Improvements of the Weil bound for Artin-Schreier curves

IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES ANTONIO ROJAS-LEON AND DAQING WAN Abstract. For the Artin-Schreier curve yq−y=f(x) defined over a finite field Fqof qelements, the celebrated Weil bound for the number of Fqr-rational points can be sharp, especially in super-singular cases and when ris divisible. In this paper, we show how the Weil bound can be significantly improved, using ideas from moment L-functions and Katz’s work on `-adic monodromy calculations. Roughly speaking, we show that in favorable cases (which happens quite often), one can remove an extra √qfactor in the error term. 1. Introduction Let k=Fqbe a finite field of characteristic p > 2 with qelements, and let f∈k[x] be a polynomial of degree d > 1. Without loss of generality, we can and will always assume that dis not divisible by p. Let Cfbe the affine Artin-Schreier curve defined over kby yq−y=f(x). Let rbe a positive integer, and let Nr(f) denote the number of Fqr-rational points on Cf. The genus of the smooth projective model of Cfis given by g= (q−1)(d−1)/2. The celebrated Weil bound in this case gives the estimate |Nr(f)−qr| ≤ (d−1)(q−1)qr 2. This bound can be sharp in general, for instance when Cfis supersingular and ris divisible. If qris not a square, Serre’s improvement [14] leads to a somewhat better bound: |Nr(f)−qr| ≤ (d−1)(q−1) 2[2qr 2], where [x] denotes the integer part of a real number x. In this paper, we shall show that if qis large compared to d(and thus the genus g= (d−1)(q−1)/2 is small compared to the field size qrwith r≥2), then the above Weil bound can be significantly improved in many cases. The type of theorems we prove is of the following nature. For simplicity, we just state one special case. Theorem 1.1. Let r≥1and p > 2. If the derivative f0is square-free and either ris odd or the hypersurface f(x1) + ···+f(xr)=0in Ar kis non-singular, then we have the estimate |Nr(f)−qr| ≤ Cd,rqr+1 2, The research of Antonio Rojas-Leon is partially supported by P08-FQM-03894 (Junta de Andaluc´ıa), MTM2007-66929 and FEDER. The research of Daqing Wan is partially supported by NSF. 1 2 ANTONIO ROJAS-LEON AND DAQING WAN where Cd,r is the constant Cd,r = r X a=0 |a−1|d−2 + r−a r−ad−1 a. Note that the constant Cd,r is independent of qand it is a polynomial in dwith degree r. Thus, for fixed dand r, our result essentially removes an extra √qfactor from Weil’s bound. The non-singularity hypothesis cannot be dropped in general, as there are cases for reven where we can have |Nr(f)−(qr+qr 2+1)| ≤ Cd,rqr+1 2, see section 4 for more details. This gives further examples that the q-factor in the Weil bound cannot be replaced by an O(√q) factor in general. As an extreme illustration, we consider the elementary case that r= 1. It is clear that N1(f) = qnf, where nfis the number of distinct roots of f(x) in Fq which is at most d. Thus, the best estimate in this case should be |N1(f)−q| ≤ (d−1)q, which is precisely what our bound gives! It is far better than the Weil bound |N1(f)−q| ≤ (d−1)(q−1)√q. For r= 2, our bound takes the form |N2(f)−q2| ≤ (d−1)2q3/2, which is better than the Weil bound |N2(f)−q2| ≤ (d−1)(q−1)q as soon as q≥(d−1)2+ 3. For r= 3, our bound takes the form |N3(f)−q3| ≤ (d−1)(d2−3d+ 3)q2, which is better than the Weil bound |N3(f)−q3| ≤ (d−1)(q−1)q3/2 as soon as q≥(d2−3d+ 4)2. Our idea is to translate Nr(f) to moment exponential sums and then calculate the associated moment L-function as explicitly as possible. Let ψbe a fixed nontrivial additive character of k. For f∈k[x], it is clear that we have the formula Nr(f) = X t∈kX x∈kr ψ(Tr(tf(x))), where kr=Fqrand Tr denotes the trace map from krto k. Separating the term from t= 0, we obtain (1) Nr(f)−qr=X t∈k?X x∈kr ψ(Tr(tf(x))). Now, Weil’s bound for exponential sums gives the estimate X x∈kr ψ(Tr(tf(x)))≤(d−1)qr 2 for every t∈k?. It follows that |Nr(f)−qr| ≤ (q−1)(d−1)qr 2. IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 3 In order to improve this bound, we need to understand the cancelation of the outer sum of (1) over t∈k?. Heuristically, one expects that the outer sum contributes another O(√q) factor instead of the trivial qfactor, if fis sufficiently “random”. This is in fact what we shall prove using the full strength of Deligne’s general theorem on Riemann hypothesis. The double sum in (1) is precisely a moment exponential sum associated to the two variable polynomial tf(x). Thus, we