ESTIMATES FOR “NONSINGULAR” MULTIPLICATIVE CHARACTER SUMS

NICHOLAS M. KATZ

1. Introduction, and the basic setting

Let k be a finite field of characteristic p and cardinality q, ψ a nontrivial ℂ^{×}-valued additive character of k, and χ a nontrivial ℂ^{×}-valued multiplicative character of k^{×}. We extend χ to a function on all of k by defining χ(0) := 0.

Recall that, given an integer n, a number α ∈ ℂ is said to be pure of weight n (relative to q) if it and all its Aut(ℂ/ℚ)-conjugates have absolute value \sqrt{q^{n}}. Such an α is necessarily algebraic over ℚ.

Fix a polynomial f(X) in k[X] of degree d ≥ 1. If d is prime to p, one has the estimate

| ∑_{x∈k} ψ(f(x))| ≤ (d − 1)\sqrt{q}.

More precisely, the associated L-function is a polynomial of degree d − 1 which is pure of weight one, i.e., all its reciprocal zeroes have absolute value \sqrt{q}.

The analogue for multiplicative character sums in one variable is this. Whatever the value of d, if f has d distinct zeroes (in some overfield of k), then one has the estimate

| ∑_{x∈k} χ(f(x))| ≤ (d − 1)\sqrt{q}.

The associated L-function is again a polynomial of degree d − 1, but in general it is mixed of weights one and zero. It is pure of weight one if and only if in addition χ^{d} is nontrivial, otherwise it has d − 2 reciprocal roots which are pure of weight one, and one reciprocal root which is pure of weight zero.

It was known to Davenport and Hasse (cf. [Dav-Ha], [Ha], [We-OSES], and [We-NS, footnote on page 498]) in 1934 that both these results followed from the then unproven Riemann Hypothesis for curves over finite fields (in the first case for the curve of affine equation Y^{q} − Y = f(X), in the second case for the curve of affine equation Y^{N} = f(X), for N the order of χ).

What happens to these results in several variables? Fix an integer n ≥ 1, and a polynomial f(X) := f(X_{1}, …, X_{n}) in k[X_{1}, …, X_{n}] of degree d ≥ 1, say

f = f_{d} + f_{d−1} + … + f_{0}

Date: April 19, 2001.

with f_{i} homogeneous of degree i. If d is prime to p and if the equation f_{d} = 0 defines a smooth, degree d hypersurface in ℙ^{n−1} (for n = 1, this means only that f_{d} is not identically 0), then Deligne proved [De-Weil I, Theorem 8.4] the estimate

| ∑_{x∈k^{n}} ψ(f(x))| ≤ (d − 1)^{n}\sqrt{q^{n}}.

Moreover, he proved that the associated L function is a polynomial (for n odd) or a reciprocal polynomial (for n even) of degree (d − 1)^{n} which is pure of weight n.

We will establish n-variable multiplicative character sum analogues of Deligne’s nvariable additive character sum results (and of the mild generalizations of Deligne’s results given in [Ka-SE, 5.1.1 and 5.4.1]). With these results, the n-variable situation almost perfectly mirrors the one-variable situation, except that in several variables our method sometimes requires the assumption that d be prime to p, an assumption which was never necessary in the one-variable case. One striking aspect of this mirroring is that the degrees of the L functions for the multiplicative character sums we consider coincide with the degrees for the corresponding additive character sums.

2. Statement of the two main theorems

Fix a finite field k of characteristic p and cardinality q. Let χ be a nontrivial ℂ^{×}-valued multiplicative character of k^{×}. Fix an integer n ≥ 1, and a polynomial f(X) := f(X_{1}, …, X_{n}) in k[X_{1}, …, X_{n}] of degree d ≥ 1, say

f = f_{d} + f_{d−1} + … + f_{0}

with f_{i} homogeneous of degree i. Suppose that

(a) the equation f_{d} = 0 defines a smooth, degree d hypersurface in ℙ_{k}^{n−1},

(b) the equation f = 0 defines a smooth hypersurface in 𝔸_{k}^{n}.

Theorem 1. In the above situation, if d is prime to p and if χ^{d} is nontrivial, we have the estimate

| ∑_{x∈k^{n}} χ(f(x))| ≤ (d − 1)^{n}\sqrt{q^{n}}.

The associated L function is a polynomial P (T) (for n odd) or a reciprocal polynomial 1/P (T)(for n even) of degree (d − 1)^{n}, and P (T) is pure of weight n.

Theorem 2. In the above situation, If χ^{d} is trivial, we have the estimate

| ∑_{x∈k^{n}} χ(f(x))| ≤ (d − 1)^{n}\sqrt{q^{n}}.

