Estimates for exponential sums with a large automorphism group

ESTIMATES FOR EXPONENTIAL SUMS WITH A LARGE AUTOMORPHISM GROUP ANTONIO ROJAS-LE ´ ON Abstract. We prove some improvements of the classical Weil bound for one variable additive and multiplicative character sums associated to a polynomial over a finite field k=Fqfor two classes of polynomials which are invariant under a large abelian group of automorphisms of the affine line A1 k: those invariant under translation by elements of kand those invariant under homotheties with ratios in a large subgroup of the multiplicative group of k. In both cases, we are able to improve the bound by a factor of √qover an extension of kof cardinality sufficiently large compared to the degree of f. 1. Introduction Let k=Fqbe a finite field with qelements. As a consequence Weil’s bound for the number of rational points on a curve over k, one can obtain estimates for character sums defined on the affine line A1 k(cf. [6],[17]). Let us describe the precise results. Let f∈k[x] be a polynomial of degree dand ψ:k→C?a non-trivial additive character. Consider the sum Px∈kψ(f(x)) (and, more generally, Px∈krψ(Trkr/k(f(x))) for a finite extension krof kof degree r). Then, if dis prime to p, we have the estimate X x∈kr ψ(Trkr/k(f(x)))≤(d−1)qr 2. If dis divisible by p, we can reduce to the previous case using the following trick. Since t7→ ψ(tp) is a non-trivial additive character, there must be some a∈ksuch that ψ(tp) = ψ(at) for every t∈k. If f(x) = adxd+ad−1xd−1+··· with d=ep, let bd∈kbe such that bp d=ad, then ψ(Trkr/k(f(x))) = ψ(Trkr/k((bdxe)p))ψ(Trkr/k(f(x)−adxd)) = =ψ(Trkr/k(bdxe)p)ψ(Trkr/k(f(x)−adxd)) = ψ(a·Trkr/k(bdxe))ψ(Trkr/k(f(x)−adxd)) = =ψ(Trkr/k(f(x)−adxd+abdxe)). We keep reducing the polynomial in this way until we get a polynomial with degree d0prime to p. Then we apply the prime to pcase and obtain an estimate X x∈kr ψ(Trkr/k(f(x)))≤(d0−1)qr 2. except when d0is zero (that is, when f=c+gp−ag for some constant cand sone g∈k[x]). If the character ψis obtained from a character of the prime subfield Fp by pulling back via the trace map, then a= 1. Partially supported by P08-FQM-03894 (Junta de Andaluc´ıa), MTM2007-66929 and FEDER. 1 2 ANTONIO ROJAS-LE ´ ON Similarly, if χ:k?→C?is a multiplicative character of order m > 1 and f∈k[x] is not an m-th power, we have an estimate X x∈kr χ(Nkr/k(f(x)))≤(e−1)qr 2≤(d−1)qr 2 where eis the number of distinct roots of f. In this article we will improve these estimates for a special class of polynomials: those which are either translation invariant or homothety invariant, that is, either f(x+λ) = f(x) for every λ∈kor f(λx) = f(x) for every λ∈k?