can use the techniques of moment Lfunctions to get improved information about the solution number Nr(f). We now briefly outline our method. Let `be a fixed prime different from p. Let Gfdenote the relative `-adic cohomology with compact support associated to the family of one variable exponential sums attached to tf(x), where xis the variable and tis the parameter on the torus Gm. Applying the `-adic trace formula fibre by fibre, we obtain X t∈k?X x∈kr ψ(Tr(tf(x))) = −X t∈k? Tr(Frobr q|(Gf)t), where (Gf)tis the fibre of Gfat t, and Frobqis the geometric q-th power Frobenius map. Alternatively, one can rewrite Tr(Frobr q|(Gf)t) = Tr(Frobq|[Gf]r t), where [Gf]rdenotes the r-th Adams operation of Gf. It is a virtual `-adic sheaf on Gm. For example, Katz [9] used the formula [Gf]r= r X i=1 (−1)i−1i·Symr−iGf⊗∧iGf. We shall use the following optimal formula from [17] given by [Gf]r= r X i=0 (−1)i−1(i−1) ·Symr−iGf⊗∧iGf. Note that the term i= 0 does not occur in the first formula, and the term i= 1 does not occur in the second formula as the coefficient becomes zero for i= 1. The coefficients of the second formula are smaller and thus lead to fewer number of zeros and poles for the corresponding L-functions. In this way, we get the smaller constant Cd,r in Theorem 1.1. It follows that Nr(f)−qr= r X i=0 (−1)i(i−1) ·X t∈k? Tr(Frobq|(Symr−iGf⊗∧iGf)t). This reduces our problem to the study of the L-function over Gmof the `-adic sheaves Symr−iGf⊗ ∧iGffor all 0 ≤i≤r. By general results of Deligne [3], we deduce that |Nr(f)−(qr+δf,rqr 2+1)| ≤ Cd,rqr+1 2, where Cd,r comes from the Euler characteristic of the components of the virtual sheaf [Gf]r, and δf,r = r X i=0 (−1)i−1(i−1) ·dimH2 c(Gm,¯ k,Symr−iGf⊗∧iGf). 4 ANTONIO ROJAS-LEON AND DAQING WAN Under the conditions of Theorem 1.1, it follows that the sheaf Symr−iGf⊗ ∧iGf has no geometrically trivial component for any 0 ≤i≤r, and thus we deduce that δf,r = 0. Our main result is somewhat stronger. We determine the weights, the trivial factors and the degrees of the L-functions of all the sheaves Symr−iGf⊗∧iGf, thus obtaining a fairly complete information about the associated moment L-function, see [5][6] and [13] for the study of moment L-functions in two other examples, namely, the family of hyper-Kloosterman sums and the Dwork family of toric Calabi-Yau hypersurfaces. Under slightly more general hypotheses, Katz’s results on monodromy group calculations [7][8] give stronger results which lead to further improvements of Theorem 1.1 (see Corollaries 4.2 and 4.6). See also [9] for a result on the average number of rational points on hypersurfaces obtained using a similar approach. The possibility of our improvement for the Weil bound in the case of ArtinSchreier curves is due to the fact that the curve has a large automorphism group Fq, which is the group of Fq-rational points on the group scheme A1. We expect that similar improvements should exist for many other curves (or higher dimensional varieties) with a large automorphism group. For example, in the last section of this paper we treat the case of Artin-Schreier hypersurfaces yq−y=f(x1, ..., xn). This method leads to similar improvements of Deligne’s bound for such hypersurfaces in many cases. As an explicit new example to try, we would suggest the affine Kummer curve of the form y(q−1) e=f(x), where eis a fixed positive integer, qis a prime power congruent to 1 modulo e, and f(x)∈k[x] is a polynomial of degree d. For r≥1 and Nr(f, e) denoting the number of Fqr-rational points on the above Kummer curve, we conjecture that for certain generic f, there is the following estimate |Nr(f, e)−qr| ≤ sd,e,rqr+1 2, where sd,e,r is a constant independent of q. We do not know how to prove this conjecture, even in the case e= 1. To conclude this introduction, we raise another open problem. In Theorem 1, we assumed that the curve Cf:yq−y=f(x) is defined over the subfield Fqof Fqr. We believe that similar improvement is also true if Cfis defined over the larger field Fqr. But we could not prove this at present. Remarks. Weil’s estimate gives both an upper bound and a lower bound for the number of rational points on a curve of genus gover the finite field Fq. Improvements for the lower bound are in general harder to get. Improvements for the upper bound can often be obtained by more elementary means. In fact, there are already several such results in the literature for large genus curves. The first result along these