Estimates for singular multiplicative character sums

Estimates for Singular Multiplicative Character Sums Antonio Rojas-Le´on University of California, Irvine Department of Mathematics Irvine, CA 92697, USA e-mail: [email protected] ∗ April 26, 2005 Abstract We give some estimates for multiplicative character sums on quasiprojective varieties over finite fields depending on the severity of the singularities of the variety at infinity. We also remove the hypothesis of non-divisibility by the characteristic of the base field in the known estimates for the non-singular case. 1 Introduction In [Ka4], Katz proved the following estimate for multiplicative character sums. Let kbe a finite field of characteristic pand cardinality q, and X/k a projective smooth scheme of dimension nendowed with ak-embedding in PN k. Let Z(resp. H) be a hyperplane (resp. a hypersurface of degree d) in PN k, and suppose that X∩H,X∩Z and X∩H∩Zare all smooth of the right codimension. Then, if V=X−X∩Zand f:V→A1 kdenotes the map f(x) = H(x)/Z(x)d, ∗Partially supported by MTM2004-07203-C02-01 and FEDER 1 for any non-trivial multiplicative character χ:k?→C?we have the estimate ¯¯¯¯¯¯X x∈V(k) χ(f(x))¯¯¯¯¯¯ ≤C·qn/2 where Cdepends only on d,nand the total Chern class of X. This article extends this result to the singular case, in the same way that [Ka2] extended the results in [Ka1] for additive character sums. Let Xbe a scheme which is projective over kand purely of dimension n≥2, embedded in PN kas the closed subscheme defined by rhomogeneous forms F1, . . . , Frof degrees a1, . . . , ar. Let Hand Zbe homogeneous forms in k[X0, . . . , XN] of degrees dand e. We will also denote by Hand Zthe hypersurfaces they define in PN k. Assume that (H, Z) is a regular sequence in the graded coordinate ring ⊕i≥0Γ(X, OX(i)) of X(If Xis Cohen-Macaulay, this just means that X∩H∩Zhas pure codimension 2 in X). For simplicity we will also assume that dand eare coprime. See the remarks at the end of section 3 for the case where they are not. Following [Ka2], we define δto be the dimension of the singular locus of X∩H∩Z, and εto be that of the singular locus of X∩Z. We also define ε0as the dimension of the singular locus of X∩H. We have the a priori inequalities (cf. [Ka2], Lemma 3) ε≤δ+ 1, ε0≤δ+ 1. since the singular locus of X∩H∩Zcontains the intersection of Z and the singular locus of X∩Hand the intersection of Hand the singular locus of X∩Z. Fix a non-trivial multiplicative character χ:k?→C?. Let V= X−(H∪Z) and f:V→Gm,k be the map defined by f(x) = H(x)e/Z(x)d. Our main result is: Theorem 1. Denote by Sthe sum Px∈V(k)χ(f(x)). a) Suppose that eis prime to pand χeis non-trivial (for instance e= 1). Let C= 3(3 + sup(a1, . . . , ar, e) + d)N+r+2. We have the estimate |S| ≤ C·q(n+δ+2)/2. Furthermore, if ε0≤δ, we have the sharper estimate |S| ≤ C·q(n+δ+1)/2. 