 c(A1 ¯ k,Lψ(xd))⊗H2n−2 c(An−1 ¯ k,¯ Q`) = = H1 c(A1 ¯ k,Lψ(xd))⊗¯ Q`(1 −n) has dimension d−1 and is pure of weight 2n−1, since H1 c(A1 ¯ k,Lψ(xd)) has dimension d−1 and is pure of weight 1 (cf. [4], 8.11). So in this case we get ¯¯¯¯¯X x∈kn ψ(f(x))¯¯¯¯¯ =O(qn−1 2). 6 3 First estimates for the rank In the next section we will prove a closed formula for Min terms of the type (d, d0, r, e, α). For now, we will give some easier bounds for it using the following ”degeneration principle”: Suppose that we have a family of polynomials of degree din nvariables whose coefficients are parameterized by the points of an irreducible variety, and suppose that the exponential sum associated to a generic element of the family is pure (in the sense of Theorem 2). Then, for every polynomial in the family whose associated exponential sum is pure, the rank of this sum is bounded above by the rank of the sum corresponding to a generic element of the family. This assertion is justified by the following theorem of Katz: Theorem 7 (cf. [16], Proposition 12) Let Gbe a ¯ Q`-sheaf on a smooth variety Sof dimension r, and suppose that there is a perverse object K∈ Db c(S, ¯ Q`) such that G=H−r(K). Then the integer valued function defined by s7→ rank G¯s on S(where ¯sis a geometric point over s) is lower semicontinuous. In other words, the rank of Gcan never increase under specialization. We will apply this result to the following situation: Sis the affine k-space parameterizing all polynomials of degree ≤din nvariables, π:S×An k→S is the projection, F∈Γ(S×An k,OS×An k) is the universal polynomial, K= Rπ!Lψ(F)[n+ dim S] and G= Rnπ!Lψ(F). By ([17], Part (1) of Theorem 3.1.2) Kis perverse, so the theorem applies to this case and we conclude that the rank of Rnπ!Lψ(F)is a lower semicontinuous function on S. As a first application of this, we can show Proposition 8 For every admissible type (d, d0, r, e, α)we have the upper bound M(p, n, d, d0, r, e, α)≤(d−1)n. Proof. It is well known (cf. [4], Lemme 8.5) that the generic rank of Rnπ!Lψ(F) on Sis (d−1)n. Therefore this is a direct consequence of the semicontinuity of the rank. ¤ Corollary 9 For every admissible polynomial f∈k[x1, . . . , xn]of degree dand every non-trivial additive character ψ:k→C?we have the estimate ¯¯¯¯¯X x∈kn ψ(f(x))¯¯¯¯¯ ≤(d−1)n·qn/2. We will now give a few inequalities that can be deduced from the degeneration principle. In all cases, the proof is the same: the subvariety of Sparameterizing all admissible polynomials of a given type is a subset of the closure of the subvariety of Sparameterizing the admissible polynomials of another type. From the degeneration principle we conclude that the rank corresponding to the former type is bounded above by the rank corresponding to the latter. 7 Notice that applying the same permutation to the ei’s and the αi’s does not change M. Therefore in the following examples we will always modify the first factors of fd, but the same inequalities will hold if we modify any of the other factors the same way. 1. Suppose that e1=Ps j=1 hjβjwith all βj’s prime to p. Then M(p, n, d, d0, r, (e1, e2, . . . , er),(α1, α2, . . . , αr)) ≥ ≥M(p, n, d, d0, r+s−1,(h1, . . . , hs, e2, . . . , er),(α1β1, . . . , α1βs, α2, . . . , αr)) (g1”becomes reducible”). 2. If r≥2 and e1=e2, then M(p, n, d, d0, r, (e2, e2, e3, . . . , er),(α1, α2, α3, . . . , αr)) ≥ ≥M(p, n, d, d0, r −1,(e2, . . . , er),(α1+α2, α3, . . . , αr)) (g1and g2”become equal”). 