lines is due to Stark [15] in the hyperelliptic case, using Stepanov’s method. Using the explicit formula, Drinfeld-Vladut and Serre [14] obtained an upper bound improvement when 2g > qr−qr/2, which in our Artin-Schreier setting becomes (d−1)(q−1) > qr−qr/2. For r > 1, this means that qmust be small compared to d. In comparison, our improvements apply when qis large compared to d. Using a geometric intersection IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 5 argument, St¨oher-Voloch [16] obtained another upper bound which in our case becomes Nr(f)≤1 2D(D+qr−1), where D= max(d, q). Acknowledgment. It is a pleasure to thank the referee for his careful reading of the first version and for his very helpful comments. 2. Cohomology of the family t7→ Pψ(Tr(tf(x))) Let k=Fqbe a finite field of characteristic p, and f∈k[x] a polynomial of degree dprime to p. Let Cfbe the Artin-Schreier curve defined on A2 kby the equation (2) yq−y=f(x) and denote by Nr(f) its number of rational points over kr:= Fqr. Fix a non-trivial additive character ψ:k→C?. It is clear that (3) Nr(f) = X t∈kX x∈kr ψ(t·Tr(f(x))) = X t∈kX x∈kr ψ(Tr(tf(x))) where Tr denotes the trace map kr→k. Fix a prime `6=pand an isomorphism ι:¯ Q`→C. Consider the Galois ´etale cover of Gm×A1(with coordinates (t, x)) given by u−uq=tf(x), with Galois group k; and let Lψ(tf(x)) be the rank 1 smooth ¯ Q`-sheaf corresponding to the representation of kgiven by ψ−1via ι. Define Kf= Rπ!Lψ(tf(x)) ∈ Db c(Gm,k,¯ Q`), where π:Gm×A1→Gmis the projection. The trace formula implies that the trace of the action of the r-th power of a local geometric Frobenius element at t∈k?on Kfis given by Px∈krψ(Tr(tf(x))). It is known [3, 3.7] that Kf=Gf[−1] for a smooth sheaf Gfof rank d−1 and punctually pure of weight 1, whose local r-th power Frobenius trace at t∈k?is then given by −Px∈krψ(Tr(tf(x))). Therefore (4) Nr(f)−qr=X t∈k?X x∈kr ψ(Tr(tf(x))) =−X t∈k? Tr(Frobr t|(Gf)t) = −X t∈k? Tr(Frobt|[Gf]r t) where [Gf]r= r X i=0 (−1)i−1(i−1) ·Symr−iGf⊗∧iGf is the r-th Adams operation on Gf. The sheaf Gfcan also be interpreted in terms of the Fourier transform. Consider the sheaf f?¯ Q`on A1 k. There is a canonical surjective trace map φ:f?¯ Q`= f?f?¯ Q`→¯ Q`, let Ffbe its kernel. It is a constructible sheaf of generic rank d−1 on A1 k. Lemma 2.1. If j:Gm,k →A1 kis the inclusion, the shifted sheaf j!Gf[1] is the Fourier transform of Ff[1] with respect to ψ. 6 ANTONIO ROJAS-LEON AND DAQING WAN Proof. Taking Fourier transform in the distinguished triangle in Db c(A1 k,¯ Q`): Ff[1] →f?¯ Q`[1] →¯ Q`[1] → we get a distinguished triangle: FTψ(Ff)[1] →FTψ(f?¯ Q`)[1] →(¯ Q`)0(−1)[0] →. where (¯ Q`)0is a punctual sheaf supported at 0. If µ:A1×A1→A1is the multiplication map, the Fourier transform of f?¯ Q`[1] is given by Rπ1!(π? 2f?¯ Q`⊗µ?Lψ)[2] = Rπ1!(Lψ(tf(x)))[2], where πi:A1×A1→A1are the projections. In particular, by proper base change j?FTψ(f?¯ Q`)[1] = j?Rπ1!(Lψ(tf(x)))[2] = Kf[2] = Gf[1]. Applying j?to the triangle above we find quasi-isomorphisms j?FTψ(Ff)[1] ∼ =Gf[1] and j!j?FTψ(Ff)[1] ∼ =j!Gf[1]. To conclude, it remains to show that the natural map j!j?FTψ(Ff)[1] →FTψ(Ff)[1] is a quasi-isomorphism. Since its restriction to Gm,k is a quasi-isomorphism, we only need to check that it induces a quasi-isomorphism on the stalks at (a geometric point over) 0, that is, that FTψ(Ff)0= 0. By definition of the Fourier transform, FTψ(Ff)0= RΓc(A1 ¯ k,Ff). We conclude by using the long exact sequence of cohomology with compact support associated to the sequence 0→ Ff→f?¯ Q`→¯ Q`→0, since Hi c(A1 ¯ k, f?¯ Q`) = Hi c(A1 ¯ k,¯ Q`) = 0 for i6= 2 and H2 c(A1 ¯ k, f?¯ Q`) = H2 c(A1 ¯ k,¯ Q`) = ¯ Q`(−1) is one-dimensional.  We can now use Laumon’s local Fourier transform theory to determine the monodromy actions at 0 and ∞for Gf. Recall that, for every character χ:k?→¯ Q? `, there is an associated Kummer sheaf Lχon Gm,k: The (q−1)-th power map Gm,k →Gm,k is a Galois ´etale cover with Galois group canonically isomorphic to k?, and one just takes the pull-back of the character ¯χto π1(Gm,k,¯η)k?. For every d|q−1, if [d] denotes the d-th power map Gm,k →Gm,k we have [d]?¯ Q`=LLχ, where the sum is taken over all characters of k?such that χdis trivial. Assume that kcontains all d-th roots of unity. The sheaf Ffis smooth on the complement Uof the set of the critical values of fin A1. Since dis prime to p, in a neighborhood of infinity the map x7→ f(x) = adxd(1 + ad−1 adx+··· +a0 adxd) is equivalent (for the ´etale topology) to the map x7→ adxd(just by making the change of variable x7→ αx, where αd= 1 + ad−1 adx+···+a0 adxd). In particular, the decomposition group D∞at infinity acts on the generic stalk of f?