The associated L function is a polynomial P (T) (for n odd) or a reciprocal polynomial 1/P (T)(for n even) of degree (d − 1)^{n}, and P (T) is mixed of weights n and n − 1: it has ((d − 1)^{n+1} − (−1)^{n+1})/d reciprocal roots which are pure of weight n, and it has ((d − 1)^{n} − (−1)^{n})/d reciprocal roots which are pure of weight n − 1.

3. Statement of a second version of the main theorems

In this section, we give a generalization which is analogous to our additive character generalization [Ka-SE, 5.1.1] of [De-Weil I, Theorem 8.4]. Fix a finite field k of characteristic p and cardinality q. Let χ be a nontrivial ℂ^{×}-valued multiplicative character of k^{×}. Fix an integer n ≥ 1.Let X/k be a projective, smooth, and geometrically connected k-scheme of dimension n ≥ 1, given with a projective embedding X ↪ ℙ_{k}^{N} := ℙ. Fix a strictly positive integer d. Denote by L the class of O_{ℙ}(1), and by c(X) the total Chern class of X. Define non-negative integers C_{0}, C_{1}, and C by

C_{0} := (−1)^{n} ∫_{X} \frac{c(X)}{(1 + dL)},

C_{1} := (−1)^{n−1} ∫_{X} \frac{Lc(X)}{(1 + L)(1 + dL)},

C := (−1)^{n} ∫_{X} \frac{c(X)}{(1 + L)(1 + dL)}.

Notice that

C = C_{0} + C_{1}.

Let Z ∈ H^{0}(ℙ, O_{ℙ}(1)) define a hyperplane (itself denoted Z) in ℙ. Let H ∈ H^{0}(ℙ, O_{ℙ}(d)) define a hypersurface (itself denoted H) in ℙ . Assume that

(a) X ∩ Z is lisse of codimension 1 in X.

(b) X ∩ H is lisse of codimension 1 in X.

(c) X ∩ H ∩ Z is lisse of codimension 2 in X.

To this data, we attach the smooth affine k-scheme

V := X − X ∩ Z,

,and the function

f := H/Z^{d} : V → 𝔸_{k}^{1}.

Theorem 3. Hypotheses and notations as above, suppose that d is prime to p, and that χ^{d} is nontrivial. Then we have the estimate

| ∑_{x∈V (k)} χ(f(x))| ≤ C\sqrt{q^{n}}.

The associated L function is a polynomial P (T) (for n odd) or a reciprocal polynomial 1/P (T)(for n even) of degree C, and P (T) is pure of weight n: all its reciprocal roots are pure of weight n.

Theorem 4. Hypotheses and notations as above, suppose that χ^{d} is trivial. Then we have the estimate

| ∑_{x∈V (k)} χ(f(x))| ≤ C\sqrt{q^{n}}.

The associated L function is a polynomial P (T) (for n odd) or a reciprocal polynomial 1/P (T)(for n even) of degree C, and P (T) is mixed of weights n and n − 1: it has C_{0} reciprocal roots which are pure of weight n, and it has C_{1} reciprocal roots which are pure of weight n − 1.

4. Statement of a third version of the main theorems

In this section, we give a generalization which is analogous to our additive character generalization [Ka-SE, 5.4.1] of [De-Weil I, Theorem 8.4]. Fix a finite field k of characteristic p and cardinality q. Let χ be a nontrivial ℂ^{×}-valued multiplicative character of k^{×}. Fix an integer n ≥ 1.Let X/k be a projective, smooth, and geometrically connected k-scheme of dimension n ≥ 1, given with a projective embedding X ↪ ℙ_{k}^{N} := ℙ. Fix an integer r ≥ 1, and two r-tuples (d_{1}, …, d_{r}) and (b_{1}, …, b_{r}) of strictly positive integers. Define

d := ∑_{i=1}^{r} b_{i}d_{i}.

Define

b := lcm(b_{1}, …, b_{r}).

Denote by L the class of O_{ℙ}(1), and by c(X) the total Chern class of X. Define a non-negative integer C by

C := (−1)^{n} ∫_{X} \frac{c(X)}{(1 + dL)Π_{i=1}^{r}(1 + d_{i}L)}.

Suppose given

Z_{i} ∈ H^{0}(ℙ, O_{ℙ}(d_{i}))

for each i, which defines a hypersurface (also denoted Z_{i}) in ℙ. Suppose given

H ∈ H^{0}(ℙ, O_{ℙ}(d))

which defines a hypersurface (also denoted H) in ℙ. Assume that

(a) All the b_{i} are prime to p.

(b) For every subset I ⊂ {1, 2, …, r}, X ∩ ⋂_{i∈I} Z_{i} is lisse of codimension #I in X.

