CHAPTER 17

GL(n) Examples

Here we work on either the split or the nonsplit form. We begin with a lisse sheaf F on a dense open set j : U ⊂ G which is geometrically irreducible, pure of weight zero, and not geometrically isomorphic to (the restriction to U of) any Kummer sheaf L_χ. We denote by G := j_? F its middle extension to G. Then the object N := G(1/2)[1] ∈ P_arith is pure of weight zero and geometrically irreducible.

Theorem 17.1. Suppose that N is not geometrically isomorphic to any nontrivial multiplicative translate of itself. Suppose further that for one of the two possible geometric isomorphisms G/k ≅ G_m/k, F(0)^unip is a single Jordan block Unip(e) for some e ≥ 1, and F(∞)^unip = 0. For n := dim(!(N)) we have

G_geom,N = G_arith,N = GL(n).

Proof. We have a priori inclusions

G_geom,N⊂ G_arith,N ⊂ GL(n),

so it suffices to prove that G_geom,N = GL(n). So we may extend scalars if necessary from k to its quadratic extension k₂, and reduce to the case where G is G_m, F(0)^unip is a single Jordan block Unip(e), and F(∞)^unip = 0. Then by the previous chapter, G_arith,N contains, in a suitable basis of !(N), the torus Diag(x, 1, 1, ….1).

The hypothesis that N is not geometrically isomorphic to any nontrivial multiplicative translate of itself insures that N is geometrically Lie-irreducible, i.e., that G⁰_geom is an irreducible subgroup of GL(n). Hence G⁰_geom is the almost product of its center, which by irreducibility consists entirely of scalars, with its derived group (:=commutator subgroup) , which is a connected semisimple group which also must act irreducibly.

Because G_geom,N is a normal subgroup of G_arith,N, so also its intrinsic subgroup is a normal subgroup of G_arith,N. Therefore the Lie algebra Lie() is a semisimple irreducible Lie subalgebra of End(!(N)) which is normalized by G_arith,N. In particular, Lie() is normalized by all elements Diag(x, 1, 1, ….1). Such elements are pseudoreflections, with determinant equal to x. Take x ≠ ±1. Our semisimple irreducible Lie subalgebra of End(!(N)) is normalized by a pseudoreflection of determinant not ±1. The only such Lie algebra is Lie(SL(n)), cf. [Ka-ESDE, 1.5]. [See [Se-Dri, Prop. 5] for another approach to this result.] Therefore = SL(n), and hence we get the inclusion SL(n) ⊂ G_geom,N. So it suffices to show that the determinant as a character of G_geom,N has infinite order.

Equivalently, we must show that “det” (N) in the Tannakian sense is geometrically of infinite order. The one-dimensional objects in <N>_arith are either punctual objects α^deg ⊗δ_a for some unitary scalar α and some a ∈ k^× or they are, geometrically, multiplicative translates of hyper-geometric sheaves H[1], cf. [Ka-ESDE, 8.5.3]. But H[1] has infinite geometric order, because its successive middle convolutions with itself are again of the same form H′ [1].

So it suffices to show that “det” (N) in the Tannakian sense is not punctual. But if it were punctual, it would have no bad characters, i.e., we would have | det(Frob_E,χ)| = 1 for every finite extension E/k and every character χ of E^×. But we have seen that Frob^ss_k, is Diag(1/√#k^e, 1, 1, …, 1). Hence det(Frob_k,) = 1/√#k^e is not pure of weight zero. Therefore “det” (N) in the Tannakian sense is not punctual, and hence it is geometrically of infinite order.

Here are five explicit examples, all on G_m/k. The first is based on the Legendre sheaf Leg introduced in chapter 15.

Theorem 17.2. Let k have odd characteristic. For any odd integer n ≥ 1, the object N := Symⁿ(Leg)((n + 1)/2)[1] in P_arith is pure of weight zero, has “dimension” n, and has

G_geom,N = G_arith,N = GL(n).