¯ Q`through the direct sum of the tame characters (ad)?Lχfor all non-trivial characters χof k?such that χd=1, where (ad) : Gm,k →Gm,k is the multiplication by admap. Since (ad)?Lχ= (a−1 d)?Lχ= ¯χ(ad)deg ⊗ Lχ, we conclude that D∞acts on the generic stalk of Ffvia the direct sum L¯χ(ad)deg ⊗Lχtaken over all non-trivial characters χof k?such that χdis trivial. Proposition 2.2. Suppose that kcontains all d-th roots of unity. The action of the decomposition group D0at 0on Gfis tame and semisimple, and it splits as a direct sum L(χ(ad)g(¯χ, ψ))deg ⊗ Lχover all non-trivial characters χof k?such that χd=1, where g(¯χ, ψ) := −Pt¯χ(t)ψ(t)is the Gauss sum. IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 7 Proof. By ([12, Proposition 2.5.3.1],[8, Theorem 7.5.4]), the local monodromy at 0 of Gfcan be read from the local monodromy at infinity of Ff. More precisely, we have LFT(∞,0)(L¯χ(ad)deg ⊗Lχ) = LLFT(∞,0)(¯χ(ad)deg ⊗Lχ). Now, for every χ, since the Fourier transform commutes with tensoring by an unramified sheaf (by the projection formula, since π? 1(αdeg) = αdeg and µ?(αdeg) = αdeg for π1and µ:A1 k×A1 k→A1 kthe projection and multiplication) we have LFT(∞,0)(¯χ(ad)deg ⊗ Lχ) = ¯χ(ad)deg ⊗LF T (∞,0)Lχ= ¯χ(ad)deg ⊗g(χ, ψ)deg ⊗ L¯χby [12, Proposition 2.5.3.1] (note that Lχcorresponds to V¯χas a representation of D∞and to V0 χas a representation of D0in the notation of [12] due to the choice of uniformizers).  For simplicity, we will assume from now on that f0is square-free and p > 2. Suppose that kcontains all roots of f0(and therefore all critical values of f). Let s∈kbe a critical value of f. The polynomial fs:= f−shas at worst double roots and kcontains all its double roots. Let gsbe the square-free part of fs(i.e. fsdivided by the product of all its monic double linear factors), which lies in k[x]. Let S0be the henselization of A1 kat s,z1, . . . , ze∈kthe double roots of fs,Sjthe henselization of A1 kat zjfor j= 1, . . . , e and Tthe union of the henselizations of A1 kat the closed points of the subscheme defined by gs= 0. We have a cartesian diagram (`jSj)`T−−−−→ A1 k   y(`jhj)`h  yf S0−−−−→ A1 k where the map hj:Sj→S0is isomorphic (for the ´etale topology) to the map x7→ bj(x−zj)2(where bjis fs(x)/(x−zj)2evaluated at zj, that is, f00(zj)/2) via the change of variable mapping the local coordinate x−zjto α(x−zj), where α∈Sjis a square root of fs(x)/bj(x−zj)2(which exists by Hensel’s lemma, since its image in the residue field kis 1), and h:T→S0is finite ´etale. In particular, the decomposition group Dsat sacts on the generic stalk of f?¯ Q`through the direct sum Lj(1⊕(bj)?Lρ)LL=Lj(ρ(bj)deg ⊗Lρ)L(e·1⊕L) where Lis unramified and ρ= ¯ρ:k?→¯ Q? `is the quadratic character. Proposition 2.3. Suppose that p > 2,f0is square-free and all its roots are in k. The action of the decomposition group D∞at infinity on Gfsplits as a direct sum Lz(ρ(bz)g(ρ, ψ))deg ⊗Lρ⊗Lψf(z)where the sum is taken over the roots of f0, bz=f00(z)/2,ρ:k?→¯ Q? `is the quadratic character and g(ρ, ψ) = −Ptρ(t)ψ(t) the corresponding Gauss sum. Proof. By ([12],[8, Theorem 7.5.4]), the local monodromy at infinity of Gfcan be read from the local monodromies of Ff. More precisely, the part of slope >1 corresponds to the slope >1 part of the local monodromy at infinity of Ff, so it vanishes. The part of slope ≤1 is a direct sum, over all critical values sof f, of Lψstensored with the local Fourier transform LFT(0,∞)applied to the action of Is on the generic stalk of Ffmodulo its Is-invariant space. Using [12, 2.5.3.1] and the fact that Fourier transform commutes with tensoring by unramified sheaves, for every root zof f0LF T (0,∞)(ρ(bz)deg ⊗Lρ) = ρ(bz)deg ⊗ g(ρ, ψ)deg⊗Lρ. So each critical value scontributes a factor Lf(z)=s(ρ(bz)g(ρ, ψ))deg⊗ Lρ⊗Lψsto the monodromy of Gfat infinity.  