(c) For every subset I ⊂ {1, 2, …, r}, X ∩H ∩⋂_{i∈I} Z_{i} is lisse of codimension 1+#I in X.

To this data, we attach the smooth affine k-scheme

V := X − X ∩ ⋃_{i∈I} Z_{i},

and the function

f := H/Π_{i=1}^{r}Z_{i}^{b_{i}} : V → 𝔸_{k}^{1}.

Theorem 5. Hypotheses and notations as above, suppose that χ^{b} is nontrivial. Then we have the estimate

| ∑_{x∈V (k)} χ(f(x))| ≤ C\sqrt{q^{n}}.

The associated L function is a polynomial P (T) (for n odd) or a reciprocal polynomial 1/P (T)(for n even) of degree C, and P (T) is pure of weight n: all its reciprocal roots are pure of weight n.

Theorem 6. Hypotheses and notations as above, suppose that χ^{b} is trivial. Then we have the estimate

| ∑_{x∈V (k)} χ(f(x))| ≤ C\sqrt{q^{n}}.

The associated L function is a polynomial P (T) (for n odd) or a reciprocal polynomial 1/P (T)(for n even) of degree C, and P (T) is mixed of weight ≤ n: each of its reciprocal roots is pure of some non-negative integer weight w ≤ n.

5. Proof of the theorems: first steps

Fix a prime number l invertible in k, an algebraic closure ℚ̄_{l} of ℚ_{l}, and a field isomorphism ι of ℚ̄_{l} with ℂ. By means of ι, we view χ as a nontrivial ℚ̄_{l}^{×}-valued character of k^{×}. We denote by L_{χ} on 𝔸_{k}^{1} the extension by zero across the origin of the lisse rank one Kummer sheaf [De-AFT] on the multiplicative group 𝔾_{m,k}. For any scheme X and any morphism f : X → 𝔸_{k}^{1}, we define the constructible ℚ̄_{l}-sheaf L_{χ(f)} := f^{⋆}L_{χ} on X: it is lisse of rank one on X[1/f], and extended by zero to all of X.

Theorem 1 is the special case of Theorem 3 in which X is ℙ_{k}^{n} := P roj(k[X_{0}, …, X_{n}]), Z is X_{0}, and H := ∑_{i=0}^{d} X_{0}^{d−i}f_{i}. See [Ka-ESES, Introduction] for the verification that the constant C of Theorem 3 is (d − 1)^{n} in this special case. Similarly, Theorem 2 is a special case of Theorem 4. Theorem 3 is visibly a special case of Theorem 5.

So it suffices to prove Theorems 4, 5, and 6. All of these concern the exponential sum

∑_{x∈V (k)} χ(f(x)) = ∑_{x∈V [1/f](k)} χ(f(x)).

Let us denote

V̄ := V ×_{k} k̄.

By the Lefschetz trace formula, we know that

∑_{x∈V [1/f](k)} χ(f(x)) = ∑_{i}(−1)^{i}T race(F_{k}|H_{c}^{i}(V̄ [1/f], L_{χ(f)})).

Theorem 5 results in the standard way (compare [Ka-SE, 5.4.1]) from Deligne’s Weil II estimate [De-Weil II, 3.3.3 and 3.3.4] and the following l-adic theorem.

Theorem 7. Hypotheses and notations as in Theorem 5, we have

(1) The natural ”forget supports” map is an isomorphism

H_{c}^{⋆}(V̄ [1/f], L_{χ(f)}) ≅ H^{⋆}(V̄ [1/f], L_{χ(f)}).

(2) The compact Euler characteristic is given by

χ_{c}(V̄ [1/f], L_{χ(f)}) = (−1)^{n}C.

(3) For i ≠ n, H_{c}^{i}(V̄ [1/f], L_{χ(f)}) = 0.

(4) The group H_{c}^{n}(V̄ [1/f], L_{χ(f)}) has dimension C, and is pure of weight n.

(5) For χ̄ the inverse character to χ, the cup-product pairing

H_{c}^{n}(V̄ [1/f], L_{χ(f)}) × H_{c}^{n}(V̄ [1/f], L_{χ̄(f)}) → H_{c}^{2n}(V̄ [1/f], ℚ̄_{l}) ≅ ℚ̄_{l}(−n)

is a Gal(k̄/k)-equivariant perfect pairing. In particular, if χ has order two, then H_{c}^{n}(V̄ [1/f], L_{χ(f)}) is Gal(k̄/k)-equivariantly self-dual toward ℚ̄_{l}(−n), by a pairing which is orthogonal for n even and symplectic for n odd.