Proof. Symⁿ(Leg)(n/2) is the middle extension of a geometrically irreducible (because Symⁿ(std₂) is an irreducible representation of SL(2)) lisse sheaf of rank n+1 on G_m{1} which is pure of weight zero. So N is a geometrically irreducible object in P_arith which is pure of weight zero. Its local monodromies at 0 and 1 are both Unip(n + 1). Because n is odd, its local monodromy at ∞ is L_χ2 ⊗ Unip(n + 1). Because the only singularity of N in G_m is 1, N is not geometrically isomorphic to any nontrivial multiplicative translate of itself. The Euler-Poincaré formula [Ray] shows that N has “dimension” n (it is tame at 0 and ∞, and has drop n at 1). The result now follows from the previous theorem. In this example, we can compute the Tannakian determinant “det” (N) explicitly. By Theorem 16.1 and Corollary 16.3, “det” (N) is geometrically isomorphic to a multiplicative translate of the shifted hypergeometric sheaf H[1] of type (n + 1, n + 1) given by H = H(Ã; , …, ; χ₂, …, χ₂). [By Lemma 19.3, proven later but with no circularity, there is in fact no multiplicative translate: the unique singularity in G_m of both N and its determinant is at 1.]

The second example, a mild generalization of the first, is based on hypergeometric sheaves of type (2, 2). We pick a character χ of k^× for a large enough finite field k. We assume χ² is nontrivial. We begin with the objects L_χ(1–x)(1/2)[1] and L_{χ(x²/(1–x))}(1/2)[1] on G_m/k. Their middle convolution is H(2)[1], for H := H(!, Ã; , χ²; χ, χ) the hypergeometric sheaf of type (2, 2) on G_m/k as defined in [Ka-ESDE, 8.2]. This sheaf H is the middle extension of its restriction to G_m {1}, where it is lisse, and pure of weight 3. Its local monodromy at 0 is ⊕ χ². Its local monodromy at 1 is Unip(2), and its local monodromy at ∞ is L_χ ⊗ Unip(2).

Theorem 17.3. Let n ≥ 2 be an integer, H := H(!, Ã; , χ²; χ, χ). If χ² has order > n, the object N := Symⁿ(H)((3n + 1)/2)[1] in P_arith is pure of weight zero, has “dimension” n, and has

G_geom,N = G_arith,N = GL(n).

Proof. Because we assume that χ² has order > n, the local monodromy of Symⁿ(H) at 0, which is the direct sum ⊕ χ²r, has Symⁿ(H)(0)^unip = Unip(1). The local monodromy of Symⁿ(H) at 1 is Unip(n + 1), and its local monodromy at ∞ is L_χn ⊗ Unip(n + 1). Notice that χⁿ is nontrivial (otherwise (χ²)ⁿ is trivial). Therefore Symⁿ(H)(∞)^unip = 0. The rest of the proof is identical to that in the Legendre case above. Just as in that case, we see that here the Tannakian determinant is geometrically isomorphic to H[1] for H the hypergeometric sheaf of type (n, n) H(Ã; χ², …, χ²n; , …, ).

Our third example works on G_m/F_p, any prime p. For any integer n ≥ 1, we will construct a lisse rank one sheaf F on A¹/F_p which is pure of weight zero and whose Swan conductor at ∞ is the integer n. We will do this in such a way that F|G_m is not geometrically isomorphic to any nontrivial multiplicative translate of itself. Then it is immediate from Theorem 17.1 that the object N := (F|G_m)(1/2)[1] in P_arith is pure of weight zero, has “dimension” n, and has

G_geom,N = G_arith,N = GL(n).