(or every λin a large subgroup of k?). For such polynomials, there is a large abelian group Gof automorphisms of A1 ksuch that f◦σ=ffor every σ∈G. On the level of `-adic cohomology, this gives an action of Gon the pull-back by fof the Artin-Schreier and Kummer sheaves associated to ψand χrespectively [1, 1.7], so they induce an action on their cohomology. The character sums can be expressed as the trace of the geometric kr-Frobenius action on this cohomology, by Grothendieck’s trace formula. The above estimates are a consequence of the fact that this action has all eigenvalues of archimedean absolute value ≤qr 2. Precisely, if Sr=Px∈krψ(Trkr/k(f(x))) (respectively Ur=Px∈krχ(Nkr/k(f(x)))) the L- functions L(ψ, f;T) := exp X r≥1 Sr Tr r and L(χ, f;T) := exp X r≥1 Ur Tr r are the polynomials det(1−T·Frobk|H1 c(A1 ¯ k, f?Lψ)) and det(1−T·Frobk|H1 c(A1 ¯ k, f?Lχ)), of degree d0−1 and e−1 respectively. Now under the action of the abelian group G, this cohomology splits as a direct sum of eigenspaces for the different characters of G. Under certain generic conditions, it is natural to expect some cancellation among the traces of the Frobenius actions on these eigenspaces, thus giving a substantial improvement of Weil’s estimate if Gis large (namely by a √#Gfactor). Compare [15], where an improvement for the Weil estimate for the number of rational points on Artin-Schreier curves was obtained using the same arguments we apply in this article. For the translation invariant case (sections 2 ans 3), we obtain this improvement using the local theory of `-adic Fourier transform [14] and Katz’ computation of the geometric monodromy groups for some families of exponential sums [7], [9]. The argument is similar to that in [15]. For the homothety invariant case (sections 4 and 5), we use Weil descent together with certain properties of the convolution of sheaves on Gm,k. Throughout this article, k=Fqwill be a finite field of characteristic p,¯ k=¯ Fqa fixed algebraic closure and kr=Fqrthe unique extension of kof degree rin ¯ k. We will fix a prime `6=p, and work with `-adic cohomology. In order to speak about weights without ambiguity, we will fix a field isomorphism ι:¯ Q`→C. We will use this isomorphism to identify ¯ Q`and Cwithout making any further mention to it. When we speak about weights, we will mean weights with respect to the chosen isomorphism ι. ESTIMATES FOR EXPONENTIAL SUMS WITH A LARGE AUTOMORPHISM GROUP 3 2. Additive character sums for translation invariant polynomials Let f∈k[x] be a polynomial. fis said to be translation invariant if f(x+a) = f(x) for every a∈k. Lemma 2.1. Let f∈k[x]. The following conditions are equivalent: (a) fis translation invariant. (b) There exists g∈k[x]such that f(x) = g(xq−x). Proof. (b)⇒(a) is clear. Suppose that fis translation invariant. If the degree of fis < q, the polynomial f(x)−f(0) has at least qroots (all elements of k) and degree < q, so it is identically zero. So fis the constant polynomial f(0). Otherwise, we can write f(x) = (xq−x)h(x) + r(x) with deg(r)< q. For every a∈kwe have then f(x+a)=(xq−x)h(x+a) + r(x+a)=(xq−x)h(x) + r(x), so (xq−x)(h(x+a)−h(x)) = r(x)−r(x+a). Since the right hand side has degree < q, we conclude that h(x+a)−h(x) = r(x+a)−r(x) = 0. r(x) is then translation invariant and therefore constant, for its degree is less than q, and his also translation invariant of degree deg(f)−q. By induction, there is t∈k[x] such that h(x) = t(xq−x). So we take g(x) = xt(x) + r. Let f∈k[x] be translation invariant, and g∈k[x] of