Because V [1/f]/k is smooth and affine, and L_{χ(f)} is lisse and pure of weight zero, parts (3) and (4) result from parts (1) and (2), cf. [Se-MSE]. Part (5) is not needed for Theorem 5, but may be useful in other contexts. It results from part (1) and Poincaré duality. The sign of the pairing is as asserted because the coefficients L_{χ(f)} are themselves orthogonally self-dual when χ has order two.

Similarly, Theorem 6 results from [De-Weil II, 3.3.3 and 3.3.4] and the following l-adic theorem.

Theorem 8. Hypotheses and notations as in Theorem 6, the groups H_{c}^{i}(V̄ [1/f], L_{χ(f)}) vanish for i ≠ n, and the compact Euler characteristic is given by

χ_{c}(V̄ [1/f], L_{χ(f)}) = (−1)^{n}C.

Thereom 6 asserts that in addition the weights are non-negative integers ≤ n. Because L_{χ(f)} is pure of weight zero, we know that H_{c}^{n}(V̄ [1/f], L_{χ(f)}) is mixed of integer weights ≤ n. The trace function of L_{χ(f)} has values which are algebraic integers, so the L function as a power series has algebraic integer coeffiecients, and constant term 1. But by the theorem we have

L(T)^{(−1)^{n+1}} = det(1 − T F_{k}|H_{c}^{n}(V̄ [1/f], L_{χ(f)}).

Therefore all the eigenvalues of F_{k} on H_{c}^{n}(V̄ [1/f], L_{χ(f)}) are algebraic integers. Hence their weights must be non-negative.

6. Further reduction steps, via the incidence variety

We now pass from V to the incidence variety X̃_{𝔸^{1}} ⊂ X × 𝔸^{1} defined as the locus of points (x ∈ X, λ ∈ 𝔸^{1}) where H − λΠ_{i}Z_{i}^{b_{i}} vanishes. This incidence variety X̃_{𝔸^{1}} is smooth over k, cf. [Ka-SE, pp. 173-174]. The second projection defines a proper flat k-morphism

f̃: X̃_{𝔸^{1}} → 𝔸^{1}

with

f̃^{−1}(λ) = X ∩ (H = λΠ_{i}Z_{i}^{b_{i}}).

We denote by X̃_{𝔾_{m}} ⊂ X × 𝔾_{m} the inverse image of 𝔾_{m} under f̃. The variety V sits in X̃_{𝔸^{1}} as an open set, with closed complement the product (X ∩ H ∩ ⋃_{i} Z_{i}) × 𝔸^{1}. The function f̃ agrees with f on V, and is the second projection on the complement. Similarly, V [1/f] sits in X̃_{𝔾_{m}} as an open set, with closed complement the product (X ∩ H ∩ ⋃_{i} Z_{i}) × 𝔾_{m}. The function f̃ agrees with f on V [1/f], and is the second projection on the complement.

Because χ is nontrivial, we have H_{c}^{⋆}(𝔾_{m,k̄}, L_{χ}) = 0. By Kunneth we get

H_{c}^{⋆}(((X ∩ H ∩ ⋃_{i} Z_{i}) × 𝔾_{m})k̄, L_{χ(f̃)}) = 0.

So the excision sequence for L_{χ(f̃)} and

V [1/f] ⊂ X̃_{𝔾_{m}} ⊃ (X ∩ H ∩ ⋃_{i} Z_{i}) × 𝔾_{m}

gives

H_{c}^{⋆}(V̄ [1/f], L_{χ(f)}) ≅ H_{c}^{⋆}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}).

The Poincaré dual of this isomorphism, applied with the inverse character χ̄, is an isomorphism

H^{⋆}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}) ≅ H^{⋆}(V̄ [1/f], L_{χ(f)}).

Moreover, if we surround the ”forget supports” map on X̃_{𝔾_{m,k̄}} with these isomorphisms,

H_{c}^{⋆}(V̄ [1/f], L_{χ(f)}) ≅ H_{c}^{⋆}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}) → H^{⋆}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}) ≅ H^{⋆}(V̄ [1/f], L_{χ(f)}),

we get the ”forget supports” map on V̄ [1/f]. Thus Theorem 7 results from

Theorem 9. Hypotheses and notations as in Theorem 5, we have

(1) The natural ”forget supports” map is an isomorphism

H_{c}^{⋆}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}) ≅ H^{⋆}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}).

(2) The compact Euler characteristic is given by

χ_{c}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}) = (−1)^{n}C.

Similarly, Theorem 8 results from Theorem 10. Hypotheses and notations as in Theorem 6, the groups H_{c}^{i}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}) vanish for i ≠ n, and the compact Euler characteristic is given by

χ_{c}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}) = (−1)^{n}C.