Here is one such construction. Let us write n = p^rd with r ≥ 0 and with d ≥ 1 prime to p. Suppose first that r = 0, i.e., n = d is prime to p. Pick a nontrivial additive character à of F_p. Choose a polynomial f_d(x) ∈ k[x] which is Artin-Schreier reduced, i.e., no monomial x^e appearing in f_d has p|e, and such that the set of e’s such that x^e occurs in f_d generates the unit ideal in Z. For instance we could take f₁(x) = x and f_d(x) = x^d – x for d ≥ 2. Then the Artin-Schreier sheaf L_Ã(fd(x)) has Swan conductor d at ∞ [De-ST, 3.5.4]. Its multiplicative translate by an a ≠ 1 in k^× is L_Ã(fd(ax)), which is not geometrically isomorphic to L_Ã(fd(x)) because their ratio is L_{Ã(fd(ax)–fd(x))}. Indeed, the difference f_d(ax) – f_d(x) is Artin-Schreier reduced, and has strictly positive degree so long as a ≠ 1; this degree is the Swan conductor at ∞ of the ratio.

Suppose next that n = p^rd with r ≥ 1 and d ≥ 1 prime to p. Then we pick a character Ã_{r+ 1} of Z/p^{r+ 1}Z = W_{r+ 1}(F_p) which has order p^r+ 1. We consider the Witt vector of length r + 1 given by v := (f_d(x), 0, 0,.., 0) ∈ W_{r+ 1}(F_p[x]). We form the Z/p^{r+ 1}Z covering of A¹ defined by the Witt vector equation z – F (z) = v in W_{r+ 1}; its pushout by Ã_{r+ 1} gives us the Artin-Schreier-Witt sheaf L_{Ãr+ 1((fd(x), 0, 0, 0,.., 0))}, whose Swan conductor at ∞ is p^rd = n, cf. [Bry, Prop. 1 and Cor. of Thm. 1]. It is not geometrically isomorphic to any nontrivial multiplicative translate of itself. Indeed, its p^r’th tensor power is just L_Ã(fd(x)) for à the additive character of F_p, viewed as p^rZ/p^r+1Z, obtained by restricting Ã_{r+ 1}.

In summary, then, we have the following theorem.

Theorem 17.4. For n = p^rd with r ≥ 0 and d ≥ 1 prime to p, form the Artin-Schreier-Witt sheaf F := L_{Ãr+ 1((fd(x), 0, 0,.., 0))}. Then the object N := (F|G_m)(1/2)[1] in P_arith is pure of weight zero, has “dimension” n, and has

G_geom,N = G_arith,N = GL(n).

In this example, the Tannakian determinant is geometrically isomorphic to a multiplicative translate of H[1] for H = L_Ã(x), the hyper-geometric H(Ã; ; ∅) of type (1, 0).

Our fourth example is this.

Theorem 17.5. Take a polynomial f[x] = of degree n ≥ 2 with all distinct roots in k. Suppose that f(0) ≠ 0, and that gcd{i|A_i ≠ 0} = 1. Then for any character χ of k^× such that χⁿ is nontrivial, the object N := L_χ(f)(1/2)[1] in P_arith is pure of weight zero, has “dimension” n, and has

G_geom,N = G_arith,N = GL(n).

Proof. Here the local monodromy at 0 is Unip(1), and the local monodromy at ∞ is L_χn. So the assertion is an immediate application of Theorem 17.1, once we show that N is isomorphic to no nontrivial multiplicative translate of itself. To see this, we argue as follows. The set S of zeroes of f is the set of finite singularities of N, so is an intrinsic invariant of the geometric isomorphism class of N. So it suffices to show that if α ∈ k^× satisfies αS = S, then α = 1. The condition that αS = S is the condition that f(αx) = αⁿf(x). Equating coefficients, we get the equations

αⁱ A_i = αⁿ A_i

for every i. Taking i = 0, and remembering that A₀ ≠ 0 by hypothesis, we find that αⁿ = 1. Then our equations read

αⁱ A_i = A_i

for every i. Using the fact that gcd{i|A_i ≠ 0} = 1, we infer that α = 1.

In this example, the Tannakian determinant is geometrically isomorphic to H[1] for H = H(Ã; ; χⁿ) ≅ L_χⁿ(1–x).