2 b) Suppose that dis prime to pand χdis non-trivial. Let C= 3(3 + sup(a1, . . . , ar, d) + e)N+r+2. We have the estimate |S| ≤ C·q(n+δ+2)/2. Furthermore, if ε≤δ, we have the sharper estimate |S| ≤ C·q(n+δ+1)/2. c) Suppose that gcd(d, p) = gcd(e, p) = 1. Let C= 3(3 + sup(a1, . . . , ar, d, e) + sup(d, e))N+r+2. We have the estimate |S| ≤ C·q(n+δ+2)/2. Furthermore, if ε≤δand ε0≤δ, we have the sharper estimate |S| ≤ C·q(n+δ+1)/2. Part (b) of the theorem is deduced from part (a) by just switching the roles of Hand Zand replacing χby ¯χ, since ¯χ(Z(x)d/H(x)e) = χ(H(x)e/Z(x)d). Part (c) follows immediately from (a) and (b). It remains to prove (a). 2 Cohomological interpretation of the sums Fix a prime `6=p, we will work with `-adic cohomology. We will pick an isomorphism ι:¯ Q`→Cso that we can freely speak of absolute values of element of ¯ Q`and weights. This also gives a way to look at a C?-valued character as a ¯ Q? `-valued character and viceversa. Given a non-trivial multiplicative character χ:k?→C?, there is an associated Kummer ¯ Q`-sheaf Lχon Gm,k (cf. [De2], 1.7) such that for every finite extension k0/k and every t∈Gm(k0) = k0?, the trace of the geometric Frobenius element in Gal(¯ k/k0) acting on the stalk of Lχat a geometric point ¯ tover tis χ(Nk0/k(t)). In particular, Lχis pure of weight zero. 3 If we denote by Lχ(f)the pull-back of Lχto Vby f, it follows from Grothendieck trace formula that X x∈V(k) χ(f(x)) = 2n X i=0 (−1)iTrace(F|Hi c(V⊗¯ k, Lχ(f))) where F∈Gal(¯ k/k) is the geometric Frobenius element. Furthermore, Deligne’s theorem (cf. [De1], Corollaire 3.3.4) implies that all eigenvalues of Facting on Hi c(V⊗¯ k, Lχ(f)) have absolute value at most qi/2. Therefore, Theorem 1 will be a consequence of the following two cohomological results: Theorem 2. With the previous notation, suppose that eis prime to pand χeis non-trivial. Then the cohomology group Hi c(V⊗¯ k, Lχ(f)) vanishes for i > n +δ+ 2. Furthermore, if ε0≤δ, it also vanishes for i=n+δ+ 2. Theorem 3. Suppose that eis prime to pand χeis non-trivial. Then we have the bound X i dim Hi c(V⊗¯ k, Lχ(f))≤3(3 + sup(a1, . . . , ar, e) + d)N+r+2. Consider the finite ´etale covering π:W→Vgiven by W:= {(x, s)∈V×Gm,k :se=f(x)} mapping to Vvia the first projection, and let g:W→Gm,k be the restriction of the second projection. We have a cartesian diagram Wπ −−−−→ V   yg  yf Gm,k [e] −−−−→ Gm,k where [e] is the e-th power map λ7→ λe. Since πis finite, we have Rπ?=π?=π!. Combining that with proper base change and the projection formula we get Hi c(W⊗¯ k, Lχe(g)) = Hi c(W⊗¯ k, g?[e]?Lχ) = Hi c(W⊗¯ k, π?f?Lχ) = = Hi c(V⊗¯ k, π?π?Lχ(f)) = Hi c(V⊗¯ k, (π?¯ Q`)⊗ Lχ(f)) = = Hi c(V⊗¯ k, (π?g?¯ Q`)⊗ Lχ(f)) = Hi c(V⊗¯ k, (f?[e]?¯ Q`)⊗ Lχ(f)) = = Hi c(V⊗¯ k, (f?Lρe=1Lρ)⊗ Lχ(f)) = = Hi c(V⊗¯ k, Lρe=1Lρ(f)⊗ Lχ(f)) = Lρe=1Hi c(V⊗¯ k, Lρχ(f)) 4 where the direct sum is taken over the set of characters of k?whose eth power is trivial. In particular, Hi c(V⊗¯ k, Lχ(f)) is a direct summand of Hi c(W⊗¯ k, Lχe(g)), so in order to prove Theorems 2 and 3 it suffices to show Theorem 4. With the previous notation, suppose that eis prime to pand χeis non-trivial. Then the cohomology group Hi c(W⊗¯ k, Lχe(g)) vanishes for i > n +δ+ 2. Furthermore, if ε0≤δ, it also vanishes for i=n+δ+ 2. Theorem 5. Suppose that eis prime to pand χeis non-trivial. Then we have the bound X i dim Hi c(W⊗¯ k, Lχe(g))≤3(3 + sup(a1, . . . , ar, e) + d)N+r+2. We will now construct a new scheme Yas the closed subscheme of PN+2 k(with coordinates X0, . . . , XN, T, U) defined by the homogeneous forms F1, . . . , Fr, Te−Zand H−UTd−1. Roughly speaking, we are adding two more variables to X, one representing the e-th root of Zand the other one the e-th root of the value of f. Then, define the incidence variety ˜ Yas the divisor in Y×P1 k(with coordinates λ0, λ1 for the second factor) given by the vanishing of λ0U−λ1T, thus ˜ Y(¯ k) = {(x0, . . . , xN, t, u),(λ0, λ1)) ∈Y(¯ k)×P1(¯ k) : λ0u=λ1t}. Let ˜g:˜ Y→P1 kbe the restriction of the projection Y×P1 k→P1 k, it is a proper map. We can embed Was a dense open subset of ˜ Y in the following way: first pick α, β ∈Zsuch that αd +βe = 1. We map the point (x, s)∈W(where x= (x0, . . . , xN)) to the pair τ(x, s) := ((x0, . . . , xN, t, u), s)∈˜ Y, where t=H(x)αZ(x)β sα,u=H(x)αZ(x)β sα−1. Notice that HαZβis a rational function of total degree 1 defined at every point of V, therefore the map is well defined. This gives an isomorphism between Wand the dense open subset of ˜ Ywhere T6= 0 and U6= 0, the inverse map being given by ((x0, . . . , xN, t, u), s)7→ (x, s). We have a commutative diagram Wτ −−−−→ ˜ Y   yg  y˜g Gm,k −−−−→ P1 k 5 where the horizontal arrows are open embeddings. We extend by zero the sheaf Lχeto all of P1 k, and take its pull-back to ˜ Yby ˜g, which we will denote by Lχe(g). Notice that its restriction to Wis just the previously defined Lχe(g). Lemma 6. There is a quasi-isomorphism RΓc(W⊗¯ k, Lχe(g))∼ →RΓc(˜ Y⊗¯ k, Lχe(g)). Proof. By excision, it suffices to show that RΓc(( ˜ Y−W)⊗¯ k, Lχe(g)) = 0. We have a decomposition of ˜ Y−Was the disjoint union of W0,W1 and W2, where (identifying Tand Uwith the divisors they define) W0= (Y∩T∩U)×P1 k W1= ( ˜ Y∩T)−U W2= ( ˜ Y∩U)−T. Since W1maps to infinity under ˜g, the sheaf Lχe(g)vanishes on W1. Similarly, W2maps to zero under ˜g, so Lχe(g)vanishes on W2too. Again by excision we deduce RΓc(( ˜ Y−W)⊗¯ k, Lχe(g))∼ →RΓc((Y∩T∩U)×P1 ¯ k,Lχe(g)). Now on (Y∩T∩U)×P1the sheaf Lχe(g)is the external tensor product ¯ Q`£Lχe, so by K¨unneth we conclude that RΓc((Y∩T∩U)×P1 ¯ k,Lχe(g)) = = RΓc((Y∩T∩U)⊗¯ k, ¯ Q`)⊗RΓc(P1 ¯ k,Lχe) = 0 since RΓc(P1 ¯ k,Lχe) = RΓc(Gm,¯ k,Lχe) = 0 when χeis non-trivial (cf. [De2], Th´eor`eme 2.7*) By the projection formula, we have RΓc(˜ Y⊗¯ k, Lχe(g)) = RΓc(P1 ¯ k,R˜g?˜g?Lχe) = RΓc(P1 ¯ k,(R˜g?¯ Q`)⊗ Lχe). Furthermore, there is a spectral sequence Ha c(P1 ¯ k,(Rb˜g?¯ Q`)⊗ Lχe)⇒Ha+b c(P1 ¯ k,(R˜g?¯ Q`)⊗ Lχe). In particular, in order to prove Theorem 4 it suffices to show that Ha c(P1 ¯ k,(Rb˜g?