3. If d > d0> d00 ≥1 are all prime to p, then M(p, n, d, d0, r, (e1, e2, . . . , er),(α1, α2, . . . , αr)) ≥ ≥M(p, n, d, d00, r, (e1, e2, . . . , er),(α1, α2, . . . , αr)) (fd0”degenerates to 0”). We can use these inequalities to give a lower bound for M. First of all, let us compute an easy case: Lemma 10 M(p, n, d, d0,1,(1),(d)) = (d−1)(d0−1)n−1. Proof. We will use the polynomial f(x1, . . . , xn) = xd 1+Pn i=1 xd0 i, which is clearly admissible of type (d, d0,1,(1),(d)). Since ψ(f(x)) = ψ(xd 1+xd0 1)ψ(xd0 2)· · · ψ(xd0 n), we have, as sheaves on An k, Lψ(f)=Lψ(xd+xd0)£Lψ(xd0)£· · · £Lψ(xd0). But Lψ(xd+xd0)(respectively Lψ(xd0)) has rank d−1 (resp. d0−1) (cf. [4], 8.11), so Lψ(f)has rank (d−1)(d0−1)n−1.¤ Corollary 11 For any admissible type (d, d0, r, e, α)we have the lower bound M(p, n, d, d0, r, e, α)≥(d−1)(d0−1)n−1. Proof. By repeated application of the first two inequalities, one shows that M(p, n, d, d0, r, (e1, . . . , er),(α1, . . . , αr)) ≥M(p, n, d, d0,1,(1),(d)). ¤ 8 4 An explicit formula for the rank This section will be entirely devoted to the proof of the following Theorem 12 For all positive integers r≤nand d1, . . . , drlet χ(n;d1, . . . , dr) denote the Euler characteristic of a non-singular complete intersection of multidegree (d1, . . . , dr)in Pn. Then for every prime pand every admissible type (d, d0, r, e, α)we have M(p, n, d, d0, r, e, α) = (−1)n+d(d0−1)n−(−1)n d0+ (−1)n(d−d0)χ(1) where χ:= X I⊆{1,...,r} 1≤|I|≤n−1 (−1)|I|−1χ(n−1; eI)−X I⊆{1,...,r} 1≤|I|≤n−2 (−1)|I|−1χ(n−1; d0, eI) and eIstands for ei1, . . . , eijif I={i1, . . . , ij}. Corollary 13 M(p, n, d, d0, r, e, α)is independent of pand of α. We will denote this number by M(n, d, d0, r, e). Of course, given dand e, there are only a finite number of possible choices for α, namely the solutions in positive integers (x1, . . . , xn) of e1x1+. . . +enxn=d. Fix a finite field kof characteristic pand let f∈k[x1, . . . , xn] be an admissible polynomial of type (d, d0, r, e, α). We already know that the rank does not depend on the polynomial chosen, so we can assume that fdoes not have any terms of degree < d0. Therefore f=fd+fd0, where fd=gα1 1· · · gαr r,giis homogeneous of degree ei,fd0is homogeneous of degree d0and g1· · · grfd0= 0 defines a divisor with normal crossings in Pn−1 ¯ k. By Theorem 2, the cohomology sheaves Hi c(An ¯ k,Lψ(f)) vanish for i6=n, so M(p, n, d, d0, r, e, α) = (−1)nχc(An ¯ k,Lψ(f)). We need to compute this Euler characteristic. The idea is to reduce this to computing the Euler characteristics of some sheaves in A1 ¯ k, which can be done by studying their local properties. Consider the object Rfd!Lψ(f)in Db c(A1 ¯ k,¯ Q`), we have χc(An ¯ k,Lψ(f)) = χc(A1 ¯ k,Rfd!Lψ(f)). By the projection formula, Rfd!Lψ(f)= Rfd!Lψ(fd+fd0)= Rfd!(Lψ(fd)⊗ Lψ(fd0)) = Lψ⊗Rfd!Lψ(fd0) We will now study the object Rfd!Lψ(fd0)in detail. By abuse of language, we will say that an object K∈ Db c(A1 ¯ k,¯ Q`) is smooth (resp. unramified, tamely ramified, totally wild) at a point t∈A1 ¯ k(resp. t∈P1 ¯ k) if so are all its cohomology sheaves. The dimension of the fiber of Kat a point t∈A1 ¯ k(resp. its generic rank, its Swan conductor at t∈P1 ¯ k) will be the alternating sum of the dimensions of the fibers of the cohomology sheaves at t(resp. of their generic ranks, of their 9 so χc(An ¯ k,Lψ(f)) = 1 + (−1)nd(d0−1)n−(−1)n d0+ (d−d0)χc(Z−D). Theorem 12 follows by excision and the inclusion-exclusion principle, since χc(Z−D) = χc(Z)−χc(D) = =χc({g1· · · gr= 0})−χc({g1···gr= 0, fd0= 0}) = =χc([ i=1,...,r {gi= 0})−χc([ i=1,...,r {gi= 0, fd0= 0}). 