ON A QUESTION OF LILLIAN PIERCE

NICHOLAS M. KATZ

Abstract. We establish estimates for certain families of character sums over finite fields, including those which arise in Lillian Pierce’s recent work [P] on estimating the 3-part of the class number of quadratic fields.

1. Introduction, the basic setting, and statement of the main result

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 define n := the order of χ. We extend χ to a function on all of k by defining χ(0) := 0. Let f(x) and g(x) in k[x] be two polynomials over k in one variable x, each of strictly positive degree. We define

d := deg(f), e := deg(g).

For each element a in k, we define a ℂ-valued function G_{a} on k by

G_{a}(z) := ∑_{x∈k} χ(−f(x) + g(z))ψ(ax).

To provide some context, recall the following lemma, which results in a by now well known way from the truth, due to Weil [We-CA], of the Riemann Hypothesis for curves over finite fields, and its application, already known to Davenport and Hasse [Dav-Ha], to the estimation of abelian character sums.

Lemma 1.1. Suppose that either χ^{d} is nontrivial, or that a is nonzero. Then we have the estimate

|G_{a}(z)| ≤ dq^{1/2}.

We are interested in the L^{2} norm of the function G_{a}, and in the inner product of G_{a} both with additive translates of itself and with additive character ”twists” of additive translates of itself. Denote by \overline{G_{a}} the complex conjugate of the function G_{a}. For a, b, c ∈ k, we define the complex character sum

Date: April 15, 2004.

I(a, b, c) := ∑_{z∈k} G_{a}(z)\overline{G_{a}}(z + c)ψ(bz).

In view of the lemma above, we have the following ”trivial estimate” for the character sum I(a, b, c).

Lemma 1.2. Suppose that either χ^{d} is nontrivial, or that a is nonzero. Then we have the estimate

|I(a, b, c)| ≤ d^{2}q^{2}.

Our main result gives conditions under which we can improve on the above ”trivial estimate”, and obtain the improved estimate

|I(a, b, c)| ≤ 2de(d − 1)q^{3/2}.

Our method requires that the characteristic p be large compared to d and to e. Fortunately, this is not a problem in Pierce’s applications, where d and e are fixed, and p varies. [Indeed, in her applications, χ is the quadratic character, f(x) is −4x^{3}, and g(z) is δz^{2} for some nonzero δ ∈ k.]

Theorem 1.3. We have the estimate

|I(a, b, c)| ≤ 2de(d − 1)q^{3/2}.

in each of the following five situations:

(1) ab ≠ 0, and e < d < p,

(2) ac ≠ 0, b = 0, e < d < p, gcd(e, d) = 1, and e(d − 1) < p,

(3) a = 0, b ≠ 0, d < p and χ^{d} is nontrivial,

(4) a = b = 0, c ≠ 0, the polynomials g(z) and g(z + c) in k[z] are relatively prime, f(x) = αx^{d} for some nonzero α ∈ k, n and d are relatively prime, and either (4a) e < n or (4b) d is prime and e < nd.

(5) a = b = 0, c ≠ 0, e < d < p, d ≥ 3, e(d − 1) < p, χ^{d} is nontrivial, the the polynomial f(x) is ”supermorse” (i.e., its derivative f′(x) has d − 1 distinct zeroes in \overline{k}, which map by f to d − 1 distinct points), and either (5a) χ is the quadratic character χ_{2}, or (5b) d > 5, or (5c) d > 3 and (χχ_{2})^{3} is nontrivial.

2. Proof of the main result: basic ideas

Recall that n is the order of χ. Thus the function G_{a} takes values in the cyclotomic field K := ℚ(ζ_{n}, ζ_{p}), and the character sum I(a, b, c) lies in K. We choose a prime number ℓ ≠ p, and a field embedding of K into \overline{ℚ}_{ℓ}. We introduce the K-valued function H_{a} on k defined by

H_{a}(t) := ∑_{x∈k} χ(t − f(x))ψ(ax).

Thus we have the tautological relation

G_{a}(z) = H_{a}(g(z)).