degree dsuch that f(x) = g(xq−x). Let ψ:k→¯ Q? `be a non-trivial additive character. The Artin- Scheier-reduced degree of f(i.e. the lowest degree of a polynomial which is Artin- Schreier equivalent to f) is q(d−1) + 1 (since g(xq−x) = adxqd +dadxq(d−1)+1 + (terms of degree ≤q(d−1))). Therefore the Weil bound for exponential sums gives X x∈kr ψ(Trkr/k(f(x)))≤q(d−1)qr 2= (d−1)qr 2+1 On the other hand, since f(x) = g(xq−x) we get, for every r≥1, X x∈kr ψ(Trkr/k(f(x))) = X x∈kr ψ(Trkr/k(g(xq−x))) = =X t∈kr #{x∈kr|xq−x=t}ψ(Trkr/k(g(t))) = =X t∈krX u∈k ψ(uTrkr/k(t))ψ(Trkr/k(g(t))) = X u∈kX t∈kr ψ(Trkr/k(g(t) + ut)). Consider the ¯ Q`-sheaf Lψ(g):= g?Lψon A1 k, where Lψis the Artin-Schreier sheaf associated to ψ. The Fourier transform of the object Lψ(g)[1] with respect to ψ[13] is a single sheaf Fgplaced in degree −1. The sheaf Fgis irreducible and smooth of rank d−1 on A1 k, and totally wild at infinity with a single slope d d−1and Swan conductor d[7, Theorem 17]. We have X x∈kr ψ(Trkr/k(f(x))) = X x∈kr ψ(Trkr/k(g(xq−x))) = X u∈kX t∈kr ψ(Trkr/k(g(t) + ut)) = (1) = −X u∈k Tr(Frobr k,u|(Fg)u) = −X u∈k Tr(Frobk,u|[Fg]r u) where [Fg]ris the r-th Adams power of Fg[4]. 4 ANTONIO ROJAS-LE ´ ON Let g(x) = Pd i=0 aixi. Recall the following facts about the local and global monodromies of the sheaf Fg: (1) Suppose that p>dand kcontains all 2(d−1)-th roots of −dad. Let u(t) = Pi≥0rit1−i∈tk[[t−1]] be a power series such that f0(t) + u(t)d−1= 0 and let v(t) = Pi≥0sit1−ibe the inverse image of tunder the automorphism k((t−1)) →k((t−1)) defined by t−17→ u(t)−1(cf. [3, Proposition 3.1]). Let h(t) = Pd i=0 bitibe the polynomial obtained from f(v(t)) + v(t)td−1∈ tdk[[t−1]] by removing the terms with negative exponent. Then, as a representation of the decomposition group D∞at infinity, we have Fg∼ =[d−1]?(Lψ(h(t)) ⊗Lρd(s0t))⊗ρ(d(d−1)ad/2)deg ⊗g(ρ, ψ)deg where ρ:k?→¯ Q? `is the quadratic character, g(ρ, ψ) = −Pt∈kρ(t)ψ(t) the corresponding Gauss sum and [d−1]?:Gm,k →Gm,k the (d−1)-th power map [5, Equation 3]. Notice that sd−1 0=−1/dad. (2) Suppose that p > 2, and let G⊆GL(V) be the geometric monodromy group of Fg, where Vis its stalk at a geometric generic point. Then by [16, Propositions 11.1 and 11.6], either Gis finite or G0(the unit connected component of G) is SL(V) or Sp(V) in its standard representation. By [7, proof of Theorem 19], for p>dthe Sp case occurs if and only if g(x+c)+d is odd for some c, d ∈k. Moreover for p > 2d−1Gis never finite by [7, Theorem 19]. See [5, Section 2] for some other criterions that rule out the finite monodromy case in the p≤2d−1 case. The determinant of Fgis computed over ¯ kin [7, Theorem 17]. In order to obtain a good estimate in the exceptional case below, we need to find its value over k. Lemma 2.2. Suppose that p>dand kcontains all 2(d−1)-th roots of −dad. Then det Fg∼ =Lψ((d−1)bd−1t+(d−1)b0)⊗ρd(−1)deg ⊗ρd−1(d(d−1)ad/2)deg ⊗(g(ρ, ψ)d−1)deg Proof. Note that the result is compatible with [7, Theorem 17], since bd−1= ad−1sd−1 0=ad−1/rd−1 0=−ad−1/dadas one can easily check. Let Dd−1 ∞⊆D∞be the closed subgroup of index d−1 which fixes 1/td−1. Since kcontains all (d−1)-th roots of unity, Dd−1 ∞is normal in D∞and the quotient D∞/Dd−1 ∞is generated by t7→ ζt, where ζ∈kis a primitive (d−1)-th root of unity. Using the previous description of the representation of D∞given by Fg, we get an isomorphism of Dd−1 ∞-representations [d−1]?Fg∼ = ∼ = d−2 M i=0 (t7→ ζit)?Lψ(h(t)) ⊗Lρd(s0t)!