7. Proof of Theorems 9 and 10, by the method of pencils

We now turn to a study of the proper flat k-morphism

f̃: X̃_{𝔸^{1}} → 𝔸^{1},

and the higher direct image sheaves (remember f̃ is proper)

R^{i}f̃_{!}ℚ̄_{l} ≅ R^{i}f̃_{⋆}ℚ̄_{l}

on 𝔸^{1}.

Proposition 11. The morphism

f̃: X̃_{𝔸^{1}} → 𝔸^{1}

is proper and smooth of relative dimension n − 1 over a Zariski open neighborhood of the origin 0 in 𝔸^{1}. Its singular fibres have at worst isolated singularities. The sheaves R^{i}f̃_{⋆}ℚ̄_{l} are all lisse in a Zariski open neighborhood of the origin in 𝔸^{1}.

Proof. The fibre over a point λ is X ∩ (H = λΠ_{i}Z_{i}^{b_{i}}), which is smooth for λ = 0, hence smooth for λ in a Zariski open neighborhood of 0. By [SGA4, Exposé XV, 2.1 and Exposé XVI, 2.1], the sheaves R^{i}f̃_{⋆}ℚ̄_{l} are all lisse in a Zariski open neighborhood of the origin in 𝔸^{1}. Whatever the point λ, the intersection f̃^{−1}(λ) ∩ Z_{1} is always the same, namely X ∩ H ∩ Z_{1}, which by hypothesis is smooth of codimension 2 in X. Therefore Z_{1} does not meet the singular locus of f̃^{−1}(λ). But Z_{1} is the intersection with X of a hypersurface in the ambient ℙ, so it meets every closed subscheme of X of strictly positive dimension. Therefore the singular locus of f̃^{−1}(λ) has dimension at most 0.

Corollary 12. For i ≥ n + 1, the sheaf R^{i}f̃_{⋆}ℚ̄_{l} is lisse on 𝔸^{1}. For i = n, denote by j : U ↪ 𝔸^{1} the inclusion of a dense open set on which R^{n}f̃_{⋆}ℚ̄_{l} is lisse. Then we have a short exact sequence of sheaves on 𝔸^{1},

0 → (punctual) → R^{n}f̃_{⋆}ℚ̄_{l} → j_{⋆}j^{⋆}R^{n}f̃_{⋆}ℚ̄_{l} → 0

and the sheaf j_{⋆}j^{⋆}R^{n}f̃_{⋆}ℚ̄_{l} is lisse on 𝔸^{1}.

Proof. This is immediate from [SGA7, Exposé I, Cor. 4.3] and the preceding result, cf. [Ka-ESES, proof of Theorem 13].

Proposition 13. The sheaves R^{i}f̃_{⋆}ℚ̄_{l} on 𝔸^{1} are all tamely ramified at ∞, and in their semisimplifications as representations of the inertia group I(∞) only characters of order dividing b occur.

Proof. This is proven in [Ka-SE, 5.4.2].

Corollary 14. For i ≥ n + 1, the sheaf R^{i}f̃_{⋆}ℚ̄_{l} is geometrically constant on 𝔸^{1}. For i = n, the sheaf j_{⋆}j^{⋆}R^{n}f̃_{⋆}ℚ̄_{l} is geometrically constant on 𝔸^{1}.

Proof. These sheaves are lisse on 𝔸^{1} and tamely ramified at ∞, so are geometrically constant.

Proposition 15. For any nontrivial χ, the groups H_{c}^{i}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}) vanish for i ≠ n.

Proof. For i ≤ n − 1, use the isomorphism

H_{c}^{⋆}(V̄ [1/f], L_{χ(f)}) ≅ H_{c}^{⋆}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}).

Since V [1/f] is smooth and affine of dimension n, and the coefficients are lisse, these groups vanish for i ≤ n − 1 by the Poincaré dual of the Lefschetz affine theorem.

For i ≥ n + 1, use the Leray spectral sequence for compact cohomology for

f̃: X̃_{𝔾_{m}} → 𝔾_{m}

and the sheaf L_{χ(f̃)} := f̃^{⋆}L_{χ},

E_{2}^{a,b} = H_{c}^{a}(𝔾_{m,k̄}, R^{b}f̃_{⋆}L_{χ(f̃)}) ≅ H_{c}^{a}(𝔾_{m,k̄}, R^{b}f̃_{⋆}ℚ̄_{l} ⊗ L_{χ}) ⇒ H_{c}^{a+b}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}).