We first interpret the function H_{a} as the trace function of an ℓ-adic sheaf H_{a} on the affine line 𝔸^{1} over k. The sheaf H_{a} turns out to be a ”middle additive convolution” in the sense of [Ka-RLS, Chapter 2]. The function G_{a} is then the trace function of the pullback sheaf

G_{a} := g^{⋆}H_{a}.

We then analyze the possible interactions of the sheaf G_{a} with various twisted additive translates of itself. We then use Deligne’s fundamental results [De-Weil II, 3.3] to obtain the asserted estimates.

3. Proof of the main result in cases (1) and (2)

What cases (1) and (2) have in common is that a ≠ 0. At the expense of replacing the originally chosen nontrivial additive character ψ by the the equally nontrivial additive character

ψ_{a}(x) := ψ(ax),

and replacing b by b/a, we reduce to the case a = 1. We denote by L_{ψ} the Artin-Schreier sheaf on 𝔸^{1} over k corresponding to ψ. We introduce the sheaf F on the w-line 𝔸^{1} over k defined by

F := f_{⋆}L_{ψ}.

We denote by L_{χ} the Kummer sheaf on 𝔾_{m} over k corresponding to χ, extended by zero across the origin. Write the function H_{1} on k as

H_{1}(t) := ∑_{x∈k} χ(t − f(x))ψ(x)

= ∑_{w∈k} χ(t − w) ∑_{x∈k,f(x)=w} ψ(x)

= ∑_{w∈k} χ(t − w)T race(F rob_{k,w}|F)

= ∑_{w∈k} T race(F rob_{k,w}|L_{χ(t−w)} ⊗ F).

This means precisely that the function H_{1} on k is the trace function, at k-rational points, of the lower ! additive convolution of the sheaves L_{χ} and F, or what is the same, of the lower ! additive convolution, say M,

M := L_{χ}[1] ⋆_{+,!} F[1]

of the perverse sheaves L_{χ}[1] and F[1] on 𝔸^{1} over k. Both of these perverse sheaves are visibly middle extensions (remember χ is nontrivial) which, as perverse sheaves, are pure of weight one. Both are geometrically irreducible: this is obvious for L_{χ}[1], and it holds for F[1] because F has generic rank d, and all its ∞-slopes are 1/d, so is already irreducible under the inertia group I(∞), cf. [Ka-GKM, 1.14], and has Swan conductor 1 at ∞. As d > 1, this irreducibility already guarantees that F[1] has ”condition P” in the sense of [Ka-RLS, 2.6.2], thanks to [Ka-RLS, 2.6.13-15]. Therefore, by [Ka-RLS, 2.9.7], the additive ! convolution

M := L_{χ}[1] ⋆_{!,+} F[1]

is itself a perverse sheaf on 𝔸^{1} over k. Looking fibre by fibre, we see that M is of the form H[1], with H a single sheaf, which is, on a dense open set, both lisse and pure of weight one. Thus M is, on a dense open set, pure of weight 2 as a perverse sheaf. We next claim that M is in fact the ”middle additive convolution” of L_{χ}[1] and F[1]. Denote by M_{⋆} the additive ⋆ convolution

M_{⋆} := L_{χ}[1] ⋆_{+,⋆} F[1].

then we know from [Ka-RLS, 2.10.10] and [Ka-MMP, 6.5.4, 3)] that the kernel Ker of the canonical ”forget supports” map from M to M_{⋆} is a perverse sheaf of the form (a lisse sheaf on 𝔸^{1})[1] which is mixed of weight ≤ 1 and which is geometrically a successive extension of various L_{ψ_{a}}[1] sheaves. Since M is generically pure of weight 2, while Ker is everywhere lisse and of strictly lower weight, it follows that Ker = 0, and hence that M is in fact the middle additive convolution

M := L_{χ}[1] ⋆_{+,mid} F[1].