⊗ρ(d(d−1)ad/2)deg ⊗g(ρ, ψ)deg ∼ = ∼ = d−2 M i=0 Lψ(h(ζit)) ⊗Lρd(s0ζit)!⊗ρ(d(d−1)ad/2)deg ⊗g(ρ, ψ)deg so [d−1]?det Fg∼ =det[d−1]?Fg∼ = ∼ = d−2 O i=0 Lψ(h(ζit)) ⊗Lρd(s0ζit)!⊗ρd−1(d(d−1)ad/2)deg ⊗(g(ρ, ψ)d−1)deg ∼ = ESTIMATES FOR EXPONENTIAL SUMS WITH A LARGE AUTOMORPHISM GROUP 5 ∼ =Lψ(Pd−2 i=0 h(ζit)) ⊗Lρd(Qd−2 i=0 (s0ζit)) ⊗ρd−1(d(d−1)ad/2)deg ⊗(g(ρ, ψ)d−1)deg ∼ = ∼ =Lψ((d−1)bd−1td−1+(d−1)b0)⊗Lρd((−1)d(s0t)d−1)⊗ρd−1(d(d−1)ad/2)deg⊗(g(ρ, ψ)d−1)deg ∼ = ∼ =Lψ((d−1)bd−1td−1+(d−1)b0)⊗Lρd(d−1)(−s0t)⊗ρd(−1)deg⊗ρd−1(d(d−1)ad/2)deg⊗(g(ρ, ψ)d−1)deg ∼ = ∼ =Lψ((d−1)bd−1td−1+(d−1)b0)⊗ρd(−1)deg ⊗ρd−1(d(d−1)ad/2)deg ⊗(g(ρ, ψ)d−1)deg since Pd−2 i=0 (ζj)i= 0 for (d−1) 6 |j,d(d−1) is even and Qd−2 i=0 ζi= (−1)d. In particular, [d−1]?(det Fg) and [d−1]?Lψ((d−1)bd−1t+(d−1)b0)⊗ρd(−1)deg ⊗ρd−1(d(d−1)ad/2)deg ⊗(g(ρ, ψ)d−1)deg are isomorphic characters of Dd−1 ∞, so there is some character χ:k?→¯ Q? `with χd−1=1such that det Fg∼ =Lχ⊗Lψ((d−1)bd−1t+(d−1)b0)⊗ρd(−1)deg⊗ρd−1(d(d−1)ad/2)deg⊗(g(ρ, ψ)d−1)deg as representations of D∞. But then \ (det Fg)⊗Lχ⊗Lψ((d−1)bd−1t+(d−1)b0)⊗ρd(−1)deg⊗ρd−1(d(d−1)ad/2)deg⊗(g(ρ, ψ)d−1)deg is a rank 1 smooth sheaf on Gm,k, tamely ramified at 0 and unramified at infinity, so it must be geometrically trivial, that is, χis trivial (since everything else is unramified at 0). Moreover, since the Frobenius action is trivial at infinity it must be the trivial sheaf. Therefore det Fg∼ =Lψ((d−1)bd−1t+(d−1)b0)⊗ρd(−1)deg ⊗ρd−1(d(d−1)ad/2)deg ⊗(g(ρ, ψ)d−1)deg as sheaves on A1 k. Proposition 2.3. Suppose that p>d, the sheaf Fgdoes not have finite monodromy (e.g. p > 2d−1) and there do not exist c, d ∈ksuch that g(x+c) + dis odd. Then we have an estimate X x∈kr ψ(Trkr/k(f(x)))≤Cd,rqr+1 2 where Cd,r =1 d−1 d−1 X i=0 |i−1|d−2 + r−i r−id−1 i unless ad−1= 0 and r=d−1, in which case there is an estimate X x∈kr ψ(Trkr/k(f(x))) −(−1)d−1q·ρd(−1)(ψ(b0)ρ(d(d−1)ad/2)g(ρ, ψ))d−1 < Cd,rqr+1 2. Proof. By [4, Section 1], we have X x∈kr ψ(Trkr/k(f(x))) = −X u∈k Tr(Frobk,u|[Fg]r u) = = r X i=0 (−1)i−1(i−1)Tr(Frobk,H1 c(A1 ¯ k,Symr−iFg⊗∧iFg))− − r X i=0 (−1)i−1(i−1)Tr(Frobk,H2 c(A1 ¯ k,Symr−iFg⊗∧iFg)). Let G⊆GL(V) be the geometric monodromy group of Fg. Under the hypotheses of the proposition, the unit connected component of Gis SL(V), so Gis the inverse 6 ANTONIO ROJAS-LE ´ ON image of its image by the determinant. By lemma 2.2, Gis SL(V) if bd−1= 0 (if and only if ad−1= 0) and GLp(V) = µp·SL(V) (since p > d, so pdoes not divide d−1) if bd−16= 0. For every i, the dimension of H2 c(A1 ¯ k,Symr−iFg⊗∧iFg) is the dimension of the coinvariant (or the invariant) space of the action of Gon Symr−iV⊗ ∧iV. By [15, Corollary 4.2], the action of SL(V)⊆Gon Symr−iV⊗∧iVhas no invariants unless r=d−1 and i=r, r −1, in which case the invariant