We can now compute the determinant of Gf: 8 ANTONIO ROJAS-LEON AND DAQING WAN Corollary 2.4. Suppose that kcontains all d-th roots of unity. If dis odd, the determinant of Gfis the Tate-twisted Artin-Schreier sheaf Lψs((1 −d)/2), where s=s1+··· +sd−1is the sum of the critical values of fand ψs(t) = ψ(st). If dis even, the determinant of Gfis Lρ⊗ Lψs⊗(ρ(ad)g(ρ, ψ))deg((2 −d)/2), where ρis the multiplicative character of order 2,g(ρ, ψ) = −Ptρ(t)ψ(t)is the corresponding Gauss sum, adis the leading coefficient of f,= 1 if d≡0or 2 mod 8 and = (−1)(q−1)/d if d≡4or 6 mod 8. Proof. The determinant of Gfis a smooth sheaf of rank one on Gm,k. At 0, it is isomorphic by proposition 2.2 to the product Nχd=1,χ6=1(χ(ad)g(¯χ, ψ))deg ⊗Lχ. For any χwe have (χ(ad)g(¯χ, ψ))deg ⊗Lχ⊗(¯χ(ad)g(χ, ψ))deg ⊗L¯χ= (g(¯χ, ψ)g(χ, ψ))deg = (χ(−1)q)deg. If dis odd, the non-trivial characters with χd=1can be grouped in conjugate pairs. Moreover, χ(−1) = χ((−1)d) = χd(−1) = 1. We conclude that the determinant at 0 is the unramified character (qd−1 2)deg =¯ Q`(1−d 2). At infinity, it is geometrically isomorphic by proposition 2.3 to the product Nz(Lρ⊗Lψf(z)) = Lψs (the hypothesis that kcontains all roots of f0is not needed for the geometric isomorphism, since it is always satisfied in a sufficiently large finite extension of k). So det(Gf)⊗Lψ−sis everywhere unramified and therefore geometrically constant. Looking at the Frobenius action at 0, it must be ¯ Q`(1−d 2), so det(Gf) = Lψs(1−d 2). If dis even, the factor at 0 corresponding to the quadratic character ρstays unmatched, so as a representation of D0the determinant is (q d−2 2)deg⊗(ρ(ad)g(ρ, ψ))deg⊗ Lρ, where =Q(d−2)/2 i=1 χi(−1) for a fixed character χof exact order d. At ∞it is geometrically isomorphic to Lρ⊗Lψs, so det(Gf)⊗Lρ⊗Lψ−sis everywhere unramified and therefore geometrically constant. Looking at the Frobenius action at 0, it must be (q d−2 2ρ(ad)g(ρ, ψ))deg, so det(Gf) = Lρ⊗Lψs⊗(ρ(ad)g(ρ, ψ))deg(2−d 2). It remains to compute the value of . We have = (d−2)/2 Y i=1 χi(−1) = χd(d−2)/8(−1) = χ((−1)d(d−2)/8). If d≡0 or 2 mod 8, d(d−2)/8 is even and therefore = 1. If d≡4 or 6 mod 8, d(d−2)/8 is odd so =χ(−1) = (−1)(q−1)/d. 3. The moment L-function of Gf. Recall the definition [4] of the moment L-function for the sheaf Gf. For a fixed r≥1, let Lr(f, ψ, T ) := Y t∈|Gm,k| 1 det(1 −Frobr tTdeg(t)|(Gf)t), where |Gm,k|denotes the set of closed points of Gm,k. It is known ([4, Theorem 1.1]) that Lr(f, ψ, T ) is a rational function, and we have the formula (5) Lr(f, ψ, T ) = det(1 −FrobkT|H1 c(Gm,¯ k,[Gf]r)) det(1 −FrobkT|H2 c(Gm,¯ k,[Gf]r)) = =Qr i=0 det(1 −FrobkT|H1 c(Gm,¯ k,Symr−iGf⊗∧iGf))(−1)i−1(i−1) Qr i=0 det(1 −FrobkT|H2 c(Gm,¯ k,Symr−iGf⊗∧iGf))(−1)i−1(i−1) . Thus, we get a decomposition IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 9 (6) Lr(f, ψ, T ) = Q(T)P0(T)P∞(T) P(T)P0(T). We now describe each of the factors in this decomposition. First, Q(T) = r Y i=0 det(1 −FrobkT|H1(P1 ¯ k, j?(Symr−iGf⊗∧iGf)))(−1)i−1(i−1) is the non-trivial factor. Notice that the dual of Gfis G−f(1), since Ffis self-dual and D◦FTψ=FT ¯ ψ◦D(1) [11, Corollaire 2.1.5] and FT ¯ ψFf[1] = [t7→ −t]?Gf[1] = G−f[1]. Therefore the dual of Symr−iGf⊗∧iGfis Symr−iG−f⊗∧iG−f(r), so the dual (in the derived category) of j?Symr−iGf⊗∧iGf[1] is j?Symr−iG−f⊗∧iG−f[1](r+1), cf. [2, 2.1]. Since P1is proper, by [2, Th´eor`eme 2.2] we get a perfect pairing H1(P1 ¯ k, j?(Symr−iGf⊗∧iGf)) ×H1(P1 ¯ k, j?(Symr−iG−f⊗∧iG−f)) 7→ ¯ Q`(−r−1) for every i= 0, . . . , r. In particular, we get a functional equation relating the polynomial Qi(T) := det(1 −FrobkT|H1(P1 ¯ k, j?(Symr−iGf⊗∧iGf))) = si Y j=1 (1 −γijT). and the corresponding polynomial Q? i(T) for −f. The functional equation is given by Q? i(T) = si Y j=1 (1 −qr+1γ−1 ij T) = =Tsiq(r+1)si (−1)siγi1···γisi si Y j=1 (1 −γijq−(r+1)T−1) = Tsiq(r+1)si csi Qi(q−(r+1)T−1) where csiis the leading coefficient of Qi(T). Therefore, Q?(T) := r Y i=0 Q? i(T)(−1)i−1(i−1) =Tsq(r+1)s cs Q(q−(r+1)T−1) where sis the degree of the rational function Q(T) and csits leading coefficient (i.e. the ratio of the leading coefficients of the numerator and denominator). By [3, Th´eor`eme 3.2.3], all reciprocal roots and poles of Q(T) are pure Weil integers of weight r+ 1. The other factors of Lr(f, ψ, T ) are the “trivial factors”: P(T) = r Y i=0 det(1 −FrobkT|H0(P1 ¯ k, j?(Symr−iGf⊗∧iGf)))(−1)i−1(i−1) and P0(T) = r Y i=0 det(1 −FrobkT|H2(P1 ¯ k, j?