How can we exploit this? We know, from [Ka-RLS, 2.9.7. 2)] and[Ka-MMP, 6.5.4, 2)] respectively, that M = H[1] is geometrically irreducible as a perverse sheaf, and is pure of weight 2. This means in turn that H is mixed of weight at most 1, and that it is the middle extension from some open dense set of a geometrically irreducible lisse sheaf which is pure of weight 1. Moreover, it is an exercise, using [Ka-RLS, 3.3.5], to compute both the generic rank of M = H[1], and all its local monodromies, given the same data for F. Recall that F is a middle extension of generic rank d, all of whose ∞-slopes are 1/d. Because d := deg(f) < p, and L_{ψ} is lisse on 𝔸^{1}, F is tamely ramified outside the point at ∞, and the sum of its drops over all \overline{k}-valued points of 𝔸^{1} is just d − 1, the total number of zeroes, counting multiplicities, of the derivative f′(x) of f at all \overline{k}- valued points of 𝔸^{1}. It is then routine to conclude, using [Ka-RLS, 3.3.5], that H has generic rank d, is tame outside ∞, and has all ∞-slopes 1/d. Moreover, the drops of H at the \overline{k}-valued points of 𝔸^{1} occur precisely at the points where F had drops, and the drops at each such point are the same for H and for F. These drop points are the critical values of the polynomial f, i.e. the values f(α) at the zeroes α of the derivative f′(x). At each critical value f(α) of f, the drop at f(α) is the sum, over all zeroes γ in \overline{k} of the polynomial f(x)−f(α), of the expression (multiplicity of γ as a zero of f(x) − f(α)) - 1. In particular, the sum of all the drops of H at finite distance is just d−1, the total number, counting multiplicity, of zeroes of the derivative f′(x).

With this information established about H, we now turn to the sheaf G := G_{1} := g^{⋆}H on 𝔸^{1}. It is mixed of weight at most 1, and on a dense open set it is lisse and pure of weight 1. It has generic rank d, it is tame at all \overline{k}-valued points of 𝔸^{1}, and all of its ∞-slopes are e/d. Its only drops at \overline{k}-valued points of 𝔸^{1} are at the inverse images by g of the points at which H drops, and at such an inverse image point, the size of the drop is unchanged. Since g is a polynomial of degree e, there are at most e(d-1) finite points at which G drops, and the sums of all its drops at finite points is at most e(d-1). Moreover, the sheaf G has no nonzero punctual sections (as this was already true of H, being a middle extension sheaf).

Let us now treat cases (1) and (2). Since e < d, G has all its ∞- slopes e/d < 1. Similarly, \overline{G}, the sheaf formed using the characters \overline{χ} and \overline{ψ}, but the same f and g, has the same ramification properties as G. The effect of any additive translation

T_{c} : z → z + c

on \overline{G} is just to translate the finite singularities, keeping the same drops. The ∞-slopes remain e/d < 1. After our reduction to the case a = 1 explained above, we must estimate the quantity I(1, b, c) This quantity is the sum of the traces of Frobenius at all the k-rational points of 𝔸^{1} of the sheaf

E_{b} := G ⊗ T_{c}^{⋆}\overline{G} ⊗ L_{ψ_{b}}.

Notice that

E_{b} = E_{0} ⊗ L_{ψ_{b}}.

This sheaf E_{0} is mixed of weight at most 2, it has no nonzero punctual sections, it is tame over 𝔸^{1}, and the sum of its finite drops is at most

2(gen.rk.(G)) ∑_{x∈𝔸^{1}(\overline{k})}drop_{x}(G) = 2de(d − 1).

All of the d^{2} ∞-slopes of E_{0} are ≤ e/d < 1.

In case (1), we have b ≠ 0, and hence all of the d^{2} ∞- slopes the sheaf E_{b} are 1. In particular, E_{b} is totally wild at ∞, so we have

H_{c}^{2}(𝔸^{1} ⊗_{k} \overline{k}, E_{b}) = 0.

We also have the vanishing of the H_{c}^{0} (because there are no nonzero punctual sections), and so the Lefschetz Trace formula gives

I(a, b, c) = −T race(F rob_{k}|H_{c}^{1}(𝔸^{1} ⊗_{k} \overline{k}, E_{b})).