The only possibly nonvanishing E_{2}^{a,b} terms have a ∈ [0, 2]. It suffices to show that E_{2}^{a,b} = 0 for a+b ≥ n+1. They vanish for b ≥ n+1, because R^{b}f̃_{⋆}ℚ̄_{l} is geometrically constant, and H_{c}^{⋆}(𝔾_{m,k̄}, L_{χ}) vanishes. For b = n, R^{n}f̃_{⋆}ℚ̄_{l} sits in a short exact sequence

0 → (punctual) → R^{n}f̃_{⋆}ℚ̄_{l} → (geometrically constant) → 0,

so we get E_{2}^{a,n} = 0 for a ≥ 1. It remains to show that E_{2}^{2,n−1} = 0. For this we use the i = n − 1 case of the fact (Proposition 11 above) that all the sheaves Rf̃_{⋆}ℚ̄_{l} on 𝔸^{1} are lisse near the origin. The required vanishing then results from the first part of the following lemma.

Lemma 16. Let F be a constructible ℚ̄_{l}-sheaf on 𝔸1 ; k̄, χ a nontrivial multiplicative character of k^{×}, and ψ a nontrivial additive character of k. Then we have the following results.

(1) If F is lisse on a Zariski open neighborhood of the origin, then

H_{c}^{2}(𝔾_{m,k̄}, F ⊗ L_{χ}) = 0.

(2) If F is lisse on a Zariski open neighborhood of the origin and is tamely ramified at ∞, then we have an equality of Euler characteristics

χ_{c}(𝔾_{m,k̄}, F ⊗ L_{χ}) = χ_{c}(𝔸1 ; k̄, F ⊗ L_{ψ}) = χ_{c}(𝔸1 ; k̄, F) − rank(Fη̄).

(3) Suppose F is lisse on a Zariski open neighborhood of the origin and tamely ramified at ∞, and that the only characters of I_{∞} which occur have order dividing b. If χ^{b} is nontrivial, then the ”forget supports” map is an isomorphism

H_{c}^{⋆}(𝔾_{m,k̄}, F ⊗ L_{χ}) ≅ H^{⋆}(𝔾_{m,k̄}, F ⊗ L_{χ}).

Proof. To prove (1), use the fact that H_{c}^{2} is a birational invariant, i.e. for any dense open set U ⊂ 𝔾_{m,k̄} we have

H_{c}^{2}(U, F ⊗ L_{χ}) ≅ H_{c}^{2}(𝔾_{m,k̄}, F ⊗ L_{χ}).

Take for U a dense open set where F ⊗ L_{χ} is lisse. Then H_{c}^{2}(U, F ⊗ L_{χ}) is the (Tate-twisted) coinvariants of π_{1}(U) acting on (F ⊗ L_{χ})η̄. These coinvariants are themselves a quotient of the coinvariants under the inertia group I_{0}. But the I_{0}- coinvarinats vanish, because as I_{0}-representation we have is simply a direct sum of several copies of the nontrivial character of I_{0} given by χ.

To prove (2), we simply apply the Grothendieck-Neron-Ogg-Shafarevic Euler- Poincaré formula. Because L_{χ} is lisse of rank 1 on 𝔾_{m,k̄} and tame at both 0 and ∞, for any constructible ℚ̄_{l}-sheaf F on on 𝔸1 ; k̄ we have

χ_{c}(𝔾_{m,k̄}, F ⊗ L_{χ}) = χ_{c}(𝔾_{m,k̄}, F).

For F lisse near 0, we have

χ_{c}(𝔾_{m,k̄}, F) = χ_{c}(𝔸1 ; k̄, F) − rank(Fη̄).

For F tame at ∞, we have

χ_{c}(𝔸1 ; k̄, F ⊗ L_{ψ}) = χ_{c}(𝔸1 ; k̄, F) − Swan_{∞}(F) = χ_{c}(𝔸1 ; k̄, F) − rank(Fη̄).

To prove (3), denote by j : 𝔾_{m,k̄} → 𝔸1 ; k̄ the inclusion. We must show that the natural ”forget supports” map

j_{!}(F ⊗ L_{χ}) → Rj_{⋆}(F ⊗ L_{χ})

is an isomorphism. At 0, F is lisse, and at ∞ its inertia representation involves only characters of I_{∞} of order dividing b. So at both 0 and at ∞, F ⊗ L_{χ} is a successive extension of nontrivial characters of the inertia group, so we indeed get

j_{!}(F ⊗ L_{χ}) ≅ Rj_{⋆}(F ⊗ L_{χ}).

Corollary 17. We have an equality of Euler characteristics

χ_{c}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}) = χ_{c}(X̃_{𝔸1 ; k̄}, L_{ψ(f̃)}) = (−1)^{n}C.

Proof. To see that the χ and ψ Euler characteristics are equal, compute each using the Leray spectral sequence as the alternating sum of the dimensions of the E_{2} terms. By Propositions 11 and 13, we may apply part (2) of the previous result to the E_{2} terms. That the ψ Euler characteristic is equal to (−1)^{n}C is proven in [Ka-SE, 5.4.1].