(Symr−iGf⊗∧iGf)))(−1)i−1(i−1) are rational functions of the same degree and pure of weight rand r+2 respectively, and vanish if Symr−iGf⊗∧iGfhas no invariants for the action of π1(Gm,¯ k) for any i. The other two are the local factors at 0: P0(T) := det(1 −Frob0T|([Gf]r)I0) = 16 ANTONIO ROJAS-LEON AND DAQING WAN Sp(V) = G. In particular, all Frobenii act trivially on Wi(r/2), and therefore they act by multiplication by qr 2on Wi⊆NrV. Therefore r Y i=0 det(1 −FrobkT|H2 c(Gm,¯ k,Symr−iGf⊗∧iGf))(−1)i−1(i−1) = det(1 −FrobkT|Wr−1(−1))(−1)r−2(r−2) det(1 −FrobkT|Wr(−1))(−1)r−1(r−1) = (1 −qr 2+1T)(−1)r−2(r−2)+(−1)r−1(r−1) = (1 −qr 2+1T)(−1)r−1= (1 −qr 2+1T)−1 since r−1 = d−2 is odd. In the case where b6= 0, G/Sp(V)∼ =µpacts on Wi. Let A= diag(ζp, . . . , ζp)∈G be a scalar matrix, where ζp∈¯ Q`is a p-th root of unity. Then the class of Agenerates G/Sp(V), so Gfixes Wiif and only if Adoes. But Aacts on Wiby multiplication by ζr p, so this action is trivial if and only if ζr p= 1, that is, if and only if pdivides r. In that case, Symr−iGf⊗∧iGf= (Symr−iGf−b/2⊗∧iGf−b/2)⊗L⊗r ψb/2= Symr−iGf−b/2⊗∧iGf−b/2, so we can apply the b= 0 case and we get again r Y i=0 det(1 −FrobkT|H2 c(Gm,¯ k,Symr−iGf⊗∧iGf))(−1)i−1(i−1) = = (1 −qr 2+1T)(−1)r−1= (1 −qr 2+1T)−1. We conclude as in corollary 4.2.  Again, the hypothesis of proposition 4.5 can be checked from the coefficients of f: After adding a constant, we may assume that b= 0. Let Af0be the companion matrix of f0, and B=f(Af0). The eigenvalues of the (d−1) ×(d−1) matrix Bare s1, . . . , sd−1, and its trace is s=b(d−1) 2. Construct the (d−1)2×(d−1)2 matrix B⊗Id−1−Id−1⊗B, whose eigenvalues are all differences si−sj. Its characteristic polynomial is then of the form Td−1h(T/2)g(T)2, where h(T) is the characteristic polynomial of B, since all non-zero roots different from si−sd−i= 2si for i= 1, . . . , d −1 appear in pairs. The hypothesis of proposition 4.5 is equivalent to the discriminant of h(T/2)g(T) being non-zero. 5. Generalization to Artin-Schreier hypersurfaces In this section we will extend corollary 3.4 to higher dimensional hypersurfaces. Since the proofs are very similar, we will only sketch them, indicating the differences where necessary. Let f∈k[x1, . . . , xn] be a polynomial of degree dprime to p,Cfthe ArtinSchreier hypersurface defined on An+1 kby the equation (7) yq−y=f(x1, . . . , xn). Denote by Nr(f) its number of rational points over kr. We have again a formula (8) Nr(f)−qnr =X t∈k?X x∈kn r ψ(t·Tr(f(x))) = X t∈k?X x∈kn r ψ(Tr(tf(x))) where Tr denotes the trace map kr→k. Assume that fis a Deligne polynomial, that is, the leading form of fdefines a smooth projective hypersurface of degree IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 17 dnot divisible by p. Applying Deligne’s bound [3] to the above inner sum, one deduces that |Nr(f)−qnr| ≤ (q−1)(d−1)nqnr 2. This is precisely Weil’s bound in the case n= 1. Our purpose of this section is to improve the above bound and obtain the estimate of the following form |Nr(f)−qnr| ≤ Cd,rqnr+1 2, for some constant Cd,r depending only on d, r and n. Define Kf= Rπ!Lψ(tf(x)) ∈ Db c(Gm,k,¯ Q`), where π:Gm×An→Gmis the projection. The trace formula implies that the trace of the action of the r-th power of a local Frobenius element at t∈k?on Kfis given by Px∈kn rψ(Tr(tf(x))). Suppose from now on that the homogeneous part fdof highest degree of fdefines a non-singular hypersurface. Then by [3, 3.7], Kfis a single smooth sheaf Gfplaced in degree n, of rank (d−1)nand pure of weight n. Therefore Nr(f)−qnr = (−1)nX t∈k? Tr(Frobr t|(Gf)t)=(−1)nX t∈k? Tr(Frobt|[Gf]r t) where [Gf]r= r X i=0 (−1)i−1(i−1) ·Symr−iGf⊗∧iGf is the r-th Adams operation on Gf. We can give an interpretation of Gfin terms of the Fourier transform like we did in the one-dimensional case. Exactly as in lemma 2.1, we can show Lemma 5.1. The object Gf[1] ∈ Db c(Gm,¯ Q`)is the restriction to Gmof the Fourier transform of Rf!¯ Q`[n]with respect to ψ. We compactify fvia the map ˜ f:X→A1 k, where X⊆Pn×A1is defined by the equation F(x0, x1, . . . , xn) = txd 0,Fbeing the homogenization of fwith respect to the variable x0, and ˜ fthe restiction of the second projection to X. Suppose that the subscheme of An kdefined by the ideal h∂f/∂x1, . . . , ∂f/∂xniis finite ´etale over k, and the images of its ¯ k-points under fare distinct. Then for every s∈¯ k, the fibre Xshas at worst one isolated non-degenerate quadratic singularity, which is located on the affine part (since the part at infinity is defined for every fibre by fd(x) = 0 and is therefore non-singular). We have a distinguished triangle Rf!¯ Q`→R˜ f?¯ Q`→R( ˜ f|X0)?¯ Q`→ where X0=X\An∼ =Y×A1,Ybeing the smooth hypersurface defined in Pn−1 by fd= 0. Since R( ˜ f|X0)?¯ Q`is just the constant object RΓ(Y, ¯ Q`), its Fourier transform is supported at 0. So Gf[1] ∼ =(FTψRf!¯ Q`[n])|Gm,k ∼ =(FTψR˜ f?