space Wiis onedimensional. If ad−16= 0, a generator ζpof the quotient G/SL(V)∼ =µpacts on Wi via multiplication by ζd−1 p, which can not be trivial since p > d. So the action of Ghas no invariants on Symr−iV⊗∧iVfor any iif ad−16= 0. In that case, since H1 c(A1 ¯ k,Symr−iFg⊗∧iFg) is mixed of weights ≤r+ 1 we get X x∈kr ψ(Trkr/k(f(x)))≤ r X i=0 |i−1|dim H1 c(A1 ¯ k,Symr−iFg⊗∧iFg)·qr+1 2. Moreover, by the Ogg-Shafarevic formula we have dim H1 c(A1 ¯ k,Symr−iFg⊗∧iFg) = −χ(A1 ¯ k,Symr−iFg⊗∧iFg) = = Swan∞(Symr−iFg⊗∧iFg)−rank(Symr−iFg⊗∧iFg)≤ ≤1 d−1rank(Symr−iFg⊗∧iFg) = 1 d−1d−2 + r−i r−id−1 i since all slopes at infinity of Fg(and a fortiori of Symr−iFg⊗∧iFg) are ≤d d−1. Suppose now that ad−1= 0 and r=d−1. As in [15, Corollary 4.2], we have r X i=0 (−1)i−1(i−1)Tr(Frobk,H2 c(A1 ¯ k,Symr−iFg⊗∧iFg)) = = (−1)r(r−2)Tr(Frobk,H2 c(A1 ¯ k,Sym1Fg⊗∧r−1Fg))+ +(−1)r−1(r−1)Tr(Frobk,H2 c(A1 ¯ k,∧rFg)) = = (−1)r−1Tr(Frobk,H2 c(A1 ¯ k,det Fg)) = = (−1)dq·ψ((d−1)b0)ρd(−1)ρd−1(d(d−1)ad/2)g(ρ, ψ)d−1= = (−1)dq·ρd(−1)(ψ(b0)ρ(d(d−1)ad/2)g(ρ, ψ))d−1 by lemma 2.2. We conclude as above using the fact that, for the two values of ifor which H2 c(A1 ¯ k,Symr−iFg⊗ ∧iFg) is one-dimensional, the sheaf Symr−iFg⊗ ∧iFg has at least one slope equal to 0 at infinity, so dim H1 c(A1 ¯ k,Symr−iFg⊗∧iFg)=1−χ(A1 ¯ k,Symr−iFg⊗∧iFg) = = 1 + Swan∞(Symr−iFg⊗∧iFg)−rank(Symr−iFg⊗∧iFg)≤ ≤1 + d d−1(rank(Symr−iFg⊗∧iFg)−1) −rank(Symr−iFg⊗∧iFg)< <1 d−1rank(Symr−iFg⊗∧iFg) = 1 d−1d−2 + r−i r−id−1 i.  ESTIMATES FOR EXPONENTIAL SUMS WITH A LARGE AUTOMORPHISM GROUP 7 Proposition 2.4. Suppose that p>d, the sheaf Fgdoes not have finite monodromy (e.g. p > 2d−1) and there exist α, β ∈ksuch that g(x+α) + βis odd (so dis odd). Then we have an estimate X x∈kr ψ(Trkr/k(f(x)))≤Cd,rqr+1 2 where Cd,r =1 d−1 d−1 X i=0 |i−1|d−2 + r−i r−id−1 i unless ad−1= 0 and r≤d−1is even, in which case there is an estimate X x∈kr ψ(Trkr/k(f(x))) −(−1)rψ(−β)rqr 2+1 < Cd,rqr+1 2. Proof. The proof is similar to the previous one. In this case, the unit connected component of Gis Sp(V), so by lemma 2.2 Gis Sp(V) if bd−1= 0 (if and only if ad−1= 0) and µp·SL(V) (since p>d, so pdoes not divide d−1) if bd−16= 0. By [10, lemma on p.62], the action of Sp(V)⊆Gon Symr−iV⊗ ∧iVhas no invariants unless r≤d−1 is even and i=r, r −1, in which case the invariant space Wiis one-dimensional. If ad−16= 0, a generator ζpof the quotient G/Sp(V)∼ =µp acts on Wivia multiplication by ζd−1 p, which can not be trivial since p>d. So the action of Ghas no invariants on Symr−iV⊗∧iVfor any iif ad−16= 0. We conclude this case as in the previous proposition. Suppose now that ad−1= 0, r≤d−1 is even and i=ror r−1. Since the coefficient of xd−1in g(x) is 0, the coefficient in g(x+α) + βis dadα, so it can only be an odd polynomial if α= 0. That is, g(x) + βis odd, or equivalently, g(−x) = −2β−g(x). Then the sheaf ψ(β)deg ⊗Fg(1/2) is self-dual: since the dual of Lψ(g)is Lψ(−g)(1), using that D◦FTψ= [−1]?F Tψ◦D(1) [13, Corollaire 2.1.5] we get that the dual of Fg=H−1(FTψ(Lψ(g)[1])) is [−1]?H−1(FTψLψ(−g)(1)) = [−1]?F−g(1) = [−1]?F2β+g(−x)(1) = ψ(2β)deg ⊗Fg(1) so ψ(β)deg ⊗ Fg(1/2) is self-dual (symplectically, since it is so geometrically by [7, Theorem 19]). In particular, the one-dimensional Sp(V)-invariant subspace of (Symr−iFg⊗∧iFg)⊗ψ(β)r·deg(r/2) is also invariant under all Frobenii. So Wiis in fact the geometrically constant sheaf ψ(−β)r·deg(−r/2). In particular r X i=0 (−1)i−1(i−1)Tr(Frobk,H2 c(A1 ¯ k,Symr−iFg⊗∧iFg)) = = (−1)r(r−2)Tr(Frobk,H2 c(A1 ¯ k,Sym1Fg⊗∧r−1Fg))+ +(−1)r−1(r−1)Tr(Frobk,H2 c(A1 ¯ k,∧rFg)) = = (−1)r−1Tr(Frobk,H2 c(A1 ¯ k, ψ(−β)r·deg(−r/2))) = (−1)r−1ψ(−β)rqr 2+1. We conclude as in the previous proposition.  8 ANTONIO ROJAS-LE ´ ON 3. Multiplicative character sums for translation invariant polynomials Let f∈k[x] be translation invariant, and g∈k[x] of degree dsuch that f(x) = g(xq−x). Let χ:k?→¯ Q? `a non-trivial multiplicative character of order m, extended by zero to all of k. Since fhas degree qd, Weil’s bound gives in this case X x∈kr χ(Nkr/k(f(x)))≤(qd −1)qr 2. On the other hand we have, for every r≥1, X x∈kr χ(Nkr/k(f(x))) = X x∈kr χ(Nkr/k(g(xq−x))) = =X t∈kr #{x∈kr|xq−x=t}χ(Nkr/k(g(t))) = =X t∈krX u∈k ψ(uTrkr/k(t))χ(Nkr/k(g(t))) = X u∈kX t∈kr ψ(uTrkr/k(t))χ(Nkr/k(g(t))). Consider the ¯ Q`-sheaf Lχ(g):= g?Lχon A1 k, where Lχis the Kummer sheaf on Gm,k associated to χ[1, 1.7], extended by zero to A1 k. Suppose that gis square-free and its degree dis prime to p. Then Lχ(g)is an irreducible middle extension sheaf, smooth on the complement of the subscheme Z⊆A1 kdefined by g= 0. Since there is at least one point where it is not smooth, it is not isomorphic to an Artin-Schreier sheaf and therefore the Fourier transform of Lχ(g)[1] is a single irreducible middle extension sheaf Fgplaced in degree −1 [8, 8.2]. We have X x∈kr χ(Nkr/k(f(x))) = X x∈kr χ(Nkr/k(g(xq−x))) = X u∈kX t∈kr ψ(uTrkr/k(t))χ(Nkr/k(g(t))) = (2) = −X u∈k Tr(Frobr k,u|(Fg)u) = −X u∈k Tr(Frobk,u|[Fg]r u) where [Fg]ris the r-th Adams power of Fg. Proposition 3.1. The sheaf Fghas generic rank d, it is smooth on Gm,k and tamely ramified at 0. Its rank at 0is d−1. If all roots of gare in k, the action of the decomposition group D∞on the generic stalk of Fgsplits as a direct sum Laχ(g0(a))deg ⊗g(χ, ψ)deg ⊗ L¯χ⊗ Lψawhere the sum is taken over the roots of f,Lψais the Artin-Schreier sheaf corresponding to the character t7→ ψ(at)and g(χ, ψ) = −Ptχ(t)ψ(t)if the Gauss sum. Proof. The generic rank of Fgis the dimension of H1 c(A1 ¯ k,Lχ(g)⊗Lψz) for generic z. Since Lχ(g)is tamely ramified everywhere and has rank one, for any z6= 0 Lχ(g)⊗Lψzis tamely ramified at every point of A1 ¯ kand totally wild at infinity with Swan conductor 1. In particular its Hi cvanish for i6= 1. By the Ogg-Shafarevic formula, its Euler characteristic is then 1 −d−1 = −d, since there are dpoints in A1 ¯ kwhere the stalk is zero. Therefore dim H1 c(A1 ¯ k,Lχ(g)⊗Lψz) = dfor every z6= 0. Similarly, it is d−1 for z= 0. Since Fgis a middle extension, it is smooth exactly on the open set where the rank is maximal, so it is smooth on Gm,k. It is tamely ramified at zero, since Lχ(g)is tamely ramified at infinity [14, Th´eor`eme 2.4.3]. Suppose now that all roots of gare