5 Examples In this section we will simplify the rank formula in several important particular cases. 1) Suppose that r= 1, so fd=gαfor some non-singular homogeneous form g∈k[x1, . . . , xn] of degree e. We will use the following well known formulas, which can be deduced for instance from [8], Expos´e XVII, 5.7.5: χ(n;d) = n+ 1 + (1 −d)n+1 −1 d χ(n;d1, d2) = n+1+ 1 d1−d2µd1 (1 −d2)n+1 −1 d2 −d2 (1 −d1)n+1 −1 d1¶ if d16=d2, and χ(n;d, d) = n+1+(n+ 1)(1 −d)n+ 2(1 −d)n+1 −1 d. Then formula (1) for the rank becomes in this case M(n, d, d0,1,(e),(α)) = = (−1)n+d(d0−1)n−(−1)n d0+ (−1)n(d−d0)(χ(n−1; e)−χ(n−1; d0, e)) = = (−1)n+d(d0−1)n−(−1)n d0+ +(d−d0)·(e−1)n−(−1)n e−1 d0−eµd0(e−1)n−(−1)n e−e(d0−1)n−(−1)n d0¶¸= = (−1)n+d0d−e d0−e (d0−1)n−(−1)n d0−ed−d0 d0−e (e−1)n−(−1)n e= 16 =1 d0−e[(d−e)(d0−1)n−(d−d0)(e−1)n] if d06=e. If d0=e, a similar computation gives M(n, d, e, 1,(e),(α)) = n(d−e)(e−1)n−1+ (e−1)n. In particular, when d=ewe are in the classical non-singular case, and the rank becomes −(d−d0)(d−1)n d0−d= (d−1)n as expected. When e= 1 we are in the other extremal case studied in the previous section and the rank is (d−1)(d0−1)n d0−1= (d−1)(d0−1)n−1 as we found there. 2) Now let ei= 1 for all i= 1, . . . , r. Then χ(n−1; eI) = n− |I|and χ(n−1; d0, eI) = n− |I|+(1 −d0)n−|I|−1 d0 for all I⊆ {1, . . . , r}. Plugging these values in formula (1) we get χ=X I⊆{1,...,r} 1≤|I|≤n−1 (−1)|I|−1(n−|I|)−X I⊆{1,...,r} 1≤|I|≤n−2 (−1)|I|−1µn− |I|+(1 −d0)n−|I|−1 d0¶= = n−1 X k=1 µr k¶(−1)k−1(n−k)− n−2 X k=1 µr k¶(−1)k−1µn−k+(1 −d0)n−k−1 d0¶= = (−1)nµr n−1¶− n−2 X k=1 µr k¶(−1)k−1(1 −d0)n−k−1 d0= =− n X k=1 µr k¶(−1)k−1(1 −d0)n−k−1 d0 so the rank is (−1)n+d(d0−1)n−(−1)n d0+ (d−d0) n X k=1 µr k¶(d0−1)n−k−(−1)n−k d0= = (d0−1)n+ (d−d0) n X k=0 µr k¶(d0−1)n−k−(−1)n−k d0. 17 In order to further simplify it, we need to assume r≤n. Then the sum becomes a complete binomial expansion: n X k=0 µr k¶(d0−1)n−k−(−1)n−k d0= =(d0−1)n−r d0 n X k=0 µr k¶(d0−1)r−k−(−1)n−r d0 n X k=0 µr k¶(−1)r−k= =(d0−1)n−r d0(d0−1 + 1)r−(−1)n−r d0(1 −1)r= (d0−1)n−rd0r−1 and the rank becomes (d0−1)n+ (d−d0)(d0−1)n−rd0r−1= = (d0−1)n−r(dd0r−1−d0r+ (d0−1)r). If r=n, all αi= 1 and d0= 1, one particular example of admissible sum is X x∈kn ψ(x1· · · xn+x1+· · · +xn−1−axn) for any a6= 0 in k. But X x∈kn ψ(x1· · · xn+x1+· · · +xn−1−axn) = =X x1,...,xn−1 ψ(x1+· · · +xn−1)X xn ψ(xn(x1· · · xn−1−a)) = =qX x1···xn−1=a ψ(x1+···+xn−1) which is just a Tate-twisted Kloosterman sum. So Theorem 1 gives a new proof of the purity of Kloosterman sums, at least in the case where the number of variables is not congruent to −1 mod p. The formula for the rank gives in this case M=d−1 = n−1, which agrees with the previously known result (cf. [6], Th´eor`eme 7.4). Similarly, if r=n,αn= 1 and d0= 1, we have the sum X x∈kn ψ(xα1 1· · · xαn−1 n−1xn+x1+· · · +xn−1−axn) = =X x1,...,xn−1 ψ(x1+· · · +xn−1)X xn ψ(xn(xα1 1· · · xαn−1 n−1−a)) = =qX xα1 1···xαn−1 n−1=a ψ(x1+· · · +xn−1) which is a Tate twist of the more general kind of Kloosterman sum considered in [12]. The rank in this case is d−1 which again agrees with the previously known value (cf. [12], Theorem 4.1.1). Therefore we can view the case r=nas a natural generalization of a Kloosterman sum: 18 Theorem 20 For any a1, . . . , an∈k?and any prime to ppositive integers α1, . . . , αnand d0< α1+···+αnsuch that α1+. . . +αnis prime to p, the sum X x∈kn ψ(xα1 1· · · xαn n+a1xd0 