Thus Theorem 10 is now completely proven. To complete the proof of Theorem 9, we must show that the ”forget supports” map

H_{c}^{⋆}(V̄ [1/f], L_{χ(f)}) → H^{⋆}(V̄ [1/f], L_{χ(f)})

is an isomorphism. To see this, compare the Leray spectral sequences for both compact and ordinary cohomology,

E_{2}^{a,b} = H_{c}^{a}(𝔾_{m,k̄}, R^{b}f̃_{⋆}L_{χ(f̃)}) ≅ H_{c}^{a}(𝔾_{m,k̄}, R^{b}f̃_{⋆}ℚ̄_{l} ⊗ L_{χ}) ⇒ H_{c}^{a+b}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)})

and

E_{2}^{a,b} = H^{a}(𝔾_{m,k̄}, R^{b}f̃_{⋆}L_{χ(f̃)}) ≅ H^{a}(𝔾_{m,k̄}, R^{b}f̃_{⋆}ℚ̄_{l} ⊗ L_{χ}) ⇒ H^{a+b}(X̃_{𝔾_{m,k̄}}, L_{χ(f̃)}).

It suffices to show that the ”forget supports” map on the E_{2} terms

H_{c}^{a}(𝔾_{m,k̄}, R^{b}f̃_{⋆}ℚ̄_{l} ⊗ L_{χ}) → H^{a}(𝔾_{m,k̄}, R^{b}f̃_{⋆}ℚ̄_{l} ⊗ L_{χ})

is an isomorphism for all (a, b). This is given by part (3) of the above lemma.

Thus we have proven Theorems 9 and 10, and with them Theorems 1, 3, 5, 6, 7, and 8. It remains only to prove Theorem 4 (of which Theorem 2 was a special case).

8. Proof of Theorem 4

Let us recall the situation. We are given a positive integer d (which is not necessarily prime to p) and a nontrivial multiplicative character χ with χ^{d} trivial. We have X/k which is projective, smooth, and geometrically connected, of dimension n, sitting in ℙ. We are given a linear form Z which defines a hyperplane Z transverse to X, and a degree d form H which defines a hypersurface which is transverse both to X and to X ∩ Z. On V := X[1/Z] we have the function f := H/Z^{d}.

Because H has degree d and χ^{d} is trivial, it makes sense to speak of the lisse, rank one Kummer sheaf L_{χ(H)} on X[1/H]. Concretely, for any linear form L, we can form the usual Kummer sheaf L_{χ(H/L^{d})} on X[1/LH]. For any two linear forms L_{1} and L_{2}, the Kummer sheaf L_{χ(L_{1}^{d}/L_{2}^{d})} on X[1/L_{1}L_{2}] is canonically trivial. This allows us to patch together L_{χ(H/L^{d})} on X[1/LH] for variable L to get the desired lisse, rank one Kummer sheaf L_{χ(H)} on X[1/H]. This Kummer sheaf agrees with L_{χ(f)} on V [1/f] = X[1/ZH]. So we have (writing X̄ for Xk̄)

H_{c}^{⋆}(V̄ [1/f], L_{χ(f)}) ≅ H_{c}^{⋆}(X̄[1/HZ], L_{χ(H)}).

Now look at the excision sequence for

X̄[1/HZ] = V̄ [1/f] ⊂ X̄[1/H] ⊃ X̄ ∩ Z̄[1/H],

⋯ → H_{c}^{i}(X̄[1/HZ], L_{χ(H)}) → H_{c}^{i}(X̄[1/H], L_{χ(H)}) → H_{c}^{i}(X̄ ∩ Z̄[1/H], L_{χ(H)}) → …, which we may rewrite as

⋯ → H_{c}^{i−1}(X̄ ∩ Z̄[1/H], L_{χ(H)}) → H_{c}^{i}(V̄ [1/f], L_{χ(f)}) → H_{c}^{i}(X̄[1/H], L_{χ(H)}) → ….

Using this excision sequence, we see that Theorem 4 results immediately from the following theorem.

Theorem 18. Let d be a positive integer, χ a nontrivial multiplicative character with χ^{d} trivial, and X/k projective, smooth, and geometrically connected, of dimension n, sitting in ℙ.

(1) Suppose given a degree d form H which defines a hypersurface which is transverse to X. Then the ”forget supports” map defines an isomorphism

H_{c}^{⋆}(X̄[1/H], L_{χ(H)}) ≅ H^{⋆}(X̄[1/H], L_{χ(H)}).