¯ Q`)⊗ Lχe) vanishes when a+b > n +δ+ 2, and when a+b=n+δ+ 2 if ε0≤δ. 6 Lemma 7. The map ˜g:˜ Y→P1 kis flat. Proof. Following ([Ka2], Lemma 9) we will show that all geometric fibers of ˜ghave the same Hilbert polynomial. Let C(X)⊂PN+1 kbe the cone over X, i.e. the subscheme defined in PN+1 k(with coordinates X0, . . . , XN, T) by the same ideal that defines Xin PN k. First of all, notice that Tis not a zero divisor in (the homogeneous coordinate ring of) C(X), and the section of C(X) it defines is isomorphic to X. Since (H, Z) is a regular sequence for Xby hypothesis, we conclude that (T, H, Z) in a regular sequence for C(X). Recall that the property of a sequence of homogeneous elements in a graded ring being a regular sequence is invariant under permutation of the elements of the sequence (cf. [BT], Lemma 23.5). The fiber of ˜gover a finite point λ∈¯ kis defined in PN+2 ¯ k(with coordinates X0, . . . , XN, T, U) by the vanishing of F1, . . . , Fr, Z−Te, H− λTdand U−λT. So in PN+1 ¯ k(which we identify with the hyperplane U−λT = 0 in PN+2 ¯ k) it is obtained from C(X) by taking the hypersurface sections Z−Te= 0 and H−λTd= 0. But (Z−Te, H −λTd) is a regular sequence in C(X) for every λ(because it is if we add T), so the Hilbert polynomial of any such fiber is given by P(m) = Q(m)−Q(m−d)−Q(m−e) + Q(m−d−e), where Qis the Hilbert polynomial of C(X). Similarly, the fiber over infinity is defined in PN+2 ¯ kby the vanishing of F1, . . . , Fr, Z, H and T, so in PN+1 ¯ k(identified with the hyperplane T= 0 in PN+2 ¯ k) it is obtained from C(X) by taking the hypersurface sections Z= 0 and H= 0. Again (Z, H) is a regular sequence in C(X), so the Hilbert polynomial of this fiber is also given by P(m) = Q(m)−Q(m−d)−Q(m−e) + Q(m−d−e). The proof of the previous lemma shows that the intersection of the fiber of ˜gover a finite point λ∈¯ kwith the hyperplane T= 0 is just X∩H∩Z, which has singular locus of dimension δ. Therefore, the fiber itself has singular locus of dimension at most δ+ 1. Similarly, the fiber over infinity is the cone over X∩H∩Z, so it has singular locus of dimension δ+ 1. From ([SGA7I], Expos´e I, Cor. 4.3) we deduce that for every λ∈P1(¯ k) the Iλ-invariant specialization map 7 (Rb˜g?¯ Q`)λ→(Rb˜g?¯ Q`)¯η(where ¯ηis a geometric generic point of P1 ¯ k and Iλthe inertia group at λ) is an isomorphism for b > n+δ+ 1 and surjective for b=n+δ+ 1. This implies that Rb˜g?¯ Q`is lisse on P1 ¯ k for b > n +δ+ 1, and that we have an exact sequence 0→ G → Rn+δ+1˜g?¯ Q`→ H → 0 where His lisse on P1 ¯ kand Gis punctual (cf. [Ka2], Theorem 13). Since P1 ¯ kis simply connected, every lisse sheaf on it is constant. In particular, for b > n +δ+ 1 and any awe get Ha c(P1 ¯ k,(Rb˜g?¯ Q`)⊗ Lχe) = (Rb˜g?¯ Q`)¯η⊗Ha c(P1 ¯ k,Lχe) = 0 since χeis non-trivial (cf. [De2], Th´eor`eme 2.7*). Similarly Ha c(P1 ¯ k,H⊗ Lχe) = 0, so from the exact sequence above we get isomorphisms Ha c(P1 ¯ k,G ⊗ Lχe)∼ =Ha c(P1 ¯ k,(Rn+δ+1˜g?¯ Q`)⊗ Lχe). Now Gis punctual, so we conclude that Ha c(P1 ¯ k,(Rn+δ+1˜g?¯ Q`)⊗Lχe) = 0 for a > 0. Since Ha cof any constructible sheaf on P1 ¯ kvanishes for all a > 2, this covers all possible cases where a+b > n +δ+ 2. The only case with a+b=n+δ+ 2 that has not yet been considered is a= 2, b=n+δ. So it remains to show that H2 c(P1 ¯ k,(Rn+δ˜g?