To end this chapter, we give a fifth example, valid in any odd characteristic p.

Theorem 17.6. Let k be a finite field of odd characteristic p. Take a polynomial f(x) = ∑ⁿ_i=0 A_ixⁱ in k[x] of prime-to-p degree n ≥ 2 with all distinct roots in k. Suppose that f is “weakly supermorse” [Ka-ACT, 5.5.2], i.e., its derivative f′(x) has n – 1 distinct zeroes (the critical points) α_i in k, and the n – 1 values f(α_i) (the critical values) are all distinct. Denote by S the set of critical values. Suppose that S is not equal to any nontrivial multiplicative translate of itself. Form the middle extension sheaf

F := f_?Q_ℓ/Q_ℓ | G_m.

Then the object N := F(1/2)[1] ∈ P_arith is pure of weight zero, has “dimension” n – 1, and has

G_geom,N = G_arith,N = GL(n – 1).

Proof. Because f is weakly supermorse, F is irreducible, of generic rank n – 1, with geometric monodromy group the symmetric group S_n in its deleted permutation representation, cf. [Ka-ESDE, 7.10.2.3 and its proof]. Because f has n distinct zeroes, 0 is not a critical value. So S ⊂ G_m, and at each point of S the local monodromy is a reflection, necessarily tame, as p ≠ 2. At ∞, the local monodromy is tame, the direct sum ⊕_{χ|χⁿ⁼, χ ≠}L_χ. Thus F is tame at ∞. Therefore the “dimension” of N is the sum of the drops of F, each one, at the n – 1 points of S. As F is not lisse at n – 1 ≥ 1 points of G_m, F is not geometrically isomorphic to any Kummer sheaf L_ρ. Thus N ∈ P_arith. It is pure of weight zero because F is a middle extension which on a dense open set is pure of weight zero. From the hypothesis that the set S of singularities of N is not equal to any nontrivial multiplicative translate of itself, it follows that N is not geometrically isomorphic to any nontrivial multiplicative translate of itself. Thus N is geometrically Lie-irreducible.

It remains to show that N has G_geom = GL(n – 1). For this, it is equivalent to show that for some ρ, N ⊗ L_ρ has G_geom = GL(n – 1) (simply because M M ⊗ L_ρ is a Tannakian automorphism of P_geom).

We choose for ρ any nontrivial character of order dividing n. For such a choice of ρ, the sheaf F_ρ := F ⊗ L_ρ has F_ρ(0)^unip = 0 (because F was lisse at 0), and F_ρ(∞)^unip = Unip(1) (because the local monodromy of F at ∞ was ⊕_{χ|χⁿ=,χ≠}L_χ). The result now follows from Theorem 17.1, using the “other” geometric isomorphism of G_m with itself (which interchanges 0 and ∞).

In this case, the Tannakian determinant of the original N is geometrically isomorphic to some multiplicative translate of H[1] for H the hypergeometric sheaf of type (n – 1,n – 1) H(Ã; , …, ; χ, χ², …, χ^n–1) for any character χ of full order n.

Remark 17.7. For any polynomial f, the trace function of the sheaf F := f_?Q_ℓ/Q_ℓ is the counting function given, for any finite extension E/k and any a ∈ E^×, by

Trace(Frob_E,a|F) = #{x ∈ E|f(x) = a}– 1.

So for any nontrivial character χ of E^×, we have

Thus for f satisfying the hypotheses of Theorem 17.6, we are saying, in particular, that these sums, as χ varies over characters of E^× with χⁿ ≠ , are approximately distributed like the traces of random elements of the unitary group U(n–1), the approximation getting better and better as #E grows.