in k, and let abe one such root. In an ´etale neighborhood of a, the sheaf Lχ(g)is isomorphic to Lχ(g0(a)(x−a)) =χ(g0(a))deg ⊗ ESTIMATES FOR EXPONENTIAL SUMS WITH A LARGE AUTOMORPHISM GROUP 9 Lχ(x−a), since g(x) = g0(a)(x−a)g(x) g0(a)(x−a)and g(x) g0(a)(x−a)is an m-th power in the henselization of A1 kat a(since its image in the residue field is 1). Applying Laumon’s local Fourier transform [14, Proposition 2.5.3.1] and using that Fourier transform commutes with tensoring by unramified sheaves, we deduce that the D∞-representation Fgcontains (LFT(0,∞) ψχ(g0(a))deg ⊗Lχ)⊗Lψa=χ(g0(a))deg ⊗ g(χ, ψ)deg ⊗L¯χ⊗Lψaas a direct summand. Since ghas ddistinct roots we obtain ddifferent terms this way, which is the rank of Fg, so its monodromy at ∞is the direct sum of these terms.  Define by induction the sequence of polynomials gn[x]∈k[x] for n≥1 by: g1(x) = g(x), and for n≥1gn+1(x) is the resultant in tof gn(t) and g(x−t). Corollary 3.2. Suppose that either mdoes not divide ror gr(0) 6= 0. Then we have an estimate X x∈kr χ(Nkr/k(f(x)))≤Cd,rqr+1 2 where Cd,r = r X i=0 |i−1|d−1 + r−i r−id i−d−2 + r−i r−id−1 i. Proof. By the previous proposition, the action of the inertia group I∞on F⊗r gsplits as a direct sum over the r-uples of roots of f M (a1,...,ar)L⊗r ¯χ⊗Lψa1⊗···⊗Lψar=M (a1,...,ar)L⊗r ¯χ⊗Lψa1+···+ar. For each (a1, . . . , ar), the character L⊗r ¯χ⊗Lψa1+···+aris trivial if and only if both L⊗r ¯χand Lψa1+···+arare trivial, that is, if and only if mdivides rand a1+··· + ar= 0. Under the hypotheses of the corollary, at least one of these conditions does not hold (since the sums a1+··· +arare the roots of gr). So F⊗r ghas no invariants under the action of I∞and, a fortiori, under the action of the larger group π1(Gm,¯ k,¯η). Since Symr−iFg⊗ ∧iFgis a subsheaf of F⊗r gfor every i, we conclude that H2 c(A1 ¯ k,Symr−iFg⊗∧iFg) = 0 for every i= 0, . . . , r. Therefore X x∈kr χ(Nkr/k(f(x))) = −X u∈k Tr(Frobk,u|[Fg]r u) = = r X i=0 (−1)i−1(i−1)Tr(Frobk,H1 c(A1 ¯ k,Symr−iFg⊗∧iFg)). Since H1 c(A1 ¯ k,Symr−iFg⊗∧iFg) is mixed of weights ≤r+ 1, we get X x∈kr χ(Nkr/k(f(x)))≤ r X i=0 |i−1|dim H1 c(A1 ¯ k,Symr−iFg⊗∧iFg)·qr+1 2. And by the Ogg-Shafarevic formula, we have dim H1 c(A1 ¯ k,Symr−iFg⊗∧iFg) = −χ(A1 ¯ k,Symr−iFg⊗∧iFg) = = Swan∞(Symr−iFg⊗∧iFg)−rank0(Symr−iFg⊗∧iFg)≤ ≤d−1 + r−i r−id i−d−2 + r−i r−id−1 i 16 ANTONIO ROJAS-LE ´ ON (r−2) + 1 = r−1 [2, Th´eor`eme 3.2.3]. So Lris pure of weight r−1 on the open set where it is smooth. Now let jW:W ,→A1 ¯ kbe the inclusion of the largest open sen on which Lr is smooth. Since Lrhas no punctual sections, there is an injection 0 → Lr→ jW ?j? WLr, let Qbe its punctual cokernel. We have an exact sequence 0→H0 c(A1 ¯ k,Q)→H1 c(A1 ¯ k,Lr)→H1 c(A1 ¯ k, jW ?j? WLr)→0 where H0 c(A1 ¯ k,Q) has weight ≤r−1. Since H1 c(A1 ¯ k,Lr) is pure of weight r, we conclude that H0 c(A1 ¯ k,Q) and therefore Qare zero, so Lris a middle extension. Now let j:A1 ¯ k,→P1 ¯ kbe the inclusion, again we get an exact sequence 0→ LI∞ r→H1 c(A1 ¯ k,Lr)→H1 c(P1 ¯ k, j?Lr)→0 with LI∞ rof weight ≤r−1, since H1 c(A1 ¯ k,Lr) is pure of weight rwe conclude that LI∞ r= 