¯ Q`)⊗ Lχe)=0 when ε0≤δ. Lemma 8. The sheaf F:= Rn+δ˜g?¯ Q`is lisse at 0∈P1 ¯ k. Proof. Let I=I0⊂Gal(¯ k(t)sep/¯ k(t)) be the inertia group at zero. If ¯ηis a geometric generic point of P1 ¯ k, the lemma states that Iacts trivially on F¯η. Therefore, it suffices to show that the I-invariant specialization map F0→ F¯ηis surjective. By ([SGA7I], Expos´e I, Cor. 4.3), this will happen if the fiber of ˜gat zero has singular locus of dimension at most δ. Such fiber ˜ Y0is given in PN+2 ¯ k(with coordinates X0, . . . , XN, T, U) by the vanishing of F1, . . . , Fr, Te−Z, H and U. We have an obvious finite projection map π:˜ Y0→X∩H, which is ´etale outside ˜ Y0−T. In particular, the singularities of ˜ Y0−Tmap to singularities of X∩H. But the singular locus of X∩Hhas dimension ε0≤δand πis finite, so the singular locus of ˜ Y0outside Thas dimension at most δ. On the other hand, a singular point of ˜ Y0in ˜ Y0∩Tmust also be a singular point of ˜ Y0∩T(cf. [Ka2], Lemma 3). But ˜ Y0∩Tis isomorphic to X∩H∩Z, so its singular locus has dimension δ. Therefore, the singular locus of ˜ Y0in Talso has dimension at most δ. This proves the lemma. 8 Let U⊂Gm,¯ kbe a dense open subset on which Fis lisse. By the birational invariance of H2 cit suffices to show that H2 c(U, F ⊗Lχe) = 0. By Lemma 8 Fis lisse at zero. Therefore if I=I0⊂Gal(¯ k(t)sep/¯ k(t)) is the inertia group at zero, Iacts trivially on the stalk F¯ηof Fat a geometric generic point ¯ηof P1 ¯ k. On the other hand, since χeis non-trivial, Lχeis totally ramified at zero, so (Lχe)I ¯η= 0. Hence (F ⊗ Lχe)I ¯η= (F¯η⊗(Lχe)¯η)I=F¯η⊗(Lχe)I ¯η= 0. In particular the coinvariants ((F ⊗ Lχe)¯η)Ialso vanish and a fortiori H2 c(U, F ⊗ Lχe) = ((F ⊗ Lχe)¯η)π1(U,¯η)(−1) = 0. This completes the proof of Theorem 4. 3 An upper bound for the sum of the Betti numbers In this section we will prove Theorem 5. The main tool will be the following bound of Katz: Theorem 9. ([Ka3], Theorem 12) Let V⊂AN kbe a closed subscheme, defined by the vanishing of rpolynomials f1, . . . , frof degrees a1, . . . , ar. Let h, h1, . . . , hs∈k[x1, . . . , xN],s≥0, be polynomials of degrees e, e1, . . . , es. Fix a non-trivial additive character ψ:k→C? and snon-trivial multiplicative characters χ1, . . . , χs:k?→C?. Let Lψand Lχjbe the corresponding Artin-Schreier and (extension by zero of) Kummer ¯ Q`-sheaves on A1 k, and denote by Lψ(h)and Lχj(hj) their pull-backs to Vby hand hjrespectively. Then we have the upper bound Pidim Hi c(V⊗¯ k, Lψ(h)⊗(Ns j=1 Lχj(hj))) ≤ ≤3(s+ 1 + supi(e, 1 + ai) + Pjej)N+r. In order to optimize the bound, we will not embed Win Y, but in a new projective scheme Y0defined in PN+1 k(with coordinates X0, . . . , XN, T) by the homogeneous forms F1, . . . , Frand Te−Z. We now embed Was a dense open subscheme of Y0by mapping (x, s), where x= (x0, . . . , xN), to (x0, . . . , xN, t)∈Y0, with t= s−αH(x)αZ(x)β(Recall that αand βare integers such that αd+βe = 9