1+· · · +anxd0 n) is cohomologically pure of weight nand rank d0n−1(Pαi−d0) + (d0−1)n. 3) The previous example includes all admissible polynomials in two variables, since every homogeneous form in two variables splits as a product of linear factors in the algebraic closure of the base field k. The condition for a polynomial to be admissible becomes much simpler in this case, it means that we can write f=fd+fd0+f0with fdhomogeneous of degree dprime to p,fd0homoge- neous and square-free of degree d0< d prime to pand f0of degree < d0, and gcd(fd, fd0) = 1. In that case, the rank of the sum Px∈knψ(f(x)) is (−1)2+d(d0−2) + (d−d0) r X i=1 χ(1; 1) = 1 + d(d0−2) + r(d−d0). The highest homogeneous form fdwill be non-singular precisely when it has d distinct linear factors, i.e. when r=d, and then the rank becomes 1 + d(d0− 2) + d(d−d0) = d2−2d+ 1 = (d−1)2, as expected. References [1] Adolphson, A. and Sperber, S. Exponential sums on An.Israel J. Math. 120 (2000) part A, 3–21. [2] Adolphson, A. and Sperber, S. Exponential sums on An, II. Trans. Amer. Math. Soc. 356 (2004) no. 1, 345–369. [3] Beilinson, A., Bernstein, J. and Deligne, P. Faisceaux pervers, in Analyse et topologie sur les espaces singuliers (I). Ast´erisque 100. Societ´e Math´ematique de France (1982). [4] Deligne, P. La Conjecture de Weil I. Inst. Hautes ´ Etudes Sci. Publ. Math. 43 (1974) 273–307. [5] Deligne, P. La Conjecture de Weil II. Inst. Hautes ´ Etudes Sci. Publ. Math. 52 (1980) 137–252. [6] Deligne, P. Application de la formule des traces aux sommes trigonom´etriques, in Cohomologie ´ Etale (SGA 4 1/2) (Springer-Verlag 1977), 168–232. [7] Deligne, P. Th´eor`emes de finitude en cohomologie `-adique, in Cohomologie ´ Etale (SGA 4 1/2) (Springer-Verlag 1977), 233–261. 19 [8] Deligne, P. and Katz, N. Groupes de Monodromie en G´eom´etrie Alg´ebrique (SGA 7 II). Lecture Notes in Mathematics 340, Springer-Verlag (1973). [9] Denef, J. and Loeser, F. Weights of exponential sums, intersection cohomology, and Newton polyhedra. Invent. math. 106 (1991), 275–294. [10] Garc´ıa L´opez, R. Exponential sums and singular hypersurfaces. Manuscripta Math. 97 (1998) no. 1, 45–58. [11] Grothendieck, A. et al. Cohomologie `-adique et Fonctions L(SGA 5). Lecture Notes in Mathematics 589 (Springer-Verlag 1977). [12] Katz, N. Gauss Sums, Kloosterman Sums and Monodromy Groups. Annals of Mathematics Studies 116 (Princeton University Press 1987). [13] Katz, N. Exponential Sums and Differential Equations. Annals of Mathematics Studies 124 (Princeton University Press 1990). [14] Katz, N. Affine Cohomological Transforms, Perversity, and Monodromy. J. Amer. Math. Soc. 6(1993) no. 1, 149–222. [15] Katz, N. Estimates for ”Singular” Exponential Sums. Int. Math. Res. Not. (1999) no. 16, 875–899. [16] Katz, N. A semicontinuity result for monodromy under degeneration. Forum Math. 15 (2003) no. 2, 191–200. [17] Katz, N. Moments, Monodromy, and Perversity: a Diophantine Perspective, preprint. [18] Katz, N. and Laumon, G. Transformation de Fourier et Majoration de Sommes Exponentielles. Inst. Hautes ´ Etudes Sci. Publ. Math. 62 (1985) 361–418. [19] Laumon, G. Transformation de Fourier, constantes d’´equations fonctionelles et conjecture de Weil. Pub. Math. IHES 65 (1987), 131–210. [20] Rojas-Le´on, A. Purity of Exponential Sums on An.Compositio Math. (to appear) Antonio Rojas-Le´on University of California, Irvine Department of Mathematics Irvine, CA 92697, USA e-mail: [email protected] 20