0, that is, Lris totally ramified at infinity. It remains to prove that Lrhas unipotent monodromy at zero. Consider the exact sequence 0→ LI0 r→H1 c(Gm,¯ k,Lr)→H1 c(A1 ¯ k,Lr)→0 which identifies LI0 rwith the weight < r part of H1 c(Gm,¯ k,Lr). Since H1 c(Gm,¯ k,Lr) = Nr i=1 H1 c(Gm,¯ k,Lχ(gi)) and H1 c(Gm,¯ k,Lχ(gi)) has d−1 Frobenius eigenvalues of weight 1 and one of weight 0, we conclude that H1 c(Gm,¯ k,Lr) has r i(d−1)ieigenvalues of weight ifor every i= 0, . . . , r. By [8, Theorem 7.0.7], an eigenvalue of weight i<ron LI0 rcorresponds to a unipotent Jordan block of size r−ifor the action of I0. So the sum of the sizes of the unipotent Jordan blocks for the monodromy of Lrat 0 is r−1 X i=0 r i(d−1)i(r−i) = r r−1 X i=0 r i(d−1)i−r r−1 X i=0 r−1 i−1(d−1)i= =r r−1 X i=0 r−1 i(d−1)i=r(1 + d−1)r−1=rdr−1 which is the generic rank of Lr. So the unipotent Jordan blocks fill out the entire monodromy at 0.  Corollary 5.2. Suppose that gis square-free of degree dprime to pand χdis not trivial. For any µ∈k?,Hi c(Vµ⊗¯ k, Lχ(H)) = 0 for i6=r−1and dim Hr−1 c(Vµ⊗ ¯ k, Lχ(H)) = rdr−1. Proof. Apply the previous proposition with (g1, . . . , gr)=(gσ)σ∈Gal(kr/k), and proper base change.  Corollary 5.3. Suppose that gis square-free of degree dprime to pand χdis not trivial. Then X x∈k? r χ(Nkr/k(f(x)))≤rdr−1(q−1)qr−1 2 ESTIMATES FOR EXPONENTIAL SUMS WITH A LARGE AUTOMORPHISM GROUP 17 Proof. Since Lχ(H)is pure of weight 0, Hr−1 c(Vµ⊗¯ k, Lχ(H)) has weights ≤r−1 for every µ. So the previous corollary together with (6) implies X Nkr/k(x)=µ χ(Nkr/k(g(x)))≤rdr−1qr−1 2 for every µ∈k?. We conclude by using (5).  Remark 5.4. The following example shows that the hypothesis χdnon-trivial is necessary. Let pbe odd, r= 2, g(x) = x2+ 1 and ρ:k?→¯ Q? `the quadratic character. Then X Nkr/k(x)=1 ρ(Nkr/k(x2+ 1)) = X xq+1=1 ρ((x2+ 1)(x2q+ 1)) = =X xq+1=1 ρ(x2+x2q+ 2) = X xq+1=1 ρ((x+xq)2)≥q−1 since x+xq= Trkr/k(x)∈kand therefore ρ((x+xq)2) = ρ(x+xq)2= 1 unless x+xq= 0, which only happens for x2=−1, that is, for at most two values of x. So we can never have an estimate of the form X Nkr/k(x)=1 ρ(Nkr/k(x2+ 1))≤C·q1 2 which is valid for all q. References 1. P. Deligne, Application de la formule des traces aux sommes trigonom´etriques dans Cohomologie Etale, S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie, SGA 4 1/2, Lecture Notes in Math 569, 168–232. 2. , La conjecture de Weil. 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Laumon, Transformation De Fourier, Constantes d’Equations Fonctionnelles Et Conjecture De Weil, Publications Math´ematiques de l’IH´ ES 65 (1987), no. 1, 131–210. 15. Antonio Rojas-Leon and Daqing Wan, Big improvements of the Weil bound for Artin-Schreier curves, arXiv:1004.2224 (2010). 16. O. ˇ Such, Monodromy of Airy and Kloosterman sheaves, Duke Mathematical Journal 103 (2000), no. 3, 397–444. 17. A. Weil, On some exponential sums, Proceedings of the National Academy of Sciences of the United States of America 34 (1948), no. 5, 204. Departamanto de ´ Algebra, Universidad de Sevilla, Apdo 1160, 41080 Sevilla, Spain E-mail: ar[email protected]