By Deligne’s fundamental result, this H_{c}^{1} is mixed of weight at most 3 (because E_{b} is mixed of weight at most 2), and so we get the estimate

|I(a, b, c)| ≤ (dimH_{c}^{1})q^{3/2} = −χ_{c}(𝔸^{1} ⊗_{k} \overline{k}, E_{b})q^{3/2},

in the case when b ≠ 0. In this case, the ”generic rank” term cancels the ”Swan_{∞}” term in the Euler-Poincare formula,

χ_{c}(𝔸^{1} ⊗_{k} \overline{k}, E_{b}) = gen.rk.(E_{b}) − ∑_{x∈𝔸^{1}(\overline{k})}drop_{x}(E_{b}) − Swan_{∞}(E_{b}),

and we find

−χ_{c}(𝔸^{1} ⊗_{k} \overline{k}, E_{b}) = ∑_{x∈𝔸^{1}(\overline{k})}drop_{x}(E_{b}) ≤ 2de(d − 1).

So in case (1) we have the asserted estimate

|I(a, b, c)| ≤ 2de(d − 1)q^{3/2}.

We now turn to the proof of case (2). The basic idea for treating this case is to pay attention to the location of the finite singularities of the sheaf G. Here a remains nonzero, b is now zero, but c is nonzero. We must estimate the quantity I(1, 0, c) This quantity is the sum of the traces of Frobenius at all the k-rational points of 𝔸^{1} of the sheaf

E_{0} := G ⊗ T_{c}^{⋆}\overline{G}.

This sheaf E_{0} is mixed of weight at most 2, it has no nonzero punctual sections, it is tame over 𝔸^{1},and the sum of its finite drops is at most

2(gen.rk.(G)) ∑_{x∈𝔸^{1}(\overline{k})}drop_{x}(G) = 2de(d − 1).

All of the d^{2} ∞-slopes of E_{0} are ≤ e/d < 1, and hence we have the inequality

Swan_{∞}(E_{0})) ≤ ed.

We rewrite the Euler-Poincare formula

χ_{c}(𝔸^{1} ⊗_{k} \overline{k}, E_{0}) = gen.rk.(E_{0}) − ∑_{x∈𝔸^{1}(\overline{k})}drop_{x}(E_{0}) − Swan_{∞}(E_{0}),

as

−χ_{c}(𝔸^{1} ⊗_{k} \overline{k}, E_{0}) = ∑_{x∈𝔸^{1}(\overline{k})}drop_{x}(E_{0}) + Swan_{∞}(E_{0}) − d^{2},

so we have the inequality

−χ_{c}(𝔸^{1} ⊗_{k} \overline{k}, E_{0}) ≤ 2de(d − 1) + ed − d^{2} ≤ 2de(d − 1),

this last equality simply because e < d. We have already noted that the sheaf E_{0} is mixed of weight at most 2, and has no nonzero punctual sections. Thus its H_{c}^{0} vanishes, and exactly as in case (1) above, it remains only to prove that

H_{c}^{2}(𝔸^{1} ⊗_{k} \overline{k}, E_{0}) = 0.

But recall that

E_{0} := G ⊗ T_{c}^{⋆}\overline{G}.

By assumption in case (2), the integers e and d are relatively prime. Therefore the d ∞-slopes of G, all being e/d, have exact denominator d, and hence [Ka-GKM, 1.14] G is irreducible as a representation of I(∞), the inertia group at ∞. Consequently, on any dense open set U of 𝔸^{1} ⊗_{k} \overline{k} on which the sheaf G is lisse, this sheaf is geometrically irreducible. As H_{c}^{2} is a birational invariant, what we must prove is that for any c ≠ 0, and for any dense open set U of 𝔸^{1} ⊗_{k} \overline{k} on which the sheaf G and its additive translate T_{c}^{⋆}G are both lisse, these two sheaves are not geometrically isomorphic. We argue by contradiction. If G and its additive translate T_{c}^{⋆}G are geometrically isomorphic on some dense open set, then their extensions by direct image to 𝔸^{1} ⊗_{k} \overline{k}, say G_{mid} and T_{c}^{⋆}G_{mid} are isomorphic middle extension sheaves on 𝔸^{1} ⊗_{k} \overline{k}, and hence they have the same set S of ”finite singularities”, i.e., the same set S of points in 𝔸^{1} ⊗_{k} \overline{k} at which they fail to be lisse. Thus the set S of finite singularities is equal to its additive translate by c. Now the additive group 𝔽_{p}c generated by c acts freely on 𝔸^{1}(\overline{k}) by additive translation, so from the stability of the set S under this free action, we infer the congruence Card(S) ≡ 0 mod p. We have already noted that the literal sheaf G on 𝔸^{1} ⊗_{k} \overline{k} has at most e(d − 1) finite singularities. Therefore the middle extension sheaf G_{mid}, whose finite singularities are among those of G, itself has at most e(d−1) finite singularities. By assumption in case (2), we have