¯ Q`[n])|Gm,k . Proposition 5.2. Suppose p > 2. Under the previous hypotheses, let z1, . . . , z(d−1)n∈ An ¯ kbe the distinct points such that ∂f ∂xi(zj) = 0 for all i= 1, . . . , n, and let si=f(zi). The action of the inertia group I∞at infinity on Gfdecomposes as a direct sum LLψsiif nis even, and L(Lρ⊗ Lψsi)if nis odd, where ρis the unique character of I∞of order 2. 18 ANTONIO ROJAS-LEON AND DAQING WAN Proof. We will obtain, for every i, a factor Lψsi(resp. Lρ⊗ Lψsi) in the local monodromy of Gfat infinity. Since the rank is (d−1)nand these characters are pairwise non-isomorphic, this will determine the action of I∞completely. Let S={si|i= 1,...,(d−1)n}, and U=A1\S. Since ˜ fis proper and smooth over U, Ri˜ f?¯ Q`is smooth on Ufor every i. Since Xscontains one isolated nondegenerate quadratic singularity for each s∈S, by [1, 4.4] the sheaves Ri˜ f?¯ Q`are smooth on A1for i6=n−1, n. In particular, their Fourier transforms are supported at 0. We conclude that there is a distinguished triangle (FTψRn−1˜ f?¯ Q`[1])|Gm,k → Gf[1] →(FTψRn˜ f?¯ Q`[0])|Gm,k → and therefore an exact sequence of sheaves (9) 0 → H−1(FTψRn−1˜ f?¯ Q`[1])|Gm,k → Gf→ H−1(FTψRn˜ f?¯ Q`[0])|Gm,k → → H0(FTψRn−1˜ f?¯ Q`[1])|Gm,k →0 since FTψRn˜ f?¯ Q`[0] can only have non-zero cohomology sheaves in degrees 1, 0 and −1. Furthermore H0(FTψRn−1˜ f?¯ Q`[1]) is punctual, so this induces an exact sequence of I∞-representations (10) 0 → H−1(FTψRn−1˜ f?¯ Q`[1]) → Gf→ H−1(FTψRn˜ f?¯ Q`[0]) →0. Let Vbe the generic stalk of Rn−1˜ f?¯ Q`. Suppose that nis odd, and let s∈S. Then by [1, 4.3 and 4.4], the inertia group Isacts on Vwith invariant space VIsof codimension 1 (the orthogonal complement of the ’vanishing cycle’ δ) and on the quotient V/VIsvia its quadratic character ρ. Moreover, Rn−1˜ f?¯ Q`is isomorphic at sto the extension by direct image of its restriction to the generic point. By Laumon’s local Fourier transform [8, Section 7.4], the action of the inertia group I∞on H−1(FTψRn−1˜ f?¯ Q`[1]) (and thus on Gfby (10)) contains a subcharacter isomorphic to Lρ⊗Lψs. Suppose now that nis even, and let s∈S. By [1, 4.3 and 4.4], there are two possibilities: if the ’vanishing cycle’ δis non-zero, the inertia group Isacts on Vwith invariant space VIsof codimension 1 (the orthogonal complement of δ) and trivially on the quotient V/VIs. Moreover, Rn−1˜ f?¯ Q`is isomorphic at sto the extension by direct image of its restriction to the generic point. By Laumon’s local Fourier transform [8, Section 7.4], the action of the inertia group I∞on H−1(FTψRn−1˜ f?¯ Q`[1]) (and thus on Gfby (10)) contains a subcharacter isomorphic to Lψs. If δ= 0, then Isacts trivially on V, and there is an exact sequence of sheaves: 0→(¯ Q`)s→Rn˜ f?¯ Q`→js?j? sRn˜ f?¯ Q`→0 where (¯ Q`)sis the punctual object ¯ Q`supported on sand js:A1− {s},→A1is the inclusion. Taking Fourier transform, we deduce a distinguished triangle Lψs[1] →FTψRn˜ f?¯ Q`[0] →FTψjs?j? sRn˜ f?¯ Q`[0] → and in particular an injection 0→ Lψs→ H−1(FTψRn˜ f?¯ Q`[0]). By (10), this gives a subcharacter isomorphic to Lψsin the monodromy of Gfat infinity.  For completeness, we determine also the monodromy of Gfat 0. IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 19 Proposition 5.3. The inertia group I0at 0acts on Gfas a direct sum LnχLχ where the sum is taken over all characters χof I0such that χdis trivial, nχ= 1 d((d−1)n−(−1)n)if χis non-trivial and nχ= (−1)n+1 d((d−1)n−(−1)n)if χ is trivial. Proof. We will show that, for every χ, the action of I0on Gfcontains nχJordan blocks for the character χ. Since these numbers add up to (d−1)n, which is the dimension of the representation Gf, this will prove that the action is semisimple and determine it completely. Let χbe non-trivial such that χd=1. Since adding a constant ato fcorresponds to tensoring Gfwith the Artin-Schreier sheaf Lψaand this does not change the monodromy at 0, we can assume that Gfis totally wild at ∞(or equivalently, that the hypersurface f(x) = 0 is non-singular). Then so is Gf⊗ L¯χ. The number of Jordan blocks associated of Lχin the representation of I0given by Gfis the dimension of the I0-invariant subspace of Gf⊗L¯χ. If j:Gm,¯ k→A1 ¯ kand i:{0} → A1 ¯ kare the inclusions, we have an exact sequence 0→j!(Gf⊗L¯χ)→j?(Gf⊗L¯χ)→i?i?j?(Gf⊗L¯χ)→0 and therefore 0→(Gf⊗L¯χ)I0→H1 c(Gm,¯ k,Gf⊗L¯χ)→H1 c(A1 ¯ k, j?(Gf⊗L¯χ)) →0. Since Gf⊗L¯χis totally wild at ∞, the latter cohomology group is pure of weight n+ 1. So the dimension of (Gf⊗L¯χ)I0is the dimension of the weight ≤npart of H1 c(Gm,¯ k,Gf⊗L¯χ). By the projection formula, Gf⊗L¯χ= (Rnπ!Lψ(tf(x)))⊗L¯χ∼ =Rnπ!