The groups H_{c}^{i}(X̄[1/H], L_{χ(H)}) vanish for i ≠ n. The remaining group H_{c}^{n} is pure of weight n, and has dimension C_{0} := (−1)^{n} ∫_{X} \frac{c(X)}{(1+dL)}.

(2) Suppose that we are given in addition a linear form Z on X which is transverse both to X and to X∩H. Then the ”forget supports” map defines an isomorphism

H_{c}^{⋆}(X̄ ∩ Z̄[1/H], L_{χ(H)}) ≅ H^{⋆}(X̄ ∩ Z̄[1/H], L_{χ(H)}).

The groups H_{c}^{i}(X̄ ∩ Z̄[1/H], L_{χ(H)}) vanish for i ≠ n − 1. The remaining group H_{c}^{n−1} is pure of weight n − 1, and has dimension C_{1} := (−1)^{n−1} ∫_{X} \frac{Lc(X)}{(1+L)(1+dL)}.

Proof. Because H is transverse to X, and χ is nontrivial, if we denote by

j : X[1/H] ↪ X

the inclusion, we have

j_{!}L_{χ(H)} ≅ Rj_{⋆}L_{χ(H)}.

[Indeed, at a point of X ∩ H the situation is etale over that at the origin in 𝔸^{n} of

(j : 𝔾_{m} ↪ 𝔸^{1}) × 𝔸^{n−1}

and the sheaf pr_{1}^{−1}L_{χ} on the source.] Hence

H_{c}^{⋆}(X̄[1/H], L_{χ(H)}) ≅ H^{⋆}(X̄[1/H], L_{χ(H)}).

Since X̄[1/H] is smooth and affine of dimension n, and the coefficients are lisse and pure of weight zero, we find that H_{c}^{i}(X̄[1/H], L_{χ(H)}) vanishes for i ≠ n and that H_{c}^{n} is pure of weight n.

Because L_{χ(H)} is lisse of rank one and trivialized by a finite etale covering of degree prime to p (namely of degree the order of χ), we have [Ka-SE, Cor. 1 of 5.5.2] the equalities (“Hurwitz’s formula”)

χ_{c}(X̄[1/H], L_{χ(H)}) = χ_{c}(X̄[1/H], ℚ̄_{l}) = χ_{c}(X̄, ℚ̄_{l}) − χ_{c}(X̄ ∩ H̄, ℚ̄_{l}).

Because H is transverse to X ∩ Z, this same argument shows that the ”forget supports” map defines an isomorphism

H_{c}^{⋆}(X̄ ∩ Z̄[1/H], L_{χ(H)}) ≅ H^{⋆}(X̄ ∩ Z̄[1/H], L_{χ(H)}),

ESTIMATES FOR “NONSINGULAR” MULTIPLICATIVE CHARACTER SUMS 13

that H_{c}^{i}(X̄ ∩ Z̄[1/H], L_{χ(H)}) vanishes for i ≠ n − 1, and that H_{c}^{n−1} is pure of weight n − 1. And just as above, we have the equalities χ_{c}(X̄ ∩ Z̄[1/H], L_{χ(H)}) = χ_{c}(X̄ ∩ Z̄[1/H], ℚ̄_{l}) = χ_{c}(X̄ ∩ Z̄, ℚ̄_{l}) − χ_{c}(X̄ ∩ Z̄ ∩ H̄, ℚ̄_{l}).

We recall now the Chern class formulas [SGA7, Exposé XVII, 5.7.5] [Ka-SE, page 163] for Euler characteristic:

χ_{c}(X̄, ℚ̄_{l}) = ∫_{X} c(X),

χ_{c}(X̄ ∩ H̄, ℚ̄_{l}) = ∫_{X} \frac{dLc(X)}{1 + dL},

χ_{c}(X̄ ∩ Z̄, ℚ̄_{l}) = ∫_{X} \frac{Lc(X)}{1 + L},

χ_{c}(X̄ ∩ Z̄ ∩ H̄, ℚ̄_{l}) = ∫_{X} \frac{dL^{2}c(X)}{(1 + L)(1 + dL)}.

Recall the constants

C_{0} := (−1)^{n} ∫_{X} \frac{c(X)}{(1 + dL)},

C_{1} := (−1)^{n−1} ∫_{X} \frac{Lc(X)}{(1 + L)(1 + dL)}.

We leave to the reader the elementary verification that we have

(−1)^{n−1}C_{1} = χ_{c}(X̄ ∩ Z̄, ℚ̄_{l}) − χ_{c}(X̄ ∩ Z̄ ∩ H̄, ℚ̄_{l}),

and

(−1)^{n}C_{0} = χ_{c}(X̄, ℚ̄_{l}) − χ_{c}(X̄ ∩ H̄, ℚ̄_{l}).

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