e(d − 1) < p.

Thus we infer that S is empty, i.e., that the sheaf G_{mid} is lisse of rank d on 𝔸^{1} ⊗_{k} \overline{k}. This now leads to a contradiction as follows. From its I(∞)-slopes all being e/d, we have already noted that G_{mid} is I(∞)- irreducible. Therefore G_{mid} is lisse of rank d, geometrically irreducible and nonconstant. So its only possibly nonzero compact cohomology group on 𝔸^{1} ⊗_{k} \overline{k} is H_{c}^{1}. Thus its Euler characteristic is non-positive. But this Euler characteristic is

rank(G_{mid}) − Swan_{∞}(G_{mid}) = d − e ≥ 1,

contradiction. This concludes the proof of case (2).

4. Proof of the main result in cases (3), (4), and (5)

What these cases have in common is that a = 0. In this situation, the relevant sheaf F on the w-line 𝔸^{1} over k we need to begin with is

F := Ker(T race : f_{⋆}\overline{ℚ_{ℓ}} → \overline{ℚ_{ℓ}}).

The function H_{0} on k is

H_{0}(t) := ∑_{x∈k} χ(t − f(x))

= ∑_{w∈k} χ(t − w) ∑_{x∈k,f(x)=w} 1

= ∑_{w∈k} χ(t − w)(−1 + ∑_{x∈k,f(x)=w} 1),

this last equality because χ is nontrivial. Thus

H_{0}(t) = ∑_{w∈k} T race(F rob_{k,w}|L_{χ(t−w)} ⊗ F).

This means precisely that the function H_{0} on k is the trace function, at k-rational points, of the lower ! additive convolution of the sheaves L_{χ} and F, or what is the same, of the lower ! additive convolution, say M,

M := L_{χ}[1] ⋆_{+,!} F[1]

of the perverse sheaves L_{χ}[1] and F[1] on 𝔸^{1} over k. Both of these perverse sheaves are visibly middle extensions (remember χ is nontrivial) which, as perverse sheaves, are pure of weight one. While L_{χ}[1] is geometrically irreducible, F[1] need not be (and indeed is certainly not in Pierce’s original question), but it is, in any case, geometrically semisimple. Because d < p, the perverse sheaf F[1] is everywhere tamely ramified. We claim it satisfies ”condition P” in the sense of [Ka-RLS, 2.6]. Indeed, by tameness it contains, geometrically, no sheaf L_{ψ_{t}} for any nonzero t. It contains no constant sheaf either, because by Frobenius reciprocity, the sheaf f_{⋆}\overline{ℚ_{ℓ}} contains just one copy of the constant sheaf. So once again by [Ka-RLS, 2.6], the additive ! convolution

M := L_{χ}[1] ⋆_{!,+} F[1]