(Lψ(tf(x)) ⊗L¯χ(t)), so H1 c(Gm,¯ k,Gf⊗L¯χ)=Hn+1 c(Gm,¯ k×An ¯ k,Lψ(tf(x)) ⊗L¯χ(t)) since Riπ!Lψ(tf(x)) = 0 for i6=n. Let Z⊂An kbe the closed subset defined by f(x) = 0 and Uits open complement. The sheaf Lψ(tf(x)) is trivial on Gm×Z, so H? c(Gm,¯ k×Z, Lψ(tf(x)) ⊗L¯χ(t)) = H? c(Gm,¯ k×Z, L¯χ(t)) = H? c(Gm,¯ k,L¯χ)⊗H? c(Z⊗ ¯ k, ¯ Q`) = 0 since χis non-trivial. By excision we get an isomorphism Hn+1 c(Gm,¯ k× An ¯ k,Lψ(tf(x)) ⊗L¯χ(t))∼ =Hn+1 c(Gm,¯ k×U, Lψ(tf(x)) ⊗L¯χ(t)). Consider the automorphism φ:Gm×U→Gm×Ugiven by φ(t, x)=(tf(x), x). Then φ?(Lψ(tf(x)) ⊗L¯χ(t)) = Lψ(t)⊗L¯χ(t/f(x)) =Lψ(t)⊗L¯χ(t)⊗Lχ(f(x)). So Hn+1 c(Gm,¯ k×U, Lψ(tf(x)) ⊗L¯χ(t))∼ =Hn+1 c(Gm,¯ k×U, Lψ(t)⊗L¯χ(t)⊗Lχ(f(x))) which, by K¨unneth, is isomorphic to H1 c(Gm,¯ k,Lψ⊗L¯χ)⊗Hn c(U⊗¯ k, Lχ(f)) (since Hi c(Gm,¯ k,Lψ⊗L¯χ) = 0 for i6= 1). The first factor is one-dimensional and pure of weight 1, so we want the dimension of the weight ≤n−1 part of Hn c(U⊗¯ k, Lχ(f)). By [10, Theorem 2.2], this dimension is nχ=1 d((d−1)n−(−1)n). Similarly, if χ=1is the trivial character, the searched dimension is the dimension of the weight ≤npart of Hn+1 c(Gm,¯ k×An ¯ k,Lψ(tf(x))). From the exact sequence . . . →Hn c({0}×An ¯ k,¯ Q`)→Hn+1 c(Gm,¯ k×An ¯ k,Lψ(tf(x)))→ →Hn+1 c(A1 ¯ k×An ¯ k,Lψ(tf(x)))→Hn+1 c({0}×An ¯ k,¯ Q`)→. . . 20 ANTONIO ROJAS-LEON AND DAQING WAN we get an isomorphism Hn+1 c(Gm,¯ k×An ¯ k,Lψ(tf(x)))∼ =Hn+1 c(A1 ¯ k×An ¯ k,Lψ(tf(x))). Now let π:A1×An→Anbe the projection, by the base change theorem we have R2π!Lψ(tf(x)) =i?¯ Q`(−1), where i:Z→Anis the inclusion of the closed set where f(x) = 0, and Riπ!Lψ(tf(x)) = 0 for i6= 2. So we need the dimension of the weight ≤n−2 part of Hn−1 c(Z, ¯ Q`). Let Zbe the projective closure of Zand Z0=Z\Z, we have an exact sequence . . . →Hn−2(Z, ¯ Q`)→Hn−2(Z0,¯ Q`)→Hn−1 c(Z, ¯ Q`)→Hn−1(Z, ¯ Q`)→. . . Since Zis smooth, Hn−1(Z, ¯ Q`) is pure of weight n−1, and therefore the weight ≤ n−2 part of Hn−1 c(Z, ¯ Q`) is the cokernel of the map Hn−2(Z, ¯ Q`)→Hn−2(Z0,¯ Q`), that is, the primitive part Primn−2(Z0,¯ Q`) of the middle cohomology group of Z0, which has dimension n1= (−1)n+1 d((d−1)n−(−1)n).  Corollary 5.4. Let s=P(d−1)n i=1 si. Over ¯ k, the determinant of Gfis the ArtinSchreier sheaf Lψsif n(d−1) is even, and the product Lρ⊗ Lψsif n(d−1) is odd. Proof. The determinant is a smooth sheaf on Gmof rank 1. At 0, its monodromy is the product of χnχfor all characters χof I0such that χdis trivial. Since the non-trivial characters (except for the quadratic one) appear in conjugate pairs, the product is trivial if dis odd, and comes down to ρnρ, which is ρor 1depending on the parity of nρ=1 d((d−1)n−(−1)n), which is congruent to nmod 2, if dis even. At infinity, its monodromy is the product of the Lψsi(resp. of the Lρ⊗Lψsi) if nis even (resp. if nis odd), which is Lψs(resp. Lρ⊗Lψs) if n(d−1) is even (resp. if n(d−1) is odd). We conclude as in Corollary 2.4.  We now give the higher dimensional analogue of Corollary 3.4: Corollary 5.5. Let f∈k[x1, . . . , xn]be a polynomial of degree dprime to pand ra positive integer. Suppose that p > 2, the highest degree homogeneous part of fdefines a non-singular hypersurface, the subscheme of An kdefined by the ideal h∂f/∂x1, . . . , ∂f/∂xniis finite ´etale over kand the images of its ¯ k-points under fare distinct. If nr is even, suppose additionally that the hypersurface defined by f(x1,1, . . . , x1,n)+···+f(xr,1, . . . ,xr,n) = 0 in Anr k= Spec k[xi,j|1≤i≤r, 1≤j≤ n]is non-singular. Then the number Nr(f)of kr-rational points on the hypersurface yq−y=f(x1, . . . , xn) satisfies the estimate |Nr(f)−qnr| ≤ Cd,rqnr+1 2 where Cd,r = r X i=0 |i−1|(d−1)n+r−i−1 r−i(d−1)n i is independent of q. The proof is identical to the one of Corollary 3.4, using Proposition 5.2. In the neven case we need the non-singularity hypothesis for any r, since the Kummer factor does not appear in the monodromy at infinity. IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 21 References [1] Deligne, P., La conjecture de Weil I, Publ. Math. IHES, 43(1974), 273-307. 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London Math. Soc., 52(1986), 1-19. [17] Wan, D., Dwork’s conjecture on unit root zeta functions, Ann. Math., 150(1999), 867-927. Department of Mathematics, University of California, Irvine, CA 92697-3875, USA E-mail address:[email protected] Department of Mathematics, University of California, Irvine, CA 92697-3875, USA E-mail address:[email protected]