How restrictive are the hypotheses imposed on the polynomial f? One knows that given any polyonomial f(x) ∈ k[x] of prime-to-p degree n ≥ 2 such that f" (x) is nonzero, then for all but finitely many values of a ∈ k, the polynomial f(x) + ax is weakly supermorse, cf. [Ka-ACT, 5.15]. For example, if n(n – 1) is prime to p, then xⁿ – nx is weakly supermorse, with μ_n–1 as the critical points, and (1 – n)μ_n–1 as the set S of critical values. In this example, the set S is equal to its multiplicative translate by any element of μ_n–1. Nonetheless, we have the following lemma, which shows that by adding nearly any constant to a weakly supermorse f, the set of its critical values will then be equal to no nontrivial multiplicative translate of itself.

Lemma 17.8. Let k be a field of characteristic p. Take a weakly super-morse polynomial f[x] = ∑ⁿ_i=0 A_ixⁱ in k[x] of prime-to-p degree n ≥ 2, with set S of critical values. For each λ ∈ k, consider the weakly super-morse polynomial f_λ(x) := f(x) – λ, with set S_λ := –λ + S of critical values. Denote by F_S(x) := Π_s∈S (x – s) the polynomial of degree n – 1 whose roots are the critical values of f. For any λ ∈ k such that both F_S(λ) and its derivative F'_S(λ) are nonzero, the set S_λ is equal to no nontrivial multiplicative translate of itself. In particular, there are at most (n – 1)(n – 2) values of λ ∈ k for which S_λ is equal to some nontrivial multiplicative translate of itself.

This results from the following elementary lemma.

Lemma 17.9. Let k be a field, S ⊂ k a finite subset consisting of d ≥ 1 elements. Denote by F_S(x) := Π_s∈S (x – s) the polynomial of degree d whose roots are the points of S. For any λ ∈ k such that both F_S(λ) and its derivative F'_S(λ) are nonzero, the set S_λ := –λ + S is equal to no nontrivial multiplicative translate of itself.

Proof. The polynomial F_Sλ(x) is just F_S(x + λ), hence we have the Taylor expansion

F_Sλ(x) = F_S(λ)+ F'_S(λ)x mod x².

Suppose F_S(λ)F'_S(λ) ≠ 0. If S_λ is equal to its multiplicative translate by some nonzero α, then F_Sλ(αx) has the same roots as F_Sλ(x). Comparing highest degree terms, we get

F_Sλ(αx) = α^dF_Sλ(x).

Comparing the nonzero (by our choice of λ) constant and linear terms of their Taylor expansions, we get α^d = 1 and α^d–1 = 1, and hence α = 1.

Corollary 17.10. Let k be a finite field of odd characteristic p. Let n ≥ 2, and assume that n(n–1) is prime to p. Consider the polynomial

Then for any a ∈ k^× with a^n–1 ≠ 1, the polynomial f(x) – a satisfies all the hypotheses of Theorem 17.6.

Proof. Here the set S of critical values of f(x) is μ_n–1, so F_S(x) = x^n–1 – 1, and F'_S(x) = (n – 1)x^n–2.

There are some special cases where no adjustment of the constant term is necessary. Here is one such.

Lemma 17.11. Let k = F_q be a finite field of odd characteristic. For any a ∈ F^×_q, the polynomial f(x) = x^q+1 – x²/2+ ax satisfies all the hypotheses of Theorem 17.6.

Proof. We have f '(x) = x^q–x+ a, so the critical values are the elements α ∈ k with α^q = α – a. No such α lies in F_q, since a ≠ 0. The value of f at such an α is

f(α) = α(α^q) – α²/2+ aα = α(α – a) – α²/2+ aα = α²/2.

If we fix one critical point α, any other is α+ b, for some b ∈ F_q. If their critical values coincide, i.e., if (α + b)²/2 = α²/2, then αb + b²/2 = 0; if b ≠ 0, this implies that α = –b/2, contradicting the fact that α does not lie in F_q. So the set S of critical values consists of q distinct, nonzero elements. So the polynomial F_S(x) has degree q, and a nonzero constant term. If S is a multiplicative translate of itself, say by ° ∈ k^×, then we get F_S(°x) = °^qF_S(x), and comparing the nonzero constant terms we get °^q = 1, and hence ° = 1.