is itself a perverse sheaf on 𝔸^{1} over k. Looking fibre by fibre, and remembering that in cases (3) and (4) χ^{d} is assumed nontrivial, we see that M is of the form H[1], with H a single sheaf, which is, on a dense open set, both lisse of rank d-1 and pure of weight one. Thus M is, on a dense open set, pure of weight 2 as a perverse sheaf. Exactly as in the previous section, this generic purity shows that M is the ”middle additive convolution” of L_{χ}[1] and F[1]. We know, from [Ka-RLS, 2.10.10] and [Ka-MMP, 6.5.4, 2)] respectively, that M = H[1] is pure of weight 2 as a perverse sheaf, and hence geometrically semisimple. This means in turn that H is mixed of weight at most 1, and that it is the middle extension from some open dense set of a geometrically semisimple lisse sheaf which is pure of weight 1. Moreover, it is an exercise, using [Ka-RLS, 3.3.5], to compute both the generic rank of M = H[1], (which in our case we know ”by inspection” to be d-1) and all its local monodromies, given the same data for F. Recall that F is an everywhere tame middle extension of generic rank d − 1. The sum of its drops over all \overline{k}-valued points of 𝔸^{1} is just d − 1, the total number of zeroes, counting multiplicities, of the derivative f′(x) of f at all \overline{k}-valued points of 𝔸^{1}. It is then routine to conclude, using [Ka-RLS, 3.3.5], that H is everywhere tame. Moreover, the drops of H at the \overline{k}-valued points of 𝔸^{1} occur precisely at the points where F had drops, and the drops at each such point are the same for H and for F. These drop points are the critical values of the polynomial f, i.e. the values f(α) at the zeroes α of the derivative f′(x). At each critical value f(α) of f, the drop at f(α) is the sum, over all zeroes γ in \overline{k} of the polynomial f(x) − f(α), of the expression (multiplicity of γ as a zero of f(x) − f(α)) - 1. In particular, the sum of all the drops of H at finite distance is just d − 1, the total number, counting multiplicity, of zeroes of the derivative f′(x).

With this information established about H, we now turn to the sheaf G := G_{1} := g^{⋆}H on 𝔸^{1}. It is mixed of weight at most 1, and on a dense open set it is lisse and pure of weight 1. It has generic rank d − 1, and everywhere tame. Its only drops at \overline{k}-valued points of 𝔸^{1} are at the inverse images by g of the points at which H drops, and at such an inverse image point, the size of the drop is unchanged. Since g is a polynomial of degree e, there are at most e(d-1) finite points at which G drops, and the sums of all its drops at finite points is at most e(d-1). Moreover, the sheaf G has no nonzero punctual sections (as this was already true of H, being a middle extension sheaf).

We can now treat cases (3), (4), and (5). The sheaf \overline{G}, the sheaf formed using the character \overline{χ}, but the same f and g, has the same ramification properties as G. The effect of any additive translation

T_{c} : z → z + c

on \overline{G} is just to translate the finite singularities, keeping the same drops. We must estimate the quantity I(0, b, c) This quantity is the sum of the traces of Frobenius at all the k-rational points of 𝔸^{1} of the sheaf

E_{b} := G ⊗ T_{c}^{⋆}\overline{G} ⊗ L_{ψ_{b}}.

Notice that

E_{b} = E_{0} ⊗ L_{ψ_{b}}.

This sheaf E_{0} is mixed of weight at most 2, it has no nonzero punctual sections, it is everywhere tame, and the sum of its finite drops is at most

2(gen.rk.(G)) ∑_{x∈𝔸^{1}(\overline{k})}drop_{x}(G) = 2e(d − 1)^{2}.

In case (3), we have b ≠ 0, and hence E_{b} is totally wild at ∞, with all breaks 1. So H_{c}^{i}(𝔸^{1} ⊗_{k} \overline{k}, E_{b}) vanishes for i ≠ 1. The dimension of the H_{c}^{1} is given by the Euler-Poincare formula, in which the ”generic rank” term is cancelled by the Swan_{∞} term, and we find

dimH_{c}^{1} = −χ(𝔸^{1} ⊗_{k} \overline{k}, E_{b}) = ∑_{x∈𝔸^{1}(\overline{k})}drop_{x}(E_{b}) ≤ 2e(d − 1)^{2}.

So in case (3) we find the estimate

|I(0, b, c)| ≤ 2e(d − 1)^{2}q^{3/2},

which is in fact a slight improvement over the asserted estimate

|I(0, b, c)| ≤ 2ed(d − 1)q^{3/2}.

In case (4), we have a = b = 0, but c ≠ 0. In this case we have

−χ(𝔸^{1}⊗_{k}\overline{k}, E_{0}) = −(d−1)^{2}+ ∑_{x∈𝔸^{1}(\overline{k})}drop_{x}(E_{0}) ≤ −(d−1)^{2}+2e(d−1)^{2} ≤ 2ed(d−1).

It remains to show that H_{c}^{2}(𝔸^{1} ⊗_{k} \overline{k}, E_{0}) vanishes, or equivalently, that on any dense open set U in 𝔸^{1} ⊗_{k} \overline{k} on which both G and T_{c}^{⋆}G are lisse (remember both are geometrically semisimple, by purity ), these two sheaves have no irreducible constituent in common. Because we are in case (4), f is just a monomial of degree d, and consequently F is, geometrically, the extension by zero of the lisse sheaf on 𝔾_{m} ⊗_{k} \overline{k} given by

F ≅ ⨁_{ρ^{d}=1,ρ nontriv.} L_{ρ}.

Hence its middle convolution with L_{χ} is given by

H ≅ ⨁_{ρ^{d}=1,ρ nontriv.} L_{ρχ}.

[Remember that χ^{d} is nontrivial, so each character ρχ which appears in the above sum is nontrivial]. The pullback by g is then given by

G ≅ ⨁_{ρ^{d}=1,ρ nontriv.} L_{(ρχ)(g(z))}.

Its additive translate by c is then

T_{c}^{⋆}G ≅ ⨁_{ρ^{d}=1,ρ nontriv.} L_{(ρχ)(g(z+c))}.

So what we must show is that for any c ≠ 0, no L_{(ρ_{i}χ)(g(z+c))} is isomorphic to any L_{(ρ_{j}χ)(g(z))}, for any two nontrivial, not necessarily distinct, characters ρ_{i} and ρ_{j} of order dividing d. Since the polynomials g(z) and g(z + c) have no common zero, this would be obvious if we knew that each L_{(ρ_{i}χ)(g(z+c))} were ramified at every zero of g(z + c), and that each L_{(ρ_{i}χ)(g(z))} were ramified at every zero of g(z). In case (4a), the fact that n and d are relatively prime implies that each character ρ_{i}χ has order at least n. As g has degree e < n, every zero of g has multiplicity at most e < n, and hence we do indeed have the asserted ramification. In case (4b), d is prime as well as prime to n, so each character ρ_{i}χ has order precisely nd. In case (4b), g has degree e < nd, and we conclude by the same ramification argument as in case (4a).

In case (5), we use the fact [Ka-ACT, 5.15] that the geometric monodromy group G_{g}eom of the sheaf H is either the symplectic group Sp(d − 1) in case (5a), or that its identity component G_{g}eom^{0} is SL(d − 1), in cases (5b) and (5c). We retain only the consequence that the sheaf H is geometrically Lie-irreducible, and hence its pullback by any nonconstant map is geometrically irreducible. So G is itself geometrically irreducible. Just as above, it suffices to show that for any choice of c ≠ 0, on any dense open set U in 𝔸^{1} ⊗_{k} \overline{k} on which both G and T_{c}^{⋆}G are lisse, these two sheaves are not isomorphic. For this, we argue by contradiction, using the argument alrady used in case (2) of paying attention to the set S of finite singularities of G_{mid}. Because e(d − 1) < p, we conclude that S is empty,i.e., that G_{mid} is lisse on 𝔸^{1}. Since G_{mid} is also tame at ∞, it must be geometrically constant. But as it has rank d − 1 ≥ 2, this geometric constancy contradicts its geometric irreducibility.

References

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[Ka-MMP] Katz, N., Moments, Monodromy, and Perversity: a Diophantine Perspective, to appear as an Annals of Math. Study, available as preprint on www.math.princeton.edu/ nmk.

[Ka-RLS] Katz, N., Rigid Local Systems, Annals of Math. Study 138, Princeton Univ. Press, 1995.

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[Ray] Raynaud, M., Caracteristique d’Euler-Poincare d’un Faisceau et cohomologie des varietes abeliennes, Expose 286 in Seminaire Bourbaki 1964/65, W.A. Benjamin, New York, 1966.

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Princeton University, Mathematics, Fine Hall, NJ 08544-1000, USA

E-mail address: nmk@math.princeton.edu